LECTURE 7 :LECTURE 7 :

THE CAPMTHE CAPM

(Asset Pricing and Portfolio (Asset Pricing and Portfolio Theory)Theory)

THE CAPMTHE CAPM

Quantitative Asset PricingQuantitative Asset Pricing

Dirk Nitzsche (E-mail : Dirk Nitzsche (E-mail : d.nitzsche@city.ac.uk)d.nitzsche@city.ac.uk)

ContentsContents

The market portfolioThe market portfolio Factor models : The CAPMFactor models : The CAPM

Equilibrium model for asset pricing Equilibrium model for asset pricing (SML)(SML)

Performance measures / Risk Performance measures / Risk adjusted rate of returnadjusted rate of return

Using the CAPM to appraise projects Using the CAPM to appraise projects

The UK Stock Market The UK Stock Market (FT All Share Index)(FT All Share Index)

FT All Share Price Index

0

500

1000

1500

2000

2500

3000

3500

01/12/1996 28/08/1999 24/05/2002 17/02/2005

Daily Share price between 1st April 1997 and 12th Nov. 2004

FT All Share Index FT All Share Index (Return)(Return)

FT All Share Pricve Index (Returns)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

01/12/1996 28/08/1999 24/05/2002 17/02/2005

BA and Newcastle Utd. BA and Newcastle Utd. Share Price Share Price

BA Share Price

0

100

200

300

400

500

600

700

800

28/10/1995 24/07/1998 19/04/2001 14/01/2004 10/10/2006

Newcastle United Share Price

0

20

40

60

80

100

120

140

160

28/10/1995 24/07/1998 19/04/2001 14/01/2004 10/10/2006

Tesco Share PriceTesco Share Price

Tesco Share Price

0

50

100

150

200

250

300

350

28/10/1995 24/07/1998 19/04/2001 14/01/2004 10/10/2006

Assumptions to Derive Assumptions to Derive the CAPMthe CAPM Assumption 1 : Assumption 1 :

– Investors agree in their forecasts of expected Investors agree in their forecasts of expected returns, standard deviation and correlations returns, standard deviation and correlations

– Therefore all investors optimally hold risky Therefore all investors optimally hold risky assets in the same relative proportionsassets in the same relative proportions

Assumption 2 : Assumption 2 : – Investors generally behave optimally. In Investors generally behave optimally. In

equilibrium prices of securities adjust so that equilibrium prices of securities adjust so that when investors are holding their optimal when investors are holding their optimal portfolio, aggregate demand equals its portfolio, aggregate demand equals its supply. supply.

The Capital Market The Capital Market LineLine

ERi = rf + [(ERm – rf)/m] iER

Standard deviation

Slope of the CML

CML

rf

Portfolio M

ERm - rf

m

Risk Premium, Beta Risk Premium, Beta and Market Portfolioand Market Portfolio Suppose risk premium on the market is a Suppose risk premium on the market is a

function of its variance. function of its variance. The market : ERThe market : ERmm – r – rff = A = A22

mm ERERii = r = rff + [(ER + [(ERmm-r-rff)/)/22

mm] ] 22ii

ii imim / / 22mm

Rem. : Portfolio risk is covarianceRem. : Portfolio risk is covarianceERERii = r = rff + + ii[ER[ERmm – r – rff]] (SML)(SML)or or ERERii –r –rff = = ii[ER[ERmm – r – rff]]

22ii = = 22

ii 22mm + var( + var(ii) )

total risk = systematic risk + nonsystematic risktotal risk = systematic risk + nonsystematic risk

Systematic and Non Systematic and Non Systematic RiskSystematic Risk

ER

Standard deviation

CML

rf

Portfolio M

Asset with systematic risk ONLY

Assets with non systematic risk

The Security Market The Security Market LineLine

Expected return and

actual return

SML

Beta, i

Q (buy)

S (sell)

P

actual return

expectedreturn

0.5 1 1.2

T (sell)

