Formulas Final Ii-sept09

Formulae Final Examination

Fixed Income Valuation and Analysis Derivatives Valuation and Analysis Portfolio Management

Table of Contents

1.

Fixed Income Valuation and Analysis

1.1

Time Value of Money ...................................................................................1

1.1.1 1.1.2 1.1.3 1.1.4 1.1.5

1.2

Time Value of Money ......................................................................................... 1 Bond Yield Measures......................................................................................... 2 Term Structure of Interest Rates .................................................................... 3 Bond Price Analysis .......................................................................................... 4 Risk Measurement.............................................................................................. 6

Convertible Bonds........................................................................................8

1.2.1

1.3

Investment Characteristics .............................................................................. 8

Callable Bonds ..............................................................................................9

1.3.1

1.4

Valuation and Duration ..................................................................................... 9

Fixed Income Portfolio Management Strategies ..................................9

1.4.1 1.4.2

2.

Passive Management......................................................................................... 9 Computing the Hedge Ratio: The Modified Duration Method................. 9

Derivative Valuation and Analysis

2.1

11

Financial Markets and Instruments .......................................................11

2.1.1

2.2

Related Markets ................................................................................................ 11

Analysis of Derivatives and Other Products .......................................13

2.2.1 2.2.2 2.2.3

3.

1

Futures ................................................................................................................ 13 Options................................................................................................................ 17 Standard Normal Distribution: Table for CDF........................................... 24

Portfolio Management

3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5

3.2 3.2.1 3.2.2

3.3 3.3.1 3.3.2 3.3.3

3.4 3.4.1

25

Modern Portfolio Theory...........................................................................25 The Risk/Return Framework .......................................................................... 25 Measures of Risk .............................................................................................. 27 Portfolio Theory ................................................................................................ 29 Capital Asset Pricing Model (CAPM) ........................................................... 30 Arbitrage Pricing Theory (APT) .................................................................... 31

Practical Portfolio Management .............................................................34 Managing an Equity Portfolio........................................................................ 34 Derivatives in Portfolio Management .......................................................... 37

Asset/Liability-Analysis and Management...........................................42 Valuation of Pension Liabilities .................................................................... 42 Surplus And Funding Ratio ........................................................................... 43 Surplus Risk Management ............................................................................. 43

Performance Measurement ......................................................................45 Performance Measurement and Evaluation .............................................. 45

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Fixed Income Valuation and Analysis

1.

Fixed Income Valuation and Analysis

1.1 Time Value of Money 1.1.1 Time Value of Money 1.1.1.1

Present and Future Value

Simple Discounting and Compounding

Present value =

Future value ( 1 + Interest rate p.a.) number of years

Future value = (Present value) ⋅ (1 + Interest rate p.a.) number of years

1.1.1.2

Annuities

The present value of an annuity is given by N

CF

Pr esent value = ∑

t =1 ( 1 +

R )t

=

CF R

 1 ⋅ 1 −  (1 + R )N 

where CF R N

constant cash flow discount rate, assumed to be constant over time number of cash flows

The future value of an annuity is given by  (1 + R ) N − 1   Future value = CF ⋅    R  

where CF R N

1.1.1.3

constant cash flow discount rate, assumed to be constant over time number of cash flows

Continuous Discounting and Compounding

Present value =

Future value e

number of years ⋅ continuous interest rate p.a.

Future value = (Present value) ⋅ e number of years ⋅ continuous interest rate p.a.

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   

Fixed Income Valuation and Analysis

1.1.2 Bond Yield Measures 1.1.2.1

Current Yield

Current yield =

1.1.2.2

Annual coupon Price

Yield to maturity

The bond price as a function of the yield to maturity is given by N

P0 = ∑

i =1

CFi

(1 + Y )

ti

=

CF1

(1 + Y )

t1

+

CF2

(1 + Y )

t2

+ ... +

CFN

(1 + Y ) t N

where

Y P0 CFi CFN N

yield to maturity current paid bond price (including accrued interest) cash flow (coupon) received at time t i cash flow (coupon plus principal) received at repayment date t N number of cash flows

Between two coupon dates, for a bond paying coupons annually, the bond price is given by  CF1 CFN  CF2 Pcum, f = Pex, f + f ⋅ C = (1 + Y ) f  + + ... +  1 (1 + Y )2 (1 + Y ) N   (1 + Y )

where

Pcum , f

current paid bond price (including accrued interest)

Pex, f

quoted price of the bond

Y f

yield to maturity time since last coupon date in years

CFi

cash flow (coupon) received at time t i

CFN

final cash flow (coupon plus principal) number of cash flows

N

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Fixed Income Valuation and Analysis

1.1.2.3

Yield to Call N

P0 = ∑

i =1

CFi

(1 + Yc )

ti

=

CF1

(1 + Yc )

+

t1

CF2

(1 + Yc )

t2

+ ... +

CFN

(1 + Yc ) t N

where

P0

current paid bond price (including accrued interest)

Yc

yield to call

CFi

cash flow (coupon) received at time t i

CFN N

cash flow (coupon plus principal) received at call date t N

1.1.2.4

number of cash flows until call date

Relation between Spot Rate and Forward Rate 1

(1 + R0,t ) = [(1 + R0,1 ) ⋅ (1 + F1,2 ) ⋅ (1 + F2,3 )...(1 + Ft −1,t )] t where

R0 ,t

spot rate p.a. from 0 to t

R0,1

spot rate p.a. from 0 to 1

Ft −1 ,t

forward rate p.a. from t − 1 to t

(1 + R0,t ) t ⋅ (1 + Ft ,t 1

1

1 2

)

t2 −t1

(

= 1 + R 0 ,t 2

)t

2

⇔ Ft1 ,t2

where

R0 ,t1

spot rate p.a. from 0 to t1

R0 ,t 2

spot rate p.a. from 0 to t 2

Ft1 ,t 2

forward rate p.a. from t 1 to t 2

1.1.3 Term Structure of Interest Rates 1.1.3.1

Theories of Term Structures

Expectations hypothesis ~ Ft1 ,t2 = E( Rt1 ,t2 ) where

Ft1 ,t2 ~ Rt1 ,t2

forward rate from t 1 to t 2

E(.)

expectation operator

random spot rate from t1 to t 2

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( (

 1 + R 0 ,t 2 =  1+ R 0,t1 

)t )t

1 2

1

 t2 −t1  −1  

Fixed Income Valuation and Analysis

Liquidity Preference Theory ~ Ft1 ,t2 = E( Rt1 ,t2 ) + Lt1 ,t2 ,

Lt1 ,t2 > 0

where

Ft1 ,t2 ~ Rt1 ,t2

forward rate from t 1 to t 2

Lt1 ,t2

liquidity premium for the time t1 to t 2

E(.)

expectation operator

random spot rate from t1 to t 2

Market Segmentation Theory ~ Ft1 ,t2 = E( Rt1 ,t2 ) + Π t1 ,t2 ,

>0 Π t1 ,t2 <

where

Ft1 ,t2 ~ Rt1 ,t2

forward rate from t1 to t 2

Π t ,t

risk premium for the time from t1 to t 2

E(.)

expectation operator

random spot rate from t1 to t 2

1 2

1.1.4 Bond Price Analysis 1.1.4.1

Yield Spread Analysis

Relative Yield Spread

Relative yield spread =

Yield bond B - Yield bond A Yield bond A

Yield Ratio

Yield ratio = 1.1.4.2

Yield bond B Yield bond A

Valuation of Coupon Bonds using Zero-Coupon Prices

Valuation of Zero-Coupon Bonds P0 =

CFt

(1 + Rt ) t

where

P0

bond price at time 0

CFt

cash flow (principal) received at repayment date t

Rt

spot rate from 0 to t

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Fixed Income Valuation and Analysis

Valuation of Coupon-Bearing Bonds N

P0 = ∑

i =1

CFi

(1 + Ri )

ti

=

CF1

(1 + R1 )

t1

+

CF2

(1 + R2 )

t2

+...+

CFN

(1 + R N ) t N

where

P0

bond price at time 0

CFi

cash flow (coupon) received at time t i

CFN

cash flow (coupon plus principal) received at repayment date t N

Ri N

spot rate from 0 to t i number of cash flows

Price with accrued interest of a bond paying yearly coupons N

Pcum, f = Pex, f + f ⋅ C = ∑

i =1

CFi

(1 + Rt ) t − f i

i

where

Pcum, f

price of the bond including accrued interest

Pex, f

quoted price of the bond

f CFi Rti

time since the last coupon date in fractions of a year cash flow at time ti spot rate from f to ti

C

coupon

Valuation of Perpetual Bonds P0 =

CF R

where

P0 CF R

current price of the perpetual bond perpetual cash flow (coupon) discount rate, assumed to be constant over time

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Fixed Income Valuation and Analysis

1.1.5 Risk Measurement 1.1.5.1

Duration and Modified Duration

Macaulay’s Duration N

D=

t i ⋅ PV( CFi ) = P i =1 N

∑

∑

i =1 N

∑

i =1

t i ⋅ CFi

(1 + Y )

ti

CFi

t1 ⋅ CF1

=

(1 + Y )

ti

(1 + Y )

t1

CF1

(1 + Y )

t1

+ +

t 2 ⋅ CF2

(1 + Y )

t2

CF2

(1 + Y )

t2

+ ... + + ... +

t N ⋅ CFN

(1 + Y )t N CFN

(1 + Y )t N

where

D P Y CFi PV(CFi) CFN N

Macaulay’s duration current paid bond price (including accrued interest) bond’s yield to maturity cash flow (coupon) received at time ti present value of cash flow CFi cash flow (coupon plus principal) received at repayment date t N number of cash flows

