FIN 360 Ch. 7

What is the expected rate of return of the market is the risk free rate is 3%, the average investor has a risk aversion (A) of 3, and the standard deviation (sd) of the market portfolio is 15%?

(𝐸(𝑟_𝑀 )= 𝑟_𝑓+𝐴𝑜_𝑀^2) = .03 + (3*.15^2) = .03 + 0675 = .0975 or 9.75%

Risk Premium of Market Portfolio

- Demand drives prices, lowers expected rate of return/risk premiums - When premiums fall, investors move funds into risk-free asset - Equilibrium risk premium of market portfolio proportional to: * Risk of market * Risk aversion of average investor

CAPM is false based on validity of its assumptions

- useful predictor of expected returns - untestable as a theory - principles still valid - investors should diversify - systematic risk is the risk that matters - well diversified risky portfolio can be suitable for wide range of investors

Hypothetical Equilibrium

All investors choose to hold market portfolio (the portfolio for which each security is held in proportion to its total market value) Market portfolio is on efficient frontier, in fact it is the optimal risky portfolio (the tangency point of the CAL to the efficient frontier) Risk premium on market portfolio is proportional to variance of market portfolio and investor's risk aversion Risk premium on individual assets Proportional to risk premium on market portfolio Proportional to beta coefficient of security on market portfolio

Mutual fund theorem

All investors desire the same portfolio of risky assets, can be satisfied by single mutual fund composed of that portfolio

Portfolio Beta

Bp = (w1*b1) + (W2*B2) + (W3*B3)

SML

E(rd) = rf + Bd [E(rm) - rf]

CAPM

E(rm) - rf = Ao^2m

𝑬(𝒓_𝑴 )−𝒓_𝒇=𝑨𝒐_𝑴^𝟐 or 𝑬(𝒓_𝑴 )=𝒓_𝒇+𝑨𝒐_𝑴^𝟐

E(rm): Expected return of the market A: degree of risk aversion Rf: risk free rate In other words, the risk premium on the market portfolio will be proportional to the variance of the market portfolio and investor's typical degree of risk aversion

Expected Returns on Individual securities

Expected return-beta relationship Implication of CAPM that security risk premiums (expected excess returns) will be proportional to beta 𝑬(𝒓_𝑫 )=𝒓_𝒇+𝜷_𝑫 [𝑬(𝒓_𝑴 )−𝒓_𝒇 ] 𝐸(𝑟_𝐷 )=expected return of asset d in equilibrium β_𝐷 = beta of asset d (risk coefficient)

Suppose the risk premium of the market portfolio is 8% with a standard deviation of 22%. What is the risk premium on a portfolio invested in 25% in GE with a beta of 0.8 and 75% invested in Delta with a beta of 1.2?

First solve, 𝜷_𝒑 𝜷_𝒑 = (W1*𝜷_𝟏) + (W2*𝜷_𝟐) = (0.25*0.8) + (0.75*1.2) = 1.1 Now if the 𝜷_𝒑 =1.1 then, 𝑬(𝒓_𝒑 )− 𝒓_𝒇=𝜷_𝑫 [𝑬(𝒓_𝑴 )−𝒓_𝒇 ] = 1.1*.08 = 8.8%

The T-bill rate is 3% and historical data for the S&P 500 index show an average excess return over T-bills of about 7.5% with a standard deviation of about 20%. Estimate the coefficient of risk aversion (A) of average investor?

From the CAPM Equation (𝐄(𝐫_𝐌 )−𝐫_𝐟=𝐀𝐨_𝐌^𝟐), we estimate the coefficient of risk aversion (A) as follows: A = (𝐸(𝑟_𝑀 )−𝑟_𝑓)/ 𝑜_𝑀^2 A = 0.075/(.2^2) = 1.88

In many ways portfolio theory and the CAPM have become accepted tools in the practitioner world.

Many investment professionals think about the distinction between firm specific and systematic risk and are comfortable with the use of beta to measure systematic risk

Implications of the CAPM that security risk premiums (expected excess returns) will be proportional to beta

Only systematic risk matters to investors who can diversify and systematic risk is measured by beta 𝑬(𝒓_𝑫 )=𝒓_𝒇+𝜷_𝑫 [𝑬(𝒓_𝑴 )−𝒓_𝒇 ] This expected return beta relationship is the most familiar expression of the CAPM (the security market line or SML)

Security Characteristic Line (SCL)

Plot of security's expected excess return over risk-free rate as function of excess return on market Required rate = Risk-free rate + B x Expected excess return of index

The Security Market Line (SML)

Represents expected return-beta relationship of CAPM Graphs individual asset risk premiums as function of asset risk Alpha Abnormal rate of return on security in excess of that predicted by equilibrium model (CAPM) 𝛼_𝑖= 𝐸(𝑟)− 〖{𝑟〗_𝑓+β_𝐷 [𝐸(𝑟_𝑀 )−𝑟_𝑓]}

What is the expected return of the market if we suppose the risk free rate is 4%, the average investor has a risk aversion (A) of 4, and the standard deviation (sd) of the market portfolio is 14%?

So the (𝐸(𝑟_𝑀 )= 𝑟_𝑓+𝐴𝑜_𝑀^2) = .04 + (4*.14^2) = .04 + .078 = .1184 or 11.84%

What is the expected return of the market if we suppose the risk free rate is 5%, the average investor has a risk aversion (A) of 2, and the standard deviation (sd) of the market portfolio is 20%. Then from the CAPM Equation (𝐸(𝑟_𝑀 )−𝑟_𝑓=𝐴𝑜_𝑀^2), we estimate expected rate of return on the market...