M

The larger is i, the larger is ERi

r

Properties of BetasProperties of Betas

Betas represent an asset’s Betas represent an asset’s systematic (market or non-systematic (market or non-diversifiable) riskdiversifiable) risk

Beta of the market portfolio : Beta of the market portfolio : mm = 1 = 1 Beta of the risk free asset : Beta of the risk free asset : = 0 = 0

Beta of a portfolio : Beta of a portfolio : pp = = wwiiii

Applications of BetaApplications of Beta

Market timing Market timing … … if you expect the market to go up you want to move if you expect the market to go up you want to move

into higher beta stocks to get more exposure to the into higher beta stocks to get more exposure to the ‘bull market’. ‘bull market’.

… … vice versa if market goes down you want less exposure vice versa if market goes down you want less exposure to the stock market and hence should buy stocks with to the stock market and hence should buy stocks with lower betas lower betas

Portfolio construction Portfolio construction … … to construct a customised portfolio. (Rem : to construct a customised portfolio. (Rem : pp = = ii))

Performance measuresPerformance measures Risk ManagementRisk Management Calculating the WACC Calculating the WACC

… … to use for the DPV for assessing the viability of a to use for the DPV for assessing the viability of a project or the value of a company. project or the value of a company.

US Companies : Betas US Companies : Betas and Sigma (7and Sigma (7thth December December 1979)1979)

Company Company NameName

BetaBeta VolatilityVolatility

Coca ColaCoca Cola 1.191.19 18%18%

Exxon CorpExxon Corp 0.670.67 18%18%

General General ElectricElectric

1.261.26 15%15%

General Motor General Motor 0.810.81 19%19%

Gillette Gillette 1.091.09 21%21%

LockheedLockheed 3.023.02 43%43%

Performance Measures / Performance Measures / Risk Adjusted Rate of Risk Adjusted Rate of ReturnReturn

Sharpe Ratio, Treynor Sharpe Ratio, Treynor RatioRatio Shape ratio (from the CML)Shape ratio (from the CML)

SRSRii = (ER = (ERii –r –rff)/)/ii

Risk is measured by the standard deviation Risk is measured by the standard deviation (total risk of security)(total risk of security)Aim : to maximise the Sharpe ratioAim : to maximise the Sharpe ratio

Treynor ratio (from the SML)Treynor ratio (from the SML)TRTRii = (ER = (ERii – r – rff)/)/ii Risk is measured by beta (market risk only)Risk is measured by beta (market risk only)Aim : to maximise the Treynor ratioAim : to maximise the Treynor ratio

Performance Performance EvaluationEvaluation CAPM can be used to evaluate the CAPM can be used to evaluate the

performance of an investment portfolio (i.e. performance of an investment portfolio (i.e. mutual fund)mutual fund)

Step 1 : Step 1 : Calculate the summary stats of the investment Calculate the summary stats of the investment portfolio (e.g. average rate of return, variance)portfolio (e.g. average rate of return, variance)

Step 2 : Step 2 : Calculate the covariance between the investment Calculate the covariance between the investment portfolio and the market portfolio and the variance portfolio and the market portfolio and the variance of the market portfolio : b = Cov(Rof the market portfolio : b = Cov(R ii,R,Rmm)/Var(R)/Var(Rmm))

Step 3 : Step 3 : Calculate Jensen’s alpha performance measure : Calculate Jensen’s alpha performance measure :

(R(Rii – r – rff) = ) = + b(R + b(Rm m - r- rff))

Obtaining BetaObtaining Beta

Estimating the BetasEstimating the Betas

Time series regression : Time series regression :

(R(Rii-r-rff))tt = = ii + + ii (R (Rmm-r-rff))tt + + itit

(alternative models also available) (alternative models also available) Decision : Decision :

– How much historical data to use (i.e. 1 How much historical data to use (i.e. 1 years, 2 years, 5 years, 10 years or what) ? years, 2 years, 5 years, 10 years or what) ?

