Investment Theory > Portfolio Management - Performance Measurement

Portfolio Management – Performance Measurement Copyright © 1996 – 2006 Investment Analytics

1

Overview „ „ „

Measuring Profitability Equity Curve Measures Portfolio Performance Measures

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 2

Measuring Profitability „ „ „

Net returns Buy and hold test Distance from the ideal

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 3

Net Returns „

Test of investment strategy „ „

Long positions when expected returns are positive Short positions when expected returns are negative n

r = ∑ pt ( yt +1 − yt ) t =1

⎧1 if ( f t +1 − yt ) > 0 ⎪ pt = ⎨− 1 if ( f t +1 − yt ) < 0 ⎪0 if ( f − y ) = 0 t +1 t ⎩ Copyright © 1996-2006 Investment Analytics

Performance Measurement

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Buy and Hold Test „

Benchmark to quantify excess returns „

Tests whether profitability is due to predictive ability or just general market conditions

c + ( yt + n − yt ) r= yt „

C is stock dividend or bond coupon

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 5

Distance From the Ideal „

Measures returns from trading system against perfect predictor d n

rd =

∑ p (y t =1 n

∑| y t =1

„

t

t +1

t +1

− yt )

− yt |

Pt as previously defined

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 6

Equity Curve Measures „ „ „ „

Drawdown Luck coefficient Stirling ratio Risk of ruin

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Performance Measurement

Slide: 7

Drawdown Equity Curve

Cumulative Returns

130 120 110 100

Drawdown

90 80 70 27

25

23

21

19

17

15

13

11

9

7

5

3

1

60

Trades

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Performance Measurement

Slide: 8

Drawdown „

Systems with large drawdowns hard to trade „

„ „

Requires lots of capital & confidence!

Smooth equity curve is desirable Usually harder to obtain than high net return

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 9

Luck Coefficient „

How much of total profit was dependent on most profitable (k) trades(s)?

l (k ) =

Maxk {r0 , r1 , ..., rn } n

∑r i =1

„

i

Large L indicates system success unlikely to be repeatable

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 10

Stirling Ratio „

Penalizes average returns for drawdown n

1 ri ∑ n i =1 s= 10 − d i „

„

di is the i-period maximum drawdown.

Can be too slow to change „

Recalculate frequently

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 11

Risk of Ruin „

Probability that capital will be depleted „

Depends on „ „ „

„

Assume: „ „ „

„

Probabilkity of successful trade p Payoff ratio (av. Win / av. Loss) Fraction of capital exposed to trading Payoff ratio is 1 We risk all capital K sequential trades

R ~ [(1-p)/p]k

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 12

Portfolio Performance Measures „

Sharpe ratio: (rp - rf) / σp „

„

Treynor’s measure: (rp - rf) / βp „

„

Excess return per unit of systematic risk

Jensen’s measure: αP = rp - [rf + βp(rM - rf)] „

„

Measures reward to total risk trade-off

The portfolio’s alpha - abnormal return above that predicted by CAPM

Appraisal ratio: αP / σ(ep) „

Abnormal return per unit of specific risk that could be diversified away using a market index portfolio

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 13

Which Measure to Use „ „ „

Suppose you have invested in a portfolio P Case 1: P is your entire investment fund Case 2: P is your active portfolio and: „

„

You are also investing in the passive market index portfolio

Case 3: P is one of many portfolios „ „

Combined in a large investment fund E.g. You are one of a number of portfolio managers

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 14

Case 1: P Is Your Entire Investment Fund „

Compare P’s Sharpe ratio with other fund: „ „

Passive index fund Professionally managed active funds

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 15

Case 2: P Is Your Active Portfolio „

Recall: S2C = S2M + [αP / σ(eP)]2 „

„

„

SC is the Sharpe ratio of the combined portfolio (M and P)

“How much does your active portfolio P add to the Sharpe ratio SM of your passive market index portfolio?” Use appraisal ratio: [αP / σ(eP)]

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 16

Case 3: P Is One of Many Portfolios „ „

P’s contribution to the entire diversified fund is αP So could use Jensen’s measure (portfolio alpha) „

„

But this takes no account of risk

Better to use Treynor’s measure: (rp - rf) / βp „

Measure P’s excess return against the systematic risk (beta) rather than the total diversifiable risk (s.d.)

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 17

Lab: Portfolio Performance Measurement „

Advise a client in choice of funds „

„

Excel lab: portfolio performance measurement „ „

„

Use different performance measures

Complete worksheet See solution worksheet

See written notes and solution

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 18

Portfolio Performance Measurement - Solution Fund P Fund Q Benchmark M Sharpe Alpha (Jensen) Beta Treynor σ(e) Appraisal ratio R2

Copyright © 1996-2006 Investment Analytics

0.43 1.63% 0.70 3.97 1.92% 0.85 91.12%

0.49 5.26% 1.40 5.38 9.35% 0.56 63.82%

Performance Measurement

0.19 0.00% 1.00 1.64 0 0.00 100.00%

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Portfolio Performance Measurement - Solution „

„

Both P & Q outperform M: „ Higher Sharpe ratios, positive alphas

Fund Q is preferred: „

If this fund is the client’s only investment „

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As one of a mix of portfolios „

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Higher Sharpe ratio than P Higher Treynor measure than P

P is preferred if used as an active fund „

In conjunction with a passive index fund „

Higher appraisal measure than Q

Copyright © 1996-2006 Investment Analytics

Performance Measurement

Slide: 20

Summary: Performance Measurement „

Appropriate testing metric depends on application „ „ „

„

Models unlikely to perform equally on every basis „ „

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Forecasting Trading system development Portfolio management E.g. with low R2 may generate significant profits Models with good statistical fit may trade badly

Moral „

Decide objective and testing strategy before modeling!

Copyright © 1996-2006 Investment Analytics

Performance Measurement

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