Salinan Terjemahan Jensen1969.docx

Risiko, Penentuan Harga Aset Modal, dan Evaluasi Portofolio Investasi Michael C. Jensen Jurnal Bisnis, Vol. 42, No. 2. (Apr. 1969), hlm. 167-247. URL yang stabil: http://links.jstor.org/sici?sici=00219398%28196904%2942%3A2%3C167%3ARTPOCA%3E2.0.CO%3B2-Z Jurnal Bisnis saat ini diterbitkan oleh Universitas Chicago Press. Penggunaan arsip JSTOR oleh Anda menunjukkan bahwa Anda menerima Syarat dan Ketentuan Penggunaan JSTOR, tersedia di http://www.jstor.org/about/terms.html. Syarat dan Ketentuan Penggunaan JSTOR menyediakan, sebagian, bahwa kecuali Anda telah memperoleh izin sebelumnya, Anda tidak boleh mengunduh seluruh jurnal atau beberapa salinan artikel, dan Anda dapat menggunakan konten dalam arsip JSTOR hanya untuk pribadi Anda, bukan -penggunaan komersial. Silakan hubungi penerbit tentang penggunaan lebih lanjut dari karya ini. Informasi kontak penerbit dapat diperoleh di http://www.jstor.org/journals/ucpress.html. Setiap salinan dari setiap bagian dari transmisi JSTOR harus mengandung pemberitahuan hak cipta yang sama yang muncul di layar atau halaman cetak dari transmisi tersebut. Arsip JSTOR adalah repositori digital tepercaya yang menyediakan pelestarian jangka panjang dan akses ke jurnal akademis terkemuka dan literatur ilmiah dari seluruh dunia. Arsip ini didukung oleh perpustakaan, perkumpulan ilmiah, penerbit, dan yayasan. Ini adalah inisiatif dari JSTOR, organisasi nirlaba dengan misi untuk membantu komunitas ilmiah mengambil keuntungan dari kemajuan teknologi. Untuk informasi lebih lanjut tentang JSTOR, silakan hubungi [email protected]. http://www.jstor.org Fri 21 Sep 16:55:36 2007

RISIKO, THE HARGA ASET MODAL, DAN EVALUASI portofolio investasi *

I. PENDAHULUAN A. RISIKO LID EVALUASI

T

HE pengembangan

Ating utama tujuan OF Kinerja PORTFOLIOS dari

penelitian ini adalah model untuk evaluasi portofolio aset berisiko. Dalam mengevaluasi kinerja portofolio, efek dari risiko yang berbeda harus dipertimbangkan. ' Jika investor umumnya menolak risiko, mereka akan lebih suka (ceteris pa- ribus) aliran pendapatan yang lebih pasti lebih sedikit

* Penelitian pada penelitian ini didukung oleh beasiswa fellowship dari US Steel Foundation, American Banking Association dan hibah penelitian dari Dana Penelitian di bidang Keuangan disediakan oleh Sekolah Bisnis Universitas Chicago. Waktu komputer yang luas di 7094 Pusat Komputasidi Universitas Chiago dibiayai oleh Sekolah Pascasarjana Bisnis, dan Sekolah Tinggi Bisnis Universitas Rochester menyediakan waktu tambahan di 360 Pusat Komputasidi Universitas California. Rochester. t Asisten profesor, Fakultas Administrasi Bisnis, University of Rochester. Saya ingin mendapatkan hutang yang besar untuk komitmen disertasi saya; Eugene Fama (ketua), Lawrence Fisher, Merton Miller (yang awalnya menyarankan bidang penelitian ini kepada saya), dan Harry Roberts, yang semuanya telah dengan murah hati memberikan waktu dan gagasan mereka dan terus-menerus memaksa saya untuk memikirkan kembali dan mempertahankan posisi saya di banyak masalah. Saya terutama berhutang pada beberapa profesor draft Fama untuk makalah ini. penetrasi saya, saya juga ingin berterima kasih kepada para anggota Lokakarya Keuangan di Universitas Chicago untuk banyak merangsang dan membantu D. Duvel, diskusi, dan M. Scholes. terutama saya

memiliki M. juga Blume, memberi manfaat bagi P. Brown, dari percakapan dengan M. Geisel, F. Black, dan Profesor Peter Pashigian, Arnold Zellner, Donald Gordon, dan Julian Keilson. Risiko, konsep kritis dalam makalah ini, akan didefinisikan dan V. dan dibahas secara luas dalam Bagian 11,111, aliran tertentu. Dalam kondisi ini, investor akan menerima risiko

tambahan hanya jika mereka dikompensasikan dalam bentuk pengembalian yang diharapkan di masa depan. Dengan demikian, di dunia yang didominasi oleh investor yang tidak mau mengambil risiko, portofolio berisiko harus diharapkan untuk menghasilkan pengembalian yang lebih tinggi daripada portofolio yang kurang berisiko, atau tidak akan dimiliki. Model evaluasi portofolio yang dikembangkan di bawah ini menggabungkan aspek risiko ini secara eksplisit dengan memanfaatkan dan memperluas hasil-hasil teoretis baru-baru ini oleh Sharpe [52] dan Lintner [37] pada penetapan harga aset modal dalam ketidakpastian. Mengingat hasil ini, ukuran portofolio "kinerja" (yang hanya mengukur kemampuan manajer untuk memperkirakan harga keamanan) didefinisikan sebagai perbedaan antara pengembalian aktual pada portofolio dalam setiap periode holding tertentu dan pengembalian yang diharapkan. pada portofolio yang tergantung pada tingkat tanpa risiko, tingkat "risiko sistematis," dan pengembalian aktual pada portofolio pasar. Kriteria untuk menilai kinerja portofolio menjadi netral, superior, atau inferior telah ditetapkan. Ukuran dari portofolio "efisiensi" juga berasal, dan kriteria untuk penghakiman ing portofolio yang akan eficient, sien superefi-, atau ine $ icient didefinisikan. Juga diperlihatkan bahwa sangat mustahil untuk menentukan ukuran efisiensi semata-mata dalam hal variabel ex post yang dapat diamati. Selain itu, terlihat bahwa ada hubungan alami antara ukuran kinerja portofolio dan ukuran efisiensi. 168 JURNAL BISNIS B. KEAMANAN HARGA GERAKAN,EFISIEN PASAR, MARTINGAL DAN IMPLIKASI MEREKA UNTUK KEAMANAN ANALISA Baru-baru ini ada minat yang besar dalam perilaku harga keamanan di "pasar efisien" dan lebih khusus dalam hipotesis martingale perilaku harga. Tampaknya ada dua bentuk hipotesis yang berbeda yang muncul dari definisi yang berbeda tentang konsep "pasar yang efisien," definisi yang jarang secara eksplisit dijelaskan. Seseorang dapat mendefinisikan pasar yang lemah efisien dalam pengertian berikut: Pertimbangkan kedatangan di pasar sepotong informasi baru mengenai nilai keamanan. Sebuah pasar lemah efisien adalah pasar yang mungkin diperlukan waktu untuk mengevaluasi informasi ini berkaitan dengan kation impli- untuk nilai keamanan. Namun, setelah evaluasi ini selesai, harga sekuritas segera menyesuaikan (dengan cara yang tidak bias) dengan nilai baru yang tersirat oleh informasi. Dalam pasar yang sangat efisien seperti itu, seri harga sekuritas masa lalu tidak akan mengandung informasi yang belum disita dalam harga saat ini. Mandelbrot [39] dan Samuel-son 1477 telah dengan ketat menunjukkan bahwa harga di pasar seperti itu akan mengikuti submartingale-nilai itu .

. . a), semua sebagai masa depan harga adalah, X (t +T), diharapkan (7

= 1,

waktu t independen dari(7 = urut 1 ,. ..

a),dan

harga masa lalu X (t - 7),

sama dengan: di mana f (r) adalah tingkat akumulasi "normal". Dengan demikian, di pasar di mana harga keamanan berperilaku sebagai bagian dari bentuk (1. I), teknik peramalan2 yang menggunakan hanya urutanmasa lalu harga diuntuk memperkirakan harga di masa depan akan gagal. Perkiraan terbaik dari harga di masa depan hanyalah harga sekarang ditambah pengembalian yang diharapkan normal selama periode tersebut. Pasar saham telah mengalami banyak penyelidikan empiris yang ditujukan untuk menentukan apakah (1.1) merupakan deskripsi yang memadai tentang perilaku serisaham harga^. ^ Bukti yang ada menunjukkan bahwa sangat tidak mungkin bahwa seorang investor atau manajer portofolio akan dapat menggunakan sejarah harga

saham masa lalu saja. (dan karenanya aturan perdagangan mekanis berdasarkan harga-harga ini4) untuk meningkatkan keuntungannya. Namun, kesimpulannya adalah harga saham mengikuti submartingale formulir (1.l)tidak menyiratkan bahwa seorang investor tidak dapat meningkatkan keuntungannya dengan meningkatkan kemampuannya untuk memprediksi dan mengevaluasi konsekuensi dari peristiwa masa depan yang mempengaruhi harga saham. Memang, telah disarankan oleh Fama [I21 bahwa keberadaan "pelaku pasar" yang canggih yang mahir mengevaluasi informasi saat ini dan memprediksi peristiwa di masa depan adalah salah satu alasan mengapa harga pasar pada suatu titik waktu mewakili estimasi yang tidak bias dari "benar" menghargai dan menyesuaikan dengan cepat, dan akurat, ke informasi baru mengenaiini nilai^. ^ Ini membawa kita ke definisi alternatif dari "pasar efisien," yaitu, di mana semua informasi masa lalu tersedia hingga waktu t disita dalamsaat ini teknik Chartingadalah salah satu contohnya. 3 Lihat terutama karya Fama [12] dan- karyakarya dicetak ulang dalam Cootner [9]. 'Untuk contoh pengujian satu kelas aturan semacam itu lihat [12]. 6 Sebagai contoh pemeriksaan iklan tersebut, lihat Fama et al. [19]

.169 RISIKO, ASET MODAL, DAN EVALUASIPORTFOLIOS hargaDalam definisi pasar yang efisien ini, bukti Mandelbrot-Samuelson menyiratkan bahwa yang erty propmartingale dapat ditulis sebagai wher e variabel pengkondisi Ot sekarang mewakili semua informasi yang tersedia pada waktu t.'j Pembaca akan mencatat bahwa (1.2) adalah bentuk yang jauh lebih kuat dari hipotesis martin- gale daripada (1.1), yang hanya dikondisikan pada harga masa lalu. seri. Dengan demikian (1.2) dapat diberi label bentuk "kuat" dari hipotesis martingale dan (1.1) bentuk ((lemah). 'Memang, jika harga keamanan mengikuti martingale dari bentuk kuat, tidak ada analis yang dapat dapatkan pengembalian di atas rata-rata dengan mencoba memprediksi harga di masa depan berdasarkan masa lalu informasi di. Satu-satunya individu yang dapat memperoleh pengembalian yang unggul adalah orang yang sesekali adalah orang pertama yang mendapatkan informasi baru yang biasanya tidak tersedia bagi orang lain di tetapi sebagaimana Roll [46] berargumen, dalam upaya untuk segera bertindak atas informasi ini, individu ini (atau kelompok individu) akan memastikan bahwa efek dari informasi baru ini dengan cepat ditingkatkan. dalam harga keamanan. Lebih jauh lagi, jika informasi baru dari jenis ini muncul secara acak, tidak ada individu yang akan dapat meyakinkan dirinya sendiri tentang penerimaan sistematis dari informasi tersebut. Oleh karena itu, ketika seseorang sesekali dapat mengubah kembali rejeki nomplok tersebut, akan un6

Lihat Roll [46] untuk disku bagian alasan yang mengarah ke (1.2). 7 Sepengetahuan saya, terminologi ini adalah karena Harry

Roberts, yang menggunakannya dalam pidato yang belum diterbitkan berjudul "Suhu Klinis vs. Statistik Harga Keamanan," diberikan pada Seminar Analisis Harga Keamanan yang disponsori oleh Pusat Penelitian Harga Keamanan di Universitas Chicago, Mei 1967. Setelah menulis makalah ini, sebuah artikel oleh Shelton [59] telah muncul yang berisi pernyataan hipotesis yang sangat mirip.

mampu menghasilkan mereka secara sistematis melalui waktu. Sementara bentuk lemah darimartingale hipotesisdibuktikan dengan baik oleh bukti empiris, bentuk kuat dari hipotesis belum dikenakan tes empiris yang luas.8 Model yang dikembangkan di bawah ini akan memungkinkan kita untuk mengirimkan bentuk kuat dari hipotesis hingga uji empiris semacam itu — setidaknya sejauh implikasinya dimanifestasikan dalam keberhasilan atau kegagalan satu kelas khusus dari analis keamanan yang sangat kaya. C. APLIKASI MODEL Model evaluasi portofolio yang dikembangkan di bawah ini akan digunakan untuk memeriksa hasil yang dicapai oleh manajer portofolio reksa dana ujung terbuka dalam upaya untuk menjawab pertanyaan berikut: 1) Apakah pola historis risiko dan pengembalian yang diamati untuk sampel portofolio aset berisiko kami menunjukkan dominasi penghindaran risiko di pasar modal? Jika demikian, apakah pola-pola ini

mengkonfirmasi implikasi model teoritis penetapan harga aset modal yang didasarkan pada asumsi keengganan terhadap risiko? 2) Apakah secarareksa dana terbuka umummenunjukkan kemampuan untuk memilih portofolio yang memperoleh pengembalian lebih tinggi dari yang mungkin diharapkan diperoleh dengan tingkat risiko mereka? Atau, apakah mereka menunjukkan kemampuan untuk memperoleh pengembalian yang lebih tinggi daripada yang bisa diperoleh dengan kebijakan pemilihan naif yang konsisten dengan teori penetapan harga aset modal? Kesimpulan utama akan: 1) Pola sejarah diamati risiko sistematis dan kembali untuk reksa dana dalam sampel konsisten dengan menyadari 8The hanya terdapat bukti pada Fama ini et pertanyaan d.1191, itu dan saya yang saya bukti menunjukkan bahwa harga keamanan menyesuaikan dengan cepat dan dengan cara yang berisi informasi baru.

170 THE JURNAL BISNIS hipotesis bersama bahwa model set harga modal sebagai- valid dan bahwa manajer reksa dana rata-rata tidak mampu untuk meramalkan harga keamanan masa depan. 2) Jika kita mengasumsikan bahwa model penetapan harga aset modal adalah valid, maka estimasi empiris kinerja dana menunjukkan bahwa portofolio dana itu "lebih rendah" setelah dikurangi semua pengeluaran manajemen dan komisi broker yang dihasilkan dalam aktivitas perdagangan. Selain itu, ketika semua biaya manajemen dan komisi perantara ditambahkan kembali ke pengembalian dana dan keseimbangan kas rata-rata dari dana diasumsikan menghasilkan tingkat tanpa risiko, portofolio dana tampaknya adil ((netral). "Dengan demikian, tampak bahwa rata-rata sumber daya yang dihabiskan oleh dana dalam upaya untuk memprakarsai harga keamanan tidak menghasilkan pengembalian portofolio yang lebih tinggi daripada yang bisa diperoleh dengan portofolio risiko yang setara yang dipilih (a) secara acak kebijakan seleksi atau (b) dengan investasi gabungan dalam "portofolio pasar" dan obligasi pemerintah. 3) Berdasarkan bukti yang dirangkum di atas, kami menyimpulkan bahwa sejauh menyangkut 115 reksa dana ini, harga efek tampaknya berperilaku sesuai dengan ((bentuk "kuat dari hipotesis martingale. Artinya, tampaknya harga sekuritas saat ini sepenuhnya menangkap efek dari semua informasi yang tersedia untuk 115 reksa dana ini. Oleh karena itu, upaya mereka untuk menganalisis informasi masa lalu lebih lanjut secara menyeluruh belum menghasilkan pengembalian yang meningkat. Meskipun hasil ini tentu saja tidak menyiratkan bahwa bentuk kuat dari hipotesis martingale berlaku untuk semua investor dan untuk semua waktu, mereka memberikan bukti kuat untuk mendukung pasar itu setiap hari dan memiliki kontak dan asosiasi yang luas dalam berbagai bidang. baik bisnis maupun komunitas keuangan. Dengan demikian, fakta bahwa mereka tampaknya tidak dapat memperkirakan pengembalian yang cukup akurat untuk memulihkan biaya penelitian dan transaksi mereka adalah bukti yang mendukung bentuk kuat dari hipotesis martingale - setidaknya sejauh subset luas informasi yang tersedia untuk analis ini prihatin. 4) Bukti juga menunjukkan bahwa, sementara portofolio dana rata-rata ((lebih rendah "dan" tidak efisien, "ini terutama disebabkan oleh pengeluaran terlalu banyak. Karena bukti menunjukkan bahwa portofolio pada rata-rata usia yang sangat terdiversifikasi dengan baik, mereka "tidak efisien" terutama karena timbangkan gen- terlalu banyak biaya. D. AN bESAR gARIS PENELITIAN Model evaluasi portofolio adalah veloped de- dalam Bagian 11-V. The tions foundadari model dibahas dalam Bagian 11,yang melanjutkan dengan tinjauan singkat tentang: (1) teori pilihan rasional di bawah ketidakpastian, (2) teori normatif pemilihan portofolio, dan akhirnya (3) model teoritis yang terkait erat dari harga aset modal di bawah kepastian un-. Bagian I11 berisi pengembangan model evaluasi di bawah asumsi yang dari riods homogen investor cakrawala pe-. "model pasar" dan kecuali bahwa con- dari "risiko sistematis" didefinisikan , dan aplikasi mereka untuk

masalah evaluasi m dibahas secara rinci. Akhirnya, ukuran portofolio ('kinerja "diperoleh dengan asumsi alternatif tentang adanya varian terbatas atau tak terbatas untuk distribusi pengembalian. 9 Misalnya,

total pendapatan yang diterima oleh pothesis.

Kita harus menyadari bahwa ini analis sangat baik endo

~ ed. ~ Selain itu, mereka beroperasi disekuritas delapan puluh enam perusahaan penasehat investasidari investasi terbuka akhir tahun fiskal perusahaan berakhir pada 1960-61 menjadi (lih. $ 32,6 Teman juta el al. di [26 , p. 4971).

RISIKO, ASET MODAL, dAN eVALUASI portofolio 171 Bagian IV berisi diskusi tentang "masalah cakrawala," solusi untuk itu, dan perluasan model evaluasi untuk sebuah dunia di mana investor memiliki heterogen geneous periode horizon. Bagian V berisi diskusi tentang kriteria evaluasi, derivasi dari ukuran "efisiensi," dan pemeriksaan hubungan antara konsep "kinerja" (didefinisikan dalam Bagian 111) dan " efisiensi. " Bagian VI hal membenci diskusi tentang (1) perkiraan empiris konsep "risiko sistematis" untuk 115 reksa dana, (2) beberapa tes empiris dari asumsi "model pasar," dan (3) penerapan model untuk evaluasi 115 portofolio reksa dana ini. Bagian VII berisi ringkasan hasil teoritis dan empiris serta implikasinya dan diskusi singkat tentang beberapa kritik utama yang tidak diragukan lagi akan muncul mengenai temuan. Pembaca yang tertarik terutama pada aplikasi empiris dari model yang didiskusikan pada Bagian VI-B dapat memperoleh rasa umum dari model tersebut dengan memeriksa Bagian 111-A dan 111-B, pemeriksaan singkat Bagian IV, dan pemeriksaan dekat Bagian V, yang menyajikan sejumlah poin penting. 11. PONDASI MODEL A. TEORI PILIHAN RASIONAL DALAM KETIDAKPASTIAN yang

diharapkan utilitas maxim.-The lem masalah.Safe_mode pilihan di bawah ketidakpastian char- acterized oleh situasi di mana seorang terbagi in- menghadapi serangkaian tindakan alternatif, dan hasil yang terkait dengan tindakan ini tunduk pada distribusi probabilitas. Kami akan mengasumsikan dalam pengembangan untuk mengikuti bahwa seorang individu yang rasional, ketika dihadapkan dengan pilihan di bawah kondisi ketidakpastian, bertindak dengan cara yang konsisten dengan maksim utilitas yang diharapkan. Artinya, dia bertindak seolah-olah dia (1) menempelkan angka (utilitas) untuk setiap hasil yang mungkin dan (2) memilih opsi itu (atau strategi) dengan nilai utilitas terbesar yang diharapkan.1 ° Masalah konsumsi-investasi.- Menerima maksim utilitas yang diharapkan sebagai fungsi objektif, masalah umum investor di dunia yang tidak pasti dapat dinyatakan sebagai maksimalisasi nilai yang diharapkan di mana Ctadalah nilai riil konsumsi pada periode t, T adalah waktu kematian ( yang, tentu saja, adalah variabel acak), WTadalah warisan, dan U adalah utilitas dari pola konsumsi seumur hidup investor. Masalah portofolio muncul dalam kerangka kerja ini ketika investor memiliki aset dalam satu periode yang tidak ingin dia konsumsi dalam periode itu, melainkan keinginan untuk melanjutkan ke periode berikutnya. Masalah portofolionya setiap saat t kemudian menjadi pemilihan kombinasi investasi yang menghasilkan utilitas maksimum yang diharapkan. Sementara masalah konsumsi-investasi jelas merupakan masalah multiperiode, tidak adanya teori pilihan multiperiode yang dikembangkan dengan baik di bawah ketidakpastian telah membuat sebagian besar peneliti berasumsi bahwa keputusan portofolio dapat diperlakukan sebagai keputusan periode tunggal untuk dipertimbangkan. dilakukan secara independen dari keputusan konsumsi." Diperlukan dan memadai con lo Sebuah

derivasi aksiomatik dari pepatah utilitas yang diharapkan diberikan oleh Von Neuman dan Morgenstern [65] dan Markowitz

[42]. dalam bab x-xiii, Markowitz[42 ] memberikan eksposisi menyeluruh dari hipotesis dan implikasinya terhadap keputusan portofolio khususnya .1 Lihat, misalnya, referensi 7, 13, 14, 22, 36, 37,40,42,43,43,51,52,61 , dan 62, yang semuanya (baik secara implisit atau eksplisit) adalah utilitas periode tunggal model kekayaan terminal, yaitu mereka menganggap masalah investor dapat

dikarakterisasi dengan maksimalisasi nilai yang diharapkan dari U (Wt +,),di mana Wt + ~adalah kekayaan terminal dari satu

portofolio karenanya.