– What data frequency to use (i.e. daily data, What data frequency to use (i.e. daily data, weekly data, monthly data or what) ? weekly data, monthly data or what) ?

– Should I use the model above or an Should I use the model above or an alternative model ? alternative model ?

Adjusted BetaAdjusted Beta

To price securities, need to obtain forecast of To price securities, need to obtain forecast of betasbetas

Estimating betas, can use historic dataEstimating betas, can use historic data– Assume betas are ‘mean reverting’ (to mean of Assume betas are ‘mean reverting’ (to mean of

market beta)market beta) High betas (High betas ( > 1) > 1) lower beta in future lower beta in future Low betas (Low betas ( < 1) < 1) higher betas in future higher betas in future

Adjusted beta = (w) estimated beta + (1-w) 1.0Adjusted beta = (w) estimated beta + (1-w) 1.0(i.e. w = 2/3 and (1-w) = 1/3) (i.e. w = 2/3 and (1-w) = 1/3) (Can test whether betas are ‘mean’ reverting and (Can test whether betas are ‘mean’ reverting and then estimate w.)then estimate w.)

Q.: Is beta constant over time ? Q.: Is beta constant over time ?

CAPM and Investment CAPM and Investment AppraisalAppraisal

CAPM and Investment CAPM and Investment Appraisal (All Equity Appraisal (All Equity Firm)Firm) Expected return ERExpected return ERii on equity (calculated from the CAPM on equity (calculated from the CAPM

formula) is (often) used as the discount rate in a DPV formula) is (often) used as the discount rate in a DPV calculation to assess a physical investment project for calculation to assess a physical investment project for an all equity financed firman all equity financed firm

We use ERWe use ERii because it reflects the riskiness of the firm’s because it reflects the riskiness of the firm’s new investment project – provided the ‘new’ investment new investment project – provided the ‘new’ investment project has the same ‘business risk’ characteristics as project has the same ‘business risk’ characteristics as the firm’s existing project. the firm’s existing project.

This is because ERThis is because ERii reflects the return required by reflects the return required by investors to hold this share as part of their portfolio (of investors to hold this share as part of their portfolio (of shares) to compensate them, for the (beta-) risk of the shares) to compensate them, for the (beta-) risk of the firm (i.e. due to covariance with the market return, over firm (i.e. due to covariance with the market return, over the past). the past).

CAPM and Investment CAPM and Investment Appraisal (Levered Appraisal (Levered Firm)Firm) What if the new project is so large it will radically What if the new project is so large it will radically

alter the debt equity mix, in the future ? alter the debt equity mix, in the future ? How do we measure the equity return ER (then How do we measure the equity return ER (then

the WACC) to be used as the discount rate ? the WACC) to be used as the discount rate ? (MM result : Equity holder requires higher return (MM result : Equity holder requires higher return

ERERii as the debt to equity ratio increases.) as the debt to equity ratio increases.) We calculate this ‘new’ equity return by using the We calculate this ‘new’ equity return by using the

‘levered beta’ in the CAPM equation as : ‘levered beta’ in the CAPM equation as :

L(new)L(new) = = UU (1 + (1-t))(B/S) (1 + (1-t))(B/S)newnew))

How Does beta (L) How Does beta (L) Vary with Debt-equity Vary with Debt-equity Ratio ? Ratio ? B/(B+S) (B/S) B/(B+S) (B/S) new new beta (L) Leverage beta (L) Leverage

effect effect

0%0% 0%0% 1.28 (= 1.28 (= UU)) 00

50%50% 100%100% 2.1 2.1 0.820.82

70%70% 233%233% 3.2 3.2 1.921.92

90%90% 900%900% 8.7 8.7 7.47.4

Above uses Above uses L(new)L(new) = = UU (1 + (1-t))(B/S) (1 + (1-t))(B/S)newnew) with t = ) with t = 0.360.36

How Does beta (L) Vary How Does beta (L) Vary with Debt-equity Ratio ? with Debt-equity Ratio ? (Cont.)(Cont.) The ‘leverage-beta’ increases with leverage The ‘leverage-beta’ increases with leverage

(B/S) and hence so does the required return (B/S) and hence so does the required return on equity ERon equity ERii given by the CAPM and hence given by the CAPM and hence the discount rate for cash flowsthe discount rate for cash flows

ERERii can then be used with the bond yield to can then be used with the bond yield to calculate WACC, if debt and equity finance is calculate WACC, if debt and equity finance is being used for the ‘new’ project. being used for the ‘new’ project.