Modified and Price Duration D 1 ∂P = P ∂Y 1 + Y ∂P D DP = − = D mod ⋅ P = P ∂Y 1+ Y D mod = −

where

D mod DP D P Y

modified duration price or dollar duration Macaulay’s duration current paid bond price (including accrued interest) bond’s yield to maturity

Price Change Approximated with Duration −D ∆P ≅ ⋅ P ⋅∆Y = − D mod ⋅ P⋅ ∆Y = − D P ⋅ ∆Y (1 + Y )

∆P

−D − DP mod ≅ ⋅ ∆Y = − D ⋅ ∆Y = ⋅ ∆Y (1 + Y ) P P

where

∆P mod

D DP D P Y

∆Y

price change of the bond modified duration price or dollar duration Macaulay’s duration current paid bond price (including accrued interest) bond’s yield to maturity small change in the bond yield

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Fixed Income Valuation and Analysis

Portfolio Duration N

D P = ∑ xi ⋅ Di i =1

where

DP

portfolio duration

xi

proportion of wealth invested in bond i

Di N

duration of bond i

1.1.5.2

number of bonds in the portfolio

Convexity 1 ∂ 2P C= ⋅ = P ∂Y 2

N

1 CFi

∑ (1 + Y ) i =1

⋅

N t i (t i + 1) ⋅ CFi 1 ⋅ 2 ∑ (1 + Y ) i =1 (1 + Y )ti

ti

CP = C ⋅ P where C

convexity P

C P Y CFi CFN

price convexity current paid bond price (including accrued interest) bond’s yield to maturity cash flow (coupon) received at time ti cash flow (coupon plus principal) received at repayment date t N

Price change approximated with duration and convexity 1 P C ⋅ ∆Y 2 2 1 1 ∆P −D ≅ ⋅ ∆Y + C⋅ ∆Y 2 = − D mod ⋅ ∆Y + C ⋅ ∆Y 2 P (1 + Y ) 2 2

∆P ≅ − D P ⋅ ∆Y +

where

∆P

D mod DP D C CP

P Y

∆Y

price change of the bond modified duration price or dollar duration Macaulay’s duration convexity price convexity current paid bond price (including accrued interest) bond’s yield to maturity Small change in the bond yield

Portfolio Convexity N

Portfolio convexity

= ∑ wi ⋅ C i i =1

where wi Ci

N

weight (in market value terms) of bond i in the portfolio convexity of bond i number of bonds in the portfolio

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Fixed Income Valuation and Analysis

1.2 Convertible Bonds 1.2.1 Investment Characteristics Conversion ratio = Number of shares if one bond is converted

Conversion price =

Face value of the convertible bond Number of shares per bond (if there is a conversion)

Conversion value = Conversion ratio ⋅ Market price of stock

Conversion premium (in %) =

1.2.1.1

Payback Analysis

PP = where PP MP CV CY DY

1.2.1.2

(MP−CV)/CV Conversion premium = (CY − DY) (CY − DY)

payback period in years market price of the convertible conversion value of the convertible current yield of the convertible = (coupon/MP) dividend yield on the common stock = dividend amount / stock price

Net Present Value Analysis NPV =

where NPV CV FV Ync Yc n

Market price of bond - Conversion value Conversion value

CV − FV ( 1+Ync ) n

−∑

FV ⋅ ( Ync − Yc ) ( 1+Ync )t

net present value conversion value face value yield on non-convertible security of identical characteristics yield on the convertible security years before the convertible is called

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Fixed Income Valuation and Analysis

1.3 Callable Bonds 1.3.1 Valuation and Duration 1.3.1.1

Determining the Call Option Value

Callable bond price = Call-free equivalent bond price – Call option price

1.3.1.2

Effective Duration and Convexity

Call adjusted Pricecall free = duration Pricecallable Call adjusted Pricecall free = Convexity Pricecallable

 Duration of   ⋅ (1 − δ ) ⋅   call - free bond    Price of   Convexity of   ⋅ (1 − δ ) −  call - free ⋅   bond  call - free bond   

   ⋅γ  

 Duration of  ⋅  call - free  bond 

where

δ γ

Delta of the call option embedded in the bond Gamma of the call option embedded in the bond

1.4 Fixed Income Portfolio Management Strategies 1.4.1 Passive Management 1.4.1.1

Immunisation

A=L DA = DL A ⋅ DA = L ⋅ DL where A L DA DL

present value of the portfolio present value of the debt duration of the portfolio duration of the debt

1.4.2 Computing the Hedge Ratio: The Modified Duration Method HR = ρ ∆S ,∆F ⋅

NF = −

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N S ⋅ S t ⋅ DSmod k ⋅ Ft ,T ⋅ D Fmod

σ ∆S S ⋅ D Smod = t σ ∆F Ft ,T ⋅ D Fmod

=−

N S ⋅ S t ⋅ DSmod k ⋅ S CTD ,t ⋅ D Fmod

page 9

⋅ CFCTD ,t

    

2

    

Fixed Income Valuation and Analysis

where HR St Ft,T

ρ ∆S ,∆F σ ∆S σ ∆F

hedge ratio spot price at time t futures price at time t with maturity T correlation coefficient between ∆S and ∆F standard deviation of ∆S

DSmod

standard deviation of ∆F cheapest-to-deliver modified duration of the asset being hedged

D Fmod NF NS k SCTD,t CFCTD,t

modified duration of the futures (i.e. of the CTD) number of futures contracts number of the spot asset to be hedged contract size spot price of the CTD conversion factor of the CTD

CTD

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Derivative Valuation and Analysis

2.

Derivative Valuation and Analysis

2.1 Financial Markets and Instruments 2.1.1 Related Markets 2.1.1.1

Swaps

Interest rate swap The swap value for the party that receives fix may be expressed as V = B1 − B2 where

V B1 B2

value of the swap value of the fixed rate bond underlying the swap value of the floating rate bond underlying the swap

B1 is the present value of the fixed bond cash flows n

B1 = ∑

K

i =1 (1 +

R0,ti ) ti

+

Q (1 + R0,tn ) tn

where

B1

value of the fixed rate bond underlying the swap

K

fixed payment corresponding to the fixed interest to be paid at time t i

Q notional principal in the swap agreement R0, t i spot interest rate corresponding to maturity t i When entering in the swap and immediately after a coupon rate reset date, the value of bond B2 is equal to the notional amount Q . Between reset dates, the value is B2 =

K* Q + t1 (1 + R0,t1 ) (1 + R0,t1 ) t1

where

B2

value of the floating rate bond underlying the swap

K * floating amount (initially known) used for the payment at date t1 , the next

Q R0 ,t1

reset date. notional principal in the swap agreement spot interest rate corresponding to maturity t1

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Derivative Valuation and Analysis

Cross Currency Interest Rate Swap The swap value may be expressed as V = S ⋅ BF − BD where

V value of the swap S spot rate in domestic per foreign currency units B F value of the foreign currency bond in the swap, denoted in the foreign currency

B D value of the domestic bond in the swap, denoted in the domestic currency 2.1.1.2 Credit Default Swaps (CDS) CDS contingent payment The payment in case of default to the buyer of the CDS may be expressed as

N ⋅ (1 − R ) where

N R

Notional amount of the CDS Recovery rate of the reference bond

Default probabilities The probability of defaulting between t i −1 to t i is

(p1 ⋅ p2 ⋅ ... ⋅ pi −1 ) ⋅ (1 − pi ) where

p i Probability of surviving over the interval t i −1 to t i without a default payment 1 − p i Probability of a default being triggered CDS Evaluation The theoretical CDS spread s is obtained by equating: Present Value of expected payments = Present Value of expected payoffs where Present Value of expected payment = T

∑ Probability of survival · Payment · discount factor t

t =1 T

t

t

+

∑ Probability of default · accrual payment in case of default · discount factor t =1

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t

t

page 12

t

Derivative Valuation and Analysis

and Present Value of expected payoffs = T

∑ Probability of default · (1 - Recovery Rate ) · discount factor t

t =1

t

t

2.2 Analysis of Derivatives and Other Products 2.2.1 Futures 2.2.1.1

Theoretical Price of Futures

Pricing Futures on Assets that Provide no Income Ft ,T = S t ( 1 + Rt ,T )T −t where

Ft ,T

futures price at date t of a contract for delivery at date T

St Rt ,T

spot price of the underlying at date t risk-free interest rate for the period t to T

General Cost of Carry Relationship Ft ,T = S t ( 1 + Rt ,T )T −t + k ( t , S ) − FV ( revenues ) where forward or futures price at date t of a contract for delivery at date

Ft ,T

T St Rt ,T

spot price of the underlying at date t risk-free interest rate for the period t to T

k (t , S ) FV(revenues)

carrying costs, such as insurance costs, storage costs, etc. future value of the revenues paid by the spot

Continuous Time Cost of Carry Relationship Ft ,T = St e

( rt ,T − y )⋅(T −t )

where

Ft ,T futures price at date t of a contract for delivery at date T St y

rt ,T

spot price of the underlying at date t continuous net yield (revenues minus carrying costs) of the underlying asset or commodity continuously compounded risk-free interest rate

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Derivative Valuation and Analysis