So the (𝐸(𝑟_𝑀 )= 𝑟_𝑓+𝐴𝑜_𝑀^2) = .05 + (2*.2^2) = .05 + .08 = .13 or 13%

Fama-French Three-Factor Model

Systematic Risk as measured by beta Firm value as measured by book to market (HML: high minus low) Size premium as measured by market cap (SMB: small minus big)

Predicting Betas

THE BETA OF THE MARKET IS ALWAYS 1 Betas of securities revert to the mean (1.0) As an empirical rule, it appears that betas exhibit a statistical property called mean reversion. This suggests that high beta (>1) securities tend to exhibit a lower beta in the future, while low beta (<1) securities exhibit a higher beta in the future

CAPM & Passive Investing

The CAPM implies that a passive strategy, using the CML as the optimal CAL is a powerful alternative to an "active" strategy

Capital Asset Pricing Model (CAPM)

The CAPM is a centerpiece of modern financial theory. It was propose by William Share who was awarded the 1990 Nobel prize in Economics

Alpha

The abnormal rate of return on a security in excess of what would be predicted by an equilibrium model such as CAPM Whenever the CAPM holds, their expected returns must lie on the SML Attractive or underpriced stocks plot above the SML

If the mean-beta relationship holds for any individual asset, it must hold for any combination of assets.

The beta of a portfolio is simply the weighted average of the betas of the stocks in the portfolios 𝜷_𝒑 = (W1*𝜷_𝟏) + (W2*𝜷_𝟐) + (W3*𝜷_𝟑) Security prices reflect public information about a firms prospects, but only the risk of the company (as measured by beta) should affect expected returns.

Applications of CAPM

Use SML as benchmark for fair return on risky asset Stocks above the SML are attractively priced Stocked below the SML are unattractive SML provides "hurdle rate" for internal projects Capital projects above the SML are attractive Capital projects below the SML are unattractive

Alpha

i = E(r) - {rf + Bd [E(rm) - rf]}

Multifactor Models

models of security returns that respond to several systematic factors

Two-factor security market line (SML)

𝐸(𝑟_𝑖 )=𝑟_𝑓+β_(𝑖𝑀 ) [𝐸(𝑟_𝑀 )−𝑟_𝑓 ]+β_𝑖𝑇𝐵 [𝐸(𝑟_𝑇𝐵 )−𝑟_𝑓 ] The risk free rate of return The sensitivity to the market index (beta) times the risk premium of the index or (𝐸(𝑟_𝑀 )−𝑟_𝑓) The sensitivity to interest rate risk times the risk premium of the T Bond portfolio or (𝐸(𝑟_𝑇𝐵 )−𝑟_𝑓)

Suppose the T-Bill rate is 3%, the risk premium of the market portfolio is 8%, and we estimate the beta 〖(𝜷〗_𝑫) of Delta Airline equals 1.2. What equation do you use and what is the expected return of Delta Airlines?

𝑬(𝒓_𝑫 )=𝒓_𝒇+𝜷_𝑫 [𝑬(𝒓_𝑴 )−𝒓_𝒇 ] = .03 + 1.2(.08) = 12.6%

Suppose the T-Bill rate is 4%, the risk premium of the market portfolio is 7%, and we estimate the beta 〖(𝜷〗_𝑫) of Delta Airline equals 1.3. What is the expected return of Delta Airlines?

𝑬(𝒓_𝑫 )=𝒓_𝒇+𝜷_𝑫 [𝑬(𝒓_𝑴 )−𝒓_𝒇 ] = .04 + 1.3(.07) = .131 or 13.1%

Suppose the return on the market is expected to be 14%, a stock has a beta of 1.2 and the T-bill rate is 6%. The SML would predict an expected return on the stock of ?

𝑬(𝒓𝒔)=〖𝒓𝒇+ 𝜷〗_𝒔 [𝑬(rm) - rf] = .06 + 1.2(.14-.06) = 15.6% However, what if one believes the stock will provide instead a return of 17% what is the alpha? 17.0 - 15.6 = 1.4%

IBM has an expected return of 15% and a beta = 1.0. ORCL has an expected return of 18% with a beta of 1.5. The market's expected return is 9% and the T-bill rate is 3%. According to the CAPM, which stock is a better buy (what is the alpha of each stock)?

𝛼_𝑖= 𝐸(𝑟)− 〖{𝑟〗_𝑓+β_𝐷 [𝐸(𝑟_𝑀 )−𝑟_𝑓]} 𝛼_𝑖𝑏𝑚= .15−[.03+1.0(.09−.03)] = 6.0% 𝛼_𝑜𝑟𝑐𝑙= .18−[.03+1.5.09−.03)] = 6.0% PICK both because each have the same positive alpha

Stock XYZ has an expected return of 12% and a beta = 1. Stock ABC has an expected return of 13% with a beta of 1.5. The market's expected return is 11% and the T-bill rate is 5%. According to the CAPM, which stock is a better buy (what is the alpha of each stock)?

𝛼_𝑖= 𝐸(𝑟)− 〖{𝑟〗_𝑓+β_𝐷 [𝐸(𝑟_𝑀 )−𝑟_𝑓]} 𝛼_𝑥𝑦𝑥= .12−[.05+1.0(.11−.05)] = 1% 𝛼_𝑎𝑏𝑐= .13−[.05+1.5(.11−.05)] = (1%) PICK XYX because its alpha is the greatest

If we own a 50% weight in MSFT with a beta of 1.2 and a 30% weight in AEP with a beta of 0.8 and a 20% weight in gold with a beta of 0.0, what is the portfolio beta?

𝜷_𝒑 = (W1*𝜷_𝟏) + (W2*𝜷_𝟐) + (W3*𝜷_𝟑) = (0.5*1.2) + (0.3*0.8) + (.2*0) = 0.6 + 0.24 + 0.0 = .84