The Capital Asset Pricing Model The Risk Return Relation Formalized

Summary As we discussed, the “market” pays investors for two services they provide: (1) surrendering their capital and so forgoing current consumption and (2) sharing in the total risk of the economy. The first gets you the time value of money. The second gets you a risk premium whose size depends on the share of total risk you take on. From this we wrote E(R) = Rf + θ We refine this to E(R) = Rf + Units × Price

Summary We need a reference for measuring risk and choose the total risk the market has to divide or the “market portfolio” as that reference. Define the market portfolio as having one unit of risk (Var(Rm) = 1 unit of risk). Other assets will be evaluated relative to this definition of one unit of risk. From E(R) = Rf + Units × Price we can see that Price = {E(Rm) – Rf}. (Note: Units = 1 for Rm.) In other words we also identified the price per unit risk (the market risk premium).

Summary The hard part is to show that any asset’s contribution to the total risk of the economy or Var(Rm) is determined not by Var(Ri) but rather by Cov(Ri, Rm). Standardize Cov(Ri, Rm) so that we measure the risk of each asset relative to our definition of one unit and we get beta: βi = Cov(Ri, Rm)/Var(Rm) The number of units of risk for asset i is βi. So E(Ri)=Rf + βi(E(Rm) – Rf)=Rf + Units × Price.

Risk and Return When we are concerned with only one asset its risk and return can be measured, as discussed, using expected return and variance of return. If there is more that one asset (so portfolios can be formed) risk becomes more complex. We will show there are two types of risk for individual assets: Diversifiable/nonsystematic/idiosyncratic risk Nondiversifiable/systematic/market risk Diversifiable risk can be eliminated without cost by combining assets into portfolios. (Big Wow.)

Diversification One of the most important lessons in all of finance concerns the power of diversification. Part of the total risk of any asset can be “diversified away” (its effect on portfolio risk is eliminated) without any loss in expected return (i.e. without cost). This also means that no compensation needs to be provided to investors for exposing their portfolios to this type of risk. Why should the economy pay you to hold risk that you can get rid of for free (or which is not part of the aggregate risk that all agents must some how share). This in turn implies that the risk/return relation is actually a systematic risk/return relation.

Diversification Example Suppose a large green ogre has approached you and demanded that you enter into a bet with him. The terms are that you must wager $10,000 and it must be decided by the flip of a coin, where heads he wins and tails you win. What is your expected payoff and what is your risk?

Example… The expected payoff from such a bet is of course $0 if the coin is fair. We can calculate the standard deviation of this “position” as $10,000, reflecting the wide swings in value across the two outcomes (winning and losing). Can you suggest another approach that stays within the rules?

Example… If instead of wagering the whole $10,000 on one coin flip think about wagering $1 on each of 10,000 coin flips. The expected payoff on this version is still $0 so you haven’t changed the expectation. The standard deviation of the payoff in this version, however, is $100. Why the change? If we bet a penny on each of 1,000,000 coin flips, the risk, measured by the standard deviation of the payoff, is $10. The expected payoff is of course still $0.

Example… The example works so well at reducing risk because coin flips are “independent.” If the coins were somehow perfectly correlated we would be right back in the first situation. Suppose all flips after the first always landed the same way, what good is bothering with 10,000 flips? With one dollar bets on 10,000 flips, for “flip correlations” between zero (independence) and one (perfect correlation) the measure of risk lies between $100 and $10,000. This is one way to see that the way an “asset” contributes to the risk of a large “portfolio” is determined by its correlation or covariance with the other assets in the portfolio.

Covariances and Correlations: The Keys to Understanding Diversification When thinking in terms of probability distributions, the covariance between the returns of two assets (A & B) equals Cov(A,B) = AB = When estimating covariances from historical data, the estimate is given by: Note: An asset’s variance is its covariance with itself.