Stock Index Futures N

Ft ,T = I t ⋅ ( 1 + Rt ,T )T −t − ∑

T

∑ wi ⋅ Di ,t j

i =1 t j =1

T −t j

⋅ ( 1 + Rt j ,T )

where

Ft ,T

futures price at date t of a contract that expires at date T

It

current spot price of the index dividend paid by stock i at date tj

Di,t j wi

Rt ,T

weight of stock i in the index risk-free interest rate for the period t to T

Rt j ,T interest rate for the time period tj until T N

the number of securities in the index

Interest Rates Future Cost of Carry Relationship

Ft ,T =

( St + At ) ⋅ ( 1 + Rt ,T )T − t − Ct ,T − AT Conversion Factor

where

Ft ,T quoted futures “fair” price at date t of a contract for delivery at date T C t ,T future value of all coupons paid and reinvested between t and T St

spot value of the underlying bond

At accrued interest of the underlying at time t Rt ,T risk-free interest rate for the period t to T AT

accrued interest of the delivered bond at time T

Theoretical Futures at the Delivery Date FT,T =

spot price of cheapest to deliver conversion factor

Forward Exchange Rates

 1 + Rdom   Ft ,T = S t   1 + R for    with continuous compounding Ft ,T = S t e

T −t

(rdom −rfor )⋅(T −t )

where

Ft ,T

forward exchange rate (domestic per foreign currency)

St

spot exchange rate (domestic per foreign currency)

R dom

domestic risk-free rate of interest for the period t to T

R for

foreign risk-free rate of interest for the period t to T

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Derivative Valuation and Analysis

rdom

continuous domestic risk-free rate of interest for t to T

r for

continuous foreign risk-free rate of interest for t to T

Commodity Futures

Ft ,T = S t ⋅ (1 + Rt ,T ) + k (t , T ) − Yt ,T where Ft,T St Rt,T k(t,T) Yt,T

2.2.1.2

futures price at date t of a contract for delivery at date T spot price of the underlying at date t risk-free interest rate for the period (T – t) carrying costs, such as insurance costs, storage costs, etc. convenience yield

Hedging Strategies

The Hedge Ratio HR = where HR ∆S ∆F NF NS

k

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N ⋅k ∆S =− F ∆F NS

N F = − HR ⋅

hedge ratio change in spot price per unit change in futures price per unit number of futures number of spot assets contract size

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NS k

Derivative Valuation and Analysis

The Perfect (Naive) Hedge

 HR = ±1  NS   N F = m k where HR NF NS

k

hedge ratio number of futures number of spot assets contract size

Minimum Variance Hedge Ratio - Hedged Profit For a long position in the underlying asset Hedged profit = (ST − S t ) − (FT ,T − Ft ,T ) where

ST St FT ,T

spot price at the maturity of the futures contract spot price at time t futures price at its maturity

Ft ,T

futures price at time t with maturity T

- Minimum Variance Hedge Ratio HR = where HR Cov(∆S,∆F) Var(∆F)

ρ∆S,∆F σ∆S σ∆F

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σ Cov(∆S , ∆F ) = ρ ∆S , ∆F ⋅ ∆S Var (∆F ) σ ∆F

hedge ratio covariance between the changes in spot price (∆S) and the changes in futures price (∆F) variance of changes in futures price (∆F) coefficient of correlation between ∆S and ∆F standard deviation of ∆S standard deviation of ∆F

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Derivative Valuation and Analysis

2.2.2 Options 2.2.2.1

Determinants of Option Price

Put-Call Parity for European and American Options PE = C E − S + D + Ke − rτ

CUS − S + Ke − rτ ≤ PUS ≤ CUS − S + K + D where

τ

time until expiry of the option strike or exercise price of the option continuously compounded risk-free rate of interest spot price of the underlying value of European call option value of European put option value of American call option value of American put option present value of expected cash-dividends during the life of the option

K

r S CE PE CUS PUS D

2.2.2.2

Option Pricing Models

Black and Scholes Option Pricing Formula The prices of European Options on Non-Dividend Paying Stocks C E = S ⋅ N (d1 ) − Ke − rτ ⋅ N (d 2 ) PE = Ke − rτ ⋅ N (− d 2 ) − S ⋅ N (− d1 )

d1 =

ln(S / K ) + (r + σ 2 / 2)τ

σ τ

, d 2 = d1 − σ τ

where

CE PE S

τ

K

σ r

N( ⋅ )

value of European call value of European put current stock price time in years until expiry of the option strike price volatility p.a. of the underlying stock continuously compounded risk-free rate p.a. cumulative distribution function for a standardised normal random

variable (see table in 2.2.3), and x

1 e − ∞ 2π

N ( x) = ∫

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−

s2 2 ds

Derivative Valuation and Analysis

European Option on Stocks Paying Known Dividends C E = S * ⋅ N (d1* ) − Ke − rτ ⋅ N (d 2* ) PE = Ke − rτ ⋅ N (− d 2* ) − S * ⋅ N (−d1* ) d1*

=

where CE PE

τi Di S

τ

K

σ r

I N( ⋅ )

(

)

ln S * / K + (r + σ 2 / 2)τ

σ τ

I

, d 2* = d1* − σ τ , S * = S − ∑ Di ⋅ e − rτ i i =1

value of European call value of European put time in years until i th dividend payment dividend i current stock price time in years until expiry of the option strike price volatility p.a. of the underlying stock continuously compounded risk-free rate p.a. number of dividend payments cumulative normal distribution function (see table in 2.2.3)

European Option on Stocks Paying Unknown Dividends When dividends are unknown, a common practice is to assume a constant dividend yield y. Then C E = S ⋅ e − yτ ⋅ N (d1 ) − Ke − rτ ⋅ N (d 2 )

PE = Ke − rτ ⋅ N (− d 2 ) − S ⋅ e − yτ ⋅ N (−d1 )

d1 = where CE PE y S

τ

K

σ r

N( ⋅ )

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ln(S / K ) + (r − y + σ 2 / 2)τ

σ τ

, d 2 = d1 − σ τ

value of European call value of European put continuous dividend yield current stock price time in years until expiry of the option strike price volatility p.a. of the underlying stock continuously compounded risk-free rate p.a. cumulative normal distribution function (see table in 2.2.3)

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Derivative Valuation and Analysis

Options on Stock Indices J I  − r ⋅τ  C E =  S − ∑ ∑ D j ,i ⋅ e j ,i  N ( d1 ) − K ⋅ e − r ⋅τ ⋅ N ( d 2 ) j =1i =1  

J I  − r ⋅τ  PE = K ⋅ e − r ⋅τ ⋅ N ( −d 2 ) −  S − ∑ ∑ D j ,i ⋅ e j ,i  ⋅ N ( − d1 )  j =1i =1  J I  − r ⋅τ  S t − ∑ ∑ D j ,i ⋅ e j ,i  j =1i =1 ln K ⋅ e −r ⋅τ   d1 =  σ⋅ τ

      1  + ⋅ σ ⋅ τ and d = d − σ ⋅ τ 2 1 2

where CE PE S K

r σ Dj,i

τ τj,i

N( ⋅ )

price of the European call at date t price of the European put at date t price of the index at date t strike price continuously compounded risk-free rate p.a. standard deviation of the stock index instantaneous return dividend paid at time ti by company j weighted as the company in the index time in years until expiry of the option time remaining until the dividend payment at time ti by company j cumulative normal distribution function (see table in 2.2.3)

Options on Futures C E = e − rτ [F ⋅ N (d1 ) − K ⋅ N (d 2 )] PE = e − rτ [K ⋅ N (−d 2 ) − F ⋅ N (−d1 )] d1 = where CE PE F

τ

K

σ r

N( ⋅ )

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ln (F / K )

σ τ

1 + σ τ , d 2 = d1 − σ τ 2

value of European call value of European put current futures price time in years until expiry of the option strike price volatility p.a. of the futures returns continuously compounded risk-free rate p.a. cumulative normal distribution function (see table in 2.2.3)

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Derivative Valuation and Analysis

Options on Currencies

CE = S ⋅ e

− r for τ

⋅ N (d1 ) − Ke − rτ ⋅ N (d 2 )

PE = Ke − rτ ⋅ N (−d 2 ) − S ⋅ e d1 = where CE PE

S

τ K σ r r for N( ⋅ )

− r for τ

⋅ N (−d1 )

ln(S / K ) + (r − r for + σ 2 / 2)τ

σ τ

, d 2 = d1 − σ τ

value of European call value of European put current exchange rate (domestic per foreign currency units) time in years until expiry of the option strike price (domestic per foreign currency units) volatility p.a. of the underlying foreign currency continuously compounded risk-free rate p.a. continuously compounded risk-free rate p.a. of the foreign currency cumulative normal distribution function (see table in 2.2.3)

Binomial Option Pricing Model The option price at the beginning of the period is equal to the expected value of the option price at the end of the period under the probability measure π , discounted with the risk-free rate.

