SINGLE INDEX MODELS

• Input list of Markowitz modeln estimates of expected returnn estimates of variancesn(n-1)/2 estimates of covariance• Reduces the number of inputs for diversification• Easier for security analysts to specialize

Advantages of the Single Index Model

Ri = E(ri) +ei (1)Ri =E(ri) + m +ei (2)σ 2

i =σ2m +σ2 (ei) (3)

Derivation of Single factor model

• Cov(Ri,Rj) = Cov (m+ei ,m+ei)= Cov(m,m)+ Cov (ei,ej)=σ2

M

Ri =E(ri) +β m+ ei (4)σ 2

i =βi2 σ 2m+ σ 2(ei) (5)

• Cov(Ri,Rj) = Cov (βim+ei , βjm+ej)= βi βjCov(m,m)+ Cov (ei,ej)=βi βjσ2

M

ßi = index of a securities’ particular return to the factorm = Unanticipated movement related to security returnsei = Assumption: a broad market index like the S&P 500 is the common factor.

Single Factor Model( )i i i ir E r m eβ= + +

Single-Index Model

• Regression Equation:

• Expected return-beta relationship:

( ) ( ) ( )t i t M iR t R t e tα β= + +

( ) ( )i i i ME R E Rα β= +

Single-Index Model Continued• Risk and covariance:

– Total risk = Systematic risk + Firm-specific risk:

– Covariance = product of betas x market index risk:

– Correlation = product of correlations with the market index

2 2 2 2 ( )i i M ieσ β σ σ= +

2( , )i j i j MCov r r β β σ=

2 2 2

( , ) ( , ) ( , )i j M i M j Mi j i M j M

i j i M j M

Corr r r Corr r r xCorr r rβ β σ β σ β σσ σ σ σ σ σ

= = =

Concept check 1Stock Capitalisation beta Mean excess returns Standard deviationA $ 3000 1 10% 40%B 1940 0.2 2% 30%C 360 1.7 17% 50%Standard deviation of the market portfolio is 25%a. What is the mean excess return of the index portfolio?b. What is the covariance between stock A and B?c. What is the covariance between stock B and the index?d. Break down the variance of stock B into its systematic and firm specific components.

a. 3000/6300*10+1940/6300*2+1360/6300*17 =9.05%b. 1*0.2* 0.252 = .0125c. 0.2 *0.252 = .0125d. Systematic =0.22 *.252 = .0025 Firm specific = 0.302 - .0025 =.0875

The set of estimates needed for the single index model

• n estimates of stock’s expected returns if the market is neutral ,αi

• n estimates of sensitivity coefficients (βi)• N estimates of firm specific variances ,σ2(ei)• 1 estimate for market risk premium, E(Rm)• 1 estimate of variance of common

macroeconomic factor , σ2m

Concept check 2• Suppose • RA =1% +0.9 RM +eA

• RB = -2%+1.1 RM +eB

• σ(eA) =30%• σ(eB) =10%• σM =20%• Find the standard deviation of each stock and covariance between

them.Stock A = 0.92 *(20)2 + 302 = 1224SD = 35%Stock B =1.12 *(20)2 + 102 = 584SD = 24%The covariance = .9* 1.1 * 202 = 396

Index Model and Diversification• Portfolio’s variance:

• Variance of the equally weighted portfolio of firm-specific components:

• When n gets large, becomes negligible

222 2

1

1 1( ) ( ) ( )n

P ii

e e en n

σ σ σ=

⎛ ⎞= =⎜ ⎟⎝ ⎠

∑

2 2 2 2 ( )P P M Peσ β σ σ= +

2( )Peσ

Figure 8.1 The Variance of an Equally Weighted Portfolio with Risk Coefficient

βp in the Single-Factor Economy

Concept check 3• Reconsider the two stocks in concept

check 2. Suppose we form an equally weighted portfolio of A and B. What will be the nonsystematic standard deviation of that portfolio?

