The Risk Of Individual Assets

The Risk of Individual Assets

Capital Market Equilibrium and the Capital Asset Pricing Model

z z

Econ 422 Investment, Capital & Finance Summer 2006 August 15, 2006

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For people holding a diversified portfolio it is the contribution of the individual asset to the portfolio’s standard deviation that matters.

z

Beta measures the sensitivity of an asset’s rate of return to variation in the market portfolio’s return. Beta for asset i can be computed as

[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.]

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The contribution of an individual asset to the portfolio’s standard deviation: Beta

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The Risk of Individual Assets (continued) z

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.

βi =

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cov(ri , rm ) σ im = 2 V (rm ) σm

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Measuring Betas The Market Model

Beta as a Measure of Portfolio Risk: Class Example z z

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?

z

z

rit = α i + β i rmt + ε it z z

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The Capital Asset Pricing Model (CAPM) z

The slope is given by β=cov(ri,rm)/V(rm)

+10%

The intercept is given by α

CAPM describes the relationship between an asset’s beta risk and its expected return as follows:

E ( ri ) = rf + β i E ( rm ) − rf -10%

+10%

-10%

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rm represents “market risk” and εi represents “firm specific” risk independent of the market. See Spreadsheet example

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Beta as a Measure of Risk for Individual Assets Asset i’s Return ri

Beta can be interpreted as the slope coefficient in a regression of the return on the ith security, ri, on the return for the market portfolio, rm. The interpretation rests on the market model:

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Market return rm

= riskfree rate + beta x market risk premium.

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The Security Market Line (SML) Describes the CAPM Relationship E(ri)

The Capital Asset Pricing Model Example

Security Market Line (SML)

z z

E(rm) Slope is the market risk premium = E(rm)-rf

rf

1.0

z

E[rmsft ] = rf + βmsft (E[Rmt ] − rf )

β

= 2% +1.61*(7.5%) = 14.075%

E ( ri ) = rf + β i E ( rm ) − rf © 2006 R.W.Parks/E. Zivot

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Note: the return predicted from the CAPM is sometimes called the “risk-adjusted” return 9

The Market Model and the Measures of Risk z

z

z

1=

and

V(rit ) = β V (rmt ) + V (ε it ) z

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R2 measures proportion of an asset’s total risk that is market risk:

Var (rit ) βi2Var (rmt ) Var (ε it ) = + Var (rit ) Var (rit ) Var (rit )

= R2 + 1 − R2

2 i

z

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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(ri). According to the market model

rit = α i + β irmt + ε it

If T-bill yield = 2% The beta for Microsoft = 1.61 (from Yahoo! See link for key statistics) The historical market risk premium =7.5% Then

Total risk = systematic market risk + unique risk The unique risk can be eliminated through diversification.

R2 =

βi2Var (rmt ) Var (rit )

, 1 − R2 =

Var (ε it ) Var (rit )

See spreadsheet for examples ECON 422:CAPM

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Portfolio Beta

Testing the CAPM

The beta of a portfolio is a weighted average of the individual asset betas

z

β p = x1 β 1 + x 2 β 2 + " + x n β n

z

x i = portfolio share for asset i

β i = beta for asset i See spreadsheet example ECON 422:CAPM

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Testing involves a number of measurement problems: » for expected returns » for beta » for the market portfolio

z z

z z

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Some Approaches Used to Address the Measurement Problems

Testing the CAPM z

Key CAPM prediction: stocks with high betas should have high average returns; stocks with low betas should have low average returns Simple test: compute average returns and betas for a bunch of stocks and see if the high beta stocks have higher average returns than the low beta stocks CAPM predicts a straight line relationship between average return and beta

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Early studies by Black Jensen & Scholes (1972) and by Fama and McBeth (1973) Measurement of betas: data from period 1 used to estimate individual stock betas. These betas used to construct “decile portfolios”: stocks with betas in the lowest 10% grouped as the first portfolio. Stocks with betas in the next lowest 10% grouped as the second portfolio etc. » Portfolio betas are more accurate than single asset betas Data from period 2 used to re-estimate betas and average returns for the 10 portfolios Look at the regression of average return on betas for the 10 portfolios

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Black, Jensen & Scholes CAPM Test (1972)

Validity of the CAPM

Average Monthly Return

z

Theoretical SML Fitted SML

rf

Low Beta

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High Beta

Beta

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What is a Value Stock? z

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The CAPM Remains Attractive

A stock that tends to trade at a lower price relative to it's fundamentals (i.e. dividends, earnings, sales, etc.) and thus considered undervalued by a value investor. Common characteristics of such stocks include a high dividend yield, low price-to-book ratio and/or low price-to-earnings ratio.

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Evidence is mixed: » Long-run average returns are significantly related to betas. » Beta is probably not a complete explanation. – Low beta stocks have earned higher rates of return than predicted by the model. » Recent results by Fama and French (1992) – Small company stocks stocks have earned higher rates of return than predicted by the model. (Size Effect) – Value company stocks have earned higher rates of return than predicted by the model. (Style Effect)

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It is simple and gives sensible answers. It distinguishes between diversifiable and non-diversifiable risk.

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Applying the CAPM to Valuation The CAPM Remains Controversial z

z

z

Recall the one-period holding period rate of return: r=

It is difficult to devise a definitive test » We don’t know how to define and measure the market portfolio. If we use the wrong market index the resulting betas are mismeasured

At time t = 0, P0 is known, but P1 and D1 are not known; they are random variables. Take expectations: E(P1 ) − P0 + E(D1) E(r) = P0

Fama & French results cast doubt on the CAPM although their study is also subject to criticisms.