Ou ⋅ π + Od ⋅ (1 − π ) 1+ R 1+ R − d 1 π= , u = eσ τ / n , d = , d < 1 + R < u u−d u O=

where

O R Ou

value of the option at the beginning of the period simple risk-free rate of interest for one period value of the option in the up-state at the end of the period

Od

value of the option in the down-state at the end of the period volatility of the underlying returns time until expiry of the option number of periods τ is divided in upward factor of the underlying downward factor of the underlying risk neutral probability

σ

τ n u d

π

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Derivative Valuation and Analysis

2.2.2.3

Sensitivity Analysis of Option Premiums

The Strike Price

∂C = −e −r⋅τ ⋅ N (d 2 ) ∂K ∂P = e −r⋅τ ⋅ N ( −d 2 ) ∂K d2 =

∂C ≤ 0) ∂K ∂P ( ≥ 0) ∂K (

ln(S / K ) + ( r − σ 2 / 2 )τ

σ τ

where value of call option value of put option current price of the underlying strike price σ volatility p.a. of the underlying returns τ time in years until expiry of the option r continuously compounded risk-free rate p.a. N ( ⋅ ) cumulative distribution function (see table in 2.2.3)

C P S K

Price of the Underlying Asset (delta ( ∆ ) and gamma ( Γ )) ∂C ∆c = = N (d1 ) (0 ≤ ∆ c ≤ 1) ∂S ∂P ∆P = = N (d1 ) − 1 (−1 ≤ ∆ P ≤ 0) ∂S n(d1 ) ∂ 2C (ΓC ≥ 0) ΓC = = 2 S ⋅σ ⋅ τ ∂S

ΓP =

d1 =

∂2P ∂S 2

=

n(d1 ) S ⋅σ ⋅ τ

= ΓC

(ΓP ≥ 0)

ln(S / K ) + ( r + σ 2 / 2 )τ

σ τ

where C P S

value of call option value of put option current price of the underlying τ time in years until expiry of the option σ volatility p.a. of the underlying returns r continuously compounded risk-free rate p.a. N ( ⋅ ) cumulative distribution function (see table in 2.2.3) n(x) probability density function:

n( x ) = N ' ( x ) =

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1 2π

x2 e 2 −

Derivative Valuation and Analysis

The Leverage or Elasticity of the Option with respect to S (omega, Ω )

ΩC =

∂C S ⋅ ∂S C

ΩP =

∂P S ⋅ ∂S P

where

ΩC elasticity of a call Ω P elasticity of a put C current value of call option P current value of put option S current price of the underlying

The Time to Maturity (theta, θ ) ∂C ∂C S ⋅σ =− =− ⋅ n(d1 ) − K ⋅ r ⋅ e − rτ N (d 2 ) (θ C ≤ 0) θC = ∂t ∂τ 2⋅ τ ∂P ∂P S ⋅σ =− =− ⋅ n(d1 ) − K ⋅ r ⋅ e − rτ [N (d 2 ) − 1] θP = ∂t ∂τ 2⋅ τ 2 ln(S / K ) + ( r + σ / 2 )τ d1 = , d 2 = d1 − σ τ σ τ where C P S K

value of call option value of put option current price of the underlying strike price τ time in years until expiry of the option r continuously compounded risk-free rate p.a. σ volatility p.a. of the underlying returns N ( ⋅ ) cumulative distribution function (see table in 2.2.3) n(x) probability density function (see formula 0 for definition)

Interest Rate (rho, ρ )

∂C = K ⋅ τ ⋅ e − r ⋅τ ⋅ N (d 2 ) ( ρ C ≥ 0) ∂r ∂P ρP = = K ⋅ τ ⋅ e − r ⋅τ ⋅ ( N (d 2 ) − 1) ( ρ P ≤ 0) ∂r ln(S / K ) + ( r − σ 2 / 2 )τ d2 = σ τ

ρC =

where value of call option value of put option current price of the underlying strike price τ time in years until expiry of the option σ volatility p.a. of the underlying returns r continuously compounded risk-free rate p.a. N ( ⋅ ) cumulative distribution function (see table in 2.2.3)

C P S K

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Derivative Valuation and Analysis

Volatility of the Stock Returns (vega, ν ) ∂C = S ⋅ τ ⋅ n( d1 ) ( ν C ≥ 0 ) ∂σ ∂P = = S ⋅ τ ⋅ n( d1 ) = ν C (ν P ≥ 0 ) ∂σ

νC = νP

d1 =

ln(S / K ) + ( r + σ 2 / 2 )τ

σ τ

where C P S K

value of call option value of put option current price of the underlying strike price τ time in years until expiry of the option σ volatility p.a. of the underlying returns r continuously compounded risk-free rate p.a. n(x) probability density function (see formula 0 for definition)

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page 23

Derivative Valuation and Analysis

2.2.3 Standard Normal Distribution: Table for CDF Numerically defines function N(x): probability that a standard normal random variable is smaller than x. Property of N(x): N(-x)=1-N(x). x 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

0 0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159 0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713 0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981 0.9987 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 1.0000

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0.01 0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186 0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719 0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982 0.9987 0.9991 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 1.0000

0.02 0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212 0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726 0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982 0.9987 0.9991 0.9994 0.9995 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.03 0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238 0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732 0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983 0.9988 0.9991 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.04 0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7704 0.7995 0.8264 0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738 0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984 0.9988 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

page 24

0.05 0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289 0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744 0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.06 0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315 0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750 0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985 0.9989 0.9992 0.9994 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.07 0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340 0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756 0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985 0.9989 0.9992 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.08 0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365 0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761 0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986 0.9990 0.9993 0.9995 0.9996 0.9997 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

0.09 0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389 0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767 0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 0.9999 1.0000 1.0000

Portfolio Management

3.

Portfolio Management

3.1 Modern Portfolio Theory 3.1.1 The Risk/Return Framework 3.1.1.1

Return

Holding Period Return Pt − Pt −1 + Rt =

J

∑ Dt j j =1

(

⋅ 1 + Rt*j ,t

)

t −t j

Pt −1

where

Rt

simple (or discrete) return of the asset over period t − 1 to t

Pt Dt j

price of the asset at date t

tj

date of the jth dividend or coupon payment

Rt*j ,t

risk-free rate p.a. for the period t j to t

J

number of intermediary payments

dividend or coupon paid at date t j between t − 1 and t

Arithmetic versus Geometric Average of Holding Period Returns Arithmetic average of holding period returns rA =

1 N ⋅ ∑ Ri N i =1

where rA

arithmetic average return over N sequential periods

Ri

holding period returns

N

number of compounding periods in the holding period

Geometric average return over a holding period using discrete compounding R A = N (1 + R1 ) ⋅ (1 + R 2 ) ⋅...⋅ (1 + R N ) − 1 where

RA

geometric average return over N sequential periods

Ri

discrete return for the period i number of compounding periods in the holding period

N

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Portfolio Management

Time Value of Money: Compounding and Discounting Compounded returns

 R  1 + Reff = 1 + nom  m  

m

where

Reff

effective rate of return over entire period

R nom

nominal return number of sub-periods

m

Continuously compounded versus simple (discrete) returns In the case no dividends paid between time t-1 and t rt = ln

Pt = ln(1 + Rt ) Pt −1

Rt = e rt − 1 where

Pt

price of the asset at date t

rt

continuously compounded return between time t − 1 and t

Rt

simple (discrete) return between time t − 1 and t

Annualisation of Returns Annualising holding period returns (assuming 360 days per year) Assuming reinvestment of interests at rate Rτ

Rann = (1 + Rτ )360 / τ − 1 where

Rann

annualised simple rate of return

Rτ

simple return for a time period of τ days convention 360 days versus 365 or the effective number of days varies from one country to another.

Note:

Annualising continuously compounded returns (assuming 360 days per year)

ran =

360

τ

× rτ

where

ran

rτ

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annualised rate of return continuously compounded rate of return earned over a period of τ days

page 26

Portfolio Management

Nominal versus Real Returns With simple returns

Rtreal = Rtnominal − I t − Rtreal ⋅ I t ≈ Rtnominal − I t With continuously compounded returns

rtreal = rtnominal − it where

Rtreal

real rate of return on an asset over period t (simple)

Rtnominal It

nominal rate of return on an asset over period t (simple) rate of inflation over period t (simple)

rtreal

real rate of return on an asset over period t (cont. comp.)

rtnominal it

nominal rate of return on an asset over period t (cont. comp.) rate of inflation over period t (cont. comp.)

3.1.2 Measures of Risk Probability Concepts Expectation value E(.), variance Var(.), covariance Cov(.) and correlation Corr(.) of two random variables X and Y if the variables take values x k , y k in state k with probability p k K

E( X ) = ∑ p k ⋅ x k , k =1

K

E(Y ) = ∑ p k ⋅ y k k =1

[

]

K

Var ( X ) = σ X2 = E ( X − E( X )) 2 = E( X 2 ) − E( X ) 2 = ∑ p k ( x k − E( X )) 2 k =1

K

Cov( X , Y ) = σ XY = E[( X − E( X )) ⋅ (Y − E(Y ))] = ∑ p k ( x k − E( X )) ⋅ ( y k − E(Y )) k =1

Corr ( X , Y ) =

σ XY σ X ⋅σ Y

K

where

∑ p k = 1 , and

k =1

pk xk

yk K

probability of state k value of X in state k value of Y in state k number of possible states

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Portfolio Management

The mean E(.), variance Var(.), covariance Cov(.) of two random variables X and Y, in a sample of N observations of xi and yi , are given by E( X ) = x =

1 N ∑ xi , N i =1

Cov( X , Y ) = σ XY =

Var( X ) = σ X2 =

1 N ∑ ( xi − x ) 2 N − 1 i =1

1 N ∑ ( xi − x ) ⋅ ( y i − y ) N − 1 i =1

where observation i

xi , yi x, y σ X ,σ Y

mean of X and Y standard deviations

σ XY N

covariance of X and Y number of observations

Normal Distribution Its probability density is given by the following function: f ( x) = where x

µ σ

1 2 ⋅π ⋅σ

⋅e

−

( x − µ )2 2⋅σ 2

the value of the variable, the mean of the distribution, standard deviation.