172 T IIE JURNAL 01;BISNIS Ditiondi mana asumsi penyederhanaan ini akan mengarah pada solusi optimal dari masalah multiperiode tak terbatas dari (2.1) telah ditentukan hanya untuk kelas fungsi utilitas yang sangat terbatas (lih. Hakansson [29] dan Mossin [ 44] .Namun, Fama [15] telah menunjukkan dalam kondisi yang sangat umum bahwa, sementara investor harus menyelesaikan masalah periode-T seperti (2.1) untuk membuat keputusan konsumsi dan investasi untuk periode 1, ia akan berperilaku karena dia adalah seorang pemaksimisasi utilitas periode tunggal yang diharapkan, yaitu, investor akan tampak seolah- olah dia memaksimalkan di mana Ct dan Wtil, masing-masing, nilai konsumsi pada periode t dan nilai terminal dari portofolio pada akhir periode t, dan variabel keputusannya adalah Ct dan xi,fraksi portofolio yang diinvestasikan dalam aset ke-i. Selain itu, jika semua aset benar-benar liquidI2 dan habis dibagi secara tak terbatas dan tidak ada pajak, 13 Fama [16] juga telah menunjukkan itu, dalam memecahkan th Dalam masalah konsumsi-investasi simultan (2.2), investor akan selalu memilih portofolio yang efisien dalam hal parameter periode tunggal. Artinya, investor akan selalu l2 Aset

adalah sempurna cair jika (a) setiappar- saat TERTENTUpembelian dan harga jual yang cal identi- dan (b) jumlah

apapun dapat dibeli atau dijual pada harga ini. Dengan demikian, biaya transaksi diasumsikan sama dengan nol. l3Dalam tes empiris yang akan datang kemudian, ini mungkin tampak sebagai pembatasan yang signifikan, bagi seorang investor dalam

kelompok pajak penghasilan marjinal tinggi tentu tidak akan acuh tak acuh terhadap bentuk (capital gain atau dividen

pendapatan) di mana ia menerima kembali. Namun, dalam praktiknya, asumsi ini mungkin tidak seketat yang kita yakini. I-Iorowitz [30], meneliti sifat-sifat model untuk peringkat reksa dana, menemukan bahwa eksplisit allo ~ Vance untuk tarif pajak yang berbeda pada pendapatan dan keuntungan modal sults kembali di efek hanya minor pada peringkat relatif dari sembilan puluh delapan dana . Namun, dalam memilih portofolio untuk investor tertentu, pertimbangan pajak ini harus diperhitungkan.

pilihlah portofolio yang efisien dalam arti bahwa untuk periode yang dipertimbangkan itu memberikan pengembalian yang diharapkan maksimum untuk tingkat risiko tertentu dan risiko minimum untuk tingkat pengembalian yang diharapkan. Ini berarti, tentu saja, bahwa kesimpulan umum yang diperoleh dari pekerjaan sebelumnya dengan utilitas periode tunggal model kekayaan terminal mengenai keputusan portofolio investor yang menghindari risiko dan karakteristik keseimbangan umum tetap berlaku ketika konsumsi dan investasi dipertimbangkan bersama. Karena fungsi utilitas Von Neuman-Morgenstern unik hanya hingga transformasi linear positif, dan karena pengembalian portofolio adalah Rt = AW, / W, = (W, + l / Wt) - 1, kita dapat mengungkapkan masalah portofolio konsumsi-investor sebagai max xi E [U (Ct, Rt) I , (2.3) dan kami menganggap U adalah monoton yang meningkat dan menyatakan

secara ketat [2.3] dalam istilah cekung dalam R (C ,, karena R, ). itu (Kami lebih nyaman dan menghindari masalah dengan skala dalam membuat perbandingan portofolio di kemudian hari.) Kami sekarang telah menetapkan dasar untuk pertimbangan model portofolio mean-variance normatif dari Markowitz [40,42] dan Tobin [61, 62], yang pada gilirannya memberikan banyak motivasi untuk model Sharpe [52] dan Lintner [37] kondisi keseimbangan umum di pasar aset modal. Seperti yang akan kita lihat, hasil ini memberikan kunci untuk penyelesaian masalah evaluasi portofolio. Dengan demikian, mari kita beralih ke tinjauan singkat dari model portofolio mean-variance. Cukuplah untuk mengatakan bahwa semua

model ini didasarkan pada adanya varian yang terbatas untuk distribusi perubahan keamanan. Empiris bekerja dengan Mandelbrot [38], Fama [12], and Roll [46], bagaimanapun, menunjukkan bahwa distribusi pengembalian 173 RISIKO, CAPIT.4L ASET, DAN E \. 'ALUATION OF I'ORTFOLIOS pada saham biasa dan obligasi tampaknya sesuai dengan anggota kelas distribusi Stabil yang rerata ada tetapi varians tidak.14 Pada saat ini kami hanya menunjukkan bahwa Fama [13, 161 telah menunjukkan bahwa dengan beberapa modifikasi sebagian besar hasil diperoleh untuk kasus khusus varian hingga juga mencakup kasus yang lebih umum di mana distribusi pengembalian diperbolehkan menjadi anggota simetris dari kelas distribusi stabil dengan rata-rata terbatas. Kami melanjutkan diskusi dalam kerangka mean-variance untuk motor dengan asumsi bahwa distribusi probabilitas dari semua pengembalian keamanan memiliki varian yang terbatas. Perluasan hasil mean-variance ke dunia yang ditandai dengan distribusi pengembalian dengan varian tak terbatas akan dipertimbangkan dalam Bagian 111-C di bawah ini. B. NORMATIF TEORI ANALISIS PORTOFOLIO yang diharapkan zbtility pepatah dandiversifikasi investments.-Markowitz [40, 421 dan Tobin [61, 621 telah menunjukkan diversifikasi yang adalah quence quence logis bagi investor menghindari risiko yang function16 tujuan dapat ditulis sebagai

.

maks E [U (Rt)] (2.4)

Secara khusus, utilitas memaksimalkan portofolio untuk setiap investor akan menjadi varian portofolio yang efisien mean dalam arti bahwa ia menawarkan varians minimum untuk tingkat tertentu keuntungan yang diharapkan dan maxi L4Dapat

dicatat di sini bahwa Gaussian atau atau - distribusi mal adalah kasus khusus dari kelas distribusi ini dengan eksponen karakteristik

a = 2. 16 Dengan

hasil Fama [15, 161 kita telah melihat bahwa kesimpulan yang diambil dari penyelidikan implikasi (2,4) juga berlaku untuk solusi

untuk (2.1). Dengan demikian, untuk kesederhanaan, dari titik ini kita akan mengabaikan keputusan konsumsi, Ct, dan menulis diskusi kita dalam hal utilitassatu periode yang fungsi pengembaliandiberikan oleh (2.4).

pengembalian yang diharapkan ibu untuk tingkat tertentu varians jika (1) utilitas investor func- tion

U'>

(2,4) 0 dan memenuhi Uff <0, dan kondisi (2) terjadinya distribusi yangbutions aset dan pengembalian portofolio adalah dari form16 yang sama dan sepenuhnya dijelaskan oleh dua parameter (lih. Tobin [61,62]). Kondisi-kondisi ini menyiratkan bahwa semua pengembalian aset harus secara normal didistribusikan agar efisiensi varians rata-rata menjadi bermakna.l7 Selain itu, Fama [16] telah menunjukkan bahwa teorema ini dapat diperluas ke kelas umum dari distribusi Stabil gejala dengan metrik terbatas. momen pertama (yang, secara kebetulan, tampaknya menggambarkan distribusi keamanan secara empiris dengan sangat baik; lih. Fama [12], Mandelbrot [38], dan Roll [46]). Tetapi kami menunda diskusi tentang hal ini pada Bagian 111-C. Gambar 1 memberikan presentasi geometris dari model mean-variance Markowitz. Membiarkan viation masa

depan - sebuah (R)menjadi kembali,. . standar de -area yang diarsir pada Gambar 1 mewakili semua kemungkinan kombinasi risiko dan pengembalian yang tersedia dari investasi dalam keamanan yang menanggung risiko. Portofolio yang terletak pada batas ABCD mewakili sekumpulan portofolio standar deviasi (atau varian rata-rata) efisien, karena mereka semua mewakili. 16 Kualifikasi

ini sangat penting dan sering diabaikan. Samuelson [48] menyajikansederhana contohdari distribusi dua parameter yang

analisisnya gagal karena alasan ini. Menarik juga untuk dicatat bahwa kelas distribusi yang Stabil (lih. Feller [23, bab. Xvii]) adalah

satu-satunya distribusi yang stabil di bawah penambahan. Yaitu, distribusi Stabil adalah satu-satunya distribusi yang jumlah-jumlah tertimbang dari variabel-variabel acak (yaitu, portofolio) akan memiliki bentuk yang sama denganmendasarinya variabel-variabel acak yang. Tetapi ini berarti satu-satunya distribusi yang dimiliki versi mean-variance dari teorema Tobin adalah normal (lih. Fama [16]). l7 Bertentanganyang berlaku umum pendapat, sangkaan sebagai- fungsi utilitas kuadrat tidaksuffi- sienuntuk menjamin bahwa utilitas

memaksimalkanpelabuhan yang folioakan mean-variance efisien (cf. Borch [5, p. 20-211). Dengan demikian, kondisi yang diberikan dalam teks adalah satu-satunya kondisi yang akan membenarkanmean- kerangkavariance.

174 JURSAL mengirimkan kemungkinan investasi yang menghasilkan pengembalian maksimum yang diharapkan untuk risiko yang diberikan dan risiko minimum untuk pengembalian yang diharapkan. Sebagai Tobin [61] telah menunjukkan, yang hidup normal pengembalian keamanan dan adanya penghindaran risiko pada bagian dari vestor in cukup untuk menghasilkan keluarga saya. ...... ...., .... . . ,. I i i OF BISNIS

vestment dalam portofolio B, menghasilkan E (RB) dan a (RB)dengan utilitas Il. Implikasi dari keberadaan aset tanpa risiko.-Portofolio B yang digambarkan pada Gambar 1 merupakan solusi optimal untuk masalah portofolio hanya dalam kasus di mana investasi dibatasi padaberisiko peningkatan utilitas yang (R) Gambar.1.-Maksimalisasi utilitas investor mengingat adanya aset bebas risiko yang

positif, kurva miring cembung miring (diwakili oleh I ,, Iz, 13)dalam bidang deviasi standar rata-rata Gambar 1. Area yang diarsir pada Gambar 1mewakili peluang yang tersedia bagi investor tanpa adanya aset tanpa risiko, dan batas set ini ABCD mewakili set portofolio efisien dalam arti Markowitz. Seorang investor terbatas hanya untuk investasi di aset berisiko yang memiliki peta ketidakpedulian tertentu yang ditunjukkan pada Gambar 1 akan memaksimalkan utilitas masa depannya diharapkan dengan in. aset Mari kita asumsikan masa depan keberadaan kembalibebas risiko RF asetseperti F, menghasilkan tertentu ditarik pada Gambar 1,18 l 8instrumen

semacam itu mungkin tunai (menghasilkan tidak kembali moneter positif),tabungan rekeningtertanggung, atau non

obligasi pemerintah pembawa kupon yang memiliki tanggal jatuh tempo bertepatan denganinvestor tanggal horizon. Dalam kasus yang

terakhir, tentu saja, investor dapat yakin untuk mewujudkan hasil hingga jatuh tempo dengan pasti jika ia memegang obligasi hingga jatuh tempo. Karena kami berasumsi bahwa investor tidak akan mengubah portofolionya dalam periode sementara,antara apa pun fluktuasi hargatidak memberinya risiko. Kami mengabaikan masalah yang terkait dengan perubahan tingkat harga umum dan akan

terus melakukannya di sisa makalah ini.

175 RISIKO, MODAL, .ETET, DAN EV: PENGIRIMAN PORTFOLIOS Seorang investor dihadapkan dengan kemungkinan investasi dalam aset bebas risiko seperti itu, serta aset berisiko, dapat membangun portofolio dari dua aset yang akan memungkinkannya untuk mencapai kombinasi risiko dan pengembalian yang terletak di sepanjang garis lurus yang menghubungkan kedua aset dalam bidang deviasi standar rata-rata (lih. Tobin [61]). Jelas, semua portofolio yang terletak di bawah titik C di sepanjang ABCD pada Gambar 1 tidak efisien, karena setiap titik pada garis RpC yang diberikan oleh merupakan solusi yang layak. Dengan demikian, investor dapat mendistribusikan dananya antara portofolio C dan sekuritas F sehingga portofolio gabungannya, sebut saja E, menghasilkan utilitas untuknya dari I2 E (RE),

> I ,. Dalam. (RE), penjumlahan, dan maksimum jika investor dapat meminjam serta meminjamkan pada tingkat RF tanpa risiko, set portofolio layak yang diwakili oleh garis RFC dan persamaan (2.5) melampaui titik

C. C . tEORI OP mODAL ASSET HARGA Sharpe [52], Lintner [36,37], dan Mos- dosa [43] dimulai dengan els modnormatif Markowitz dan Tobin memiliki opment teori yang sama oped keseimbangan pasar modal dalam kondisi risiko. Asumsi berikut mendasari ketiga model: (1) semua investor memiliki periode horizon yang identik; (2) all investors may borrow as well as lend funds at the risk- less rate of interest; and (3) investors have homogeneous expectations regard- ing expected future return and standard deviation of return on all assets and all covariances of returns among all assets. Sharpe observed that investors would at- tempt to purchase only those assets in portfolio C and the riskless security F of Figure 1. Thus, we have a situation in which the market for capital assets would be out of equilibrium unless C is the "market portfolio," that is, a port- folio which contains every asset exactly in proportion to that asset's fraction of the total value of all assets. Conceptu- ally, if the market were out of equilibri- um the prices of assets in C would be instantaneously bid up and the prices of assets not in C would fall until such time as all assets were held. In equilibrium, all investors who se- lect ex ante efficient portfolios will have mean standard deviation combinations which lie along the line R@ in Figure 2, their individual location determined by their degree of risk aversion. Sharpe [52] has asserted that in equilibrium the effi- cient set may be tangent to RFQ at multiple points as in Figure 2. However, whether or not this ever occurs,1g the market portfolio M must always be one of the tangency points (cf. Fama [14] and Fama and Miller [20]). Most important, however, is the re- sult that in equilibrium the expected re- turn on any e$cient portfolio E will be linearly related to the expected return on the market portfolio in the following manner :20 The concept of systematic risk.-In addition, Sharpe, Lintner, and Mossin 19

While it is possible for multiple tangency points to exist, it is highly improbable that this would ever occur. The existence of

multiple tangency points would require that the returns on one or more indi- of vidual the market securities portfolio were perfectly M. correlated with those 20 For reasons which will become clear below, we choose to write equations like (2.6) (and [2.7] below) recognizing only two of the conditioning variables explicitly on the left-hand side. These variables be- come crucial to distinctions we wish to maintain below.

176 THE JOURNAL OF BUSINESS have shown that if the capital market is in equilibrium the expected return on any individual security (or portfolio) will be a linear function of the covariance of its returns with that of the market port- folio.21 The function is: diversified portfolios and can assume that the capital markets are in equilibri- um, (2.7) implies that the relevant meas- ure of the riskiness of any security (or portfolio) is the quantity cov(Rj, RM), and the market price per unit of risk is [E(RAT)- RF]/u~(RAT). We shall see in Section I11 that this result, (2.7), will become the foundation of the portfolio evaluation model disFIG.2.-Possible configuration of the "efficient set," given equilibrium in the capital markets

It is important to note that (2.6) holds only for efficient portfolios and (2.7) holds for any individual security

or any portfolio regardless of whether it is effi- cient. Thus, as long as we are concerned with risk in the context of efficiently We note here that Jack Treynor also had inde- pendently arrived at these results at about the same time as Sharpe and Lintner.

Unfortunately, his ex- cellent work [64] remains unpublished.

cussed there. Thus, a detailed discussion of the implications of (2.7) is given in Section 111. 111. THE SINGLE-PERIOD HOhIOGE- NEOUS HORIZON MODEL The reader will recall that one of the in the derivation of

vestors the asset are pricing one-period expected was that utility in177 RISK, ChPITtiL ASSETS, '\NUEVALUATION OF PORTFOLIOS maximizers having a common horizon the Sharpe-Lintner asset pricing model date. The assumption of identical invest- indicates that the expected return on any or decision horizons is admittedly un- asset (or portfolio of assets) is given realistic, but for the moment a7e pro- ceed with the development of the model within this context. It will be shown in Section IV that the asset pricing model and the portfolio evaluation model based on it can be extended to a world in which investors have horizon periods of differ- ing lengths and trading of assets is al- lowed to take place continuously through time. A. A STANDARD OF COMPARISON A major problem encountered in de- veloping a portfolio evaluation model is the establishment of a norm or standard for use as a bench mark. The discussion of Section I1 points to a natural stand- ard-the performance of the market portfolio, M. As long as the market is in equilibrium we know that ex ante this portfolio must be a member of the effi- cient set. Ex post, of course, this port- folio will not dominate all others, since in a stochastic model such as this, real- ized returns will seldom be equal to ex- pectations. The market portfolio also offers an- other interpretation as a standard of comparison, since it represents the results which could have been realized (ig- noring transaction costs) by one par- ticular naive investment strategy, that is, purchasing each security in the mar- ket in proportion to its share of the total value of all securities. Thus, the concept of the market port- folio provides a natural point of com- parison. However, as mentioned earlier, we cannot compare returns on portfolios with differing degrees of risk to the same standard; but this problem may be re- solved by reference to the asset pricing model discussed in Section 11.Recall that

qRp]

by (2.7) E[R~~E(RM) , Bli =

= RP

f [E(R~)-R~l

CoV .2(RM)

(.R~,RM) (2.7) Let us define

TRD1) COV (R~,RM) (3.1)

so that we now measure the risk of any security22j relative to the risk of the market portfolio. (The term PIj will henceforth be referred to as the "system- atic" risk of the jth asset or portfolio, and the first subscript, here 1, will be used to distinguish between three alter- native interpretations of the coefficients.) Thus, if the asset pricing model is valid and the capital market is in equilibrium, the expected one-period return on any asset (or portfolio of assets) will be a linear function of the quantity PI as por- trayed in Figure 3. The point M represents the expected return and systematic risk of the market portfolio, and the point RF represents the return on the risk-free asset. Since we are measuring risk relative to the risk of the market portfolio, it is obvious that the risk of the market portfolio is unity, since Thus, conditional on the expected re- turns on the market portfolio and the risk-free rate, (2.7) gives us the relation- ship between the expected returns on any asset (or collection of assets) and its level of systematic risk Pjj.

