ECMC49Y Capital Asset Pricing Model Date: Jun 10, 2006 [1] State the CAPM equation. E(Ri) = Rf + (E[Rm]-Rf)βi where βi=Cov(Rm,Ri)/Var(Rm) [2] May 2002 Midterm Question 1 (Posted in 2005 Review Question) The CAPM equation is: E(Ri) = Rf + (E[Rm]-Rf)βi where βi=Cov(Rm,Ri)/Var(Rm) The standard deviation of asset i’s return is given as √Var(Ri). Holding only asset i indeed gives you the risk which can be represented by the standard deviation even though we are in CAPM world. This risk as defined as the standard deviation has two components: the systematic risk component and the non-systematic risk component. If you fully understand the properties of variances as given in the notes for portfolio theory, you will be able to derive the exact expression of this standard deviation from the CAPM equation, which is equal to σi = √[(βi)2σ2

m + σ2ε],

where σ2ε represents the unsystematic risk component of asset i and (βi)2σ2

m represents the systematic risk component. If you don’t know how to get this equation, it is completely fine. This problem is tricky because if you talk about the risk of any risky asset, you are referring to its total risk, which always consists of both its systematic risk and its unsystematic risk. CAPM however says that does not mean that the market rewards anyone for bearing unsystematic risk. Thus, you may quickly respond to this question by referring to the beta as the true measure of risk. This is not true. Risk is referred to as the total risk, whereas risk that gives you higher E(R) is the risk as measured by the asset’s beta. CAPM world does not dis-regard the unsystematic risk component of any risky asset. All it is saying is that those risk are not going to be rewarded no matter how much you bear. [3] 2002 Midterm Question 3, with an additional assumption: Var(Rm) = 10%. In (a), with this additional assumption, I ask what are the E(R) and Standard deviation of the 3 portfolios. You can look at Prof. Hejazi’s solution. You should definitely try to understand the diagram provided in his solution. The following is my re-interpretation of his solution.

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(a) (i) E(R) = 3%, Standard deviation (σ) = 0%, beta = 0. (ii) E(R) = 10%, Standard deviation (σ) = √(10%), beta = 1. (iii) Now you borrow $300 from the capital market at the risk-free interest rate. Invest $400

into the market portfolio. E(Rp) = 400%x(E(Rm)) + (-300%)x(Rf) = 19%. Right now, your portfolio’s beta is 4. Var(Rp) = (400%)2xVar(Rm) = 160%. An alternative way to calculate this is to use the equation of the capital market line in the Portfolio theory diagram. You should try this.

(b) First of all, it is not possible to compare across individuals using their utility or expected utility. Those are abstract concepts. I can say that my utility now is 10. And you say yours is 20 now. Can I really say I am worse off than you? Certainly not. Comparison of utility is not possible. Comparing using ex post realized returns is not rational. There are 2 reasons. 1) Ex post return can be different from ex ante return, while all we are referring to in this question is expected return which is ex ante. 2) There is no point comparing realized returns if risk is not taken into account. Second, different individuals take different portfolios. Given their own preferences, they must have chosen the one that gives him the highest expected returns. This is the first equilibrium condition of CAPM that I referred to in class. Given that they have already maximized their expected utility by choosing the right portfolio for themselves, comparing which portfolio is better is not logical because everyone’s unique risk-preference gives him his own ranking of which portfolio is the best and he should have already chosen the one. [4] Oct 26, 2005 Midterm Question 4. [5] Feb 28, 2006 Midterm Question 4. [6] The Two Fund Separation Theorem predicts everyone’s portfolio consists of the risk-free asset and a risky portfolio. Under what conditions does everyone choose the SAME risky portfolio? Homogeneous belief, asset markets frictionless, perfect information, all assets are publicly tradable and perfectly divisible, no market imperfections, perfect competition in all market. Basically, these are the CAPM assumptions. [7] The following table shows the returns on the market, a risk-free asset and risky asset j in 4 possible states of realization.

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State Probability Asset j’s Return (Rj) Risk-free asset Return Market Return (Rm) A 0.3 0% 6% 5% B 0.1 -30% 6% -15% C 0.4 +20% 6% 15% D 0.2 +50% 6% 20% Calculate (a) Market expected return, (b) Variance of market return, (c) Asset j’s expected return, (d) Covariance of Asset j and the market, (e) Write down the security market line, (f) What is the expected return of asset j as calculated from the security market line equation in (e)? (a) 10%, (b) 1%, (c) 15%, (d) 2.15%, (e) The equation for the Security Market Line is the CAPM equation:

E(Ri) = Rf + (E[Rm]-Rf)βi

Substitutes the numbers: E(Ri) = 6% + (10%-6%)(2.15%/1%) (f) βi = Cov(Rm,Rj)/Var(Rm) = (2.15%/1%) =>14.6% is the expected return as calculated from the security market line equation. Since expected return from (c) is 15%, which is higher than the required rate of return from CAPM, the price of asset j should appreciate so that in equilibrium its expected return falls back to 14.6%. [8] Does the market portfolio in a CAPM world have unsystematic risk? No, the market portfolio does not have any unsystematic risk. It is on the efficient set. And it is efficient only if for the same risk-level, the portfolio gives the highest expected return. Only a well-diversified portfolio, with all unsystematic risk being diversified away, can have no room for improving expected return given a risk level. [9] Does the investment portfolio of any individual investor in a CAPM world have unsystematic risk? No, any individual investor would have already invested in a efficient portfolio which composed of the market portfolio, which is well-diversified, and the risk-free asset, which does not have risk by definition. This is intuitive since if anyone of them is not holding an efficient portfolio, it can always choose to diversified away the remaining unsystematic risk to make himself better off. Thus, in equilibrium, their portfolios should have no unsystematic risk.

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