Solve for P0 : P0 =

Multi-factor models have been developed to compete with the CAPM e.g. the Arbitrage Pricing Model. Ch 11

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P1 − P0 + D1 P0

E(P1 ) + E(D1) E(P1) + E(D1) = 1+ E(r) 1+ rf + β[E(rM ) − rf ]

using the CAPM relation: E(r) = rf + β[E(rM ) − rf ] 21

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Applying the CAPM to Valuation (continued) P0 =

z

z z

E[D1] = 5, g = 0.10, rf = 0.03 β = 1.5, E[rm] – rf = 0.075

E[r] = rf + β(E[rM ] − rf ) = 0.03+1.5(0.075) = 0.1425

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.

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Example: Stock valuation using CAPM

E ( P1 ) + E ( D1 ) 1 + rf + β [ E (rM ) − rf ]

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Constant growth: P0 =

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E[D1] 5 5 = = =117.64 (E[r] − g) 0.1425−0.1 0.0425

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Using the CAPM to Determine the Discount Rate for Risky Projects z

z

The Company Cost of Capital

For risk-free projects you would use a riskfree interest rate for discounting a project’s cash flow. For risky projects you discount the expected cash flow by a risk adjusted interest rate, using the CAPM.

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The average rate of return for the entire company’s assets is called the company cost of capital. If the project under consideration has the same risk as these assets, then the company cost of capital is an appropriate discount rate. But the project may have a different risk. We would then use a discount rate matching the project’s risk.

Project Discount Rate and the Company Cost of Capital Required return

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Determining Project Risk E[r] = 2% + β*(7.5%)

Security market line, CAPM

Company Cost of Capital

Category

Project Beta

Discount Rate

Speculative ventures

3.73

30%

New Products

2.4

20%

Rf

Expansion of 1.73 = estimated existing business beta from data Cost 1.06 improvement

Project Beta

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15% (company cost of capital) 10%

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Estimating the Cost of Capital for an All-Equity Firm: Example

Estimating the Cost of Capital for an All-Equity Firm z

z z z

For an all-equity firm the riskiness of the stock is the same as the riskiness of the company’s assets. Stock betas are easy to obtain or estimate. Use the CAPM to estimate the company cost of capital: E (rA ) = rf + β E [ E (rM ) − rf ] This is the discount rate for project with the same riskiness as the firm’s current assets. © 2006 R.W.Parks/E. Zivot

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z

z z z z

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A Firm’s Capital Structure: Relative Amounts of Debt and Equity Assets

Debt

D E D E βD + βE = βD + βE D+E D+E V V D E Note that since D + E = V , + = 1. V V If we apply the CAPM to the three components, it must be the case that

βA =

Buildings Equity

E

Total Liabilities

=V=D+E

Patents V=

Total Assets

E(rA ) =

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If the risk of the company's current assets is β A then

D

Receivables

Equipment

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The Risk of A Company’s Assets is Shared by Its Owners

Liabilities Cash

For Microsoft MSFT I found the following information: (From Yahoo Finance, key statistics) Microsoft is an all equity firm, i.e. it has essentially no debt, debt/equity ratio is 0. Its stock beta is 1.61 rf= 1.75%=current T-Bill yield [E(rM) - rf]=expected market risk premium=7.5%, E(re)=rf + β[E(rM) - rf] =1.75%+1.61[7.5%]=13.825%

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D E E (rD ) + E (rE ). V V

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The Riskiness of the Equity Owners Rises When There is Debt

Examples of Asset Betas All equity firm: D/V = 0, E/V =1

z

βA = 0 ⋅βD + 1⋅βE = βE

Consider the special case when the debt is risk-free: D E βD + βE V V If β D = 0,

βA =

Firm with small debt: D/V = 0.1, E/V =0.9

β

A

= 0 .1 ⋅ β D + 0 .9 ⋅ β E

βE = βA

Note: Equity beta, βE, is estimated from stock returns (easy); Debt beta, βD, may be estimated from returns on corporate debt (not so easy) © 2006 R.W.Parks/E. Zivot

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z

BE

D βA E

z z

BA Debt is risk free

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The Discount Rate for a Project in a Leveraged Firm

The Riskiness of the Equity Owners Rises When There is Debt (continued)

βE = βA +

V E+D D = βA = βA + βA E E E

Equity beta are easy to determine. Asset betas would not be easy to estimate directly. Use the equity beta, and undo the effect of the leverage. Estimate or look up the equity beta, estimate the debt beta if it is not zero. Then use the beta relationship to get the asset beta: D E βD + βE D+E D+E Then use the CAPM to determine the corresponding expected return.

βA =

D/E

E (rA ) = rf + β A [ E (rM ) − rf ] Use this rate as the discount when project has same risk as company's existing assets. ECON 422:CAPM

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The Discount Rate for a Project in a Leveraged Firm: Example z

The Discount Rate for a Project in a Leveraged Firm: Example Note:

Sea Star Inc. has a debt/equity ratio of 0.1, an equity beta of 1.21 and a debt beta of 0.11. The current risk-free interest rate is 4%. The expected market risk premium is 8.5%. Find the discount rate for a project with the same risk as the company’s current assets.

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D = 0.1 ⇒ D = 0.1 × E E 0.1 × E 0.1 × E 1 D = = = D + E 0.1 × E + E 1.1 × E 11 1 10 E D = 1− = 1− = D+E D+E 11 11 37

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The Discount Rate for a Project in a Leveraged Firm: Example Therefore,

1 10 1 10 βD + βE = .11+ 1.21 = .01+1.10 = 1.11 11 11 11 11 Then use the CAPM:

βA =

E(rA ) = rf + β A ⎡⎣E(rM ) − rf ⎤⎦ = 4% +1.11*8.5% = 13.435%

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