However, 22 Henceforth, we shall use the terms "asset" and "security" interchangeably. 178 THE JOURNAL OF BUSINESS since expectations can be observed only in Part C under the assumption that the with error, these results will be much distributions of returns conform to the more useful if they can be translated into infinite variance (but finite mean) mem- a relationship between ex post realiza- bers of the symmetric Stable family of ti on^.^^

We now show how this may be distributions. ARA.3.-The relationship between the expected return on any asset (or collection of assets) and systemat- ic risk (61) as implied by the capital asset pricing model.

accomplished by utilizing the additional structure imposed on the asset pricing model by the assumptions of what Blume [4] and Fama [14, 161 have called the "market model." In Part B we consider the model under the assumption that the distributions of returns are normal and 53

For a discussion of the problems and issues which can arise around just this question regarding ex ante relationships, see West [66] and

Sharpe [58]. While the criticisms raised by West are legitimate, we shall see below that the problems can be com- pletely surmounted

in that we can derive explicit re- lationships between ex post variables which still yield testable results. This same issue also arises in the debate contained in [8] and [28]. B. SYSTEMATIC RISK IN THE CONTEXT OF THE GAUSSIAN MARKET MODEL The

model.-The market model was originally suggested by Markowitz [42, p. 1001and analyzed in considerable de- tail by Sharpe 151, 52, 571, who referred to it as the "diagonal model." Simply stated, the model postulates a linear re- lationship between the returns on any security and a general "market factor."24 s4 The

model described by equations (3.2) and (3.3) is slightly different from the diagonal model originally proposed by Markowitz,

analyzed by

RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS 179 That is, we express the returns on the jth security as where the "market factor" n is defined such that E(n) = 0, bj is a constant, nand ej are all normally distributed ran- dom variables, and N is the total num- ber of securities in the market. The fol- lowing assumptions are made regarding the disturbance terms ej: E(ej)=OE(e,a)=O jj = = ll , , 2 2 ,...,N(3.3b) ,...,N(3.3a) Now let Vj be the total value of all units of the jth security outstanding. Then rj =

vj/

x

i-1 N

V, is the fraction of the jth security in the market portfolio defined earlier. The returns on the market portfolio RM are given by As Blume [4] and Fama [16]have pointed out, the market factor n is unique up to a linear transformation, and thus we can always change the scale of n such that Hence, with no loss of generality, we as- sume this transformation and reduce (3.4) to Now we saw earlier that the measure of systematic risk is cov(Rj, RM). By direct substitution from (3.2) and

+ bjr + ej,] [E(RM) 4- a 4- xxiei] 1

(3.5) into the definition of the covariance, cov (Rj,R~) = cov [E(R~) and Sharpe, and empirically tested by Blume [4]. The model is

dom where variable I is some uncorrelated index of market with I,and returns, Aiand uj is Bj a ran- are constants. The differences in specification in (3.2) are necessary in order to avoid the overspecification pointed out by Fama [I41which arises if one chooses to interpret the market index I as an average of se- curity returns or as the returns on the market port- folio M (cf. Lintner [37]and Sharpe [52, 561). That is, if I is some average of security returns, then the assumption that uj is uncorrelated with I cannot hold, since I contains uj. a

(3.6)

26

Reproducing Fama's argument directly, if we have the untransformed

XX,~;z market 1 factor ,

+ +

T* and j where the bj*are defined by Ri = E(Ri) b:~* ei, we can create Now Ki = E(Kj)

+ b , ~+

ej,

where

and

180 THE JOURNAL OF BUSINESS Hence, restating the results of the capital asset pricing model given in (2.7) in terms of the parameters of the market model, we have2" where (.) refers to the arguments in brackets on the RHS of (3.8). Now de- fine which is the measure of systematic risk in the context of the Gaussian market model. All previous discussion the interpretation of pli also regarding applies to P2j. However, (3.9) can be considerably simplified by noting that we can invoke several approximations and thereby elim- inate the strictly unobservable market factor nfrom the expression. The results of King27 [34] and B l ~ m [4] e ~ imply ~ that the market factor naccounts for approxi- mately 50 per cent of the variability z6 This is essentially the same expression as Lint- ner [37] arrived at, but as we have seen, and as Fama [14] has already shown, the

results of Sharpe origi- nally stated in (2.7) are in no way inconsistent with (3.8). 27 King

examined sixty-three securities in the pe- riod June, 1927, to December, 1960, by methods of factor analysis. He found that the

market factor on the average accounts for approximately 50 per cent of the variability of the monthly returns on the in- dividual securities, and various industry factors ac- count for another 10 per cent. We have ignored these industry factors in constructing the model, since they are relatively unimportant and their inclusion would introduce a great deal of additional com- plexity. Blume, using regression analysis, also finds that a market index accounts for an average of 50 per cent of the variability of the

28

monthly returns on 251 securities in the period January, 1927, to De- cember, 1960.

of u2(Rj) individual = bbj2u2(r) security

+ u2(ej), returns.29 and since Since the average bj is equal to unity,

the results of King and Blume imply that u2(ej) is roughly the same order of magnitude as u2(n). Let us examine the expression for u2(Rnf) in light of these facts. The last term on the RHS of (3.7) can be approximately expressed as

where - u3(e) is the average variance of the disturbance terms. Recall that since xiis the ratio of the value of the jth security to the total value of all securities, it must on the average be on the order of 1/N, where N is the total number of distinct securities in the market. Since there are more than 1,000 securities on the New York Stock Exchange alone, xj will be much smaller than 1/1000 on the aver- age,30 and thus (3.10) will be minute rel- ative to u2(n). Hence, Substituting for u2(n) in (3.9), we have 29

There is some indication in Blume's results, however, that this proportion may be declining in recent times.

ao There

are some firms, of

course, for which r, is much larger than 1/1000. Data obtained from Stand- ard & Poor's indicates that as of December 31,1964, the

four largest firms on the New York Stoclr Ex- change and their percentages of the total values of the Standard & Poor Composite 500 Index were: AT & T., 9.1 per cent; General hlotors, 7.3 per cent; cent. could Thus, take IBM,

k1964 the 3.7 largest &s per

.091, cent; value and and that even DuPont, the this fraction is an 2.9 over: per x, statement, since the 500 securities were obviously not

the total universe-which, of course, includes all other exchanges, unlisted securities, and debt in- struments as well.

181 RISK, CAPITAL ASSETS, AND El-ALUATIOX OF PORTFOLIOS For simplicity, let us define Substituting for E(Rj) from (3.9) and (3.8) into (3.2), we have Adding and subtracting zjn and on Rj = the RF(~- RHS of P2j) (3.14)

+P~~E(RM)

gives Noting definition that of RM Pzj from bj (3.5) ing, we get Rj = RF(~- P2j)

+ zj, and using simplify- the

+RAIP~~

Now we have an ex post relationship in which all the important variables are meas~rable.~~By assumption (3.3a), a1 Note that z our previous arguments j ~will be trivially u2(ei) g u2(r) small, r since u2(R,w) by and xi is on the average less than 1/1000.

Thus, and is unimportant. Note that

~zjxxiei i will be unimportant also, since by assumption the ej are independently distributed random variables with E(e,) = 0. We have already seen that the variance of this weighted average (given by [3.10]) will be minute. But since and its variance is extremely small, it is unlikely that it will be very different from zero at any given time.

E(ej) = 0. Thus, eliminating zjn and from (3.16) by the arguments of note 31 above, we see that to a very close approx- imation the conditional expected return on the jth security is given by Equation (3.17) is an important result. I t gives us an expression for the expected return on security j conditional on the ex post realization of the return on the market portfolio.32 Recall that equation (2.7), the result of the capital asset pricing model, provides only an expres- sion for the expected return on the jth security conditional on the ex ante ex- pectation of the return on the market portfolio. This result (eq. [3.17]) be- comes extremely important in consider- ing the empirical application of the mod- We now have shown that we can ex- plicitly use the observed realization of the return on the market portfolio with- out worrying about using it as a proxy az Of course, as far as the algebraic manipulations are concerned, we do not need the market model to get this result. However, the

implications of the re- sults derived in the absence of the market model are not consistent with the observed behavior of the world. That is, consider the forn~ulation in which T always equals zero. The ex post returns on the mar- ket portfolio would be given by But in the discussion above, we saw that the last term has expectation equal to zero and an infinitesi- mal variance. Thus, this

formulztion implies that the realized returns on the market portfolio mould never differ from the expected returns by any amount of consequence-a result clearly contradicted by the behavior of real world prices.

53

See, for example, the discussions in references 8, 28, 58,

and 66 regarding the problems associated with testing models stated in terms of ex ante rela- tionships on ex post empirical data.

182 THE JOURNAL OF BUSINESS for the expected return and without worrying about devising an ad hoc expec- tations-generating scheme. The measure of portfolio performance in the context of Gaussian distributions.- Using equations (3.16) and (3.17), we can now define an ex post measure of port- folio performance as But by our previous arguments, the quantity zjr will be minute. In addition, the likelihood of pzjZxie$ ever

being much different from zero is extremely small, since its expected value is equal to zero and its variance is close to zero (cf. n.31 above). By these arguments, we may ig- nore these terms in (3.18), and we have to a close approximation Figure 4 gives a geometric presenta- tion of these concepts. The point M represents the realized returns on the market portfolio, and of course its sys- tematic risk (plotted on the abscissa) is unity.34 The point Rp represents the re- turns on the risk-free asset, and the equation of the line RFMQ is Let turns the Ri point on any i represent portfolio i the and ex let post p2ibe reits level of systematic risk. Then the vertical distance combination of any between portfolio the risk-return i and the line the performance RFMQ in Figure of portfolio 4 is our i. measure of The measure 62 may also be interpret84

Note that 63 merely represents the specXc ex- pression for risk in the context of the infinite vari- ance market model and will

be defined below.

ed in the following manner: Let FMi be a portfolio consisting of a combined in- vestment in the risk-free asset

F and the market gree a2; may of portfolio risk be interpreted p2; as M the offering portfolio as the the difference same i. Now dein return realized on the ith portfolio and the return RFM~which could have been ket portfolio earned on FMi. the If equivalent 82; 2 0, risk the port- marfolio i has yielded the investor a return greater than or equal to the return on a combined investment in M and F with an identical level of systematic risk. It should be noted that since (3.18) is stated in terms of the observed return on the market portfolio, the performance measure 82i allows for the actual rela- tionship between risk and return which existed during the particular holding pe- riod examined. A discussion of the

criteria to be used in judging a portfolio's performance will be postponed until Section V, at which time the entire model will have been de- veloped. Meanwhile, in the next section we shall consider the extension of the model to a world in which the distribu- tions of security returns are non-Gauss- ian members of the Stable class. C. SYSTEMATIC RISK AND THE STABLE MARKET MODEL The model.-As mentioned earlier, there is considerable empirical evidence (Fama [12], Mandelbrot 1381, Roll [46]) indicating that distributions of security returns conform to the members of the Stable class of distributions which have finite means but infinite variances. How- ever, Fama [13] has shown that the mar- ket model can be used to develop a port- folio model analogous to the mean-vari- ance models of Markowitz, Tobin, and Sharpe in the context of a market in which returns are generated by non183 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS Gaussian finite mean members of the Stable family of distributions. Moreover, Fama [16] has also demonstrated that the Sharpe-Lintner capital asset pricing mod- els can be generalized to a market characterized by returns with infinite vari- ance distributions. The following discus- sion draws heavily on his extension of the asset pricing model. The reader is re- ferred to Fama-[l6] for proofs. We begin with a few brief On the pa- rameters of Stable distribution^.^^ EX POST RETUQNS

R

eters, Stable a, p, distributions 6, and y. The have parameter four param- a is called has range the 0

characteristic < a i 2. exponent and The special case of the Stable distribution with a = 2 is the Gaussian or normal distribution and is the only distribution with finite sec- ond- and higher-order moments. The fl with range - -1 5 36 For a much more complete description of prop- erties of Stable distributions, see the Appendix in Fama [12] and the references therein. ARA.4.-A measure of the ex post performance of a portfolio i

184 THE JOURNAL OF BUSINESS distribution. tion tribution /3 5 is 1 determines symmetric, is skewed When left, the when P = skewness and 0 /3 the > when 0 distribu- the of /3 < dis- the 0 the distribution is skewed right. We as- sume in the following discussion (as does Fama) that we are dealing

with sym- metric p = 0.

distribution^,^^ and therefore The parameter 6 is the location param- eter tions of with the /3 distribution, = 0 and 1 < and a for _< 2,6 distribu- is the expected tions with value 0 < or a 5 mean. 1, the For mean distribudoes not exist, but for distributions with = 0, 6 is the median. As Fama [13] has shown, diversification is meaningless in a tions market with a 1 characterized 1. In addition, by Fama distribu- [12] and Roll [46] find that estimates of the characteristic exponent a for common stocks lowing a

> 1. and Thus, that US we 1 <

Treasury also a _< assume 2 bills indicate in the fol- and therefore 6(R) The = final E(R). parameter y(y >

0) defines the scale or dispersion of the Stable dis- tribution. For the Gaussian distribution with a = 2, y = +(r2 where (r2 is the variance. earlier, when Unfortunately, a < 2 the as mentioned variance does not exist and analytical solutions for the exact definition of y are known only for several special cases; for example, for the Cauchy case (a = I), y is exactly equal to the semi-interquartile range.37 Fama and Roll [21] have demonstrated that 36 The

assumption of symmetry seems to be satis- fied quite well by the empirical distributions of se- curity returns stated in terms of

continuously com- pounded IV that the rates. solution Furthermore, to the "horizon" we shall see problem in Section im- plies that all returns must be measured as continu- ously compounded rates in order for the model to hold. Thus, the assumption of symmetry seems quite appealing. 3' Defined as half the difference between the .75 and -2.5fractiles.

for a in the range 1 < a 1 2, y corre- sponds approximately to the semi-inter- quartile range raised to the a power. The Stable market model again con- sists of equation (3.2) : with all the variables defined as before. However, in place of (3.3) sumed that T and ej ( j = 1, it 2, is .. now

.,N) asare independently distributed symmetric Stable variables all characteristic exponent having a, (1 < the a _< same 2). The location parameters of n and ej are, respectively, E(ej) = 0 (j = 6(n) 1, 2, = . E(n) .

. ,N), = 0, and 6(ej) their =

dispersion y(ej) conditions, (j = 1, the parameters 2, location .

. . ,N). parameter are Under y (n) these of

and Rj is E(Rj)and the scale parameter of Ri is given by Y(R~)= Y(n)\bjIa4-~(ej). (3.211 By the same arguments as in the finite variance case, the return on the market portfolio is given by and the scale parameter of the distribu- tion of the returns on the market port- folio is More significantly, Fama has demon- strated (by arguments directly analogous to those of Sharpe and Lintner presented earlier) that, given the assumptions of the Stable market model (and the previ- ously stated assumptions necessary to the Sharpe-Lintner model), the expected return on any security j will be given by 185 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS Equation (3.23) is directly comparable to the results given in (3.8), which were arrived at under the assumption

+ Y(RM)

of finite variances. (Note that in case a = 2> [3.23] reduces directly to [3.8].) Now define P3j = Y(a)bj

Y(ej) IxjIa-' , (3.24) which is the measure of systematic risk in the context of the Stable market mod- garding el. As before, PI, and all pzjalso previous applies discussion to P3,. reHence, we see that by making use of the characteristics of the market model, the capital asset pricing model can be extended to the case of infinite variance distributions where the concept of a co- variance is undefined. However, as in the finite variance case (and at the expense again of some degree of approximation), the expression for systematic risk (3.24) can be consider- ably simplified. As before, the results of King [34] and Blume [4] indicate that on the average the terms y ( ~ and ) y(ej) are of about equal size. Likewise as before, the average xj is on the order of 1/N, N being very large. Hence, the last term on the RHS of (3.22) is approximately equal to where - y (e) is the average scale parameter of have evidence the assumed disturbance (cf. Fama a > 1 terms. [12]) and indicates since But empirical since 1.6 we 5 a _< 1.9, this term will be small relative to y(n). Thus, Y(RM)

and substituting for y(n) in (3.24)) we have P3j bj

= 7(a) , (3.26)

+ (ej)1 xj I "-I Y(RM) (3.27)

Letting vj = 7(ej)1 xj1 a-1 Y(RM) ' may transform (3.23) from an ex ante relationship by Gaussian

. arguments -

into an ex post

relationship identical to those for the case examined earlier. The re- sult is Similarly, we also define the analogous conditional expected return on the jth security (or portfolio) as The measure of portfolio performance in the context of non-Gaussian Stable distri- butions.-The measure of portfolio per- formance in the context of infinite vari- ance Stable distributions is directly analogous to the finite variance situation and is given by 63j = = Rj Rj - - E(RjlR~f,P3j) [RF(~- B3j)

+ R~P3j.l (3.30)

Again, our previous arguments indicate that vjr and

will be extremely small and hence can be ignored, leaving the result 83jgg ej . (3.31)

Since the purpose of all the above has been to arrive at a measure of perform- ance, we shall consider the quantities 62j and 83j very closely in determining cri- teria for judging the performance of a portfolio. Our goal is to arrive at criteria for judging a portfolio's performance to 186 THE JOURNAL OF BUSINESS be superior, neutral, or inferior.38 Given to other perplexing questions. For in- the stochastic nature of the

model, it is stance, is the existence of a discrete hori- not surprising that this becomes a proba- zon interval consistent with a world in bilistic problem. However, in view of the which trading takes place almost con- fact that we are still working within the tinuously? If so, may we arbitrarily context of a single-period model, con- choose the beginning of the portfolio sideration of these questions will be post- evaluation period (ie, the beginning of poned until we have considered the mul- the horizon interval) to be any point in tiperiod model in Section IV. calendar time? How do we go about esti- mating the length of the horizon inter- IV. THE MULTIPERIOD HETEROGE- val? NEOUS HORIZON MODEL Thus,

the problem really consists of A. THE HORIZON PROBLEM the fact that the el imply that the simple linear relation-

assumptions of the mod- ~h~ reader will recall that, in deriving results of the capital asset pricing

ships of equations (2.71, (3.81, and (3.23)

the

model given in (2.7), (3.8), or

(3.23), it hold (if at all) only for holding periods was assumed that all investors had hori- of a particular length, and we wish to be zon periods of identical length. hi^ imcourse, that all trading in the 011

these results to evaluate portfolio

performance over all holding periods. We linear rela-

time as long as the returns R~ and R~ the '(~ro~er"

crucial

dimension of a period

model is valid, the following holds for the jth Or portfolio:

The

zon intervals among investors is

to the portfolio evaluation problem,

since we want to be able to evaluate a

portfolio's performance over any horizon

are expressed in terms of

For the moment, let us assume that

assets are priced as if the "true" horizon

interval in the market were H-periods in

will also apply

ing takes place almost continuously and

in which investors most certainly have

different (and overlapping) horizon pe- riods.

difficulty caused by unequal hori-

now intend to show that the

tionships of equations (2.71, (3.8), and

(3.23) hold for any arbitrary length of

plies, of

market takes place only at the beginning

and end of this horizon ~h~ ques-

tion we now face is whether this theory

to a market in which trad-

able to use the evaluation model based

length-where the

is some small, arbitrary interval. We

know then that if the capital asset ~ric-

interval. ~ ) ~ t

ing

if equations (2.71, (3.81, and (3.23) hold only for a particular dis-

crete horizon interval, then equations E(~R,.) H ~ F ( l- pj) = (3.18) and (3.30)

defining a measure of (4.1)

+E(HRM)~~ portfolio performance also hold only for where that

horizon interval. Furthermore, con- sideration of this horizon problem39 leads the expected

E(H&) = E(A~Wj/Wj)=

H-period rate of return for the jth se- as Formal definitions of these terms will be pro- curity; vided in

Section V. E(HRM)

and HRF are similar rates of re- as The existence of what is here called the "hori- turn for the

market portfolio M and zon problem" became clear after several discussions the riskless security F; and with the members of the Finance Workshop. Iam es- pecially indebted to Professor Fisher for helping me On the

to see the problem. particular

@j =

fij' Or P3j) depending

context in which we choose

187 RISK, CAPITAL ASSETS, AND EVALUATION OF PO RTFOLIOS to interpret the concept of systematic risk.40 By equation (3.16) and the arguments given in note 31, we also know Now consider the rate of return NR on ~ the jth security over an arbitrary N- period holding interval (where we as- sume N to be an integral multiple of H) : As long as (1) RF ~and E(Rnft) are con- stant through successive H-period inter- vals, (2) the RM and ~ ejtare distributed independently through time,41 and (3) the ejtand RM are ~ inde~endent,~~ the expected N-period returns, E(NR~), are given by

+B~E(HRM)I~'' where X 1

= H/N.

pansion certainly The reader of non-linear the will RHS note will in that

pj, involve since (4.4) @' is the most and excross-product terms containing Pi. Hence, it is clear that the simple linear relation40

Since the distinction between the three alter- native interpretations of pi are unimportant to the discussion and 3 used here, in

Section we simply 111. ignore the subscripts 1,2, 4' See Fama 1121, Roll 1461, and the papers re- printed in Cootner [9]for evidence on the serial in- dependence of security returns. that

42

cov By the (RMI,ejr) construction =

xiu2(ei).But

of RM (eq. since

[3.5])we xi is on know the average ance term smaller for the than sake 1/1000, of simplicity we ignore in deriving this covari-

(4.4). Thus, there is a slight degree of approximation in equation (4.4). ship of (4.1) will hold only for a holding interval which is H-periods in length. This is the essence of the "horizon prob- lem." E(HR~)= Solving = [I (1 (4.4)

+E(NR~)]'- - B~)HRP for E(HRj), P~E(HRM).