Computing and Annualising Volatility Computing volatility

σ =

1 N ∑ (rt − r ) 2 , N − 1 t =1

r=

1 N ∑ rt N t =1

where

σ N rt = ln

standard deviation of the returns (the volatility) number of observed returns

Pt continuously compounded return of asset P over period t Pt −1

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Portfolio Management

Annualising Volatility Assuming that monthly returns are independent, then

σ σ ann = 12 ⋅ σ m = τ τ where

σ ann

annualised volatility

σm στ τ

volatility of monthly returns volatility of returns over periods of length τ length of one period in years

3.1.3 Portfolio Theory 3.1.3.1

Diversification and Portfolio Risk

Average and Expected Return on a Portfolio - Ex-Post Return on a Portfolio P in period t N

R P ,t = ∑ xi Ri ,t = x1 R1,t + x 2 R2 ,t + L + x N R N ,t i =1

∑ xi = 1 , and

where

R P,t

return on the portfolio in period t

Ri, t

return on asset i in period t

xi

initial (at beginning of period) proportion of the portfolio invested in asset

i N

number of assets in portfolio P

- Expectation of the Portfolio Return N

E( R P ) = ∑ xi E( Ri ) = x1E( R1 ) + x 2 E ( R2 ) + K + x N E( R N ) t =1

where

E( R P ) E( Ri )

expected return on the portfolio

xi N

relative weight of asset i in portfolio P

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expected return on asset i number of assets in portfolio P

page 29

Portfolio Management

Variance of the Portfolio Return N N

N N

i =1 j =1

i =1 j =1

Var( R P ) = σ P2 = ∑ ∑ xi x j σ ij = ∑ ∑ xi x j ρ ij σ i σ j where

σ P2

variance of the portfolio return

σ ij

covariance between the returns on assets i and j

ρ ij

correlation coefficient between the returns on assets i and j

σi , σ j

standard deviations of the returns on assets i and j

xi

initial proportion of the portfolio invested in asset i

xj

initial proportion of the portfolio invested in asset j

N

number of assets in portfolio P

3.1.4 Capital Asset Pricing Model (CAPM) 3.1.4.1

Capital Market Line (CML) E( R P ) = r f +

E( R M ) − r f

σP

σM

where

E( R P ) rf

expected return of portfolio P risk free rate

E( RM )

expected return of the market portfolio

σM σp 3.1.4.2

standard deviation of the return on the market portfolio standard deviation of the portfolio return

Security Market Line (SML)

[

]

E( Ri ) = r f + E( R M ) − r f ⋅ β i

βi =

Cov( Ri , R M ) Var( R M )

where

E( Ri )

expected return of asset i

E( RM ) rf

expected return on the market portfolio risk free rate

βi

beta of asset i

Cov( Ri , R M ) covariance between the returns on assets i and market portfolio Var( R M ) variance of returns on the market portfolio

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Portfolio Management

Beta of a Portfolio N

β p = ∑ xi β i i =1

where

βP βi

beta of the portfolio

xi N

proportion of the portfolio invested in asset i

3.1.4.3

beta of asset i number of assets in the portfolio

International CAPM

(

)

E [ri ] − r f = β i ⋅ E [rM ] − r f +

K −1

∑ γ i ,k

k =1

(

⋅ E [s k ] + r fk − r f

)

where expected return of asset i expected return on the market portfolio

E( ri ) E( rM )

βi

beta of asset i

rf

continuous compounded risk-free rate in the domestic country

sk

return on the exchange rate of country k

r fk

continuous compounded risk-free rate in country k

K and

number of countries considered

γ i,k =

Cov[ri , s k ] Var [s k ]

3.1.5 Arbitrage Pricing Theory (APT) N

E(Ri ) ≈ R f + ∑ λ j β ij j =1

where

E(Ri ) expected return on asset i Rf

risk free rate

λj

expected return premium per unit of sensitivity to the risk factor j

β ij

sensitivity of asset i to factor j

N

number of risk factors

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Portfolio Management

3.1.5.1

One Factor Models

Single-Index Model

Rit = α i + β i ⋅ Rindex , t + ε it where

Rit

return on asset or portfolio i over period t

αi βi Rindex ,t

intercept for asset or portfolio i

ε it

random error term ( E(ε it ) = 0 )

sensitivity of asset or portfolio i to the index return return on the index over period t

Market Model Rit = α i + β i ⋅ RMt + ε it

Market model in expectation terms E( Rit ) = α i + β i ⋅ E( RMt ) where

Rit

return on asset or portfolio i over period t

αi βi

intercept for asset or portfolio i

RMt

return on the market portfolio

ε it

random error term ( E(ε it ) = 0 )

sensitivity of asset or portfolio i to the index return

Covariance between two Assets in the Market Model or the CAPM Context 2 σ ij = β i ⋅ β j ⋅ σ M

where

σ ij

covariance between the returns of assets i and j

βi βj

beta of portfolio i beta of portfolio j

2 σM

variance of the return on the market portfolio

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Portfolio Management

Decomposing Variance into Systematic and Diversifiable Risk In the case of a single security 2 σ i2 = β i2σ M + σ ε2 123 {i market risk

residual risk

where

σ i2 :

total variance of the return on asset or portfolio i

2 β i2σ M :

market or systematic risk (explained variance)

σ ε2 :

idiosyncratic or residual or unsystematic risk (unexplained variance)

i

Quality of an index model: R 2 and ρ 2

2

R =

β i2 ⋅ σ I2 σ i2

=

β i2 ⋅ σ I2 β i2 ⋅ σ I2 + σ ε2 i

=1−

σ ε2

i

σ i2

= ρ iI2

where

R2

coefficient of determination in a regression of Ri on R I

σ i2 :

total variance of the returns on asset i

β σ :

Market index or systematic risk (explained variance)

σ ε2 :

idiosyncratic or residual or unsystematic risk (unexplained variance)

ρiI

correlation between asset i and the index I

2 i

2 I

i

3.1.5.2

Multi-Factor Models

Multi-Index Models ri = α i + β i1I1 + β i 2 I 2 + ... + β in I n + ε i where

Ri

return on asset or portfolio i

β ij

beta or sensitivity of the return of asset i to changes in index j

Ij

index j

εi

random error term number of indices

n

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Portfolio Management

Portfolio variance under a multi-index model (every index is assumed to be uncorrelated with each other) σ P2 = β P2 ,1 ⋅ σ 12 + K + β P2 ,n ⋅ σ n2 + σ ε2P where

σ P2

variance of the portfolio

σ i2 β P2 , jσ 2j

variance of the asset or portfolio i

σ ε2P :

residual risk number of indices

systematic risk due to index j

n

3.2 Practical Portfolio Management 3.2.1 Managing an Equity Portfolio Active Return R AP,,tB = RtP − RtB where

R AP,,tB

active return in period t

RtP

return of the portfolio in period t

RtB

return of the benchmark in period t

Tracking Error

TE P , B = V ( R AP , B ) where TE P , B V R AP , B

(

)

tracking error variance of the active return

The Multi-Factor Model Approach Asset excess return NF

Ri = ∑ x i , j F j + ε i j =1

where: Ri xi, j Fj

εi NF

excess return of an asset i (i = 1, …, N) exposure (factor-beta respectively factor-loading) of asset i to factor j excess return of factor j (j = 1, … , NF) specific return of asset i (residual return) number of factors

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Portfolio Management

Portfolio excess return R P = x P' ⋅ F + ε P where: N

x'P = ( x P ,1 ,K , x P ,NF ), x P , j = ∑ wiP ⋅ xi , j (j=1,…,NF) i =1

and

xi, j

exposure (factor-beta respectively factor-loading) of asset i to factor j

xP, j

exposure of the portfolio to factor j

F = ( F1 ,..., FNF )

is the NF x 1 vector of factor returns,

wiP

is the weight of asset i in the portfolio N

ε P = ∑ wiP ⋅ ε i i =1

specific return of the portfolio, where ε i is the specific

return of asset i number of factors number of assets in the portfolio

NF N

Volatility of the portfolio N

V (R P ) = x'P ⋅ W ⋅ x P + s P2

s P2 = ∑ (wiP ) 2 ⋅ s i2 i =1

where:

x'P W s i2

1 x NF vector of portfolio exposures to factor returns covariance matrix of vector F, i.e. of the factor returns variance of asset i specific return

s P2 N

variance of the portfolio’s specific return number of assets in the portfolio

Tracking error N

TE P,B = (x'P − x'B ) ⋅ W ⋅ (x P − x B ) + ∑ (wiP − wiB ) 2 ⋅ si2 i =1

where:

TE P ,B tracking error of the portfolio with respect to the benchmark x'P 1 x NF vector of portfolio exposures to factor returns x'B

1 x NF vector of benchmark exposures to factor returns

W

covariance matrix of vector F, i.e. of the factor returns

wiP wiB s i2

weight of asset i in the portfolio

N

weight of asset i in the benchmark variance of asset i specific return number of assets in the portfolio

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Portfolio Management

Forecasting the tracking error N ~ ~ TE P,B = (x'P − x'B ) ⋅ W ⋅ (x P − x B ) + ∑ (wiP − wiB ) 2 ⋅ ~ si 2 i =1

where:

~ TE P ,B forecasted tracking error of the portfolio with respect to the benchmark x'P 1 x NF vector of portfolio exposures to factor returns x'B ~ W

