CAPM Finance

CAPM

Finance 100

Prof. Michael R. Roberts

Topic Overview z Investing with a risk-free asset » Borrowing and lending » The market portfolio » The Capital Market Line (CML) z The Capital Asset Pricing Model (CAPM) » The Security Market Line (SML) »Beta » Project analysis z Alternative Models

1 1 Efficient Portfolios with Multiple Risky Assets

Investors Efficient E[r] prefer Frontier

Portfolios Asset 1 of other Portfolios of assets Asset 2 Asset 1 and Asset 2

Minimum-Variance Portfolio

Introducing a Risk-Free Asset z Suppose we introduce the opportunity to invest in a riskfree asset. » How does this alter investors’ portfolio choices? z The riskfree asset is characterized by

» Zero variance: var(rf) = 0. » Zero covariance with all assets: cov(rf, rj) = 0 for all j. z What is the expected return and variance of a portfolio consisting of a fraction (1-α) of the riskfree asset and a fraction α of the risky asset (or portfolio of risky assets)?

2 2 Risk and Return with a Risk-Free Asset

z Expected Return

Er[ Pj] =+−αα Er[ ] (1 ) r f z Variance and Standard Deviation

222 Var[] rPP==σ ασ j ⇒= σ P ασ j z Hence, the risk-return tradeoff is:

(E[rj ]− rf ) E[]rP = rf + σ P σ j » The slope is called the Sharpe-Ratio

Risk and Return with a Risk-Free Asset An Illustration

Expected Return z The line represents all portfolios Asset j (α=1) depending on α

E(rj)

rf Riskfree asset (α=0)

Lending Borrowing Standard Deviation 0 σj

3 3 Portfolios with the Risky Asset

Line 3 Expected

Return Line 2 T

Line 1 IBM

r f Dow Utility Index

Standard Deviation

The Capital Market Line

z The Capital Market Line (CML) gives the tradeoff between risk and return for portfolios consisting of the riskfree asset and the tangency portfolio T. The equation for the CML is:

Er()Tf− r Er()pfp=+ r σ σ T » Tangency portfolio has the highest Sharpe ratio of any portfolio z CML is the new efficient frontier » Tangency portfolio is the efficient portfolio of risky assets z Every investor should be on the CML » Preferences will determine where on the CML

4 4 Decomposition of Risk

Line 3 Expected Return T Line 1 A IBM

r f

Systematic Diversifiable Standard Risk Risk Deviation

Decomposition of Return

Expected Line 3 Return P

E(rM ) − rf σ p σ M

r f

σ Standard P Deviation

5 5 What is the Tangency Portfolio?

z To identify the tangency portfolio, we need for every risky investment » expected returns, » volatilities, and » correlations. z But, these quantities are tough to forecast and different people may have different beliefs » How can we identify the “T” portfolio? z This is where the CAPM comes in » It let’s us identify the efficient portfolio (“T”) without knowledge of the expected return of each security.

The CAPM

z Assume: » Investors can buy and sell all securities at competitive market prices (no taxes, transaction costs), and borrow and lend at the risk-free interest rate » Investors hold only efficient portfolios of traded securities – portfolios yielding the maximum expected return for a given level of volatility » Investors have homogenous expectations regarding the expected returns, volatilities, and correlations of securities. z Then, » All investors will identify the same portfolio as having the highest Sharpe return (homogenous expectations) » All investors will hold this same portfolio (hold only efficient portfolios) » Since everybody security is owned by someone, the sum of all investor’s portfolios must equal the portfolio of all risky securities, i.e., The Market Portfolio! – This is just demand (all investors demand eff. portfolio) = supply (market portfolio)

6 6 The CAPM (Cont.)

z The CAPM tells us that the market is the efficient portfolio, which everyone is holding z Thus, the relevant measure of risk of an asset is no longer volatility but covariation with the market

Cov (rj , rM ) » Which implies that the return per unit of risk is simply the ratio:

E (r j )− r f

Cov ()r j , rM z This measure must be the same for all assets! Er( ) − r Er( ) − r Af> Bf Cov( rAM,, r ) Cov( rBM r ) » Consider what would happen if this was not the case…

Security Market Line

z If the risk-return tradeoff is the same for all assets, then it is equal to that of the market: Er( ) − r Er( ) − r Er( ) − r Af= Bf= Mf Cov( rAM,, r ) Cov( rBM r ) Var( rM ) z This relation enables us to derive the relation between risk and expected return for individual stocks and portfolios called the Security Market Line (SML)