1 we have (4.5) But now what are HRP and E(HRM) on the RHS of (4.5) in terms of observable N-period rates? Under the assumptions of constant expectations and independ- ence E(HRM)= through HRF= time, [l [1

+NRFI' +E(NRM)I" we have

- 1 , 1 . (4.6) (4.7) Hence, rewriting (4.5) in terms of the po- tentially observable quantities given in (4.6) and

(4.7), we have: [I X f E(NR~)J" [(I

1) . (4.8) Now this relationship still holds if we di- vide ['

+ + NRF)' +E(NRM)I'- 1 -

= 11

(1 - Bj)

+E(NR')IA both sides - by 1 = A:

(1 - X

Define E(R;) X = [I

+ +E(NRj)I' E(NRM)I" X

X 1 - l (4.10) and R; and E(R;) likewise. The trans-

formed rates, R*, are just nominal N- period rates with H-period compounding intervals,43 and in terms of this notation (4.9) becomes 43

Note that +E(~Rj)I'-

X

=

$E(HRj) , (4.10a)

where, as before, E(HR,)is the expected rate of re-

1% THE JOURYAL OF BUSINESS Thus, equations (4.9) or (4.11) tell us that the simple linear relationship of (2.7)) (3.8), and (3.23) will hold for returns cal- culated over a holding period of any length as long as we state the returns in terms of the "proper" compounding in- terval. But of course this result is empiri- cally meaningless unless we can somehow determine the "proper" compounding interval, that is, unless we can deter- mine H. We shall now turn our attention to this question. B. SOB COKSIDERATIONS REGARDING THE

"MARKET HORIZON INTERVAL" There

are sev eral arguments which lead us to conclude that the market horimaking portfolio decisions and trading in the market. In addition, if we require the market to be in equilibrium at each instant, it must follow that the resulting "market horizon" is instantaneous. That is, prices behave as though all investors had instantaneous horizon periods. On the basis of these arguments, let us consider the limit of (4.9) as the length of the horizon interval, H, goes to zero. Since, through the use of L'Hospital's rule,45

limit x+o

-= xX

- X 1 log, x , (4.12) we have limit H-+O [l

+ E(NR~)]~'~ HIN - 1 = loge [I +

E(~7Rj)l zon interval is instantaneous. First, within the strict confines of the assump- tions of our model regarding the perfect liquidity of all assets (ie, transaction costs are zero), all investors will have instantaneous horizons44 as long as port- folio evaluations are costless. Although these zero cost assumptions most cer- tainly are not met in the real world, it may very well turn out that market prices behave as though they were ful- filled. Second, an instantaneous "market horizon" would also be consistent with the assumption that an infinite number of investors all have non-zero horizon periods, but these horizon periods are distributed such that at every instant of time a large number of investors are turn on the jth security expressed in terms of an H- period compounding interval. Thus, the quantity (N/N)E(HRj) is just the expected nonzkal N-period rate of return on the jth security under the assump- tion of an H-period compounding interval. 44 See Fama and Miller [20] for a discussion of this point.

and hence E(R;) for H/N log, [I close

+ E(NR~)]. to zero

(4.14)

Thus, as long as the market horizon in- terval, IT, is very small, we may use the log form (4.13) as a very good

approxi- mation to (4.9).46 45

L'Hospital's rule: limit k-10

= limit h-+O xX

logex

------ xX - 1 = limit x-+o dh

= log,

x.

46

The reader is urged to remember that the loga- rithm is a very good approximation to the

transfor- mation given by (4.10), since a literal interpretation of the logarithmic form will lead to difficulty in in- terpreting We also arguments note that log, presented [I

+ E(NR?)] later. may be given an intuitive interpretation in the same manner as

Whereas E(R7) is the nominalN-period (return under the (seen. assumption 43 below), of log,(l an H-period

+ NR compounding ~ )is

also the N-period interval rate of return but expressed in terms of continuous compounding or an infinitely small compounding interval.

189 RISK, CAPIT.\L ASSETS, AND EVALUATION OF PORTFOLIOS

An aggregatioz problem.-Consider now the nominal N-period expected re- turns [E(R~)] on a portfolio consisting of K securities where in (4.10). Let yj E(Ri)is (J' = 1, 2, . . defined . ,K) as be the fraction the jth security of the and portfolio let f(Rj)= invested R;. in Then, noting that and that we have from (4.9) and (4.10) that Since the expansion of the bracketed terms on the RHS of (4.15) will involve cross-product terms containing the pj, it portfolio, is clear that P,, ual coefficients the systematic risk of the is a pi, function the riskless of the rate, individ- and the expected return on the ma rket port- folio. p,, In fact, the risk of the portfolio, will be

strictly stationary through time only if the assets of the portfolio are continuously redistributed to maintain the fractions yj at their original values. Hence, one must be extremely careful about aggregating the risk coefficients of individual securities to obtain a portfolio. However, given the P, risk of (which can be estimated for the portfolio as a whole), we may E(RZ) = (1 - write

+ PpE(RL) . (4.16) Thus, as (4.16) shows, the expected N- period returns on the portfolio can be expressed as a linear function of the risk- free rate and the expected return on the market portfolio as long as we express these N-period rates in terms of a com- pounding interval of H-periods. We shall now consider some arguments regarding the length of the "market-horizon inter- val," H. C. IMPLICATIONS OF THE HORIZON SOLUTION FOR THE MEASURE OF SYSTEMATIC RISK One of the most important implica- tions of the horizon solution given above and the restatement of the capital asset pricing results in the form of (4.1 1) is the fact pj, may that be the used measure for a of holding systematic period risk, of any length, N. That is, we shall now show that as long as H, the "market horizon," is the instantaneous, estimate of systematic the expected risk, value pj, will of be independen t of the length of time (N) over which the sample returns are calcu- lated47 value pj. and This will result, be equal of course, to the implies true that we can use a given measure of risk for a portfolio to evaluate the portfolio's performance over a horizon of

any length. In addition, it means that we need only obtaining concern ourselves the "best" with estimate the problem of pi for of any security or portfolio j, and any in- vestor, regardless of his decision horizon, will be able to use that measure in ar- riving at an optimal portfolio decision. The Gaussian case.-Consider first the definition of systematic risk in the con- text of Gaussian distributions. Recalling the specifications of the market model That is, given the total calendar time interval of observation (and ignoring the effects on sampling error for the moment), the solution implies the ex- pected value of the estimate of will be independent of the length of the subintervals over which the sample returns are calculated, be they 47

daily, month- ly, quarterly, etc.

190 THE JOURNAL OF BUSINESS given in (3.2) and (3.3), it was shown that But since it was also shown in Section I11 that a2(RM)r a2(r), (3.9) reduces to which is the form used in deriving the measure of portfolio performance. We are

Pzj,

concerned given by

cov here (Rj, with &)/a2(&). the estimate We of emphasize that (3.9) and (3.12) were de- rived strictly within the confines of a single-period model within which the relevant covariances and variances refer to the properties of the set of probability distributions on one-period returns. With- in this context, of course, there is no am- biguity regarding the interpretation of

cov

{ cg [exP

H&)]

H, the market horizon; it is determined by the length of the period. value We of now the wish following to consider estimate, the expected b2j, de- rived from a sample of N-period returns observed over time:48 First note E(P2j) that = 1 -- cov NRj (Rf,R&) (from

+ a2(R*M)

. which (4.17) R; is derived) is given by

NIH

for integer N/H. Furthermore, we know that limit H+O = NIH

n (1 + ~Rjk)

k-1

Using (4.10)) (4.12)) (4.17), and (4.18), the assumptions of stationarity and serial independence, and taking the limit, we have

limit H-tO EG~,) = limit H-tO

H-+O

-

ex^

(x

H/N

{ [ex, - 4 (limit

- HIS "I"

If~i*)]~'~'

COY

(x

HRMR)]

f

- 1 [exp

(:g

.RMk)] f If'" - 1 , limit H-+O - [exp

1 H/N

(z

HRM~)]~'~

1

HIN -

1 u2{

- 11 J

limit [ exp ( Z ~ ~ ~ k ) ] " ~ - l ]

H-tO H/N 48

We assume, of course, that the probability distributions generating the sample observations are stationary.

191 RISK, CAPI'l'&kL ASSETS, AND EVALUATION OF PORTFOLIOS

Thus, as long as the sample data are transformed pected value according of the estimates to (4.10), of the

pzj will ex- be independent of the length of time over which the returns are calculated. If this result is empirically true, it is ex- tremely important. Our earlier results imply that we can evaluate portfolios over a horizon of any length, even if different from the market horizon. The resul ts of (4.19) imply that we may also estimate the systematic risk of the port- folio without regard for the particular horizon interval for which we intend to use it. Hence, we may calculate the measure of systematic risk on the basis of the most efficient sample available, whether it be daily, monthly, or yearly data,49 and all investors, regardless of horizon length, can use it in evaluating and selecting portfolios. lem The of infinite estimating variance Psi in case.-The the context prob- of non-Gaussian Stable distributions re- duces to the same result as above, except that the economic interpretation is slightly different from that in the finite variance case. The difficulty arises, of course, because the covariance is not de- fined in this context. However, Wise 1681 has shown (for the case of non-stochastic regressors) that as long as a > 1, the leastsquares estimates of bj in the Stable market model are unbiased and consist- ent although not efficient. In addition, Monte Carlo evidence presented by Fama and Babiak [17] suggests that the use of leastsquares procedures in Stable models like that of (3.2) is not complete49

Given the total calendar length of the sample interval, purely statistical considerations would in- dicate using the smallest

observation interval pos- sible in order to maximize the number of observa- tions. However, gathering daily data will usually be far more expensive than gathering monthly or quar- terly data, and one has to take these costs into con- sideration when deciding on the "optimal" sample size and interval of observation.

ly inappr~priate.~~ Thus, in light of this evidence, we define our estimate of bj to be b. 1 -- C

t"' T

(R;t -

e)(a:7

(4.20)

t=l

3') T C(nt

- T*)~

where T is the total number of observa- tions and the barred variables represent mean values.

By the arguments given in the model, derivation Paj g bj. of Thus, the all Stable we need market now is a measure of the market factor n.But King [34, p. 1901 found that explicit esti- mates of the market factor (obtained with factor analytic techniques) were correlated .97 with the Standard & Poor Index for the period 1927-60. As we shall see in Section VI, the Standard & Poor Index is the index which most close- ly meets the definition of M, the market portfolio, so on this basis we rewrite (4.20) as

bj C(R~~

T

-R:)(R$,- &) ;'=I

~ ( R - Z 1 ~ 7 ~ ) ~ (4.21) t=l D. TI- MEASURE OF PORTFOLIO PERFORMANCE

The discussion above indicates that the measure of risk derived in the con- text of a single-period homogeneous hori- zon model will extend quite readily to a world in which trading takes place continuously and where investors have heterogeneous horizon periods. All we need do is restate (3.18) and (3.30) in 60 There has been very little investigation into the properties of alternative estimators in stable models such as ours, and until additional insights are ob- tained, we are forced to proceed with least-squares procedures. However, recent work by Robert Blatt- berg and Thomas Sargent ("Regression with Pare- tian Disturbances: Some Sampling Results'' [un- published mimeographed paper, Carnegie-Mellon University, April, 19681)indicates that least-squares procedures may be quite acceptable for small sam- ple sizes.

192 THE JOURNAL OF BUSIXESS terms of the transformed returns R* given by (4.10). As shown in Section

IV-C for Bzj above, and Paj the are estimating identical-the procedures only difference being in the interpretation of the result. Thus, to simplify the exposi- tion we shall henceforth cussion in terms of script either 2 pzj or or 3)) Paj where as pi pj (without may couch represent the a dis- subthe reader pleases. At any point where confusion may arise, we shall revert to the explicit notation.

By applying the arguments of Sec- measure horizon tion

"

= I11

R: solution - to of E(R; performance

equation IRb,p,) is obtained (4.11), implied as the revised by the where to either 6,; is to defined be interpreted in (3.18) analogously or as aSi de- fined pretation in (3.30)) of

pi. The depending variable on ej' the is analo- intergous to the disturbance ej defined in equations (3.2) and (3.3). V. THE ELTALUATION CRITERIA AND THE CONCEPT AND MEASURE- MENT OF EFFICIENCY A. THE EVALUATION CRITERIA A

measure of portfolio performance which provides a measure of a manager's ability

to pick "winners" was developed in the preceding sections, culminating in the final form given by equation (4.22). The problem we address at this point is the determination of the criteria by which we judge the performance of any particular portfolio. In Part B of this sec- tion we shall derive a measure of a port- folio's "efficiency," and in Part C we shall discuss the relationship between the measures of efficiency and performance. Since all the assumptions made in Sec- tion I11 regarding the disturbance terms ej also apply to ej*, we are led quite natu- rally to the following criteria for the evaluation of an estimate (or series of estimates) 6j; for a particular portfolio over some time period t. Criterion for n neutral it^)."--A port- folio's performance will be defined as neutral if its historical returns are equal to those which the capital asset pricing model implies it should have earned given its level of systematic risk. For- mally, this means the results should meet the following conditions: That is, we expect the portfolio to ex- perience returns through successive hold- ing periods which will cause it to fluctu- ate randomly about the market line RFMQ portrayed in Figure 4. Thus, a neutral portfolio is one

on which the returns are no better or worse than those which could have been earned by a comparable nai've FM portfolio. A neutral portfolio may also be interpreted as one which does no better or worse than that which could have been achieved by a randomly selected port- folio with identical systematic risk. Criterion for "superiority."-A superi- or portfolio will be defined as one which, through successive holding periods, re- alizes returns such that E(6;) > 0 . ( 5.3)

Thus, a superior portfolio is defined as a portfolio whose returns are consistent- ly greater than those implied by its level of systematic risk. Hence, the returns on such a portfolio would be greater than RISK, CAPITAL ASSETS, AND EI~rlLUrlTIONOF PORTFOLIOS 193 those which could have been earned by a random selection buy-and-hold policy or by a na'ive investment in an FM port- folio having identical systematic risk. Recalling our earlier discussion in Sec- tion I regarding the martingale hypoth- esis, it is clear that (5.3) also defines the criterion for judging a portfolio manager to be a superior analyst. A portfolio man- ager who possesses superior economic in- sight and thus the ability (1) to forecast some of the factors affecting future dis- turbances (e:) for particular securities or (2) to make better than average fore- casts of the future realizations on the market factor T , will be able to create a portfolio which consistently dominates the market line RFMQ of Figure 4. We might mention here that the existence of portfolios satisfying (5.3) is inconsistent with the strong form of the martingale hypothesis given by (1.2). Criterion for "inferiority."-We define an inferior portfolio to be one which, through successive holding periods, re-. alizes results such that it is consistently dominated51 by the market line RFMQ of Figure 4 and thus has The martingale property of security price movements implies that the best estimate of future prices (barring superi- or information) is merely the present price plus a normal expected return. Since any na'ive investor or portfolio 51

The exact meaning of "consistently domi- nated" is left undefined at this point and will be con- sidered below in the context of the

empirical results. I t will suffice to say at this point that a portfolio can be above or below the efficient boundary either be- cause of random factors or because the portfolio is systematically better or worse than the market port- folio. In addition, if one is examining many port- folios, it is reasonable to expect some of them to be consistently better or worse during the sampling pe- riod for purely random

reasons. A detailed discussion of this point is contained in Jensen [32].

manager in the market could easily fol- low this forecasting procedure and ex- pect, on the average, to do as

well as the market as a whole, we conclude (if the strong form of the martingale hypothesis is correct) that an inferior portfolio can exist only because the portfolio managers pursue activities which generate ex- penses. These expenses must be paid out of income, and thus the portfolio returns are reduced. It should be noted when evaluating mutual funds that there are expenses generated in the provision of services which benefit shareholders (the provision of bookkeeping services is an example), and the value of these benefits to share- holders should be taken into considera- tion. However, there may be other un- necessary expenses generated which cause the returns to be lower than ex- pected. For example, there may very well be portfolio managers who pursue ac- tivities such as attempting to forecast security prices (and trading securities on the basis of these forecasts) while they are unable to increase returns enough to cover their research and commission ex- penses. B. THE CONCEPT AND MEASUREMENT OF EFFICIENCY TIze concept of e$iciency.-The reader is cautioned to beware of confusing the above definitions of performance with the concept of e$iciency in the Markowitz- Tobin-Sharpe sense. An efficient port- folio is one which provides maximum ex- pected return for a given level of "risk" and minimum "risk" for a given level of of expected return. It is important to note here that "risk" in the definition of efficiency refers to the total risk of the portfolio and not just its systenzatic risk (which must always be less than or equal to a portfolio's total risk). Under the as194 THE JOURNAL OF BUSINESS sumptions stated in Section 11, it was shown that any efficient52 portfolio r will satisfy where a(x,)/a(RM) is the total relative risk of the portfolio E. Recall that the re- sults of the capital asset pricing model (given in [2.7]) merely state the returns which should be expected on any asset given its level of systematic risk. We emphasize that if the capital asset pric- ing model is valid, (2.7) applies to any asset or portfolio. On the other hand, (2.6a) will be satisfied only by efficient portfolios as portrayed in Figure 2. The boundary of the opportunity set, the line RFMQ in Figure 2, is given by equation (2.6a). The only portfolios satisfying the requirements for efficiency lie along this line, and (in the absence of superior in- formation about future security returns) all other feasible portfolios lie to the right and below this line. It should be noted that

we are now abandoning the assumption of homo- geneous expectations. The reader should now interpret the opportunity set por- trayed in Figure 2 as the set which would be determined by knowledge of only the parameters of the market model for each security and the parameters of the dis- tribution on the market Any investor or portfolio manager in possession of information which enables him to (correctly) form expectations on a and ej which are non-zero will be able to form portfolios which dominate the na'ive 6%

We shall assume throughout the following dis- cu~sionthat security returns are normally distribut- ed. Stable We distributions

shall deal with at the the end case of Section of non-Gaussian V. 63 That is, we assume knowledge of only E(Ri), j3j, E(ej)= 0, u2(ej), E(r) = 0, and u2(r).

no-superior-information opportunity set. We shall henceforth use the word e$cient to refer to this "na'ive" concept of effi- ciency and i11 particular will not use it to refer to the set of "dominant" portfolios which any i~zdividualinvestor might ob- serve given any special information he might have regarding the future realizations of the market factor a and the dis- turbances ej. Within this context, then, (2.6a) gives us the expected returns on any efficient portfolio r conditional on the expected returns on the market portfolio and the total relative risk of the portfolio. (The reader is reminded that it is implicitly assumed in this definition of efficiency that

.. . ,N].)