1 x NF vector of benchmark exposures to factor returns

wiP wiB ~ si 2

weight of asset i in the portfolio

N

3.2.1.1

is the forecast covariance matrix of vector F, i.e. of the factor returns weight of asset i in the benchmark is the forecast of the variance of asset i specific return. number of assets in the portfolio

Active management

Excess Return and Risk Expected active return N ~ ~ ~ ~ ~ R AP,B = R P − R B = ∑ (wiP − wiB ) ⋅ ( Ri − R B ) i =1

where

wiP

weight of asset i in the portfolio

wiB

weight of asset i in the benchmark expected return of asset i expected return of the portfolio

~ Ri ~ RP ~ RB

N

expected return of the benchmark number of assets in the portfolio

Expected tracking error ~ T E AP,B = where ~ Ci , j

N

~

∑ (wiP − wiB ) ⋅ Ci, j ⋅ (w Pj − w Bj )

i =1; j =1

is the forecast of the covariance of asset i and j returns

wiP

weight of asset i in the portfolio

wiB

weight of asset i in the benchmark number of assets in the portfolio

N

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Portfolio Management

Information Ratio ~ R AP,B ~ P,B IR A = ~ P,B TE A

where ~ IR AP ,B information ratio for portfolio P with respect to benchmark B ~ R AP ,B the expected active return of the portfolio ~ T E AP ,B the expected tracking error

3.2.2 Derivatives in Portfolio Management 3.2.2.1

Portfolio Insurance

Static Portfolio Insurance Portfolio Return rPC + rPD = r f + β ( rMC + rMD − r f ) where rPC rPD rMC rMD rf

β

capital gain of the portfolio dividend yield of the portfolio price index return dividend yield of the index risk-free rate portfolio beta with respect to the index

The protective put strategy NP = β ⋅

where NP S0 I0

β k

S Portfoliovalue =β⋅ 0 Index level ⋅ Option contract size I0 ⋅ k

number of protective put options initial value of the portfolio to be insured initial level of the index portfolio beta with respect to the index option contract size

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Portfolio Management

Initial Value of Insured Portfolio (per unit of option contract size) V0 = S 0 + β ⋅ P( I 0 ,T , K ) ⋅

where V0 S0 I0

S0 I0

initial total value of the insured portfolio initial value of the portfolio to be insured initial level of the index portfolio beta with respect to the index put premium for a spot I0, a strike K and maturity T

β

P(I0,T,K)

Floor

f =

Φ V0

where f

insured fraction of the initial total portfolio value floor [= minimum final portfolio value, capital + dividend income] initial total value of the insured portfolio

Φ V0

Paying Insurance on Managed Funds   K S  VT = (1 − β)(1 + rf ) + β ⋅ rMD + β ⋅  ⋅ S0 = f ⋅  S0 + β ⋅ P( I 0 , T, K ) 0  I0  I0    Strike price

K=

I0 β

   P( I 0 , T , K )   − (1 − β)(1 + rf ) − β ⋅ rMD  f ⋅ 1 + β ⋅ I0    

where VT S0 I0

β

f rMD rf P(I0,T,K)

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total final value of the insured portfolio initial value of the portfolio to be insured initial level of the index portfolio beta with respect to the index insured fraction of the initial total portfolio value dividend yield of the index risk-free rate put premium for a spot I0, a strike K and maturity T

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Portfolio Management

In Case of Insurance Paid Externally  K VT = ( 1 − β )( 1 + r f ) + β ⋅ rMD + β ⋅  ⋅ S 0 = f ⋅ S 0 I0   Strike price K=

I0

β

[ f − ( 1 − β )( 1 + r f ) − β ⋅ rMD ]

where VT S0 I0

total final value of the insured portfolio initial value of the portfolio to be insured initial level of the index portfolio beta with respect to the index insured fraction of the initial total portfolio value dividend yield of the index risk-free rate

β

f rMD rf

Dynamic Portfolio Insurance Price of a European Put on an Index Paying a Continuous Dividend Yield y Black&Scholes Model P(S t , T , K ) = K ⋅ e

S ln t K d1 = 

− r f ⋅(T −t )

(

⋅ N (− d 2 ) − S t ⋅ e − y⋅(T −t ) ⋅ N ( − d1 )

  + (r f − y )⋅ (T − t ) 1  + ⋅σ ⋅ T − t 2 σ ⋅ T −t

)

d 2 = d1 − σ ⋅ T − t

where

P ( S t , T , K ) put premium for a spot St, a strike K and maturity T St K rf y

σ

T−t N(.)

index spot price at time t strike price risk-free rate (continuously compounded, p.a.) dividend yield (continuously compounded, p.a.) volatility of index returns (p.a.) time to maturity (in years) cumulative normal distribution function

Delta of a European Put on an Index Paying a Continuous Dividend Yield y ∆ P = e − y⋅(T −t ) ⋅ [N (d1 ) − 1] where

∆P y T−t

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delta of a put dividend yield (continuously compounded, p.a.) time to maturity (in years)

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Portfolio Management

Dynamic Insurance with Futures * N − r ⋅(T * −t ) N F = −e y⋅(T −T ) ⋅ e f ⋅ [1 − N (d1 )] ⋅ β ⋅ S k

where

N F number of futures T* T

β

NS k

maturity of the futures contract maturity of the replicated put risky asset beta with respect to the index number of units of the risky assets futures contract size

Constant Proportion Portfolio Insurance (CPPI) Cushion where ct Vt

Φt

ct = Vt – Φt cushion value of the portfolio floor

Amount Invested in Risky Assets At = NS,t ⋅ St = m ⋅ ct where At NS,t St m ct

total amount invested in the risky assets at time t number of units of the risky assets unit price of the risky assets multiplier cushion

Amount Invested in Risk-Free Assets Bt =Vt – At where Bt Vt At

value of the risk-free portfolio at time t value of the total portfolio at time t value of the risky portfolio at time t

3.2.2.2

Hedging with Stock Index Futures

Hedging when Returns are Normally Distributed (OLS Regression) ∆S t ∆F = α + β ⋅ t + εt St Ft ,T

HR = β ⋅

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St Ft ,T

Portfolio Management

where

∆St St α β ∆F t Ft,T

εt

HR

changes in spot price at time t spot price at time t intercept of the regression line slope of the regression line changes in the futures price at time t futures price at time t with maturity T residual term hedge ratio

And the number of futures contracts to use: N ⋅S NF = −β ⋅ S t k ⋅ Ft,T where

β

NF NS St Ft,T

k

slope of the regression line number of futures number of spot assets spot price at time t futures price at time t with maturity T contract size

Adjusting the Beta of a Stock Portfolio

HRadj = ( β actual − β target ) ⋅ N F = ( β target − β actual ) ⋅ where HRadj

βactual βtarget

St Ft,T NF NS k

St Ft ,T

N S ⋅ St k ⋅ Ft ,T

hedge ratio to adjust the beta to the target beta actual beta of the portfolio target beta of the portfolio spot price at time t futures price at time t with maturity T number of futures contracts number of the spot asset to be hedged contract size

3.2.2.3 Hedging with Interest Rate Futures Hedge Ratio B ⋅ D Bmod σ HR = ρ ∆B ,∆F ∆B = 0 σ ∆F F0 ,T ⋅ D Fmod where HR

ρ ∆B ,∆F σ

hedge ratio is correlation between bond portfolio and futures value is volatility of portfolio and futures returns respectively

D mod B0 F0 ,T

is modified duration of bond portfolio and futures respectively is the value of the bond portfolio at time 0 is the value of the futures at time 0

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Portfolio Management

Adjusting the Target Duration

HR =

S 0 ⋅ (D St arg et − D Sactual ) F0,T ⋅ D F

where HR

S0

hedge ratio is the value of the spot price at time 0

D St arg et

is the target duration

D Sactual F0,T

is actual duration

DF

is the duration of the futures (i.e. of the CTD)

is the futures price at time 0

And the number of futures contracts to use: Error! Objects cannot be created from editing field codes.