Cov( rAM, r ) Er( Af) =+ r (Er()Mf − r) =+ r fAMfβ ( Er() − r) Var() rM where

Cov( rAM, r ) βA = Var() rM

7 7 Capital Asset Pricing Model

z Under the CAPM assumptions of » Investors can buy and sell all securities at competitive market prices (no taxes, transaction costs), and borrow and lend at the risk-free interest rate » Investors hold only efficient portfolios of traded securities – portfolios yielding the maximum expected return for a given level of volatility » Investors have homogenous expectations regarding the expected returns, volatilities, and correlations of securities. we get the following conclusions: 1. The market portfolio is the efficient (tangency) portfolio » Best return-volatility combinations lie on the CML (supply = demand) 2. The risk premium for any security is proportional to its beta with the market »SML

Capital Asset Pricing Model Illustration

Expected Return

Expected Market Return Expected market risk premium

Risk free rate

0 0.5 1.0 Beta Expected Risk free Beta Expected market = + x return rate factor risk premium

8 8 CML vs SML

E ( r ) − r E ( r ) = r + σ M f p f p E (rA ) = rf + β A (E (rM )− rf ) σ M E(r) CML E(r)

SML

MM E(rM ) E(r ) IBM IBM

rf rf

σ IBM,M/σ M σ M σ IBM σββ IBM 1.0

The SML and Mispriced Stocks

z Suppose for a particular stock:

Cov( rjM, r ) Er( jf) <+ r []Er( Mf) − r Var( rM )

z Remember the definition of expected returns:

11 0 EP( jjj+− D) P Er( j ) = 0 Pj

z Then P0 falls, so that E(rj) increases until disequilibrium vanishes and the equation holds!

9 9 The SML and Mispriced Stocks

Expected Return z Stock j is overvalued at X:

Y » price drops, » expected return rises. X z Stock j is E(r )-r M f undervalued at Y: rf » price increases » expected return falls

Alpha

z Alpha is the difference between a stock’s expected return and its required return according to the SML ⎡ ⎤ αβssfsMf=−+Er( ) ( r⎣ Er( ) − r⎦) z Non-zero alpha implies that the market portfolio is inefficient and not all stocks lie on the SML

10 10 Alpha Illustration

Using Regression Analysis to Measure Beta Excess Returns for Cisco and S&P500

11 11 Using Linear Regression to Estimate Beta

z The regression model is:

(Rrif −=+ ) αβ i iMktf( R −+ r) ε i

» αi is the intercept term of the regression. » βi(RMkt –rf) represents the sensitivity of the stock to market risk.

» εi is the error term and represents the deviation from the best-fitting line and is zero on average. z Given data for rf , Ri , and RMkt , statistical packages for linear regression can estimate βi. » Use S&P500 as proxy for market » Use yields on long-term treasury bonds as proxy for risk-free » Use 5-years of monthly or 2 years of weekly returns data

Using Linear Regression to Estimate Beta (Cont.) z Taking expectations ER[] =+ r β ([ ER ] − r) + α ifiMktf N i Expected return for i from the SML Distance above / below the SML

» Since E[εi] = 0: » αi represents a risk-adjusted performance measure for the historical returns.

»If αi is positive, the stock has performed better then predicted by the CAPM.

»If αi is negative, the stock’s historical return is below the SML.

12 12 Variance Decomposition z Recall the regression model:

(Rrif −=+ ) αβ i iMktf( R −+ r) ε i z What does this relation imply about the variance of the asset return? Var(RR )=+β 2 Var Var ε ii ( M) ( i) Market Firm-Specific Risk Risk

Beta and Standard Deviation

2 Var(RRii )=+β Var( M) Var (ε i)

Risk of a Market risk Specific risk =+ Share (Variance) of the share of the share

Beta of Risk of This is the major share x market element of a share's risk

Risk of a Market risk of Specific risk of portfolio =+the the portfolio portfolio

Beta of Risk of This is negligible Portfolio x market for a diversified portfolio

13 13 Beta and Volatility

Company beta (β) SD (σi) Specific Risk (σε) Cisco Systems 1.60 0.46 0.32 Citicorp 1.40 0.33 0.17 Digital Equipment Corp. 1.20 0.48 0.42 Duke Power 0.75 0.42 0.39 General Motors 1.10 0.29 0.19 IBM 1.05 0.34 0.27 Microsoft 1.20 0.43 0.36 Nationsbank Corp. 1.25 0.27 0.09 NIKE 1.05 0.40 0.34 Northwest National Gas 0.45 0.24 0.22 Space Labs Medical 0.80 NA NA

Portfolio Betas and Expected Returns Example z What is the beta and expected rate of return of an equally-weighted portfolio consisting of Exxon (beta = 0.46) and Polaroid (beta = 0.96)?

z How would you construct a portfolio with the same beta and expected return, but with the lowest possible standard deviation ?

z Use the figure on the following page to locate the equally-weighted portfolio of Exxon and Polaroid. Also locate the minimum variance portfolio with the same expected return.