E[a]= 0 and E[ej]= 0, = 1, 2, Let us now consider the derivation of an expression for the expected return on any

ejicient portfolio E conditional on the realized returns on the market port- folio Adding rather pea

+ than e,

to the both expected sides of returns. (2.6a)) we have and since for all ejicient portfolios we have, from (3.2) and the fact that 6, b, (by the arguments given in Sec- tion 111)) that to a close

approximation Using (5.7)) we can write (5.5) as 195 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS But since, by (3.5) and the arguments of note 31 below, rRM- E(RM) , (5.9) we can substitute into (5.8) and arrive at Note that where rj is the product-moment correla- tion coefficient between the returns on the jth portfolio and the returns on the market portfolio. B

Using (5.1 1)) adding and subtracting P.RF on the RHS of (5.10) and rearrang- ing, R. rRF we have

+

(RM- for all RF)P~

efficient portfolios Now, since E(ej) = 0 for all j by (3.3a), we have E(e,) = 0 and EIREI rRF E(RM))RM,P~,U(R~)/~(RM)]

+ [E(Rnr) + (RM- - RPI@.(: RF)P€ - 1) (5.13) Equation (5.13) is an important result. It gives us the expected return on any efficient portfolio E conditional on the realized returns on the market portfolio, its systematic risk, and its total relative risk. But note also that we are left with a term involving E(RM) which indicates that we cannot define efficiehcy without taking into account the ex ante expected returns on the market portfolio. In

considering this result, note that the first two terms on the RHS of (5.13) are identical to those in (3.17) used in the definition of "performance." These two terms tell us what the portfolio should earn given its level of systematic risk. However, if the portfolio is also to be efficient, its returns must be higher by an amount given by Let us define a perfectly diversified port- folio as one for which the total risk of the portfolio is equal to its systematic risk, and hence one for which rj = 1. Now the quantity is just the increment in the portfolio's risk (measured, of course, in a relative sense) which is due to the lack of perfect diversification. In the absence of transactions costs, a rational manager would never hold an imperfectly diversified portfolio54 unless he believed he could forecast future se- curity prices to some extent. If he be- lieved he could forecast future prices successfully, it would most certainly be rational to sacrifice some

diversification and concentrate some of the portfolio's holdings in those select securities with the highest expected "abnormal" re- But to the extent that the man- ager accepts additional risk in acting on his forecasts, he must earn higher re- turns to compensate for it or the port- folio will be inefficient in the sense that a perfectly diversified FM portfolio with the same (higher) level of total risk would ous arguments, earn higher pj[(l/rj) returns. - By 11 represents our previthe incremental risk due to the lack of defined s4 That in Section is, anything 111. other than an FM portfolio E(q) That > 0 are is, those largest. for which the manager believes

196 THE JOURNAL OF BUSINESS perfect [E(RM)-RF] diversification, in (5.14) is and the the expected term premium per unit of risk. Thus, (5.14) represents the additional returns which must be earned by an imperfectly diversi- fied portfolio in order for it to be efficient. Before going on to define an explicit measure of efficiency, let us digress brief- ly to provide an intuitive interpretation of the foregoing concepts and issues. In considering the definition of a measure of efficiency, one is tempted to simply replace the term E(RM) in equation (2.6a) with RM and interpret the resulting ex- pression as one defining the expected re- turns on any efficient portfolio condition- al on the realized returns on the market p~rtfolio.~Vhatis, it is tempting to simply relabel the vertical axis in Fig- ure 2 as R instead of E(R) and to inter- pret the line RFMQ as representing the locus of points about which all efficient portfolios will scatter. It is clear from equation (5.12) that the realized returns on all efficient portfolios will not scatter about such a simple straight line in the ex post return and risk plane, since the expected returns on the

market portfolio also appear in the equation. To see the issues more clearly, consider the three situations portrayed in panels A, B, and C of Figure 5, in which the ex post re- turns of a hypothetical portfolio k are plotted against its systematic risk, Pk, and total relative risk, u(Rk)/a(R~). The panels differ only in the assumed values of the realized returns on the market port- folio. In panel A it is assumed that RM = RM RM> E(RM), < E(Rnf), E(R,M). in panel and B it in is panel assumed C, that that

Let us now consider our hypothetical portfolio k with a level of systematic risk of .5 and managed by an individual who or implied 66 Indeed in something references similar 3, 7, 8, to 27, this 28, is 54, suggested 55, 58, and 66.

attempts to forecast the future prices of individual securities. In attempting to incorporate his forecasts into the port- folio, the manager is forced to accept additional (and diversifiable) risk in the portfolio. We assume for illustrative pur- poses that this results in a total relative risk of .75 = u(Rk)/u(RM). NOW equation (5.13) (with [5.15]) indicates that in order for this portfolio to be efficient the manager's forecasting efforts must in- folio crease by the an expected amount returns equal to on [E(R.,f) the port- - RF] (.25),

which is simply the amount of incremental (and diversifiable) risk in the portfolio multiplied by the ex ante price per unit of risk. For illustrative purposes, let us also assume that our hypothetical manager actually cannot forecast any

better than a random selection policy, and thus he is reimbursed in the market only for the amount of systematic risk he has taken (in this case, .5). Let us also assume (without loss of generality) that the error

terms, e, are zero for all portfolios we shall consider in our example. Now the points labeled a in panels A, B, and C denote the ex post returns, Rk, and systematic risk (.5) of

our portfolio under the three different assumptions re- garding the value of the realized return on the market portfolio M. The points labeled b in the figures denote the ex post returns and total relative risk (.75) of the portfolio k. The points labeled c in the figures denote the ex post returns, RI, earned by all imperfectly diversijied efi- cient portfolios with a total relative risk of .75 and a systematic risk of 0.50. Fi- nally, the points labeled d in the figures denote the ex post return, R,, of all perfectly diversi$ed

e8cient portfolios with a total relative (and systematic) risk of 0.75. Di. panel A, where we have assumed RjM= E(Rnf), there is no difficulty at all 4J wl

zcL

5 +J x a, we:

198 THE JOURNAL OF BUSINESS in interpreting the diagram. All efficient portfolios (whether perfectly diversified or not) will scatter along the line RFQ when their returns are plotted against their total relative risk. It is clear that our portfolio k (which by assumption is inefficient) at point b appears to be in- efficient, since it is dominated by point c, d. assumed However, R, in < panel E(R,), B, a where simple we inter- have pretation of the

"opportunity set" given by the solid line RFQ is not valid. That is, all efficient portfolios will not lie along perfectly this line; diversified only those portfolios (ie, for which which pj are = u[Rj]/a[R~]) will lie along this line. The point b in panel B again denotes the ex post returns and total relative risk of our hypothetical portfolio. But, contrary to the situation in panel A, point b appears to dominate point d, which represents the ex post return and risk of a perfectly diversified efficient portfolio. This im- pression is misleading. Point b looks bet- ter than point d only because the real- ized returns on the market portfolio were below the risk-free rate. The realized re- turns on the manager's imperfectly di- versified portfolio were higher on the perfectly diversified because pk = .5 while P, portfolio than those p, = .75. It is clear that portfolio k cannot be efficient, since it is dominated by a perfectly diversified portfolio with identical re- turns and total relative risk of 0.5. There- fore, regardless of the realized returns on the market portfolio, if the imperfectly diversified portfolio k is to be efficient, the manager's forecasting ability must be good enough to reimburse the holders of the portfolio for the additional diversifi- able risk taken. This increment in return is precisely the quantity given by equa- tion (5.14), and the dashed line acZ in panel B denotes the ex post returns which must be earned by any imperfectly diversified portfolio with systematic risk equal to .5 in order for it to be effi- cient. The slope of acZ, of course, is de- termined by the ex ante risk premium per unit of risk. The difference between .75 and .5 is

the incremental risk, and the difference between RI and RI, is the in- cremental return necessary to compen- sate for this risk. (The reader will note that the line acZ is just one of an entire family of such lines emanating from every The point case in on which the line R, RFQ.)

>

E(R,) is por- trayed in panel C. Again, the level of systematic risk and the opportunity set RFQ determine the ex post return Rk on our portfolio, and the ex ante risk premi- um determines the slope of the line acZ. The point c represents the point at which an imperfectly diversified efficient port- folio would lie with returns RI. The reader will note that if our hypo- thetical portfolio with a total relative risk of .75 were perfectly diversified (ie, @k would = .75 have also), earned and therefore returns R, efficient, < RI; in it the situation

portrayed in panel B. Therefore, one might be tempted to con- clude that the investor was actually better off with the imperfectly diversified and inefficient portfolio with returns R,. In a sense this is true, but one must be very careful about giving the manager credit for this situation, which must be due solely to good luck. That is, if he were forecasting R,w to be less than Rp, he would certainly have been far better off to hold only the riskless asset rather than hold an imperfectly diversified port- folio. In addition, as previously men- tioned, it is misleading to compare the imperfectly and perfectly diversified portfolios along the return dimension. It is clear that the holder of the imperfectly diversified portfolio k could have earned the same returns Rk with a perfectly diversified portfolio with total relative 199 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS risk of .5 (rather than .75). Thus, the in- vestor gained nothing from accepting this needlessly higher level of risk. Moreover, if we consider the point d in panel diversified C, which efficient represents portfolio the with perfectly P, = cr(R,)/a(R,) = .75, it is clear that the returns than Rk.Hence, R, on such in a this portfolio case, the are investor greater is not better off for having accepted the higher diversifiable risk for which he is not compensated. In addition, the reader should note that in all three panels, both points c and d represent the locations of e$icient port- folios with the same degree of total rela- tive risk. Their returns will be coincident only when R, = E(R,), and the dif- ferences are due solely to the random and unpredictable factors determining the returns on the market portfolio. The I important and p are point ex ante is that efficient both by portfolios defini- tion; yet they may have vastly different RM3 ex post E(RM). returns depending on whether Now we shall consider the definition of a measure of efficiency and the evalu- ation criteria to

be applied to it, and in Part C of this section we shall consider the relationship between the measures of performance and efficiency. A measure of e$ciency.-Utilizing (5.13) and taking account of the horizon solution, ciency, yj", let as us define a measure of effiCriterion for "e$iciency."-The argu- ments above imply that an efficient port- folio can be defined as one which through successive holding periods realizes re- turns such E(y:) that5'

= E(e:.) = 0

. (5.17)

For the moment we shall ignore the prob- lems associated with obtaining empirical estimates of E(R&),

which of course are necessary for the estimation of y* We shall consider this point below.

.

Criterion for "ine$ciency."-An in- efficient portfolio will be defined as one for which E(y;) < 0 (5.19) As noted above, it is perfectly possible that a manager is able to forecast se- curity prices to some extent and still manage to create an inefficient portfolio. That is, it is possible that he might not earn returns sufficiently higher than a buy-and-hold policy to adequately com- pensate the holder of the portfolio for the additional risk taken due to the lack of perfect diversification. Criterion for "supere$ciency."--4 portfolio will be defined to be supereffi- cient if E(yf) > 0

. (5.20)

One may question the possible existence of a superefficient portfolio, since we usually think of the efficient set of port- folios as dominating the set of feasible where, as before, e; is defined analogous- portfolios. However, recall that earlier lv to eauations (3.2) and (3.3). We con- we defined eBciency in terms of the op- sider now the criteria folio to be ('e$icient,7' supereficient. " for judging a port- "ineficient," or

"

Note also that since r3

GZ e*

we know also that

E(y;,,y;,,,) = E(e;,,e;,,) = 0 . (5.18)

200 THE JOURNAL OF BUSINESS portunity set which would be determined by knowledge of just the parameters of inefficient. E(y3j*)<0 also. That That is, is, if by E(6j') (5.21) < 0, and then the the market model for each security and fact that the last term on the RHS of the parameters of the distribution on the market factor. Hence, it is certainly pos- know

always

(5.21)

that must yj*

< 6,: be positive,6g we

always. sible for a manager with superior infor- mation or insight to create portfolios which dominate this "naive" opportu- nity set. C. THE RELATIONSHIP BETWEEN THE MEAS- URES OF EFFICIENCY AND PERFORMANCE The case of perfectly diversijied portfolios.-The concept of efficiency is ex- tremely important, and it behooves us to investigate its relationship to the measure of portfolio performance, 6*, suggested above. We have seen that a portfolio may be classified as inferior, neutral, or superior, and its classification depends on the manager's forecasting ability and the amount of expenses gen- erated in the management of the port- folio. If a portfolio is either inferior or neutral, we can make unambiguous in- ferences regarding its efficiency. From the definition of the measure of perform- ance, 6*, given by (4.22), and the defini- tion of the measure of efficiency, y*, given by (5.16), we see that The second term on the RHS of (5.21) is just the adjustment for the diversifiable risk in the portfolio and must be taken into account in measuring efficiency. Consider for the moment the case of a perfectly diversified portfolio. Since for such a portfolio rj = 1, we know the last term on the RHS of (5.21) is zero. Thus, for a perfectly diversified portfolio, yj* = 6;, and the measure of perform- ance is also a measure of efficien~y.~~ The case of inferior portfolios.-If a portfolio is inferior, then it must also be

The case of superior portfolios.-The only case in which some ambiguity exists between the measure of performance, 6*, and the inference regarding the efficiency of forecaster a portfolio with is in E(6*) the > case 0. of a superidr We can see from (5.21) and the definition of effi- ciency (5.17) that the superior portfolio will also be an efficient portfolio if That is, if the positive benefits of the forecaster's ability are just large enough to offset the effects of any imperfect diversification (represented by the differ- ence between rj and unity), the portfolio will be efficient. In the situation where we define the portfolio to be supereffi- cient, since the benefits from the superior forecasting ability are more than enough to offset the effects of the imperfect diversification. Finally, in the situation where 68

One might ~vonder at first whether a perfectly diversified portfolio can possibly be inefficient. The ansver to such a question is yes, since all a manager of a perfectly diversified portfolio need do to make it inefficient is to generate expenses and therefore low- er its returns. 69

Since, under the assumption of risk aversion, E(RM) must be greater than risky ri 2 0al~vays assets and, (cf. as Blume an empirical [4] RF and or Fama no fact, one et Sj would al.[19]). 2 0 hold and

201 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS the portfolio is inefficient, since the bene- fits from the superior forecasting ability are not large enough to offset the effects of imperfect diversification. It should be noted that, while a port- folio satisfying (5.24) is inefficient in and of itself, it most surely is a desirable in- vestment if treated as a single asset in the context of an efficiently diversified portfolio. izes that ~(6,:) That > is, 0 the may investor combine who an real- in- vestment in that portfolio with invest- ments in

other assets and hence create a portfolio which is in a sense supereffi- cient. In effect, as soon as an investor realizes the superiority of a manager's forecasting ability, he may treat that ability as an additional asset in the op- portunity set and thereby enable the effi- cient set (as viewed by himself) to shift upward and to the left.60 It is also interesting to note that this discussion regarding efficiency implies an economic justification for two very dif- ferent types of funds: (1) funds which concentrate on maintaining perfectly diversified efficient portfolios and (2) special purpose funds that concentrate on being superior forecasters and perhaps ignore the diversification function en- tirely. Of course, the investor must re- alize these differences and treat them accordingly in building his own personal portfolio. The perfectly diversified efficient fund (with the proper risk level) is an appropriate investment for the in- vestor's entire wealth stock. On the other hand, the special purpose fund need not be perfectly diversified (and in general cannot be) and may not be efficient as well, so that while it is a desirable asset to be included in the investor's total portfolio, it is not an appropriate invest- ment for his entire wealth stock. (Of course, there is little if any justification for the existence of special purpose funds in the absence of superior forecasting ability.) The concept of eBciency in the context of non-Gaussian Stable distributions.- Fama [16] has shown (using arguments analogous to those of Section 11) that, in the context of non-Gaussian finite mean symmetric Stable distributions, an effi- cient portfolio E must satisfy E(R:IE(RL), (5.25)

+ [E(R;f) - Rbl yllQ(RZ), M

where is the dispersion parameter de- fined in Section 111-C. Furthermore, by arguments analogous to those given in of Section the ex VB, post we returns can put R: and (5.25) R;: in terms 60

Of course, there is some question as to why a doubts that a superior mutual fund portfolio will manager with such superior ability

would sell his ever be found. However, if the superior manager talents for anything less than their full value. This were a risk

averter, he might find it advantageous to would imply that none of the benefits would be sell his talents for something less than their full ex- passed on to the fund investor and raises serious pected value in return for a more stable income flow.

202 THE JOURNAL OF BUSINESS Equation (5.26) is analogous to (5.10), except that all the random variables are Stable stated variates in Section fulfilling 111-C the and assumptions P, = pz, rb,. Unfortunately, there is no simple re- lationship and p, comparable between to (5.11). y"a(~:)/y'/a(~&) The lack of such a relationship prevents further simplification of (5.26) to a form like that the fact of (5.13). that6' pj From rb,, (3.21), we know (3.26), that and and it is clear that only when y(ej) = 0. Thus, given Stable distributions, a perfectly diversified port- folio E must have e: = 0 always, and this is equivalent to r, = 1, given Gaussian distributions. We can also see from (5.27) that just as in the case of Gaussian distributions, the total relative risk of an imperfectly diversified portfolio will always be great- ter These than arguments its systematic imply risk. (1) that a*, the measure of performance, is also a meas- ure of efficiency fied then portfolios, ~ ( y * < ) for all perfectly (2) that if ~(6')< diversi- 0, 0, that is, if the portfolio is inferior it must also be inefficient, and (3) diversified then that ~(6;) if (ie, > the 0 yl/a(~~)/r'/a(~;) portfolio must hold is in imperfectly order > pj), for the portfolio to be efficient.

"See pp. 184-85. VI. AN APPLICATION OF THE MODEL TO THE EVALUATION OF MUTUAL FUND PORTFOLIOS A. THE EMPIRICAL ESTIMATION OF THE WKET MODEL AND SYSTEWTIC RISK The

data.-The sample consists of the portfolios of the 115 open-end mutual funds listed in Table 1. The funds includ- ed were all those for which net asset and dividend information was available in Wiesenberger's Investment Companies [67] for the ten-year period 1955-64.62 Annual data were gathered for the period 1955-64 for all 115 funds, and as many additional annual observations as pos- sible were collected for these funds in the period 1945-54.63 For this earlier period, ten years of complete data were obtained for fifty-six of the original 115 funds. Definitions of the variables.-The fol- lowing are the exact definitions of the 62The data were gathered primarily from the 1955 and 1965 editions of Wiesenberger [67], but some data not available in these editions were taken from the 1949-54 editions. Data on the College Re- tirement Equities Fund (not listed in Wiesenberger) were obtained directly from annual reports. The last three digits of the identification numbers assigned to the funds correspond to the number of the page on which the fund is listed in the 1965 edition of Wies- enberger [67]. The College Retirement Equities Fund was arbitrarily

assigned the number 1000. All per share data were adjusted for stock splits and stock dividends to represent an equivalent share as of the end of 1964. 6SThe reader is cautioned to remember, in in- terpreting the empirical results to follow, that these 115 funds do not actually represent 115 independent observations. That is, a mutual fund group com- posed of a number of separate funds with differing objectives (ie,

growth, income, and balanced) are often under identical management. In these cases, the fund strategies may very well not be independ- ent. For instance, it is not uncommon to find the common stock portion of a balanced fund almost identical to the

portfolio of a stock fund run by the same manager. In this event, we certainly do not have two independent observations. In addition, there is some indication that the fund groups do not choose strategies independently of one another; that is, there may be some funds which in essence "fol- low the leader."

TABLE 1 LISTINGOF 115 OPEN-END MUTUAL FUNDSIN THE SAMPLE NUPIBER CODE^ FUND 140 0 ABERDEEN FUND 141 0 AFFILIATED FUND, INC. 142 2 AMERICAN BUSINESS SHARES r INC. 144 3 AMERICAN MUTUAL FUND, INC. 145 4 ASSOCIATED FUND TRUST 146 0 ATOPICS, PHYSICS + SCIENCE FUND, INC. 147 2 AXE HOUGHTON FUND A* INC. 2148 0 AXE

- HOUGHTON FUND B* INC. 1148 2 AXE -

- HOUGHTON STOCK FUND, INC.

150 3 BLUE RIDGE MUTUAL FUND* INC. 151 2 BOSTON FUND, INC. 152 4 BROAD STREET INVFSTING CORP. 153 7 BULLOCK FUND, LTD. 155 0 CANADIAN FUND, INC. 157 0 CENTURY SHARES TRUST 158 0 THE CHANNING GROWTH FUND 1159 0 CHANNING INCOME FUND, INC. 2159 3 CHANNING BALANCFD FUND 160 3 CHANNING COMMON STOCK FUND 162 0 CHEMICAL FUND, INC. 163 4 THE COLONIAL FUND, INC. 164 0 COLONIAL GROWTH + ENERGY SHARES* INC. 165 2 COKMONWEALTH FUND

- PLAY C 166 2 COMMONWEALTH INVESTMENT CO. 167 3 COMMONWEALTH STOCK FUND 168 2 COMPOSITE FUND,

INC. 169 4 CORPORATE LEADERS TRUST FUND CERTIFICATES, SERIES 181 171 3 DELAWARE FUND, TNC. 172 0 DE VEGH MUTUAL FUND* INC. (NO LOAD) 173 0 DIVERSIFIED GROWTH STOCK FUND, INC. 174 2 DIVERSIFIED INVcSTMENT FUND, INC. 175 4 DIVIDEND SHARES, INC. 176 0 DREYFUS FUND INC. 177 2 EATON + HOWARD RALANCED FUND 178 3 EATON + HOWARD 5TOCK FUND 180 3 EQUITY FUND, INC. 182 3 FIDELITY FUND, INC. 184 3 FINANCIAL INDUSTRIAL FUND, INC. 185 3 FOUNDERS MUTUAL FUND 1186 0 FRANKLIN CUSTODIAN FUNDS, INC. FUNDS, INC.