See Cuthbertson and Nitzsche (2001) See Cuthbertson and Nitzsche (2001) ‘Investments : Spot and Derivatives Markets’‘Investments : Spot and Derivatives Markets’

Variants of the CAPMVariants of the CAPM

Zero Beta CAPMZero Beta CAPM

ER

ERz

M

beta

ER

ERz

M

sigma

Z

Portfolio Z is not the minimum variance portfolio.

Zero Beta CAPM Zero Beta CAPM (Cont.)(Cont.) Two factor model : Two factor model :

ERERii = ER = ERZZ + (ER + (ERmm – ER – ERZZ) ) ii

Portfolio s has smallest variance : Portfolio s has smallest variance : ss

22 = X = Xzz22zz

22 + (1 – X + (1 – Xzz))22mm22

∂∂ss22/∂X/∂Xzz = 2X = 2Xzzzz

22 – 2 – 2mm22 + 2X + 2Xzzmm

22 = 0 = 0

Solving for XSolving for Xzz : X : Xzz = = mm22 / ( / (mm

22++zz22) )

Zero Beta CAPM Zero Beta CAPM (Cont.)(Cont.) Since Since mm

22 and and zz22 must be positive must be positive

– Positive weights on both assets (M and Z)Positive weights on both assets (M and Z)

Since ERSince ERzz < ER < ERmm – Portfolio ‘S’ (Z and M) must have higher Portfolio ‘S’ (Z and M) must have higher

expected return than Zexpected return than Z Since the minimum variance portfolio Since the minimum variance portfolio

has higher ER and lower sigma than Z, has higher ER and lower sigma than Z, Z cannot be on the efficient portion of Z cannot be on the efficient portion of the efficient frontier. the efficient frontier.

Zero Beta CAPM Zero Beta CAPM (Cont.)(Cont.)

Z

S

M

sigma

S : minimum variance portfolio

ER

The Consumption The Consumption CAPM CAPM Different definition of equilibrium in Different definition of equilibrium in

the capital marketthe capital market Key assumption : Key assumption :

– Investors maximise a multiperiod utility Investors maximise a multiperiod utility function over lifetime consumptionfunction over lifetime consumption

– Homogeneous beliefs about asset Homogeneous beliefs about asset characteristics characteristics

– Infinitely lived population, one Infinitely lived population, one consumption goodconsumption good

The Consumption The Consumption CAPM (Cont.)CAPM (Cont.)

EEtt(r(rtt – r – rff) + ½(s) + ½(s22tt(r(rii

**) = -cov) = -covtt(m,r(m,rii**))

where where rrii** = ln(R = ln(Rii

**) ) M = ln(M) M = ln(M) M = M = (C(Ct+1t+1/C/Ctt))--

Excess return on asset-i depends on the Excess return on asset-i depends on the covariance between rcovariance between rii

** and consumption. and consumption. The higher is the ‘covariance’ with The higher is the ‘covariance’ with

consumption growth, the higher the ‘risk’ consumption growth, the higher the ‘risk’ and the higher the average return. and the higher the average return.

References References

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2004) ‘Quantitative Financial (2004) ‘Quantitative Financial Economics’, Chapters 7 Economics’, Chapters 7

Cuthbertson, K. and Nitzsche, D. Cuthbertson, K. and Nitzsche, D. (2001) ‘Investments : Spot and (2001) ‘Investments : Spot and Derivatives Markets’, Chapter 10.3Derivatives Markets’, Chapter 10.3

END OF LECTUREEND OF LECTURE