• σ2(ep) =0.52 [ .302 + .102 ]= .0250

• σep = .158 =15.8%

Estimating the single index Model• Security Characteristic Line• RHP (t) =αHP +βHP RS&P500(t) +eHP(t)

Figure 8.2 Excess Returns on HP and S&P 500 April 2001 – March 2006

Figure 8.3 Scatter Diagram of HP, the S&P 500, and the Security Characteristic

Line (SCL) for HP

Excel Output: Regression Statistics for the SCL of Hewlett-Packard

H0: α =0

Calculation of Risk • σ 2HP = .7162/59 =.012 per month• Monthly SD =11%• Annualized standard deviation =38.17 (11 *sqrt

12)• β2

HPσ2S&P500 =.3752

• σ 2eHP =.3410/58 =0.0057796 per month• Monthly SD of HP’s residual =7.67%Alternatively, Directly the square root of MS for

residual• Annualised residual SD = 7.67 *sqrt 12 =26.6%

Alpha and Security Analysis

• Single index model provides the framework it provides for macro economic and security analysis.

• Macroeconomic analysis is used to estimate the risk premium and risk of the market index

• Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ2 ( e i )

Alpha and Security Analysis Continued

• The market-driven expected return is conditional on information common to all securities

• Security-specific expected return forecasts are derived from various security-valuation models – The alpha value distills the incremental risk

premium attributable to private information• Helps determine whether security is a good or

bad buy

Single-Index Model Input List

• Risk premium on the S&P 500 portfolio• Estimate of the SD of the S&P 500 portfolio• n sets of estimates of

– Beta coefficient– Stock residual variances– Alpha values

Optimal Risky Portfolio of the Single-Index Model

• Maximize the Sharpe ratio– Expected return, SD, and Sharpe ratio:

1 1

1 1

12 21 1 1

2 2 2 2 2 22

1 1

( ) ( ) ( )

( ) ( )

( )

n n

P P M P i i M i ii i

n n

P P M P M i i i ii i

PP

P

E R E R w E R w

e w w e

E RS

α β α β

σ β σ σ σ β σ

σ

+ +

= =

+ +

= =

= + = +

⎡ ⎤⎛ ⎞⎡ ⎤= + = +⎢ ⎥⎜ ⎟⎣ ⎦ ⎝ ⎠⎢ ⎥⎣ ⎦

=

∑ ∑

∑ ∑

Optimal Risky Portfolio of the Single-Index Model Continued

• Combination of:– Active portfolio denoted by A (assuming

additional firm specific risk)– Market-index portfolio, the (n+1)th asset

which we call the passive portfolio and denote by M (efficient diversification)

– Modification of active portfolio position:

– When

0*

01 (1 )A

AA A

wwwβ

=+ −

* 01,A A Aw wβ = =

The Information Ratio

• The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy):

22 2

( )A

P MAes s α

σ⎡ ⎤

= +⎢ ⎥⎣ ⎦

Steps for forming optimal risky portfolio

• Compute initial position of each security in the active portfolio as vi

0 =αi/σ2(ei)• Scale those initial position to force portfolio

weights to sum 1 wi =wi0 /∑i=1to n wi

0

• Compute alpha of active portfolio αA=∑i=1to n wiαi

• Compute beta of active portfolio:βA=∑ i=1 to n wiβi

• Compute residual variances of active portfolioσ2(eA) = ∑ ito n wi

2 σ2(ei)

Steps for forming optimal risky portfolio

• Compute initial position in active portfolio:W0

A =[αi/σ2(ei)/E(RM)/σ2M] (Note: E(RM) is the risk premium)

Adjust the initial position in the active portfolio

Optimal risky portfolio now has weights W*M= 1-W*

ARisk premium on of optimal risky portfolioE(Rp) =(W*

M+W*A βA) E(RM) +W*

A αACompute the variance of the optimal risky portfolioσ2p = (W*

M+W*A βA) 2 σ2

M +[w*A σ(eA)]2

0*

01 (1 )A

AA A

wwwβ

=+ −

Question 17(a)