- UTILITIES SERIES 2186 0 FRANKLIN CUSTODIAL

- COMMON STOCK SERIES

187 3 FUNDAMENTAL INVFSTORS, INC. 188 2 GENERAL INVESTOQS TRUST 189 0 GROWTH INDUSTRY SHARES, INC. 190 4 GROUP SECURITIES COMMON STOCK FUND 1191 0 GROUP SECURITIES

-

- AEROSPACE - SCIENCE FUND 2191 2 GROUP SECURITIES - FULLY ADMINISTERED FUND

192 3 GUARDIAN MUTUAL FUND, INC. (NO LOAD1 193 3 HAMILTON FUNDS, INC. 194 0 IMPERIAL CAPITA!. FUND, INC. 195 2 INCOME FOUNDATION FUND, INC. 197 1 INCORPORATED INCOME FUND 198 3 INCORPORATED INVESTORS 200 3 THE INVESTMENT COMPANY OF AMERICA 201 2 THE INVESTORS MUTUAL, INC. 202 3 INV ESTORS STOCK FUND, INC. 203 1 INVESTORS SELECTIVE FUND, INC. 205 3 INVESTMENT TRUST OF BOSTON

TABLE I-Continued 206 2 ISTEL FUN,>, INC. 2C7 3 THE JOHNSTON MUTUAL FUYC IN:. (YO-LOAD1 208 KEYSTONE INCOME COMMON STOCY FIJhD 15-21 2209

3 KEYSTONE HIGH-GQADE C9YtJON STOCK FUND (5-1) 1209 4

0 KEYSTONE GROWTH COMMON STOCK FUND (5-31 210 O KEYSTONE LCWER-PRICED

COMMON STCCK FUND (5-41 1211 1 KEYSTONE INCOME FUND-(<-I) 2211 0 KEYSTONE GROWTH FUND I<-21 1212 1 THE KEYSTONE BOhlD FUND (8-3) 2212 1 THE KEYSTONE BOhlD FUND (E-41 215 2 LOOMIS

- SAYLES MUTUAL FUND* INC. (NO LOAD) 216 0 MASSACHUSETTS

IhIVESTORS GROWl-H STOCK FUND, IRC. 217 3 MASSACHUSETTS IhlVESTORS TRUST 218 2 MASSACHUSETTS LIFE FL!ND 219 4 MUTUAL INVESTING FOUNDATION, VIF FlJND 220 2 MUTUAL INVESTMEhlT' FUND, INC. 221 0 NATIONAL INVFSTORS CORPORATION 222 4 NATIONAL SECUQITIES STOCK SERIES 1223 0 NATIONAL SECURITIES

- G9OWTH STOCK SERIES 2223 1 NATIONAL SECURITIES - INCOME SERIES

224 1 NATIONAL SECURITIES - DIVIDEND 5ERIFS 225 2 NATION-WICE SECURITIES COMPANY, INC. 226 2 NEW ENGLAND FUND 227 4 NORTHEAST INVESTORS TRUST (NO LOAD) 231 3 PHILADELPHIA FWD, INC. 232 4 PINE STREET FUN09 INC. (NO LCAD) 233 3 PIONEER FUND, IMC. 234 0 T. ROWE

PRICE GROWTH STQCK FUND, INC. (NO LOAD1 235 1 PURITAN FUND, IVC. 236 2 THE GEORGE PUTNAM FUN3 OF BOSTON 239 2 RESEARCH IYVESTTNG CORP. 240

2 SCUDDERI STEVENS + CLARK BALANCED FUND, INC. (NO LOAD) 241 3 SCUDDERI STEVENc + CLARK COMPCN STOCK

FUND, INC. (NO LOAD1 243 3 SELECTED AMERICAN SHARES, INC. 244

2 SHAREHOLDERS' TRUST OF BOSTON 245 3 STATE STREET INVESTMENT

CORPORATION (NO LOAD1 246 2 STEIN ROE + FARhlHAV BALANCE3 FUND* INC. (NC) LOAD) 247 0 STEIN ROE + FARMHAM INTFRNATIONPL FUND* INC. (NO LOAD1 249 0 TELEVISION-ELECTRONICS FUND, INC. 250 O TEXAS FUND, INC. 251 3 UNITED ACCUMULATIVE FUND 252 4 UNITED INCOME FOND 253 O UNITED SCIENCE FUND 254 1 THE VALUE LINE INCOME FUND, INC. 255 0 THE VALUE LINE FUND, INC. 256 4 WASHINGTON MUTUAL INVESTORS FUND, INC. 257 2 WELLINGTON FUND, INC. 259 3 WISCONSIN FUND, INC. 260 2 COMPOSITE BOND AND STOCY FUND, INC. 1261

3 CROWN WESTERN - DIL ! ERSIFIEDFUND ( D - 2 ) 2261 2 DODGE + COX BALANCED FUND (NO LOAD1 2262 2 FIDUCIARY

MUTUAL INVE STING COPPANY, INC. 263 4 THE KNICKEREOCKrR FUND 267

4 SOUTHWESTERN INVESTORS, INC. 1268 2 WALL STREET

INVCSTING CORPORATION 2268 2 WHITEHALL FUND, INC. 1000 0 COLLEGE RETIREMCNT EQUITIES FUND 1

Wiesenberger classification as to fund investment objectives: 0 =growth, 1 =iccbme, 2 = balanced, 3 = growth income and 4 =income growth.

RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS 205

variables used in the estimation pro- cedures: St = level of the Standard 8: Poor Composite 500 Price Index64 at the end of year t. D, = estimate of dividends received on the market portfolio in year t as measured by annual obser- vations on the four-quarter moving average65 of the divi- dends paid by the companies in the

Composite 500 Index (stat- ed on the same scale as the level of the Standard & Poor 500 In- R,, = log. dex).

(F)

= the

estimated annual . continuously compound- ed rate of return on the market portfolio66M for

year t. NAjt = per share net asset value of the jth fund at the end of year t. IDjt = per share "income" dividends paid by the jth fund during year t. G4

Obtained from Standard & Poor [60]. 66 Obtained from Standard & Poor (601. Since the use of this moving average introduces

measurement errors in the index returns, it would be preferable to use an index of the actual dividends, but such an in- dex is not available. AS the capital asset pricing model implies, the market portfolio M is conceptually well defined as a portfolio consisting of an

investment in each secu- rity outstanding in proportion to its share of the total value of all securities. However, no exactly equivalent index of market performance actually exists for the time period under consideration, al- though the new New York Stock Exchange Index provides a very good index of the returns on the market portfolio in recent times. The Standard & Poor 500 Composite Index,

a value-weighted index, represents the closest approximation to such a meas- ure that is available for the period covered by this study. Since these 500 securities represent the larg- est companies listed on the New York Stock Ex- change, we use it as the best approximation available for the returns on our market portfolio M. Prior to March 1, 1957, the Standard & Poor Index was based on

only ninety securities (fifty industrials, twenty rails, and twenty utilities), and hence for the earlier period the index is a poorer estimate of the returns on the market portfolio.

CGjt = per share "capital gains" distri- butions paid by the jth fund

(-annual NAjt year + CGjt IDj,

R;, = log. the during

NA continuously

+

t.

i,t-1

com- pounded rate of return on the jth fund during year t (adjusted for splits and stock dividend^).^^ nj number nj

of the

of

jth

yearly

fund;

tions

= the

< 20.

observa-

10 I The empirical estimates.-It was shown estimator in Section for IV-C pj that regardless we can of use whether the same we assume Gaussian distributions or sym- metric non-Gaussian finite mean Stable distributions. Keeping these alternative interpretations in mind, let us define bj, the estimate of systematic risk for the jth portfolio obtained from annual data, as

where bj

=

I

variances nite distributions b2,under bsj j = the assumption (a = 2) under the

assumption variance symmetric with 1, 2, ..., 115, of finite of infi- 1 < a < Stable

2, and

87

Note that while most funds pay dividends on a quarterly basis, we treat all dividends as though they were paid as of

December 31only. This assump- tion, of course, will cause the measured returns on the fund portfolios on the average to be below what they would be if dividends were considered to be re- invested when received, but the data needed to ac- complish this are not easily available. However, the

206 THE JOURNAL OF BUSINESS

That is, the estimate of systematic risk for the jth fund is obtained from all the data available, and the

number of sample observations varies from ten to twenty. Also, as was shown in Section 111-B, un- der tions the pzj assumption is given by of Gaussian (3.12) and distribu- in the case tions of PSj nonGaussian is given by (3.27). Stable The distribu- argu- ments to a very in close that section approximation also indicate /3zj and that Paj are equal to the slope coefficient bj in the market Henceforth, we shall resulting bias should be quite small. In addition, the same bias is incorporated into the measured returns on the market portfolio. Since it was argued that the second term on the RHS of both (3.12) and (3.27) is trivially small, the question may arise as to whether

this term will also be trivially small for a porljolio of securities. It will be, portfolio since of the K fraction securities xj will in on (3.12) the average and (3.27) for a be equal to 1/N, just as in the case of an individual security. To see this, let yj be the fraction of the jth portfolio invested in the ith security and let hi be the fraction of the ith security in the market portfolio. Then the portfolio disturbance term ej is given by K

ej =

Cyihie: i=l (6.2)

where ei is the disturbance for the ith security. Un- der these conditions, the last term on the RHS of (3.12) is given by where is the average variance of the disturb- ance terms for the individual securities and 1/N is the average weight hi. But where, as before, N is at least on the order of 1,000

use the general notation pj (without the following subscript) to denote the esti- mate may interpret of

systematic the measure risk, and either the reader as

pzj or PSj, depending on his inclination to accept

the evidence regarding the infi- nite variance properties of the security returns (cf. references 4, 12, 19, 38, and 46). The estimates of systematic risk, bj, for all 115 funds are given in Table 2 (col. 5) along with various other statistics which we shall discuss below. Figure 6 presents a frequency distribution of the coefficients. In addition, Table 3 pre- sents a summary of the regression statis- tics for the sample of 115 funds.69 We make no inferences from these statistics here except to note the following charac- teristics: 1) The average b coefficient for the 115 funds is only .840. This implies that the funds are fairly conservative in their investment policies-in general, offering investors portfolios with smaller system- atic risks than the market portfolio (which implicitly represents a systematic risk of unity70). Hence, any

attempt to lated Section

Bg

In as 111, the

2j

= context the

xj*- a in of JiX& Table the market and 3 is is the presented model intercept

discussed only calcu- as in a matter of information for the interested reader. We should mention that the errors in variables

attenuation bias will cause our estimated coefficients to be smaller than the true coefficients, but it is very doubtful that these measurement errors are large enough to explain the total difference between .840 and 1.000. That is, let Iiwt, ut,and b: be, respective- ly, the true index return, unbiased measurement error, only I&t and = the Iht true shown if (cf. we Johnston observe [33, chap. vi]) that Plim pj g Plim bj = Plim

c6v

+ut, coefficient. it is easily Then,

a2(I>)

(Rj,Ik) and aP(e')is approximately the same size as ~(RM). (See the arguments given on p. 180.) The argu- ments for the non-Gaussian

Stable case are anal- Thus, the ratio of the variance of the measure- ogous to these and need not be repeated. ment error to the variance of the true index, [02(u)/ TABLE 2 MEASURESOF PERFORMANCE, 6*, AND EFFICIENCY, r* (SEE P. 240 FOR DEFINITIONOF r*),FOR 115 MUTUALFUNDSIN THE PERIOD 1955-64 ALONG WITH VARIOUSOTHER STATISTICS(FUNDS ARE RANKEDFROM HIGHTO LOW ON THE BASISOF 6*) RANK ID. CODE R* ri

i/; r 6* Y* 10 1964 (1) (2) (3) (4) (5) (6) (7) (8) (9)

1 ll F 6 0 1.4C2 C.53.? 0.762 0.7C6 C.627 0.427 2 176 C 1.469 0.954 1.074 C.RP8 0.323 0.216 3 Z le 6 0 1.231 C.792 C.91R 0 . et 3 C.229 0.117 4 I t 9 4 1.13R C.712 0.750 0.949 0.207 0.173 5 225 2 C.537 C.45C 0.554 O.PP5 0.2C4 0.147 6 267 4 1.CR3 6.656 0.740 O.PF6 C.2C2 0.127 7 162 0 1.225 C. 8 1 9 1.005 C.Fl5 C.159 0.034 8 25C 0 I.16E( C.762 C.809 0.942 C.1$4 0.152 9 2262 2 C.555 C.54@ 0.583 0.940 0.171 0.139 10 246 2 C.561 C.566 0.603 C.'??FI 0.161 0.127 11 234 0 1.2?5 C. F7 P 0.940 0.934 0.156 0.101 12 ZC6 2 l.Ct?t? C.716 C.754 C.950 0.154 0.121 13 192 3 1.144 C.79C 0.821 0.9C2 C. 144 0.116 14 227 4 1.ClC C.66C 0.697 0.947 0.135 0.102 15 233 3 1.102 C.758 0.856 CP?6 0.131 0.C44 16 151 2 C.S46 C.5$3 0.668 C.FP8 0.152 0.055 17 226e 2 c.891 C.537 0.555 0.968 0.116 0.100 18 207 3 c.589 c.671 0.754 CP~O c. CS ~ 0.~22 1 9 2 6 0 2 C.773 0.435 0.52R 0.R24 0.Ce9 0.C07 2 0 175 4 1.C67 C.768 C.793 C.S69 O.Ce7 0.065 2 1 22 1 0 1.240 C.97C 1.094 0.887 0.Ce0 -0.030 2 2 142 2 c.e22 a.5ce 0.634 o.Pc1 a. c74 -0.~39 2 3 2 15 2 C.e4P C.548 0.645 0.@50 0.C64 -0.023 24 14 1 C 1.154 C.P92 1.052 O.@4R 0.Ct3 -0.079 2 5 218 2 C . PC9 C.512 0.554 0.924 0.C57 0.02C 26 152 4 I.CP6 C.828 0.R65 C .S57 C.C52 0.C19 27 144 3 1.116 Cf!65 0.880 0.983 0.C50 0.036 2 8 201 2 C.@54 0.5eh 0.673 0.541 0. C36 0.003 2 9 177 2 C.F31 0.562 0.596 0.S43 0.034 O.OC4 30 257 2 C.FSC 0.585 0.614 0.952 0.C33 0.007 31 168 2 CE!57 0.594 0.627 0.948 0.C31 0.002 32 126@ 2 C.554 C.7C6 0.732 0.964 0 .C29 O.CC5 33 157 0 1.013 C.774 1.035 0.74A C.C28 -0.205 34 256 4 1.194 C.9e7 1.006 0.9El 0.C19 0.002

TABLE 2-C

~-

~

.........................................

~ h ~ ~ w e ~ ~ v ~ w ~ ~ vr ~ ~ lu ~ ~ m ~ ~ v ~ - c w ~ r m ~ r e ~ ~ ~

~ u ~ u w ~ ~ ~ ~ ~ a um a vm \ ~ r ~ m + ~ a u ~ mm ~ ~ ~ ~ ~ rp u ~ ro m ~ ~ c mm ~ m ~ um v ~ u ~ mmn ~ ~ va O 1 O 1 O 1 O 1 O 1 G 1 O 1 O 1 O 1 O 1 O 1 V 1 O 1 O 1 G 1 G 1 O 1 O 1 C 1 O 1 O 1 V 1 O 1 O 1 O 1 O 1 O 1 O 1 O 1 V 1 O 1 O 1 V 1 G 1 G 1 O 1 O 1 O 1 O I O I H l ra uu

TABLE 3 SUMMARYOF REGRESSIONSTATISTICSFOR THE SAMPLE OF 11.5 MUTUALFUNDS

Exmm VALUES MEAN MEDIAN MEW lnx VALUE VALUE

---

-. --.

. I . ............ .923

1

ABSOLUTE DEmT1ora Minimum

/

Maximum

1 1 I .943

.620

.988

,046

-

-

elb . . ... 063 - 0 3 ,699 ,990 213 n..

.......... 17.0 19.0

10.0 20.0 3.12 !I

I I I Defined

b First-order

as autocorrelation of residuals. The average 43 is .074.

BETA

FIG. 6.-Frequency distribution (half-sigma intervals) of the estimates of systematic risk for 115 mutual funds using all data available in

the period 1945-64.

211 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS compare the performance of mutual funds to such a market index without explicitly allowing for the trade-off be- tween higher risks and higher returns will be biased against the funds. 2) The in column 7 correlation of Table 2 coefficients (and for which listed a fied portfolios. The median value (see Table 3) is 0.943. In conjunction with this, we note that the average mean ab- solute deviation of the residuals is .038 and the average standard deviation of the residuals, &(e), is .052. 3) It is of special interest to note in CORRELATION COEFFICIENT

turns FIG. on 7.-Frequency 115 mutual funds distribution and the returns (half-sigma on the intervals) market of portfolio the

distribution is presented in Figure 7) are in general quite high (with an average of .923), indicating that the funds on the average hold well-diversicorrelation M in the coefficients period 1945-64. rj between the re- frequency

s'(Zh)l,would cent in order true R& Note will average cause that to any s have explain b' measurement was to the 1. be average approximately errors fi of in .840if the 19 index per the consistent underestimation of the sys- tematic risk of the portfolios and will therefore tend to cause the portfolio's performance to appear better than it actually is.

Table 3 that the average first-order auto- correlation only -.063. of Thus, the residuals, it would seem

p(e,', e;-*), that on is the average the model is well specified with regard to this factor. However. it should ge noted that there are some rela- tively minimum

+.590.

large But of with extreme -.699 such and small values, a

maximum sample namely, sizes, of a the standard error of estimate of the serial correlation coefficient is quite large,

212 THE JOURNAL OF BUSINESS and these observations are within the range of what we could expect given a sample of 115 funds.71 Since the empirical evaluation tests to follow estimates will of be pj crucially obtained dependent from these on "re- the gressions," it is extremely important that the model be well specified and sta- tionary through time and that the esti- mates of the parameter /3 be invariant to the length of the time interval over which the sample returns are calculated. The remainder of this section is devoted to an evaluation of the estimates of the market model with specific reference to these problems. In order to test the model, the follow- ing scatter diagrams were calculated for every tenth fund in the sample: (1) fund return versus market return, (2) residual versus t

+ 1 versus market residual return, in t, (3) and

residual (4) residu- in al versus time. The diagrams for the Colonial Fund, which is fairly typical of the sample as a whole, are given in Fig- ure 8. In general, for the sample as a whole, panel a indicates that the linearity assumption is valid, and panel b indi- cates that the residuals appear to be un- correlated with the market returns. There is some slight evidence,72 as in panel d of Figure 8, that the model may not be stationary through time for all funds. We shall present more evidence on this point below after consideration of the invariance of the estimates to the length of the time interval over which the returns are calculated. Stability of the estimate of systematic 71

The t statistic (cf. Johnston [33, p. 331) for testing the significance of p is given by

risk.-It was pointed out in Section IV-C that the solution to the horizon problem implies efficient, that p, will the be estimate invariant of to the the risk length coof the time interval over which returns are measured-as long as the market horizon H is close to zero and the returns are rates. stated That as is, continuously let Npjbe the compounded risk coeffi- cient for the jth fund estimated from a time series of N-period returns73 (prop- erly transformed by [4.10]); then where the prescript N indicates the length of the time interval over which each of the sample returns is calculated. The arguments in Section IV imply that the market horizon period is very likely to be nearly instantaneous, and under these conditions the estimates based on the natural logarithms of the observed return erty: lbj data = 2bi will = have .