3.3 Asset/Liability-Analysis and Management 3.3.1 Valuation of Pension Liabilities

Lt = α t ⋅ Wt ⋅ RFt ,T ⋅ Π t ,T ⋅ a x ⋅ d t ,T where Lt T

Wt

value of liabilities at time t time of retirement pension benefit accrual factor at time t salary of the future retiree at time t

RFt ,T

revaluation factor at time t

Π t ,T

retention factor at time t

ax dt ,T

annuity factor for a (future) retiree retiring at age x discount factor between time t and retirement age T

αt

Annuity factor w −x

ax = ∑

px,t

t t =0 (1 + r)

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Portfolio Management

where ax annuity factor for a (future) retiree retiring at age x w ending age of the mortality table px ,t probability of a person aged x to survive until age x+t r

(deterministic) interest rate

3.3.2 Surplus And Funding Ratio 3.3.2.1

Surplus SPt = At − Lt

where SPt surplus at time t At Lt

3.3.2.2

value of assets at time t value of liabilities at time t

Funding Ratio

A FRt = t Lt where FRt funding ratio at time t At value of assets at time t Lt value of liabilities at time t 3.3.3 Surplus Risk Management 3.3.3.1

One Period Surplus Return RSP =

SP1 − SP0 = FR0 ⋅ R A − R L L0

where RSP one period surplus return SPt surplus at time t

value of liabilities at time 0 FR0 funding ratio at time 0 R A return on assets RL return on liabilities L0

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Portfolio Management

Mean surplus return

µ SP = FR0 ⋅ µ A − µ L where

µSP mean surplus return

FR0 funding ratio at time 0

µ A mean return on assets µL mean return on liabilities Surplus risk σ SP = (FR0 ⋅ σ A ) 2 − 2 ⋅ (FR0 ⋅ σ A ) ⋅ σ L ⋅ ρ AL + σ L2

where

σ SP surplus risk FR0 funding ratio at time 0

σ A risk (volatility) on assets σ L risk (volatility) on liabilities ρAL correlation coefficient of assets and liabilities

3.3.3.2

Shortfall Risk P[RSP ≤ SPmin ] ≤ α

α

shortfall risk RSP one period surplus return SPmin minimal (threshold) surplus

Shortfall constraint for normally-distributed surplus return

µSP ≥ SPmin + zα σ SP µSP

mean surplus return SPmin minimal (threshold) surplus α tolerated level of shortfall risk zα α-percentile of the standard normal distribution

σ SP

surplus risk

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Portfolio Management

3.4 Performance Measurement 3.4.1 Performance Measurement and Evaluation 3.4.1.1

Risk-Return Measurement

Internal Rate of Return (IRR) N

CFt

CF0 = ∑

t =1 ( 1 +

where CF0 CFt IRR N

IRR )t

initial net cash flow net cash flow at the end of period t internal rate of return (per period) number of periods

Time Weighted Return (TWR) Simple Return TWRt / t −1 =

MVend ,t − MVbegin,t MVbegin,t

=

MVend ,t MVbegin,t

−1

where TWRt/t−1 simple time weighted return for sub-period t MVbegin,t market value at the beginning of sub-period t market value at the end of sub-period t MVend,t

Continuously Compounded Return  MVend ,t twrt / t −1 = ln  MVbegin ,t  where twrt/t−1 MVbegin,t MVend,t

   

continuously compounded time weighted return for sub-period t market value at the beginning of sub-period t market value at the end of sub-period t

Total period simple return N

1 + TWRtot = ∏ ( 1 + TWRt / t-1 ) t =1

where TWRtot TWRt / t −1

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simple time weighted return for the total period simple time weighted return for sub-period t

page 45

Portfolio Management

Total Period Continuously Compounded Return N

twrtot = ∑ twrt / t-1 t =1

where

twrtot

continuously compounded time weighted return for the total period

twrt / t −1

continuously compounded time weighted return for sub-period t

Money Weighted Return (MWR) Gain or Loss Incurred on a Portfolio Gain = (Ending Market Value − Beginning Market Value) − Net Cash Flow

Net Cash Flow (NCF) NCF = (∑ Ct + ∑ Pt + ∑ Et) − (∑ Wt + ∑ St + ∑ Dt + ∑ Rt) where NCF Ct Pt Et Wt St Dt Rt

net cash flow effective contributions purchases immaterial contributions measured by the expenses they generate effective withdrawals sales net dividend and other net income reclaimable taxes

Average Invested Capital (AIC) Average Invested Capital = Beginning Market Value + Weighted Cash Flow Dietz Formula AIC = MVbegin + where AIC MVbegin NCF

1 ⋅ NCF 2

average invested capital market value at the beginning of the period net cash flow

Dietz formula MWR =

( MVend − MVbegin ) − NCF MVbegin +

where MWR MVbegin MVend NCF

1 ⋅ NCF 2

money weighted return market value at the beginning of the period market value at the end of the period net cash flow

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Portfolio Management

Value Weighted Day (VWD) VWD j =

∑ CF j , i ⋅ t i ∑ CF j , i

where VWDj value weighted day of the total cash flow of type j (contributions, purchases, sales, ...) CFj,i i-th cash flow of type j (contributions, purchases, sales, ...) ti day when the i-th cash flow takes place

Day Weighted Return AIC = MVbegin +

∑ cash flow i

t end − t i ⋅ CFi t end − t begin

= MVbegin + ( pC ⋅ ∑ Ci + p P ⋅ ∑ Pi + p E ⋅ ∑ Ei ) − ( pW ⋅ ∑ Wi + p S ⋅ ∑ S i + p D ⋅ ∑ Di + p R ⋅ ∑ Ri ) 14444444444444444 4244444444444444444 3 =WCF where AIC average invested capital MVbegin market value at the beginning of the period ti time of cash flow I tbegin time corresponding to the beginning of the period time corresponding to the end of the period tend CFi cash flow effective contributions Ci purchases Pi Ei immaterial contributions measured by the expenses they generate Wi effective withdrawals Si sales net dividend and other net income Di Ri reclaimable taxes WCF weighted cash flow pC, pP, pE, pW, pS, pD and pR are the weights calculated as follows:

weight = p j = where j CFj,i VWDj

∑ ( t end

− t i ) ⋅ CF j ,i

( t end − tbegin ) ⋅ ∑ CF j ,i

=

t end − VWD j t end − tbegin

various cash flow types (contributions, purchases, expenses etc.) i-th cash flow of type j value weighted day

Day weighted return MWR =

( MVend − MVbegin ) − ((∑ Ci + ∑ Pi + ∑ Ei ) − (∑ Wi + ∑ S i + ∑ Di + ∑ Ri ) )

MVbegin + (( pC ⋅ ∑ Ci + p P ⋅ ∑ Pi + p E ⋅ ∑ Ei ) − ( pW ⋅ ∑ Wi + p S ⋅ ∑ S i + p D ⋅ ∑ Di + p R ⋅ ∑ Ri ) )

where MVend

market value at the end of the period

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Portfolio Management

3.4.1.2

Risk Adjusted Performance Measures Sharpe Ratio or Reward-to-Variability Ratio

RVAR P =

Treynor Ratio or Reward-to-Volatility Ratio

RVOLP =

Information Ratio (Appraisal Ratio)

ARP =

where

rP rM rf

average portfolio return

αP

Jensen’s alpha Active Return (Excess return) portfolio beta portfolio volatility Active Risk (standard deviation of the tracking error)

βP σP σε

3.4.1.3

average market return average risk-free rate

Relative Investment Performance

Elementary Price Indices Pt / 0 =

pt ⋅ B = ( 1 + Rt / 0 ) ⋅ B p0

where

Pt / 0

elementary price index at time t with basis at time 0

p0

price of the original good at time 0

pt

price of the unchanged good at time t

Rt / 0

index return for the period starting at 0 and ending at t index level at the reference time

B

Copyright © ACIIA®

σP rP − r f

βP

α P = (rP − r f ) − β P ⋅ (rM − r f )

Jensen’s α

α

rP − r f

page 48

α σε

Portfolio Management

Price-Weighted Indices n

∑ pt Ut / 0 =

j =1 n

j

⋅B=

∑ p0

j

j =1

p1t + pt2 + L + ptn p1 + p 02 + L + p 0n 10444 2444 3

⋅B=

p1t + pt2 + L + ptn ⋅B Dt / 0

n

Dt / 0 = ∑ ctj/ 0 ⋅ p 0j j =1

where

Ut /0

price-weighted index at time t with basis 0 number of securities actual time reference time (the index basis) security identifier divisor

n t 0 j

Dt / 0 j

j

pt , p 0

price of security j at time t, respectively at time 0

ctj/ 0

adjustment coefficient for a corporate action on security j (=1 at time 0)

B

index level at the reference time

Equally Weighted Price Indices Arithmetic average of the elementary price indices: Pt / 0 =

P 1 + Pt 2/ 0 + L + Pt n/ 0 1 n j ⋅ ∑ Pt / 0 = t / 0 n j =1 n

where

Pt / 0 n Pt j/ 0

arithmetic average of the elementary price indices number of elementary price indices pj = j t j elementary index for security j where: ct / 0 ⋅ p 0

ptj , p0j price of security j at time t, respectively at time 0 ctj/ 0

adjustment coefficient for a corporate action on security j (=1 at time 0)

Geometric average of the elementary price indices: 1/ n

 n  Pt / 0, g =  ∏ Pt j/ 0   j =1   

(

= Pt1/ 0 ⋅ Pt2/ 0 ⋅ L ⋅ Ptn/ 0

)

1/ n

= n Pt1/ 0 ⋅ Pt2/ 0 ⋅ L ⋅ Ptn/ 0

where

P t / 0 ,g geometric average of the elementary price indices n number of elementary price indices p tj j Pt / 0 = j elementary index for security j ct / 0 ⋅ p 0j ptj , p0j price of security j at time t, respectively at time 0

ctj/ 0

adjustment coefficient for a corporate action on security j (=1 at time 0)

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Portfolio Management

Capital Weighted Price Indices (Laspeyres Indices) n

PILt / 0 =

j j ∑ pt ⋅ q 0

j =1 n

j j ∑ p0 ⋅ q0

=

n

j ∑ w0 j =1

⋅ Pt /j 0

=∑

j =1

j =1

pt

j

q 0j j

n

∑ p0 ⋅ q0 i

i

⋅ Pt /j 0

i =1

Pt j/ 0 =

where PILt/0

p0j ⋅ q0j

n

pt

j

p 0j

= 1 + Rt j/ 0

Laspeyres capital-weighted index actual price of security j number of outstanding securities j at the basis

p0

price of security j at the basis

w0j

weight of security j in the index at the basis i.e. relative market capitalization

Rt j/ 0

of security j at the basis return in security j between the basis and time t

Index Scaling Btk Itoriginal scaled ⋅ Bt k = Itoriginal ⋅ It / 0 = / 0 / 0 Itoriginal Itoriginal k /0 k /0 where