12% – 8% = 4%αD = 12% – [8% + 1.0(16% – 8%)] = – 4.0%

17% – 8% = 9%αC = 17% – [8% + 0.7(16% – 8%)] = 3.4%

18% – 8% = 10%αB = 18% – [8% + 1.8(16% – 8%)] = – 4.4%

20% – 8% = 12%αA = 20% – [8% + 1.3(16% – 8%)] = 1.6%

E(ri ) – rfα i = ri – [rf + βi(rM – rf ) ]

Expected excess returnAlpha (α)[extra market expected return]

Stocks A and C have positive alphas, where as stocks B and D have negative alphas.The residual variances are:

σ2(eA ) = 582 = 3,364σ2(eB) = 712 = 5,041σ2(eC) = 602 = 3,600σ2(eD) = 552 = 3,025

Question 17(b)[Treynor –Black technique]

wi =wi0 /∑i=1to n wi

0Initial position

1.0000–0.000775Total

1.7058-4/3025=–0.001322D

–1.21813.4/3600=0.000944C

1.1265-4.4/5041=–0.000873B

–0.61421.6/3364=0.000476A

With these weights, the forecast for the active portfolio is:α = [–0.6142 * 1.6] + [1.1265 * (– 4.4)] – [1.2181 * 3.4] + [1.7058 * (– 4.0)]

= –16.90%β = [–0.6142 * 1.3] + [1.1265 *1.8] – [1.2181 * 0.70] + [1.7058 * 1] = 2.08

σ2(e) = [(–0.6142)2 * 3364] + [1.12652 * 5041] + [(–1.2181)2 *3600] + [1.70582 * 3025]= 21,809.6

σ(e) = 147.68%

Question 17(b)

• The negative position is justified for the reason stated earlier.• The adjustment for beta is:

• Since w* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:

• 1 – (–0.0486) = 1.0486

05124.023/8

6.809,21/90.16/]r)r(E[)e(/w 22

MfM

2

0 −=−

=σ−

σα=

0486.0)05124.0)(08.21(1

05124.0w)1(1

w*w

0

0 −=−−+

−=

β−+=

The optimal risky portfolio has a proportion w* in the active portfolio, computed as follows:

Question 17(c)• The information ratio for the active portfolio is computed as

follows:• A = α /σ(e)= –16.90/147.68 = –0.1144• A2 = 0.0131• Hence, the square of Sharpe’s measure (S) of the optimized risky

portfolio is:

• S = 0.3662• Compare this to the market’s Sharpe measure:• SM = 8/23 = 0.3478• The difference is: 0.0184

1341.00131.0238ASS

222

M2 =+⎟

⎠⎞

⎜⎝⎛=+=

Question 17(d)[Exact makeup of the complete portfolio]

• βP = wM + (wA × βA ) = 1.0486 + [(–0.0486) ×2.08] = 0.95

• E(RP) = α P + βPE(RM) = [(–0.0486) × (–16.90%)] + (0.95 × 8%) = 8.42%

5685.094.5288.201.0

42.8y =××

=

In contrast, with a passive strategy:

5401.0238.201.0

82 =

××=y

This is a difference of: 0.0284

( ) 94.5286.809,21)0486.0()2395.0()e( 22P

22M

2P

2P =×−+×=σ+σβ=σ

%00.23P =σ

Question 17(d)

• The final positions of the complete portfolio are:

[sum is subject to rounding error]100.00%– 4.71%0.5685 × (–0.0486) × 1.7058 =D3.37%0.5685 × (–0.0486) × (–1.2181) =C– 3.11%0.5685 × (–0.0486) × 1.1265 =B1.70%0.5685 × (–0.0486) × (–0.6142) =A59.61%0.5685 × l.0486 =M43.15%1 – 0.5685 =Bills

THE CAPITAL ASSET PRICING MODEL

• It is the equilibrium model that underlies all modern financial theoryProvides benchmark rate of returnEducated guess as to E(r) on assets that have not yet been traded in the market place.

