Capital Market Equilibrium and the Capital Asset Pricing Model

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1 Capital Market Equilibrium and the Capital Asset Pricing Model Econ 422 Investment, Capital & Finance Spring 21 June 1, 21 1 The Risk of Individual Assets Investors require compensation for bearing risk. We have seen that the standard deviation of the rate of return is an appropriate measure of risk for one s portfolio. Standard deviation is not the best measure of risk for individual assets when investors hold diversified portfolios. 2 1

2 The Risk of Individual Assets (continued) For people holding a diversified portfolio it is the contribution of the individual asset to the portfolio s standard deviation that matters. [If your portfolio involved only one asset, e.g. young Bill Gates, the portfolio standard deviation would be the standard deviation of the single asset.] 3 The contribution of an individual asset to the portfolio s standard deviation: Beta Beta measures the sensitivity of an asset s s rate of return to variation in the market portfolio s return. Beta for asset i can be computed as β cov( r, r ) = i m i V( r = m) σ i σ im 2 m 4 2

3 Beta as a Measure of Portfolio Risk: Class Example Suppose you hold an equally weighted portfolio with 99 assets What happens to the portfolio variance if a new asset, say IBM, is added to the portfolio? 5 Measuring Betas The Market Model Beta can be interpreted as the slope coefficient in a regression of the return on the ith security, r i, on the return for the market portfolio, r m. The interpretation rests on the market model: r = α + β r + ε it i i mt it r m represents market risk and ε i represents firm specific risk independent of the market. That is, cov(r mt, ε it ) =. 6 3

4 Beta as a Measure of Risk for Individual Assets Asset i s is Return ri +1% The slope is given by β=cov(ri,rm)/v(rm) The intercept is given by α -1% -1% +1% Market return rm 7 The Capital Asset Pricing Model (CAPM) CAPM describes the relationship between an asset s beta risk and its expected return as follows: Er ( ) = r + β Er ( ) r i f i m f = riskfree rate + beta x market risk premium. 8 4

5 The Security Market Line (SML) Describes the CAPM Relationship E(ri) E(rm) rf Security Market Line (SML) Slope is the market risk premium = E(rm)-rf 1. β Er ( ) = r + β Er ( ) r i f i m f 9 The Capital Asset Pricing Model Example Annual T-bill yield = 1% Beta for Microsoft =.93 (from Yahoo! See link for key statistics) Historical market risk premium =7.1% Then Er [ ] = r + β ( ER [ ] r) msft f msft mt f = 1% +.93*(7.1%) = 7.63% Note: the return predicted from the CAPM is sometimes called the risk-adjusted return 1 5

6 CAPM and Efficient Portfolios Example: Starbucks Stock E[R sbux ]=.1, SD(R sbux )=2.2 E[R m ] =.15, SD(R m ) =.1, r f =.3 Find the beta for SBUX Find the efficient portfolio of T-bills and the market with the same expected return as SBUX 11 The Market Model and the Measures of Risk The total risk of an asset, held alone, i.e. not as part of a diversified portfolio, would be measured by its variance, V(r i ). According to the market model rit = αi + βirmt + εit and 2 V( r ) = β V( r ) + V( ε ) it i mt it Total risk = systematic market risk + unique risk The unique risk can be eliminated through diversification. 12 6

7 Portfolio Beta The beta of a portfolio is a weighted average of the individual asset betas β p = x1β1 + x2β xnβ n xi = portfolio share for asset i β = beta for asset i i 13 Example Microsoft has a beta of 1.2 Starbucks has a beta of.8 Portfolio weights are x msft =.25, x sbux =.75 Portfolio beta =.25(1.2) +.75(.8) =

8 Applying the CAPM to Valuation Recall the one-period holding period rate of return: P1 P+ D1 r = P At time t=, P is known, but P and D are not known; they are random variables. Take expectations: EP ( 1) P+ ED ( 1) Er () = P Solve for P : 1 1 EP ( 1) + ED ( 1) EP ( 1) + ED ( 1) P = = 1 + Er ( ) 1 + r + β[ Er ( ) r] f M f using the CAPM relation: Er ( ) = r + β[ Er ( ) r] f M f 15 Applying the CAPM to Valuation (continued) P EP ( ) + ED ( ) 1 [ ( ) ] 1 1 = + rf + β E rm rf To value a future risky cash flow, discount the expected value of the cash flow to present value using the risk-adjusted expected return based on the CAPM. 16 8

9 Example: Stock valuation using CAPM E[D 1 ] = 5, g =.1, r f =.3 β = , E[r m ] r f =.7575 E[ r] = r + β( Er [ ] r ) = (.75) =.1425 f M f ED [ ] 5 5 ( E [ r ] g ) Constant growth: P = = = =

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