Correlation Coefficients Covariances are difficult to interpret. Only the sign is really informative. Is a covariance of 20 big or small? The correlation coefficient, , is a normalized version of the covariance given by: Correlation = CORR(A,B) = The correlation will always lie between 1 and -1. A correlation of 1.0 implies ... A correlation of -1.0 implies ... A correlation of 0.0 implies ...

Risk and Return in Portfolios: Example Two Assets, A and B A portfolio, P, comprised of 50% of your total investment invested in asset A and 50% in B. There are five equally probable future outcomes, see below. In this case: VAR(RA) = 191.6, STD(RA) = 13.84, and E(RA) = 16%. VAR(RB) = 106.0, STD(RB) = 10.29, and E(RB) = 12%. COV(RA,RB) = 21 CORR(RA,RB) = 21/(13.84*10.29) = .1475. VAR(RP)=84.9, STD(Rp)=9.21, E(Rp)=14%=½ E(RA) + ½ E(RB) Var(Rp) or STD(RP) is less than that of either component!

What risk return combinations would be possible with different weights? Asset A • ½ and ½ portfolio • Asset B •

What risk return combinations would be possible with a different correlation between A and B? Asset A • Asset B •

Symbols: The Variance of a Two-Asset Portfolio For a portfolio of two assets, A and B, the portfolio variance is: Or, For the two-asset example considered above: Portfolio Variance = .52(191.6) + .52(106.0) + 2(.5)(.5)21 = 84.9 (check for yourself)

For General Portfolios The expected return on a portfolio is the weighted average of the expected returns on each asset. If wi is the proportion of the investment invested in asset i, then Note that this is a ‘linear’ relationship.

For General Portfolios The variance of the portfolio’s return is given by: Not simple and not linear but very powerful.

In A Picture (N = 2) Var(RA) Cov(RA, RB) Cov(RB, RA) Var(RB) Portfolio variance is a weighted sum of these terms.

In A Picture (N = 3) Portfolio variance is a weighted sum of Var(RA) Cov(RA,RB) Cov(RA,RC) Cov(RB,RA) Var(RB) Cov(RB,RC) Cov(RC,RA) Cov(RC,RB) Var(RC) Portfolio variance is a weighted sum of these terms.

In A Picture (N = 10) Portfolio variance is a simple weighted sum of the terms in the squares. The blue are covariances and the white the variance terms.

In A Picture (N = 20) Which squares are becoming more important?

It is important to note that the level of correlation or equivalently the level of systematic risk is a choice you make.

Implications of Diversification Diversification reduces risk. If asset returns were uncorrelated on average, diversification could eliminate all risk. They are actually positively correlated on average. Diversification will reduce risk but will not remove all of the risk. So, There are effectively two kinds of risk Diversifiable/nonsystematic/idiosyncratic risk. Disappears in well diversified portfolios. It disappears without cost, i.e. you need not sacrifice expected return to reduce this type of risk. Nondiversifiable/systematic/market risk. Does not disappear in well diversified portfolios. Must trade expected return for systematic risk. Level of systematic risk in a portfolio is an important choice for an individual.

DIVERSIFICATION ELIMINATES UNIQUE RISK Nonsystematic risk Systematic/Market risk Number of securities 25 50 Portfolio Standard Deviation Note: this level is a choice Diversification is costless!!

Nonsystematic/diversifiable risks Examples Firm discovers a gold mine beneath its property Lawsuits Technological innovations Labor strikes The key is that these events are random and unrelated across firms. For the assets in a portfolio, some surprises are positive, some are negative. On average, across assets, the surprises offset each other if your portfolio is made up of a large number of assets.

Systematic/Nondiversifiable risk We know that the returns on different assets are positively correlated with each other on average. This suggests that economy-wide influences affect all assets. Examples: Business Cycle Inflation Shocks Productivity Shocks Interest Rate Changes Major Technological Change These are economic events that affect all assets. The risk associated with these events is systematic (system wide), and does not disappear in well diversified portfolios.