I tscaled /0

scaled index at time t with base 0 and level Btk at scaling time tk

I toriginal /0

unscaled original index at time t with basis 0

I toriginal /0

unscaled original index at scaling time tk with basis 0

Btk

scaling level (typically 100 or 1000)

k

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Portfolio Management

Index Chain-Linking I tchained = /0

I tnew /0 I tnew k /0

⋅ I told = f t / tk ⋅ I told k /0 k /0

where

I tchained /0

chained index level at time t, with original basis in 0

I tnew /0 new I tk / 0

new index computed at time t new index computed at chain-linking time tk

I told k /0

old index level at chain-linking time tk

f t / tk

chaining factor at time t, with chain-linking time tk = 1 + Rt / tk

I tchained = f t / t −1 ⋅ f t −1 / t −2 ⋅ ... ⋅ f 2 / 1 ⋅ f1 / 0 ⋅ B0 /0 = ( 1 + Rt / t −1 ) ⋅ ( 1 + Rt −1 / t −2 ) ⋅ ... ⋅ ( 1 + R2 / 1 ) ⋅ ( 1 + R1 / 0 ) ⋅ B0 where Rt/t-1 elementary index return for period t B0 index level at reference time 0

Sub-Indices General Index I tgeneral = /0

Ki

Ki

∑ wt/ 0t ⋅ I t/ 0t

segments K ti

where

I tgeneral index level for the general index at time t with reference time 0 /0 Ki

I t/ 0t

Ki

wt/ 0t

index level for the segment Ki at time t with reference time 0 the relative market capitalization of the segment Ki at time t

Sub-Index Weight w0K =

∑ w0j =

j∈K

∑

j∈K

p0j ⋅ q0j n

∑ p0i ⋅ q0i i =1

where

w0K

the relative market capitalisation of the segment K

w0j

the relative market capitalisation of security j

∑

the sum over all securities in segment K

∑

the sum over all n securities in the general aggregated index

j∈K n

i =1

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Portfolio Management

Performance Indices Elementary Performance Index perf

perf

perf

I t / 0 = f t / s ⋅ I s / 0 = (1 + Rt / s ) ⋅ I s / 0 = (1 + Rt / s ) ⋅ (1 + R s / s −1 ) ⋅ (1 + R s −1/ s −2 ) ⋅ ... ⋅ (1 + R2 / 1 ) ⋅ (1 + R1/ 0 ) ⋅ B0

where I tperf /0

index level at time t

I sperf /0

index level at time s

ft/s Rt/s B0

compounding or chain-linking factor elementary index performance from time s until time t index level at reference time 0

Compounding Factor in the Presence of Income

∑ ( pt f ti/ s =

j

j∈K si

+ d sj+1 ) ⋅ q sj

∑ p sj ⋅ q sj

j∈K si

where j

security identifier in segment i

f ti/ s j qs

compounding factor for segment i number of outstanding shares of security j at previous closing/chaining time s

psj

cum-dividend price of security j at previous closing time s

ptj

ex-dividend price of security j at actual time t

d sj+1

dividend detached from security j on day s+1

Compounding Factor in the Presence of Subscription Rights j j ∑ ( pt + rs +1) ⋅ qsj

fti/ s =

j∈K si

∑ psj ⋅ qsj

j∈K si

where

f ti/ s compounding factor for segment i j

qs

number of outstanding shares of security j at previous closing/chaining time s

psj

cum-right price of security j at previous closing time s

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Portfolio Management

ptj

ex-right price of security j at actual time t

rsj+1 price quoted for the right detached from security j on day s+1

Multi-Currency Investments and Interest Rate Differentials Simple Currency Return CBC/LC 1 + C BC / LC = where CBC/LC SBC/LC,t SBC/LC,t-1

S BC / LC ,t S BC / LC ,t −1

simple currency return spot exchange rate at the end of the period spot exchange rate at the beginning of the period

Forward Exchange Rate Return CF,BC/LC 1 + C F ,BC / LC = where CF,BC/LC SBC/LC,t FBC/LC,T Rf,BC Rf,LC

FBC / LC ,T S BC / LC ,t

=

1 + R f ,BC ⋅ ( T − t ) 1 + R f ,LC ⋅ ( T − t )

simple currency forward return spot exchange rate at time t frward exchange rate at time t with expiration in T risk-free rate on the base currency risk-free rate on the local currency

Unexpected Currency Return EBC/LC 1 + C BC / LC = where CBC/LC EBC/LC CF,BC/LC St Ft-1,1 Rf,BC Rf,LC

S t 1 + R f ,BC ⋅ = (1 + E BC / LC ) ⋅ (1 + C F ,BC / LC ) Ft −1,1 1 + R f ,LC

simple currency return unexpected currency return simple currency forward return spot price at the end of the period forward rate at the beginning of the period for one period risk-free rate on the base currency risk-free rate on the local currency

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Portfolio Management

3.4.1.4

Performance attribution analysis

Attribution Methods Based on Simple Linear Regression Jensen’s α α P = RP − RE = ( RP − RB ) + ( RB − RE ) 14243 Net selectivity

where

14243 Diversification

σP ⋅ ( RM − R f ) σM . RE = R f + β P ⋅ ( RM − R f ) RB = R f +

and

αP RP RM Rf RE RB

σP σM

Jensen’s alpha actual portfolio return actual market return risk-free rate portfolio return at equilibrium with beta equal to βP portfolio return at equilibrium with volatility equal to σP portfolio volatility market volatility

Fama’s Break-up of Excess Return R P − R f = ( R P − R B ) + ( R B − R E ) + ( RE − RM ) + ( RM − R f ) 14243 14243 14243 14243 Net selectivi ty Diversific ation return premium return premium above market risk

where RP Rf RB RE RM

actual portfolio return risk-free rate portfolio return at equilibrium with volatility equal to σP portfolio return at equilibrium with beta equal to βP actual market return

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page 54

for market risk

Portfolio Management

3.4.1.5

Special Issues

Performance Evaluation of International Investments Simple Return 1 + R Dt = (1 + R Ft ) ⋅ (1 + st ) where RDt RFt st

simple rate of return denominated in domestic currency simple rate of return denominated in foreign currency relative change (depreciation or appreciation) in the value of the domestic currency

Continuously Compounded Return rDt = rFt + s tcc where rDt

rFt

continuously compounded rate of return denominated in domestic currency continuously compounded rate of return denominated in foreign

currency stcc

relative change (depreciation appreciation) in the value of the domestic currency continuously

compounded

or

Variance of Continuously Compounded Returns

[ ]

[

Var [rDt ] = Var [rFt ] + Var s cc + 2 ⋅ Cov rFt , s cc t t where Var [] ⋅ Cov[] ⋅ rDt rFt s tcc

]

the covariance operator. the covariance operator. continuously compounded rate of return denominated in domestic currency continuously compounded rate of return denominated in foreign currency continuously compounded relative change (depreciation or appreciation) in the value of the domestic currency

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Portfolio Management

A Single Currency Attribution Model by Brinson & al. VA = R − I = ∑ wP,j ⋅ RP,j − ∑ wI,j ⋅ RI,j

= ∑ ((wP,j − wI,j) ⋅RI,j + wI,j ⋅ (RP,j −RI,j) + (wP,j − wI,j) ⋅ (RP,j − RI,j)) where VA R I ∑ RP,j RI,j wP,j wI,j

value added portfolio return benchmark return sum over every market in the portfolio and in the benchmark portfolio return in each market index return in each market portfolio weight in each market at the beginning of the period index weight in each market at the beginning of the period

Practitioners’ Break-up of Value Added VA = R − I = ∑ (wP,j − wI,j) ⋅ (RI,j – I) + ∑ wP,j ⋅ (RP,j − RI,j) where VA R I ∑ RP,j RI,j wP,j wI,j

value added portfolio return benchmark return sum over every market in the portfolio and in the benchmark portfolio return in each market index return in each market portfolio weight in each market at the beginning of the period index weight in each market at the beginning of the period

Multi-Currency Attribution and Interest Rate Differentials Value Added in Base Currency vaBC = rBC − iBC where vaBC rBC iBC

value added in base currency portfolio return in base currency(continuously compounded) benchmark return in base currency(continuously compounded)

Base Currency Adjusted Market Return rBC,adj = rLC + rf,BC − rf,LC where rBC,adj rLC rf,BC rf,LC

adjusted local market return in base currency local market return in local currency risk-free rate in base currency at the beginning of the period risk-free rate in local currency at the beginning of the period

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Portfolio Management

Unexpected Currency Return eBC/LC = cBC/LC − rf,BC + rf,LC where eBC/LC cBC/LC rf,BC rf,LC

unexpected currency return actual currency return risk-free rate in base currency at the beginning of the period risk-free rate in local currency at the beginning of the period

Break-up of Value Added in Base Currency vaBC = (∑ wP ⋅ rBC,P,adj − ∑ wI ⋅ rBC,I,adj) + (∑ wP ⋅ eP − ∑ wI ⋅ eI) where vaBC ∑ rBC,P,adj rBC,I,adj eP eI wP wI

value added in base currency sum over every market in the portfolio and in the benchmark adjusted portfolio return of the local market in base currency adjusted index return of the local market in base currency unexpected portfolio currency return unexpected passive currency return portfolio weight in each market at the beginning of the period index weight in each market at the beginning of the period

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