14 14 Example Illustration

E(r) E(r) CML SML

M M 13.8% 13.8%

11.3%

5.2% 5.2%

σM σβ0.71 1.0

Efficient Portfolio Example

z The S&P500 Index has a standard deviation of about 12% per year. z Gold mining stocks have a standard deviation of about 24% per year and a correlation with the S&P500 of about ρ = 0.15. z If the yield on U.S. Treasury bills is 5.2% and the

market risk premium is [E(rM)-rf] = 8.6%, what is the expected rate of return on gold mining stocks?

15 15 Efficient Portfolio Example (Cont.)

z What is the beta for gold mining stocks?

z What is the expected rate of return on gold mining stocks ?

z What portfolio has the same expected return as gold mining stocks, but the lowest possible standard deviation?

Illustration Example (Cont.)

E(r) E(r) CML SML

M 13.8% M 13.8%

P Gold M. 7.8% 7.8%

5.2% 5.2%

3.6% 24% σβ0.3 1.0

16 16 Using the CAPM for Project Valuation Example z Suppose Microsoft is considering an expansion of its current operations. » Initial cost = $100 million » Future net cash flow = $25 million per year for 20 years. » If Microsoft’s equity beta is 1.2, what is the appropriate risk-adjusted discount rate for the expansion project if the market risk premium is 8.6% and the risk-free rate is 5.2%?

» What is the NPV of Microsoft’s investment project?

Company Risk vs Project Risk z The company-wide discount rate is the appropriate discount rate for evaluating investment projects that have the same risk as the firm as a whole. z For investment projects that have different risk from the firm’s existing assets, the company-wide discount rate is not the appropriate discount rate. z In these cases, we must rely on industry betas for estimates of project risk.

17 17 Company Risk vs Project Risk Example z Suppose Microsoft is considering investing in the development of a new airline. » What is the appropriate risk-adjusted discount rate for evaluating the project if the airline industry beta = 1.8 ? (Recall: rf = 5% and rM = 13.6%)

» Suppose the project offers a 17% internal rate of return. Is the investment a good one for Microsoft ?

Company Risk vs Project Risk Illustration

Required Return SML

Project-specific Discount Rate Project IRR A

Company-wide Discount Rate

Company Beta Project Beta β

18 18 Some Estimates of Beta

Project Valuation Rules

z Risk of an investment project is given by the project’s beta. » Can be different from company’s beta » Can often use industry as approximation z The SML provides an estimate of an appropriate discount rate for the project based upon the project’s beta. » Same company may use different discount rates for different projects » Must account for effect of leverage…later z This discount rate is used when computing the project’s net present value.

19 19 Alternative Models of Systematic Risk z If the market portfolio is efficient then securities should have zero alphas z Researchers have identified a number of characteristics that can used to “pick” portfolios that produce high average returns (i.e., positive alpha)

The Size Effect Excess Return of Size Portfolios 1926-2005 z Stocks with lower market caps have higher average returns

20 20 The Book-to-Market (BM) Ratio Excess Return of BM Portfolios 1926-2005 z Stocks with higher BM ratios have higher average returns

Momentum Strategy z Buying stocks that have had past high returns (and shorting stock that have had past low returns) z If market portfolio is efficient, past returns should not predict future returns (or alphas) z Research shows that this strategy generates alpha

21 21 Implications of Positive Alpha

z If the CAPM is correct Æ investment with positive alpha is a positive NPV investment and investors should RUN to such strategies z The existence of positive alpha suggests: » Investors are systematically ignoring positive NPV investments (i.e., arbitrage opportunities) – They either don’t know about the opportunities or, – The costs to implement the strategies are greater than the NPV of undertaking them » Positive alpha strategies contain risk that investors are unwilling to bear but the CAPM does not capture – I.e., market portfolio is inefficient

Summary z Optimal investments depend on trading off risk and return » Investors with higher risk tolerance invest more in risky assets » Only risk that can’t be diversified counts z If investors can borrow and lend, then everybody holds a combination of two portfolios » The market portfolio of all risky assets » The riskless asset – Covariance with the market portfolio counts z In equilibrium, all stocks must lie on the security market line » Beta measures the amount of non-diversifiable risk » Expected returns reflect only market risk » Use these as required returns in project evaluation z Multifactor models represent a controversial extension

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