. . the =

Nbj.

following propIn order to test the validity of these arguments, the risk coefficients 2Pj were estimated from ten observations of two- year returns, z ~ j * ,for those fifty-six funds in the sample having a full twenty years of data a ~ ailable These . ~ ~ estimates are given in column 3 of Table 4 along with the estimates based on annual data, ISj, in column 2. The differences between the coefficients are given in column 4 of Table 4. Unfortunately, it is difficult to specify formal tests of the differences, since the errors are certainly not inde- pendent. Hence, we are forced to rely on 73

As opposed to H-period returns. 72 Which, given the small sample sizes, is very weak. 74 Of course, the returns on the market portfolio,

%R&,were also translated to the two-year dimension

.247

@=.890 f=.894 n= 20

.124

FUND RETURN RESIDUAL

213 .124 ----,183

-.163 -006 .309 .466 ---, 247

-.163 -.006 309 .466

.151

MARKET RETURN

.151 MARKET RETURN FIG. 8.--Scatter diagrams for testing the adequacy of the assumptions of the market model for the Colonial Funds Inc. (identification number 163)

pletne)=--,44 .247 .247 .124

RESIDUAL IN + +1 RESIDUAL

214 - 124

-,124 M, 247

-247

-.124 .124 ----.247 L

1944 247 L

1949 .0 RESIDUAL IN +

1959 1965 1954 TIME

(d)

(c) FIG. 8.-Continued TABLE 4 COMPARISON OF ESTIMATES RISK OBTAINED FROM ONE- AND OF SYSTEMATIC 'TWO-PERIOD DATAFOR FIFTY-SIXFUNDS

8 =-

--/

,,

2 Hean absolute (

216 THE JOURNAL OF BUSINESS informal descriptive measures of the

zbjagainst relationships. ,pj,and Figure the 9 presents reader will a

plot note of that a 45-degree line would represent per- fect correspondence between the two estimates. The correlation between the two estimates is 39. For the fifty-six funds the averages for the two estimates the opportunity set implied by equation (4.11) also seems to indicate that the log transformation is appropriate. But for the moment,. we accept the log form as appropriate and proceed to an ex- amination of the stationarity of the risk measure through time. Also, since the evidence indicates that the estimates 1 B A

-- Coefflclent Estimated onl-Eerlod Uzta FIG.9.-Scatter diagram of estimates of systematic risk derived from one- and

two-period data

are = A301 and 2 p = -8299, and thus the a verages differ by only .0002. The mean absolute difference between the two estimates for the fifty-six funds is only .lo. Given the differences in sample sizes, these differences seem small, and they seem to support the implications of the theory and the assumption of an in- stantaneous horizon period quite well. We shall see below that the linearity of are stable and ~6~ = pi, we henceforth drop to the the measure preceding of risk subscript only as N pj. and refer Stationarity of the measure of systematic risk.-If the concept of systematic risk is to be of practical use in evaluating and selecting portfolios, it must be stationary through time. That is, the investor select- ing a portfolio must be able to use past historical data to obtain estimates which 217 RISK, CAPITAL ASSETS, AhD EVALUATION OF PORTFOLIOS will be a good indication of future risk. Furthermore, in evaluating portfolios, we must be able to assume that the riski- ness of the portfolio has not changed over the period under consideration. Blume [4], in a detailed examination of the market model, finds that for 251 in- dividual (approximately securities, equal the to our coefficients pj) are ap- bi proximately stationary over the thirty- four-year period 1927-60. This is an ex- tremely important result and indicates that the systematic risk of a portfolio of securities (each representing a constant fraction of the portfolio) will also be sta- tionary through time. However, this re- sult is not sufficient to establish the stationarity of risk in a sample of fully man- aged portfolios such as the mutual funds under examination here. If the managers so choose, they can substantially change the level of systematic risk in either di- rection by altering the proportions of high- and low-risk securities in the port- folio. In order to test the stationarity of the riskiness of the mutual funds, the twen- ty-year parts. The period risk 1945-64 coefficient was split pj was into then two estimated in each of the two ten-year pe- riods for the fifty-six funds having a The complete estimates twenty years of and data bj, available. 65-64 for all fifty-six funds are given in Table 5 along with the difference between the estimates. The average coefficient in the latter period was 303, as compared with 364 for the

bj,45-54

earlier period. The mean ab- solute difference between the estimates is .I 1 and seems quite small. of the

average values of p, r, A p, * summary and PA2 is given in Table 6, and the reader will note that, except *A

for the autocorrelation of residuals, the average coefficients are quite similar in the two sample periods. Figure 10 presents a scatter diagram of b55--64 against 445-54, and again it should be noted that a 45-degree line represents perfect correspondence be- tween the two estimates. The correlation between the two estimates is75 -74 and while the relationship is not perfect, the scatter does show evidence that the funds tend to maintain their level of riskiness through time. A detailed ex- amination of Figure 10 indicates that (1) the marginal variance of the esti- mates for the earlier period seems larger and (2) there seems to be a hint of curvi- linearity. In addition, an excmination of Table 5 indicates thirty-four observa- tions for which the difference (b55-64 - b45-54) is negative. Under the assumption that the two coefficients are actual- ly identical, the expected number of negative values in column 4 of Table 5 is 28. Using the normal approximation to the binomial distribution, the stand- ard v"G= deviation

v"(56)(.5)(.5) of the distribution = 3.75, and is a the = excess number of negative values is thus within 2 SD of the expected number. Given these results, we shall proceed un- der the assumption that the risk coefficients are stationary. Hence, in the fol- lowing evaluation of fund portfolios, the risk coefficient is estimated

from all data available on each fund in order to mini- mize the sampling error in the estimates. Summarizing the empirical results so far we have found that the evidence sup- ports: (1) the assumptions of the market model regarding (a) the linear relation- ship between fund returns and the mar- ket factor and (b) the independence of The correlation between the estimates is re- duced considerably by the presence of the outlier in the middle right hand side of the

'6

figure. This point represents the Investment Trust of Boston and is caused entirely by one observation. Wiesenberger's data indicates the fund earned 177 per cent in 1945 and while suspect, the author was unable to confirm this to be an error. Thus the observation was left in the sample. TABLE 5 COMPARISON OF ESTIMATESOF SYSTEMATIC RISK IN THE TWOTEN-YEAR PERIODS1945-54 AND 1955-64 FOR FIFTY-SIXFUNDS A

ID NUMBER 55-64

$45-54 ($55-64

- 845-54)

140 0.935 0.902 0.033 141 0.729 1.071 -0.342 142 0.394 0.637 -

0.243 145 0.869 0.812 0.057 147 0.789 0.885 -0.097 1148 0.651 0 902 -0.252 2148 0.722 1.192 -0.469 151 0.512 0.683 -0.171 152 0.759 0.897 -0.138 153 0.888 1.040 -0.152 157 0.819 0.727 0.091 162 0.890 0.743 0.146 163 0.881 0.914 -0.033 166 0.648 0.723 -0.076 169 0.751 0.677 0.074 171 1.037 1.001 0.036 174 0.726 0.91 5 -0.189 175 0.776 0.769 0.006 177 0.548 0.577 0.029 178 0.865 0.798 0.067 180 0.909 0.927 -0.018 182 1.049 1.004 0.045 184 1.104 1.136 -0.033 185 1.050 0.953 0.097 187 1.024 1.082 -0.058 188 0.652 0.677 -0.025 190 0.900 1.047 -0.146 2191 0.689 0.711 -0.022 195 0.657 0.664 -0.006 198 1.179 1.328 -0.149 200 0.985 0.934 0.051 201 0.606 0.572 0.035 205 0.998 1.762 -0.764 215 0.524 0.575 -0.051 216 1.027 1.099 0.072 217 0.970 0.953 0.017 219 0.842 0.811 0.0 31 220 0.805 0.835 -0.030 221 0.864 1.090 -0.225 222 1.109 1.280 -0.171 2223 0.826 0.970 -0.144 225 0.482 0.515 -0.033 22 b 0.377 0.661 -0.284 233 0.816 0.683 0.133 236 0.735 0.684 0.051 240 0.665 0,550 0.115 241 0.997 0.907 0.090 243 0.944 0.940 0.003 245 0.879 0.768 0.111 251 0.961 1.055 -0.094 252 0.921 0.983 -0.062 257 0.584 0.595 -0.012 259 0.813 0.793 0.020 260 0.422 0.472 -0.050 2261 0 -664 0.610 0.054 263 0.746 0.888 -0.143

-

'55-64 = .803 56

I I'i ,4,-54 - 'i,ss-sh I Mean absolute difference = = .I1 56

219 RISK,CAPITAL ASSETS,AND EVAILUATIONOF POR'l'FOLlOS

residuals and the market factor; (2) the theoretical argument regarding the sta- bility of the estimate of systematic risk; and (3) the assumption of stationary risk levels for fund portfolios. The first result TABLE 6 AVERAGE VALUES OF SELECTED STATISTICS ASSOCIATEDWITH TEMATIC RISK IN THE THE ESTIMATES TWOTEN-YEAR OF SYS-

PERIODS

I

indicates that our estimates of systemat- ic risk are valid. The second result re- garding the invariance of the estimates to the length of the time interval over which the sample returns are calculated indicates that the risk coefficients may be used for a horizon interval of any length. Finally, the third result indicates that the future risk of a portfolio may be estimated from past data and that in general a more efficient estimate of a portfolio's risk may be obtained by using all past data.76 Thus, we now continue 76

Of course, in applying the model to any par- ticular portfolio, the investor who believes that the risk of that portfolio has changed will

be well ad- vised to devote the necessary resources to obtaining recent data (on a monthly, weekly, or daily basis) in order to obtain

estimates of the present riskiness of the portfolio.

ARA.10.-Scatter diagram of the estimated risk coefficients obtained for fifty-six funds in the two ten-year periods 1945-54 and 195564.

220 'I'HE JOURNAL OF BUSINESS to a consideration of the performance of mutual fund portfolios. B. THE EVALUATION OF MIlTUAL FUND PORTFOLIOS The hypotheses.-We turn now to a brief review of the questions originally posed in the Introduction before we moved to a detailed analysis of the un- derlying issues. The reader will recall that these questions involved (1) an ex- amination of the hypothesis of the pre- dominance of risk aversion among in- vestors and the validity of the concept of systematic risk as implied by the capital assets pricing model and (2) an evaluation of the historical performance of mutual fund portfolios with specific regard to the ability of mutual fund managers to predict future security prices. 1. Risk aversion, and the measure of systenzatic risk.-There are two essential issues involved here, and it is conceptu- ally impossible to test both simultane- ously. That is, we would like to know (1) whether the security markets are domi- nated by investors who are averse to risk and (2) whether the coefficient is a valid measure of risk. Clearly now, we must assume that one of these is true in order to test the other. If the market is not dominated by risk averters, no meas- ure of "risk" will be positively related to returns. On the other hand, if we do not have an "appropriate" measure of risk, the absence of a positive relationship be- tween risk and return implies nothing about the predominance of risk aversion in the capital markets. However, since we continually observe people behaving as though they were averse to risk77 (ie, generally holding diversified multi-asset portfolios and buying insurance), we shall assume the former and test the latter.78 Unfortunately, it proves difficult to specify formal tests of the measure of risk p. Thus, we must be content at this point to judge its adequacy in terms of its apparent consistency with the impli- cations of the assumption of dominant risk aversion on the part of investors. For instance, we would conclude that the measure was inconsistent with our assumption of risk aversion if we found that a plot of /3 versus realized returns on the fund portfolios yielded a negative or zero slope in a period during which re??

At least risk aversion is generally observed when the risk of substantial losses exists (as there most certainly is in the case of non-

trivial invest- ments in securities). However, there appears to be some stituations, usually involving a high proba- bility of small losses in conjunction with a small probability of large gains, in which people often be- have as though they were risk lovers. For a discus-

sion of these points, see Friedman and Savage [25] and Markowitz [41]. We might note at this point that LatanC [35] and Sharpe [54] have found evidence of a positive re- lationship between risk and return in the capital markets. LatanC examined the differences in returns earned on common stocks, commodities, and bonds and finds a

positive relationship between the riski- ness and the returns on these instruments. Sharpe examined the relation between risk (measured by the standard deviation of annual returns) and the arithmetic mean of annual returns on mutual fund portfolios over the period 1954-63. We note here, however, that this type of test is subject to several deficiencies. As M. Miller has pointed out, any

tests involving the relationship between the arithmetic mean and second moment of the distribution of re- turns are positively biased if the distribution of re- turns is positively skewed. Indeed, as shown by Cra- mer mean, [lo, 2, p. and 3481, second the sample moment covariance p2, of any between distribution the is

cov (2,p2) = -n-1 /*3 9 where ps is the third moment of the distribution and n is the sample size. Since the distributions of annual value p3 > 0, is returns and m

+

any and must attempt the be minimum skewed to test (since the is zero), assumption the maximum we know of risk aversion by

examining the relationship between the arithmetic average and u is subject to this bias.

221 RISK, CAPITAL ASSETS, AND EVALULYYION OF PORTFOLIOS alized market returns were above the empirical c~unterpart'~ of the perform- riskless rate. ance measure diagrammed in Figure 4. Figure 11 presents a scatter diagram The returns plotted on the ordinate are of the returns (measured net of manage- the natural logarithm of the ten-year ment expenses but gross of loading wealth relatives (assuming reinvestment charges) of the 115 funds plotted against of dividends at the end of the year). In their respective ,d coefficients calculated terms of our previous notation, by (6.1). Thus, for the tenyear holding period 1955-64, Figure 11 represents the 79 The symbols with which the fund portfolios are plotted denote the Wiesenberger classification 10 YEARS 1955-1964 1.61 -

:/

"++";

+ Y .92

++ +/I,-

+ Y TX ,.*+

+

X "?+++

- zzz z =*+: "'

x .69 -

+

Y

.23

- +YX XZ-

- - - - SYMBOL GROWTH

GROWTH-INCOME BALANCED INCOME-GROWTH INCOME CODE -.oo I I II I .OO .30 .60 .90 1.20 1.50

SYSTEMATIC RISK -)

FIG.11.-Scatter diagram of risk and (net) return for 115 open-end mutual funds in the ten-year period 1955-64. 1.80

THE JOURNAL OF BUSlNESS

The returns 10~jr,1964 for all of the funds are listed in column 4 of Table 2 along with the Wiesenberger classification code (col. 3). The points and M in Figure 11 are, respectively, the ex post experience of the risk-free asset

and the free log,(l market return

+ .03)1° portfolio. is =

,OR;

.296, = The loge(l where ten-year

+ 3.0 riskper =

cent was the yield to maturity of a ten- year government bond in 1955.80 The of the investment objectives of each fund based upon its stated investment emphasis (see key to Fig. 11). They are plotted in this manner in order to illustrate the correspondence between the fund man- agers' statements regarding the objectives of the fund (the basis of

the Wiesenberger classifications) and the measure of systematic risk. Wiesenberger (1961 edition, p. 134) defines growth funds to be those whose primary objective is the long-term growth of capital and for which the "risk of higher. price depreciation .

. than in

declining periods is normally for many others." Growth-income funds are those which combine an emphasis on long- term growth

of capital with a consideration of "in- come and/or relative stability." Income-growth funds combine an emphasis on current income with the possibility of long-term capital growth. Income funds are defined as th ose whose "primary objective is the most generous possible current income," and the balanced funds are those which "place more em- phasis on relative stability and continuity of

income than do those in the preceding groups." On the basis of these dehitions, it is our guess that the classifica- tions as listed in Fig. 11 represent declining riskiness from growth to balanced. While the patterns of Fig. 11 seem to lend some credence to the belief that the fund managers do have an idea of the amount of risk their portfolios contain, the classifications for individual funds are certainly not perfectly consistent with the measure of systematic risk. Table 7 presents the average and median for each of the classifications and seems to indicate that on the average there is some corre- and spondence our measure between of risk. the Wiesenberger The average

6 classifications of the groups declines consistently with our ordering of the classi- fications, but we note

that median values are not nearly as consistent.

estimated return on the market portfolio, loR&, was 1.187 in this period.81 Thus, the market line lo ~ ; M in ~ Figure 11 given by (6.5), represents the possible expected combi- nations of systematic risk and return conditional on the actual realized returns on the market portfolio which were available to an investor with a ten-year planning horizon in 1955. The patterns observed in Figure 11 seem to confirm the adequacy of our measure of risk. We observe a positive relationship between the realized returns on these portfolios and their systematic risk,82 which is exactly what we would predict if: (1) investors were averse to risk, demanding premiums (in the form of higher expected returns) for accepting This is a truly risk-free rate only in the case of a non-coupon-bearing bond, since in the case of cou- pon payments this formulation implicitly involves invested the assumption (at time that of receipt) all interest at the payments rate RF.How- are reever, we expect this error to be relatively small, since it ignores only the differential interest earned on the coupon payments, which are themselves quite small. 81 The la^$ is, of course, a ten-year rate of re- turn. Dividing I&& by 10 yields an average annual rate of 11.87

per cent compounded continuously. 82 AS mentioned earlier, we would never expect to see a perfect relationship in ex post data, but it is very probable that some of the scatterings of points are due to sampling error in our estimates of pi as well as the disturbance terms ej. In

practice, this sampling error could be reduced by using monthly or weekly data in the estimation of pi, but the data are not available in sufficiently convenient form to warrant utilization in this study.

223 KlSK, CriPr'rAL ASSE'I'S, AND EVALUATION 01;POK'I'FOLIOS increased risk; and (2) investors' expec- tations regarding risk were on the aver- age correct and the funds did not sub- stantially alter their risk levels too often.83 Thus, there seems to be some empiri- the cal as use well of as p as theoretical the measure justification of risk. Be- for

fore turning to the second point men- tioned above (the evaluation of fund of his lifetime consumption pattern, it was shown in Section I1 that the inves- tor's portfolio choice can be character- ized by a single-period utility of terminal wealth model. We reiterate this point since it has implications regarding the relevant measure of returns. In the context of a single-period utility of terminal wealth model, returns must always be stated in terms of total dollar AVERAGEAND RETURNFOR hfEDIAN VALUESOF VARIOUSCLASSESOF SYSTEMATICRISK FUNDS~ ASD NET

No. OF

!

Balanced. Income-growth Income. Growth-income. Growth. ........ ........ .......

.. . a The average and median values of 8 for the sample as a whole are given in Table 3. The over-all average return for the sample io~;sar = .955 and the median value is ,961. b As

classified by Wiesenberger 1671.

performance), let us briefly consider the problems associated with the measure- ment of returns. The

measurement of returns.-Given the assumption that the goal of the in- vestor is to maximize the expected utility 85

If all funds held neutral portfolios and there were no measurement (or sampling) errors in our gression estimates of of 18:on the ~ '

s we , ,$i, would expect to find a re- yielding coefficients close to those in (6.5),the equation of the market line. We can readily

observe that the funds do not all appear to hold neutral portfolios, and we know there are sampling errors in our estimates of the P's. For those interested, however, the estimated regression equa- tion is: In addition, the reader will note that the weakness of these results is considerably influenced by a few outliers.

amounts or (as in this study) a transfor- mation of total returns (AW/W) over the entire interval of the investor horizon period. We emphasize this point, since most empirical studies utilizing the con- cepts of risk and return have measured returns as an arithmetic average of annu- returnss4 which is inconsistent with the underlying utility model. If the dis- tribution of annual returns is skewed, there is and direct between an arithmetic average of annual returns and terminal wealth, For example, assume that the proba- bility distribution of annual returns is log normal with mean log8 = p and variance a2. (Here we are letting R = s4 Cf. references

7, 27, 31, 54, and 55. Also see n. 78 for a discussion of other problems associated with the use of the arithmetic

average of returns.

224 1

+ r, where r is the THE JOUKNAL annual return.)

OF BUSINESS

While there is a direct monotonic re- lationship between terminal wealth, WT, and p, WT = Wt-exp[p.(T t)],there is no such simple relationship between terminal wealth and an arithmetic av- erage of annual returns. Consider the mean expected R calculated value E(R) from of an arithmetic a sample of observations drawn from this log nor- mal [I, p. distribusi. 81 show that Aitchison E(R) = and exp Brown (p

+ c2/2). Thus, the

arithmetic mean of a sample from a log normal distribution is a function of the variance of that distribution as well as the mean, p. Hence, an equal investment in two port- folios having the same arithmetic aver- age of returns over several years will not yield identical values of terminal wealth if their variances differ. 2. The performance of mutual fund portfolios.-Since there does seem to be some empirical as well as theoretical jus- tification for the use of /3 as the measure of risk, we turn to an analysis of the risk- return performance of mutual funds in the period 1945-64. In particular, we address ourselves to the following ques-

tions: (1) Have the mutual funds on the average provided investors with returns greater than, less than, or equal to the returns implied by their level of system- atic risk and the capital asset pricing model? (2) And have the funds in gener- al provided investors with efficient port- folios? In attempting to evaluate the funds' performance, particular attention must be given to the treatment of loading charges, management fees, and expenses in calculating fund returns and to the treatment of commission expenses in cal- culating returns on the market port- folio M. Obviously, in evaluating fund performance from the investor's point of view, the effects of these transaction costs on his returns must be con ~ idered , ~ ~ but we defer explicit consideration of these costs for the moment. One can argue that the loading charge (which is generally a pure salesman's commission) actually represents payment for a real economic service, that is, convincing small, uninformed investors of the value of equity investment. Accepting this, the test of a fund management's perform- ance involves only a test of its ability to earn returns sufficiently greater than our naive FM policy to cover the non-load- ing-charge expenses of the fund-that is, management fees and brokerage ex- penses. Therefore, the fund returns plot- ted in Figure 11 were calculated net of all management fees, brokerage com- missions, and other expenses incurred by the funds but gross of (ignoring) loading charges to the invest ~ rs . ~ ~ In considering the appropriate meas- ure of return on the market portfolio M, one can also argue that the brokerage commissions are-analogous to the load- ing charges and represent payments by the investor for real economic services. The loading charges can be quite substantial, ranging from zero for the "no load" funds (of which there are thirteen in the sample) to

the much more usual charge of 8-83 per cent of the original invest- ment. The loading charges on the so-called front-end load contractual plans may be substantially higher, often exceeding 30 per cent of the purchase cost if discontinued after two years of their life, 50 per cent if discontinued after one year. 86 Mutual funds also provide investors with a certain amount of non-monetary returns in the form of bookkeeping services. The investor holding shares in a mutual fund rather than his own diversified portfolio avoids a significant amount of bookkeep- ing involved in clipping and mailing bond coupons, cashing and recording dividend checks throughout the year, and calculating capital gains and/or losses on any sales executed during the year. Mutual funds generally pay dividends quarterly, and at the end of each year the investor receives a statement detailing the respective amounts he must declare as income and capital gains in his income tax returns. In view of the fact that brokers will also provide similar services, allowance for non-monetary returns has not been incorporated into the analysis.