Measuring Systematic Risk How can we estimate the amount or proportion of an asset's risk that is diversifiable or non-diversifiable? The Beta Coefficient is the slope coefficient in an OLS regression of stock returns on market returns: Beta is a measure of sensitivity: it describes how strongly the stock return moves with the market return. Example: A Stock with  = 2 will on average go up 20% when the market goes up 10%, and vice versa.

Betas and Portfolios The Beta of a portfolio is the weighted average of the component assets’ Betas. Example: You have 30% of your money in Asset X, which has X = 1.4 and 70% of your money in Asset Y, which has Y = 0.8. Your portfolio Beta is: P = .30(1.4) + .70(0.8) = 0.98. Why do we care about this feature of betas? It shows directly that an asset’s beta measures the contribution that asset makes to the systematic risk of a portfolio! Also note that this is a linear relation.

Recap: What is Beta? (1) A measure of the sensitivity of a stock’s return to the returns on the market portfolio. (2) A standardized measure of a stock’s contribution to the risk of a well diversified portfolio. (3) A measure of a stock’s systematic risk (again, per unit risk or relative to the risk of the market portfolio).

The Capital Asset Pricing Model (CAPM) Given that some risk can be diversified, diversification is easy and costless, rational investors diversify, There should be no premium associated with diversifiable risk. The question becomes: What is the equilibrium relation between systematic risk and expected return in the capital markets? The CAPM is the best-known and most-widely used equilibrium model of the risk/return (systematic risk/return) relation.

CAPM Intuition: Recap E[Ri] = RF (risk free rate) + Risk Premium = Appropriate Discount Rate Risk free assets earn the risk-free rate (think of this as a rental rate on capital). If the asset is risky, we need to add a risk premium. The size of the risk premium depends on the amount of systematic risk for the asset (stock, bond, or investment project) and the price per unit risk. Could a risk premium ever be negative?

The CAPM Intuition Formalized Number of units of systematic risk (b) Market Risk Premium or the price per unit risk The expression above is referred to as the “Security Market Line” (SML).

Using the CAPM to Select a Discount Rate Three inputs are required: (i) An estimate of the risk free interest rate. The current yield on short term treasury bills is one proxy. Practitioners tend to favor the current yield on longer-term treasury bonds but this may be a fix for a problem we don’t fully understand. Remember to adjust the market risk premium accordingly. (ii) An estimate of the market risk premium, E(Rm) - Rf. Expectations are not observable. Generally use a historically estimated value. (iii) An estimate of beta. Is the project or a surrogate for it traded in financial markets? If so, gather data and run an OLS regression. If not, you enter a very fuzzy area.

The Market Risk Premium The market is defined as a portfolio of all wealth including real estate, human capital, etc. In practice, a broad based stock index, such as the S&P 500 or the portfolio of all NYSE stocks, is generally used. The Market Risk Premium Is Defined As: Historically, the market risk premium has been about 8.5% - 9% above the return on treasury bills. The market risk premium has been about 6.5% - 7% above the return on treasury bonds.

Problems The current risk free rate is 4% and the expected risk premium on the market portfolio is 7%. An asset has a beta of 1.2. What is the expected return on this asset? Interpret the number 1.2. An asset has a beta of 0.6. What is the expected return on this asset? If we invest ½ of our money in the first asset and ½ of our money in the second, what is our portfolio beta and what is its expected return? Relative to the first asset our portfolio has a smaller expected return, why? Does this mean the first asset is better than the portfolio?

Problem The current risk free rate is 4% and the expected risk premium on the market portfolio is 7%. You work for a software company and have been asked to estimate the appropriate discount rate for a proposed investment project. Your company’s stock has a beta of 1.3. The project is a proposal to begin cigarette production. RJR Reynolds has a beta of 0.22. What is the appropriate discount rate and why?