• Derived using principles of diversification with simplified assumptions

• Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Capital Asset Pricing Model (CAPM)

• Individual investors are price takers• Single-period investment horizon• Investments are limited to traded financial

assets• No taxes and transaction costs

Assumptions

• Information is costless and available to all investors

• Investors are rational mean-variance optimizers

• There are homogeneous expectations

Assumptions Continued

• All investors will hold the same portfolio for risky assets – market portfolio

• Market portfolio(M) contains all securities and the proportion of each security is its market value as a percentage of total market value

• Not only will the market portfolio be on the efficient frontier ,but it also will be tangency portfolio to the optimal CAL (CML).All investors hold M as their optimal risky portfolio, differing only in amount invested in it vs.risk free asset.

Resulting Equilibrium Conditions

• Risk premium on the market portfolio depends on the average risk aversion of all market participants and its risk.

• Risk premium on an individual security is a function of its covariance with the market.

βi =Cov(ri,rm)/σ2M

Risk premium on Individual security:E(ri) –rf =Cov(ri,rm)/σ2

M[E(rm)-rf] =βi[E(rm)-rf]

Resulting Equilibrium Conditions Continued

The Efficient Frontier and the Capital Market Line

All assets have to be included in the market portfolio .The only issue is the price at whichInvestors will be willing to include a stock in their optimal risky portfolio

Market Risk Premium•The risk premium on the market portfolio will be proportional to its risk and the degree of risk aversion of the investor:

2

2

( )

where is the variance of the market portolio and

is the average degree of risk aversion across investors

M f M

M

E r r A

A

σ

σ

− =

Y =E(rp) –rf /A σ2P

Return and Risk For Individual Securities

• Bordered covariance Matrix• wGE[w1Cov(r1,rGE)+ w2Cov (r2,rGE)

+……wGECov(rGE,rGE)+…wnCov(rn,rGE)]• It is Covariance of GE with market portfolio• GE’s contribution to variance =wGECov(rGE,rM)• If Cov of GE with rest of the market is negative-

“negative contribution” to the portfolio risk• If Cov is positive then positive contribution to

overall portfolio risk.

• The individual security’s contribution to the risk of the market portfolio

i.e wGECov(rGE,rM)=GE’s Contribution to the variance

• An individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

i.e wGE[E(rGE) –rf] =GE contribution to the risk premium

Return and Risk For Individual Securities

Using GE Text Example

• Covariance of GE return with the market portfolio:

• Therefore, the reward-to-risk ratio for investments in GE would be:

1 1( , ) , ( , )

n n

GE M GE k k k k GEk k

Cov r r Cov r w r w Cov r r= =

⎛ ⎞= =⎜ ⎟

⎝ ⎠∑ ∑

( ) ( )GE's contribution to risk premiumGE's contribution to variance ( , ) ( , )

GE GE f GE f

GE GE M GE M

w E r r E r rw Cov r r Cov r r

⎡ ⎤− −⎣ ⎦= =

Using GE Text Example Continued• Reward-to-risk ratio for investment in market

portfolio:

• Reward-to-risk ratios of GE and the market portfolio:

• And the risk premium for GE:

• E(rGE) =rf +βGE[E(rM)-rf]

2

( )Market risk premiumMarket variance

M f

M

E r rσ

−=

2

( ) ( ( )( , )GE f M f

GE M M

E r r E r rCov r r σ

− −=

2

( , )( ) ( )GE MGE f M f

M

Cov r rE r r E r rσ

⎡ ⎤− = −⎣ ⎦

Expected Return-Beta Relationship

• w1E(r1) =w1rf+w1β1[E(rM)-rf]+ w2(Er2) =w2rf +w2β2[E(rM)-rf]+…….=………….+wnE(rn) =wn rf +wnβn [E(rM)-rf]E(rp) =rf +βp [E(rM)-rf]