225 IIISK, CAPITAI, ASSETS, AND EVALUATION OF PON'I'FOLIOS If we accept this argument for the mo- ment, the appropriate measure of re- turns on the standard of comparison is the performance of the market port- folio M, ignoring commission expenses. Moreover, while a small investor could not have purchased such a portfolio without incurring high brokerage com- missions, most mutual funds could have purchased this portfolio without incur- ring much larger transaction costs than those incurred in purchasing their actual portfolios. In addition, given that all of our funds started and ended the evalua- tion period with fully invested port- folios (not cash), it would be inappro- priate to charge commissions on the market portfolio alone. Therefore, the point M in Figure 11 represents the risk- return results for the market portfolio calculated without adjustment for com- missions. The scatter of points in Figure 11 gives us a visual impression of the ability of mutual fund managers to choose securi- ties well enough to recoup the expenses they incur in attempting to forecast fu- ture prices. The scatter, generally below and to the right of the line ,&;MQ, in- dicates that in this ten-year period the funds in general provided investors with lower returns than they could have re- alized by a combined investment in port- folios F and M yielding the same degree of risk. The average value of the differ- ence in returns between the fund

port- folio and a comparable FM portfolio, 61 6,: (as < defined 0 for seventy-two in eq. [4.22]), funds is -.089, and 6; with > 0 for only forty-three funds. Thus, on the average, these 115 mutual funds earned 8.9 per cent less (compounded continu- ously) than their comparable FM port- folios over the ten-year period.87 The performance measures, 6j', for each of the funds are given in column 8 of Table 2, and Figure 12 presents a frequency distribution of the estimates. The funds are ordered from high to low on the basis of 6j' in Table 2. We caution the reader to be extremely careful about interpreting these measures without taking into ac- count the sampling errors. Indeed, we shall show below that there is very little evidence that the funds which appear superior were anything more than just lucky in this period. On the other hand, there is evidence that some of the large negative performance measures were not due to chance (see also Jensen [32]). There is one other matter which war- rants our attention at this point. An ex- amination of columns 5 and 8 of Table 2 indicates that the measure of perform- ance 6; measure is of negatively risk pi (the correlated productmoment with the correlation coefficient between them is -.68). This result is not surprising and merely reinforces the hypothesis that much of the variability in the estimates pling of 6* is error due in to the random estimates factors of pj. or sam87

One might also wish to interpret the fund per- formance in the following manner: Consider a port- folio consisting of an equal

dollar investment in each of the fund portfolios and another portfolio con- sisting of an equal dollar investment in the comparable FM portfolios. One might legitimately ask: How well did the portfolio consisting of investments in the funds perform with respect to the FM port- folio? In order to answer this, we calculate the aver- age terminal wealth ratio of the funds (= 2.639)

and take the difference between this and the average terminal wealth ratio of the FM portfolio calculat- ed (1 by

5~

115

1 116 XP[R;(l -

+

BFM) =

Pi)

Thus, the difference, 2.639 - 2.895 = -.256, indi- cates that the terminal value of $1.00 invested in the fund portfolios was 25.6 cents

less than the terminal value of $1.00 invested in the comparable Fhf portfolios. Or the terminal value of the fund portfolio was (25.6/2.895) = 8.9 per cent lower than the PM portfolios. (It should be noted that in gen- eral 6 will not be equal to the percentage difference in the average wealth ratio, as happens to be the case here.)

226 2 THE JOURNAL OF BUSINESS

To see this more clearly, let us re- arrange (4.22) and write it as 10'R* -,RZ, = 6;

+ Pj
giving explicit recognition (by the pre- script 10) to the fact that we are consid- arguments ering a tenYear of Section IV period. indicate Now that the length, turns (6.6) - holds for a time interval of any so let us consider the annual re- li; (where the subscript 1 denotes the year of again to replace

observation), 6; with ej*, using adding (4.22) an

the

,qt-pi,

'?j, =

and Vi

+ Pj(lRLt - lR;t) + lei, (6'7)

as a regression equation applying to the annual observations over the ten-year pe- ri ~ d ~NOW . ~ by the additivity of the con- tinuously compounded rates it is clear that I& =

x

lR:t t=l

10 7

and likewise for lo^; and Thus, by substitution from (6.,) into (6.6), we have

88

Equation (6.1) can be used directly for evalu- ating the performance of portfolios, and it has many convenient properties (the most

important of which is the relatively straightforward tests of the signifi- cance of the performance measures which it allows). However, since the formulation given by (6.7) is derived and discussed in detail in Jensen [32], we shall not pursue these issues here. We refer the in- terested reader to that discussion. DELTA* FIG.12.-Frequency distribution (half-sigma intervals) of the performance measure 6* for 115 funds for the ten-year period 1955-64.

227 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS Thus, 6:. = lev,

+C

10

,elt t=l (6.9) for the ten-year holding interval under consideration in Table 2. Now, as is clear from (6.9) and

as is explained in detail in Jensen [32], q,is a measure of the fore- casting ability of the portfolio manager and has the same properties as 6;. If the manager has no superior knowledge of future has, q, security > 0. prices, qj = 0, and if he The reason for deriving (6.9) is to show explicitly that the measure of per- formance 6j' is (except for a scale factor) equal to the intercept qj plus a sum of random variables, lei, all with zero ex- pected pling error values. in Pion Now the the measure effects of of sam- per- formance we let t,bt 8j' = become l~Lt- perfectly ,R;~, clear. If it is easily shown (cf. Johnston [33, p. 161) that the correlation Bj (and therefore between between the estimates gj and fjj bj) and is given by In addition, since the average a2(lej') for our sample of 115 funds is very that prising close r($, to that zero

r) the (.0032), estimates -1. Thus, we of see it 8' is by and not (6.10) P sur- in Table 2 are

highly negatively correlated, Of course, the cross-sectional observations are not perfectly correlated, since they are not all generated by the same process; (6.10) applies to repeated samples for a given fund, not a cross- section of different funds. Returning now to the

question of fund performance, the impression gained from an examination of the scatter of points in Figure 11 and the estimates of 6; given in Table 2 is that on the average these mutual funds have not

done as well as our very simple and naive policy of com- bining an investment in the market port- folio with an investment in government bonds. Now, as long as the capital asset pricing model is valid, the only possible reason for the existence of an inferior portfolio is the unnecessary generation of expenses by the

fund managers. These expenses are borne by the fund and there- fore reduce the portfolio's returns. In view of this, let us examine the funds in somewhat more detail and give special consideration to these expenses. The forecasting ability of mutual fund managers.-On the basis of the results considered above, there seems little doubt that the fund managers on the average were unable to predict future security prices well enough to increase returns sufficiently to cover their re- search and commission expenses. Before reaching a final conclusion regarding the 228 THE JOURNAL OF BUSINESS managers' forecasting success, let us con- sider the possibility of any predictive ability at all-even if insufficient to cover research and transactions costs. In order to examine this question, let us replicate the analysis of Figure 11 using as our measure of fund returns the total returns gross of all expenses except brokerage commissions.89 That is, let us hypothetically return to the funds all re- sources spent for security analysis, book- keeping services, etc., and if the man- agers have any forecasting ability at all, such ability ought to cause the average 6' to be positive. Figure 13 represents the results of the analysis using the gross returns of the funds, and the reader should note that the funds appear to scatter much more equally on either side of the market line loR~MQ.gO The average value of 6; cal- culated on the basis of the gross returns was which 6; The >

Oe91

-.025 6; average < 0 with and 6* of fifty-eight fifty-seven -.025 taken funds for at which face for value would indicate that on the average the fund portfolios were inferior. But we must recall that these returns were cal- culated without taking commission ex- penses into account. Data gathered by 8"t would be desirable to measure the returns gross of brokerage commissions as well as all other expenses, but unfortunately exact commission data is unavailable. We shall consider the effects of these commissions below, using estimates of their average size for a sample of funds. 90 The gross returns were calculated by adding to the annual net returns the annual expense ratios given by Wiesenberger (+ 100). The expense ratios are the ratio of total annual expenses, except inter- est, taxes, and brokerage commissions, to the aver- age total net asset value of the fund (X 100). 91 The average gross terminal wealth ratio for the funds was 2.813. Thus, the percentage difference in the the 2.895)/(2.895 average comparable terminal = -.081/2.895 risk wealth FM portfolios ratio = -.0028

for was the funds (see (2.813 n. and 87 - above).

Friend et al. [26] indicate that the weight- ed average portfolio turnover rate for mutual funds in the period 1953-58 was about 20 per ce nt.g2 Adding the broker- age expenses on these transactions (un- der the assumption that the average commission expense was 1 per cent) would increase the returns by about .002 per year, or about .02 for the ten-year period. This comes very close to account- ing for the average 6' of -.025. One other small bias against the funds would account for the remainder of this differ- ence. The standard of comparison, the FM portfolios, implicitly assumes a fully invested portfolio, but since the mutual funds face stochastic cash inflows and outflows, they must maintain a cash bal- ance to meet them. On the average,

the funds appear to hold about 2 per cent of their total net assets in cash.g3 If we assume that the funds had earned 2.96 per cent on these balances (equivalent to the riskless rate of 3 per cent com- pounded annually), this would increase their returns by another .0059 for the ten-year period. These adjustments in- dicate that the actual average 6' (gross of all expenses) is about +.0009, which is consistent with the hypothesis that before deduction of expenses the funds held neutral portfolios. Thus, on the basis of these results (net and gross), we conclude that in the ten- year period 1955-64 mutual fund mana- gers in general showed no evidence of an ability to predict the future performance of securities. That is, they did not as a whole show evidence of superior analyti- cal or forecasting ability in spite of the n2

Actually 19.8 per cent (see Friend et al. [26, p.2121). $3 Cf. Friend et el. [26, pp. 120-271. The data presented cover four dates in the period 1952-58 and indicate percentages of 2.67 in 1952,2.03 in 1955 and 1957, and 1.72 in 1958.

RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS 229

considerable resources devoted to these cording to the strong form of the mar- 96 tingale hypothesis given

by equation It is appropriate at this point to re- (1.2). They are, however, consistent with mind the reader that these results do not the joint hypothesis (1) that the capital '(prove" that security prices behave acasset pricing model is valid and (2) that 94

This evidence should not be construed to imply predict security prices. Sharpe [55, pp. 132-331, that there are no particular funds

which satisfy the analyzing the performance of thirty-four mutual requirements of superior analysts. We consider these funds in the period 1954-63, found a negative corre- questions below. lation between fund performance and expense ratios. 96 There is another fragment of evidence bearing Our results also tend to support this, but are not directly on the relationship between

performance, nearly as strong as his. The regression of the meas- expenses, and the ability of portfolio managers to ure of performance, 67,on Ei, the average ratio of 10 YEARS 1955-1964 1.61

++ 1.38 1.15 .92 -

.69 .46 SYMBOL CODE Y

,(/

- GROWTH

- GROWTH-INCOME .23 - X

- INCOME Z -.oo j II I II X

+ IOR'V +

-

INCOME-GROWTH

BALANCED

.OO .30 .60 .90 1.20 1.50 1.80 SYSTEMATIC

RISK -,L?

ARA.13.-Scatter diagram of risk and (gross) return for 115 open-end mutual funds in the period 195544

230 THE JOURNAL OF BUSINESS security prices behave according to the strong form of the martingale hypothesis (at least as far as these fund managers are concerned). If the reader is willing to as- sume that either (1) or (2) above is true, then the results provide a strong piece of evidence in favor of the other hypoth- esis. Ideally, we should like to test the validity of the capital asset pricing model by using individual assets or unmanaged portfolios. Given the results of these tests, we would then be in a position to make a much stronger statement regard- ing the apparent validity of the strong form of the martingale hypothesis.

I t is well to pause at this point to con- sider the effects of possible changes in the riskiness cussed earlier, of some the of measure the funds. of risk, As fi, dis- is invariant to the length of the time inter- val over which the returns are measured. Thus, in practice we would prefer to have monthly, weekly, or even daily data over the recent past for use in estimating the present risk of a portfolio. But, given our purposes and the unavailability of such data in convenient form, we have used annual data over the past ten to twenty years. Hence, it is very likely that the all expenses (except interest, taxes, and commis- sions) to net assets in the period 1955-64, yields: where the 67 are measured net of expenses. These results, although weak, are at least cunsist- ent with the hypothesis that mutual fund managers as a group cannot forecast prices any better than the average investor in the market. Hence, the more re-

sources they devote to forecasting and the more commission expenses they incur in implementing the trading advice of their research departments, the smaller will be their ex post returns. We note that the opposing hypothesis (ie, that prices can be pre- dicted implies and a positive profits relationship increased by between buying good 67 and advice) E, as long as funds devote resources to research and trad- ing only to the point where expected marginal reve- nue equals the marginal cost.

risk of some of the funds in our sample may have changed over time, and there is a legitimate question as to the possible effects of this factor on our results. The analysis of Figure 11 was also per- formed for the ten-year period 1945-54 on fifty-six of the 115 funds for which data were available. The scatter diagram of net returns and risk for these funds is given in Figure 14.96 The analysis utiliz- ing gross returns could not be replicated for this period, since sufficient expense data were not available. For comparison purposes, the results for these fifty-six funds in the period 1955-64 appear in Figure 15. Figure 14 seems to imply that the measure of systematic risk is also appro- priate for this earlier period. The scatter indicates a positive relationship between risk and return, as the theory of capital asset prices implies.g7 It appears that after expenses the funds' returns were much farther below the returns on possible FM combinations in the period 1945-54 than the later pe- riod fifty-six 1955-64. funds The was average -.I35 in 6' the for earlier these 96

The risk-free rate of interest in this period was taken to be 2.1 per cent (the ten-year yield to ma- turity ten-year on riskless

government return, bonds I&$ in = 1945) log, resulting (1

+

I&F) in = a

log, (1.230) = .208. The return on the market port-

folio, of 1.328 the 8. market was 1.536 line, over &$Me, this period. is I&* The = equation .208

+ tion

97

on Although the

estimated again we regression wish to place of 18; no interpreta- on fij, we present the results here for whatever interpretation the reader may desire to make for himself: I t should be noted that the outlier in the upper right comer of Figure 14 (the Investment Trust of Boston again) is the major

reason for the much higher corre- lation in this case as compared with the results seen. shown 75 in note above 83 for

above a discussion for the period of this 1955-64. fund.) (Also RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS 231

period and -.076 in the later period, The analysis was also performed for with forty-three and thirty-five funds the entire twenty-year period 1945-64, having 6,: < 0, respectively, in the two and the scatter of risk versus net return periods. 98 for the fifty-six funds for which data were available is given in Figure 16. The 98The average terminal wealth ratio for these risk-free rate was taken to be 2.1 per fifty-six funds over the period 1945-54 was 3.347 and cent,99

giving z&i = log, (1.021)20= that for the comparable risk FM portfolios (see n. 87

above) was 3.838. Thus, the percentage difference in the average terminal wealth ratios was (3.347 - cent less than the returns which could have been 3.838)/3.838 = -.128. Hence, the returns on an

obtained by holding a comparable risk FM portfolio.

an equal dollar

investment in the shares of these 99 The yield to maturity of a twenty-year govern- fifty-six funds in the period 1945-54 was 12.8 per

ment bond as of January, 1945. 10 YEARS 1945-1954

SYSTEMATIC RISK

-

EIG. 14.-Scatter diagram of risk and (net) return for fifty-six open-end mutual funds in the ten-year period 1945-54.

232 THE JOURNAL OF BUSINESS .416. The return on the market portfolio light of the previous results. A more in= 2.723,

z ~ ~ L was log, (15.23)

or an an- teresting facet of the scatter of Figure 16 nual return of 13.61 per cent compound-

is its apparent linearity. The reader will ed continuously. Thus, the equation of the market line is 2oR* = .416

+ 2.307 b.

lo0 Again

we present the regression results of

2uR: on

Oifor the interested reader. The results are: The

average b* for this twenty-year R: = + 1,1338j period was -.196, with thirty-nine funds 20

4: < 0 and seventeen for (.162) (.190) n = 56. which 4: > O.loO* lol

'01

' r = .63

for which

The average terminal wealth ratio for these

These results are not surprising in fifty-six funds over the twenty-year period 1945-64

SYMBOL CODE +

- GROWTH Y - GROWTH-INCOME X - INCOME-GROWTH X - INCOME Z - BALANCED -.OO

I,..

.90 1.20 1.50 1.80 .OO .30

.60

SYSTEMATIC RISK -

2

FIG.15.-Scatter diagram of risk and (net) return over the ten-year period 1955-64 for the fifty-six mu- tual funds existing in the period 1945-54.

233 RISK, CAPITAL ASSETS, AND EVALUATION OF PORTFOLIOS remember that it was argued in Section IV that the linear opportunity set will hold only for the variable R*

defined as was 9.01, and that for the comparable risk FM port- folios (see n. 87 above) was 11.45. Thus, the per- centage ratios was

di@eerence (9.01 - in 11.45)/11.45 the average = terminal -27, wealth and the of cent returns these less on funds than an the in equal this returns dollar twenty-year which investment could period have in were the been shares 27 per ob- tained by holding a comparable risk FM portfolio.

[(I

+ R)"IN - I]/(H/N). Several argu- ments were also presented which lead us to believe that H, the

"market horizon" interval, &own that is very

mated by linearity R* log, (1 of be

forH/N close to close zero. to I t zero, was also the parent

+ the R).Hence, very scatter of approxi- the Figure ap- ing l6 is our arguments piece regarding of evidence the length

20 YEARS 1945-1964 v

2.69 - U)u V)

-.OO

B 25 0 0-

LL

2.24 -

k z I -0

2 1.79 -

C*

C--,-A

.00 .30 Y Z + X X

-----

SYMBOL GROWTH

GROWTH-INCOME BALANCED INCOME-GROWTH INCOME CODE

.60 .90 1.20 1.50 1.80 SYSTEMATIC RISK

-)

FIG.16.-Scatter diagram of risk and (net) return over the 20-year period 1945-64 for fifty-six mutual funds.

234 THE JOURNAL OF BUSINESS of the "market horizon" interval. How- measurement ever, due to the errors presence in both of

24:

substantial and br, we are unwilling to place great emphasis on these arguments at this time.lo2 10%

These measurement errors arise primarily from sampling errors in the estim ation of and the inability to measure true gross

returns of the funds (ie, before deduction of brokerage commissions and management expenses). Thus, the errors can be re-

The twenty-year scatter of risk versus returns Figure 17. is plotted That is, in the arithmetic returns (1 form

+&)

in plotted on the abscissa are the twenty- year wealth relatives WZO/WO= exp duced, and work in progress on these problems at this time should allow us to obtain much better tests of the adequacy of the

capital asset pricing model and the horizon solution. 20 YEARS 1945-1964

-.Go I II 1 I

I

.OO .30 .60 SYSTEMATIC .90 RISK

-/

1.20 1.50 1.80

FIG.17 -Scatter twenty-year period 1955-64. diagram Dashed of risk versus line represents arithmetic the return "market (wealth line" relatives) which would for fifty-six be given funds by over a naive the interpretation of the capital asset pricing

model.

235 KLSK, CAPl'L'AL ASSETS, ANU EVALUATION 01.'POKTE'OLIOS (&*). There is a suggestion of non- linearity. The scatter certainly does not conform (1 (1 well = = to 1.52 1.52 expressed the

+ + &F)MZ z&)

+ + dashed (15.23 13.716 as

market - , line 1.52)b which would be given by a naive inter- pretation of the capital asset pricing model. On the other hand, they do seem line to conform (1

+ zoRp)MQ, more closely103 which is to given the by

solid = exp (.416

+ 2.3078) ,

which is the inverse of the limit of (4.13). It might also be recalled from the dis- cussion in Section IV-B that this curve is the limiting form of the market line as H, the "market horizon," goes to zero. Thus, if the arguments given in Section IV-B regarding the length of the market hori- zon are erroneous and the "true" market horizon is actually significantly greater than zero (but less than twenty years), then the "true" market line (given by the inverse of the function defined in [4.10] applied between (1 (1

+ + 2ORp)MQ 2dF) to the

+ lie ~oRF)MZ note somewhere the that points and a "true" horizon interval of H equal to one month is visibly will different not yield from a market [I + and [4.11]) passing line M. will (1 (But through

line 2aF]MQ.) which Fund performance from the investor's point of view.-An evaluation of mutual fund performance from the investor's point of view must allow explicitly for the effects of transaction costs and load- ing fees on returns in going from cash to portfolio and back to cash at the horizon date. Thus, in calculating the net returns 10' But

again, the existence of measurement errors prevents the formulation of firm conclusions on this point.

to the investor for this analysis, explicit allowance was made for the actual load- ing charge for each fund104 in 1955 as well as all other expenses incurred by the fund. The brokerage commissions on the purchase and sale of the market port- folio were assumed to be 1 per cent each way. The average 6' calculated under

these assumptions 1955-64 for the was -.146, with ten-year 6* < period 0 for eighty-nine of the 115 funds. The scat- ter of points is given in Figure 18. A question of consistency through time. -We observed earlier that we expect the ex post results of neutral portfolios to be randomly distributed about the market line. We have also seen that the funds in our sample on the average held somewhat inferior portfolios in both pe- riods. It also appears that some funds hold neutral portfolios, and we might legitimately ask the question: Do individual funds consistently hold either su- perior or inferior portfolios? One way to address this question is to examine the relationship between the performance measure,1056{, for each fund in the two ten-year periods previously analyzed. The regression of 6l1955-64 on 6~:1945--54 for the fifty-six funds observed in both periods yields

the following results: Thus, for this sample of fifty-six funds, positive correlation exists between the performance measures in the two periods, indicating that some funds may be conlo4

That is, the ten-year returns net of transac- tions costs were calculated as log,xi(l +I&), where xi is defined as the ratio of the net

asset value to the offering price per share of the jth fund and

lfij

is the return net of all expenses for the jth fund.

all management expenses and brokerage commissions but gross of (excluding) loading charges.

106 Calculated

net of