   # Lecture 7 - Capital Asset Pricing Model

### From Last Time

• The risk of holding an asset is that the actual returns may differ from the expected returns. Standard deviation measures dispersion around the mean. A low standard deviation means that all the possible returns are right around the mean, so there is little risk that the actual return will be much different than the expected return. When the standard deviation is high, there is a lot of variability in the possible returns. Many of them are far from the mean. So, there is more risk that the actual return will be much different than the expected return.

### Types of Risk

Diversification reduces risk. There are two types of risk:
1. market or systematic risk - common to all assets of a certain type, e.g. the business cycle can increase or decrease returns on stocks collectively
2. idiosyncratic or unsystematic risk - a unique risk that assets carry that does not affect the market as a whole, e.g. the price of Microsoft stock is affected by the government's anti-trust lawsuit Diversification can eliminate idiosyncratic risk but not systematic risk. Adding stocks to a portfolio reduces the variability of the portfolio's return by reducing the idiosyncratic risk. The average annual variability of the entire portfolio of stocks on the NYSE is 19.2%. This represents the risk that cannot be reduced through diversification. It is the market or systematic risk.

### Beta

Beta is a measure of systematic risk. It measures the responsiveness of a stock's expected return to changes in the value of the whole stock market.

Let
Rp = return on a portfolio of n assets
Ri = return on asset i = 1, ..., n
Xi = proportion of the portfolio held in asset i

(1) Rp = X1R1 + X2R2 + ... + XnRn

The expected return on this portfolio equals

(2) E(Rp) = E(X1R1) + E(X2R2) + ... + E(XnRn)

A measure of the risk for this portfolio is the standard deviation of the portfolio's return or its squared value, the variance of the portfolio's return, sp2.

(3) sp2 = E[Rp - E(Rp)]2

This simplifies to

(4) sp2 = X1s1p + X2s2p + ... + Xnsnp,

where sip equals the covariance of the return on asset i with the portfolio's return. It is a measure of how the return on asset i changes with changes in the return of the whole portfolio.

Equation (4) tells us that the contribution to risk of asset i to the portfolio is Xisip. So, the proportion of risk contributed by asset i equals

(5) (X1s1p)/sp2

s1p/sp2 tells us about the sensitivity of asset i's return to the portfolio's return. The higher the ratio, the more the value of asset i moves with changes in the value of the portfolio and the more asset i contributes to portfolio risk.

Let the subscript m denote the market portfolio, the portfolio with all possible risky assets.

(6) betai = s1m/sm2

An asset's beta measures the riskiness of a stock relative to the risk of the market as a whole. If beta equals 1, the stock is just as risky as the market as a whole. When the market moves up by 10%, this stock will, on average, move up by 10%. If a stock has a beta less than 1, then when the market moves up by 10%, the stock will move up by less than 10%.

The beta of a stock can be estimated by statistical methods to determine the sensitivity of movements in the stock's return to changes in the return on the stock market as a whole. There are many investment advisory services that can provide estimates of the beta of a stock.

### Equilibrium in the Asset Market

Risk adjusted return equals the expected return minus the cost of risk.

In equilibrium, all assets should have the same risk-adjusted rate of return. If one asset had a higher risk-adjusted rate of return than another, everyone would want to hold the asset with the higher risk-adjusted return. In equilibrium the risk-adjusted rates of return must be equalized.

Let
E(Rm) = expected return on the market portfolio of risky assets
sm = standard deviation of the market return
Rf = rate of return on the risk-free asset
p = price of risk (measures the tradeoff between return and risk)
Bi = beta for asset i

(7) p = [E(Rm) - Rf]/sm

Bi gives the relative amount of risk in asset i. The total amount of risk in asset i is Bism.

cost of risk = total amount of risk x price of risk
cost of risk = Bismp
cost of risk = Bism[E(Rm) - Rf]/sm

(8) cost of risk = Bi(E(Rm) - Rf)

Consider two assets i and j that have expected returns E(Ri) and E(Rj) and betas Bi and Bj. Then in equilibrium

(9) E(Ri) - Bi(E(Rm) - Rf = E(Rj) - Bj(E(Rm) - Rf

This equilibrium condition must also hold between asset i and the risk-free asset with Bf = 0:

(10) E(Ri) - Bi(E(Rm) - Rf = Rf)

Rearranging (10):

(11) E(Ri) = Rf + Bi(E(Rm) - Rf)

### Capital Asset Pricing Model

Equation (11) says that the expected return on any asset is equal to the risk-free return plus the cost of risk. This is the main result of the Capital Asset Pricing Model.

According to the CAPM, the two most important characteristics of an asset are its risk and return. Beta provides a measure of the relative risk of an asset. The different returns on an asset are entirely explained by the differences in their relative risks or betas. The higher the beta, the higher the expected return on that asset.

### Mutual Funds: Active Management or an Index Fund?

The CAPM can be used to compare different investments with respect to the risk and return, for example, mutual funds. Studies have found that funds with high betas generally have high returns to compensate people for bearing greater risk.

A stock index measures the average returns on a given day of a certain group of stocks, e.g. Dow-Jones Index, S&P 500. An index fund is a mutual fund that holds the stocks that make up such an index. The advantage is that you get the average performance of the stocks in the index with low management fees. The beta of an index fund is approximately 1 since it holds a broad base of risky assets. It holds nearly all the stocks in the market as a whole. Plot the expected return and beta of a S&P index fund and draw a line connecting it to Rf. You can get any combination of return and risk along this line by deciding how much to invest in the risk-free asset and the index fund. If you were to plot the betas and returns of actively managed mutual funds on this graph, you would f ind that the vast majority of risk/return combinations offered by mutual funds are below the line.

Here is a short article by Nobel Prize winning economist William Sharpe on active versus passive investing.

### Dow Jones Industrial Average Over Time     David A. Latzko 318 COB Department of Business and Economics Wilkes University Wilkes-Barre, PA 18766 phone: (717) 408-4718 fax: (717) 408-4917 dlatzko@wilkes.edu wilkes1.wilkes.edu/~dlatzko 