Expected Return-Beta Relationship• CAPM holds for the overall portfolio because:

• This also holds for the market portfolio:

• βM =Cov(rM,rM) /σ2M =1

• In an efficient market investors receive high expected returns only if they are willing to bear risk

P

( ) ( ) andP k kk

k kk

E r w E r

wβ β

=

=

∑

∑

( ) ( )M f M M fE r r E r rβ ⎡ ⎤= + −⎣ ⎦

Concept check• Suppose that the risk premium on the

market portfolio is estimated at 8% with a standard deviation of 22%.What is the risk premium on a portfolio invested 25% in GM and 75% in Ford, if they have betas of 1.10 and 1.25 respectively?

• βp = .75*1.25+ .25* 1.10 = 1.2125• The portfolio risk premium = 1.2125 * 8 = 9.7%

The Security Market Line• SML is the graphical representation of exp return-

beta relationship• Reward or risk premium on individual assets

,depends on the contribution of the individual asset to the risk of the portfolio.

• CML Vs SML• SML provides a benchmark for the evaluation of

investment performance• “Fairly priced” assets plot exactly on SML• CAPM is also useful in capital budgeting decisions.

Figure 9.2 The Security Market Line

Figure 9.3 The SML and a Positive-Alpha Stock

Question• Stock XYZ has an expected return of 12% ,β=1.

Stock ABC has expected return of 13% and β=1.5.The market expected return is 11% and rf= 5%. Acc. To CAPM which stock is a better buy? What is alpha of each stock?

• α=E(r) – { rf + β[E(rm) –rf] }αXYZ =12-[5+ (11-5)] = 1%αABC = 13 –[ 5+ 1.5(11-5)] =-1%ABC plots below SML ,while XYZ plots above

Question• The risk free rate is 8% and the expected return

on market portfolio is 16%.A firm considers a project that is expected to have beta of 1.3.What is the required rate of return? If the expected IRR of the project is 19% , should it be accepted ?

• α=E(r) – { rf + β[E(rm) –rf] }= 8+1.3(16-8) =18.4%(Projects hurdle rate)Any project with an IRR equal to or less than 18.4

% should be rejected.

The Index Model and Realized Returns• To move from expected to realized returns—use

the index model in excess return form:

• Cov(Ri,Rm) = Cov (βiRm+ei ,Rm)= βi Cov(Rm,Rm)+ Cov (ei,Rm)=βi σ2

M

βi = Cov(Ri,Rm) / σ2M

• The index model beta coefficient turns out to be the same beta as that of the CAPM expected return-beta relationship

i i i M iR R eα β= + +

The CAPM and Reality• Is the condition of zero alphas for all stocks as

implied by the CAPM met– Not perfect but one of the best available

• Is the CAPM testable-The market portfolio is efficient-Alpha values are zero– Proxies must be used for the market portfolio

• CAPM is still considered the best available description of security pricing and is widely accepted

The Economic validity of the CAPM

• In regulated utilities for approving pricesfor providing fair return to shareholders• In legal settings for finding the discount

rates

Econometrics and the Expected Return-Beta Relationship

• It is important to consider the econometric technique used for the model estimated

• Statistical bias is easily introduced– Miller and Scholes paper demonstrated how

econometric problems could lead one to reject the CAPM even if it were perfectly valid

– Estimation of beta coefficients when residuals are correlated (GLS Vs.OLS)

– Alpha ,beta .residuals are time variant (ARCH)

Three Elements of Liquidity

• Sensitivity of security’s illiquidity to market illiquidity:

• Sensitivity of stock’s return to market illiquidity:

• Sensitivity of the security illiquidity to the market rate of return:

1( , )

( )i M

LM M

Cov C CVar R C

β =−

3( , )

( )i M

LM M

Cov C RVar R C

β =−

2( , )

( )i M

LM M

Cov R CVar R C

β =−