Chapter 11 Notes (Continued)

The risk and return story thus far… Total return = expected return + unexpected return The unexpected return stems from the possibility of an unanticipated event. Total risk = the variance or, more commonly, the standard deviation of an investment’s return

Risk and Return (Continued) Systematic (market) and unsystematic (idiosyncratic) risks Market risks are unanticipated events that affect almost all assets to some degree because the effects are economy-wide Idiosyncratic risks are unanticipated events that affect single assets or small groups of assets Diversification can eliminate all, or nearly all, unsystematic risks; it cannot eliminate systematic risks, however.

Q10. Returns and Standard Deviations Consider the following information: Your portfolio is invested in 30 percent each in A and C and 40 percent in B. What is the expected return of the portfolio? What is the variance of this portfolio? The standard deviation? State of Economy Probability of State of Economy Rate of Return if State Occurs Stock A Stock B Stock C Boom 0.15 0.35 0.45 0.33 Good 0.50 0.12 0.10 0.17 Poor 0.25 0.01 0.02 -0.05 Bust -0.11 -0.25 -0.09

Q10 Answer Easiest way to do this is to first calculate the portfolio’s return in each of the four states. Return in “Boom”, for example, is: (0.30×0.35)+(0.40×0.45)+(0.30×0.33)=0.384. Then the portfolio’s expected return is: The portfolio’s variance is: Standard deviation is the square root of the variance. In this case, expected portfolio return is 10.41%, variance is 0.0219, and standard deviation is 14.80%. {See Excel spreadsheet}

Systematic Risk Principle Q: If one can costlessly eliminate idiosyncratic risk, does one get compensated for holding assets that have high idiosyncratic risk? A: No. Because unsystematic risk can be freely eliminated by diversification, the systematic risk principle states that the reward for bearing risk depends only on the level of systematic risk.

Beta The level of systematic risk in a particular asset, relative to the average, is given by the beta, β, of that asset. An asset with a beta coefficient of 1.0 has the same systematic risk as an average asset in a particular market. A risk-free asset has a beta of 0. Portfolio Betas: Unlike calculating portfolio risk, calculating betas for a portfolio is straightforward – simply take the weighted average of individual asset betas, using portfolio weights. This is the same method used to calculate portfolio expected return.

Q11 Calculating Portfolio Betas You own a stock portfolio invested 15% in Stock Q, 25% in Stock R, 40% in stock S, and 20% in Stock T. The betas for these four stocks are .85, .91, 1.31, and 1.76, respectively. What is the portfolio beta?

Q11 Answer The portfolio beta is: In this case, the portfolio beta is 1.231.

Reward to Risk Ratio The reward-to-risk ratio for Asset i is the ratio of its risk premium, E(Ri)—Rf, to its beta, βi. In a well-functioning market, this ratio is the same for every asset; if it weren’t, then arbitrage would force convergence across assets. Therefore, when asset expected returns are plotted against asset betas, all assets plot on the same straight line, called the security market line (SML).

Security Market Line From the SML, the expected return on Asset i can be written: E(Ri) = Rf + [E(RM)—Rf]×βi This is the capital asset pricing model (CAPM). The expected return on a risky asset thus has three components. The pure time value of money, which is the risk-free rate Rf. This is what you get if you simply wait for your money. The market risk premium, [E(RM)—Rf]. This is the reward the market offers for bearing an average amount of systematic risk, in addition to waiting. The beta for that asset, βi. This is the amount of systematic risk present in a particular asset, relative to an average asset.

CAPM and SML CAPM works for a portfolio of assets, as well. Just substitute the portfolio beta. The SML illustrates what the CAPM predicts the expected total return should be for various values of beta. The actual expected total return depends on the price of the asset. If an asset’s price implies that the expected return is greater than that predicted by the CAPM, that asset will plot above the SML. This means that the asset’s price is lower than the CAPM suggests it should be.

Q17 Using CAPM A stock has a beta of 1.15 and an expected return of 10.4 percent. A risk-free asset currently earns 3.8 percent. What is the expected return on a portfolio that is equally invested in the two assets? If a portfolio of the two assets has a beta of 0.7, what are the portfolio weights? If a portfolio of the two assets has an expected return of 9 percent, what is its beta? If a portfolio of the two assets has a beta of 2.3, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain.

Q17 Answer What is the expected return on a portfolio that is equally invested in the two assets? E(Rp) = 0.5*10.4% + 0.5*3.8% = 7.1% If a portfolio of the two assets has a beta of 0.7, what are the portfolio weights? βp = wstock*βstock + wrf*βrf  wstock = βp / βstock = 60.87% If a portfolio of the two assets has an expected return of 9 percent, what is its beta? Since (10.4% – 3.8%)beta of 1.15, then (βp/1.15) = (5.2%/6.6%). βp=0.906. If a portfolio of the two assets has a beta of 2.3, what are the portfolio weights? How do you interpret the weights for the two assets in this case? Explain. The stock has a weight of 200%. The investor is using leverage, borrowing at the risk-free rate and investing all of the borrowings into the stock. (Note that the standard margin requirements use precisely this 2:1 leverage.)

Q25. Returns and Standard Deviations Consider the following information on a portfolio of three stocks: If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the expected return of the portfolio? The variance? The standard deviation? If the expected T-bill rate is 3.75 percent, what is the expected risk premium on the portfolio? State of Economy Probability of State of Economy Rate of Return if State Occurs Stock A Stock B Stock C Boom 0.25 0.02 0.32 0.60 Normal 0.10 0.12 0.20 Bust 0.15 0.16 -0.11 -0.35

Q25 Answer If your portfolio is invested 40 percent each in A and B and 20 percent in C, what is the expected return of the portfolio? The variance? The standard deviation? {See Excel spreadsheet} If the expected T-bill rate is 3.75 percent, what is the expected risk premium on the portfolio? Expected risk premium = 13.33% – 3.75% = 9.58%