1. (Relationship between Sharpe ratio and optimal capital allocation) In Momentum Crashes, Daniel and Moskowitz base their "dynamic strategy on a relationship they derive for the Sharpe ratio and the optimal allocation to a risky asset. Their derivation based on a one-period application of Markowitz's mean-variance framework. In this problem, you are to rederive the same relationship between the Sharpe ratio and the optimal allocation to a risky asset in a multi-period investment setting. Consider a risky investment, with known return distribution: in any one investment period, the return has known mean of u and volatility o; and returns over successive periods are independent of each other. You can make an investment in this risky asset or keep your capital in cash for T successive investment periods, where T is fixed and known. At the very beginning of your T successive investments, you get to decide the fraction (allocation), w, of your capital that you allocate to the risky investment; you choose this allocation only once (before you start making investments) and you do not change it in-between your successive investments (you apply the same allocation in each investment period). You want to choose the allocation w so that you maximize your final capital after your T successive investments. Define the growth rate, g(w), of your capital as the rate defined implicitly by your initial capital and your final capital: final capital = initial capital * e9(W)T The growth rate, g(w), depends on your allocation, w, to the risky investment (the larger/smaller w is, the larger/smaller your bet on the risky investment is). To maximize your capital after your T successive periods, you need to choose your w so as to maximize g(w). (a) Assuming cash earns nothing (its return is 0), approximate g(w) by an expectation of a (suitably chosen) random variable. (b) Use an approximation Daniel and Moskowitz use in their derivation to derive an approximation for g(w) in terms of u and o (and w). (Hint: You may want to use the second order Taylor expansion for In(1 + x) around x = 0.] (c) Use your g(w) approximation from (b) to derive an optimal allocation w* that maximizes g(w). (Hint: Your optimal allocation should agree with the one Daniel and Moskowitz obtain.) (d) Use the optimal allocation w* from (c) to derive an expression for the optimal growth rate, g* = g(w*), of your capital. Can you recognize this expression as a familiar metric? (e) Repeat steps (a)-(d) with the assumption that cash earns a rate r per investment period. (f) Where did you use the independence of returns (over successive periods) in your derivation? Consider now the same setting but suppose that, instead of a single risky investment, you have at your disposal a number of risky assets with known return distributions. Now, you get to decide the allocation, w, of your capital to all the risky investments (together) as well as the allocation of your risky portfolio (how much of the capital you are putting at risk you allocate to each of the risky investments). As before, you choose these allocations once (before you begin investing) and you do not change them in-between successive investment periods. (g) Use your result from (d) (or (e)) above to argue that your allocation decisions reduce to maximizing the Sharpe ratio of your risky portfolio. [Note: For this part, your argument does not depend on the rate cash earns.) 1. (Relationship between Sharpe ratio and optimal capital allocation) In Momentum Crashes, Daniel and Moskowitz base their "dynamic strategy on a relationship they derive for the Sharpe ratio and the optimal allocation to a risky asset. Their derivation based on a one-period application of Markowitz's mean-variance framework. In this problem, you are to rederive the same relationship between the Sharpe ratio and the optimal allocation to a risky asset in a multi-period investment setting. Consider a risky investment, with known return distribution: in any one investment period, the return has known mean of u and volatility o; and returns over successive periods are independent of each other. You can make an investment in this risky asset or keep your capital in cash for T successive investment periods, where T is fixed and known. At the very beginning of your T successive investments, you get to decide the fraction (allocation), w, of your capital that you allocate to the risky investment; you choose this allocation only once (before you start making investments) and you do not change it in-between your successive investments (you apply the same allocation in each investment period). You want to choose the allocation w so that you maximize your final capital after your T successive investments. Define the growth rate, g(w), of your capital as the rate defined implicitly by your initial capital and your final capital: final capital = initial capital * e9(W)T The growth rate, g(w), depends on your allocation, w, to the risky investment (the larger/smaller w is, the larger/smaller your bet on the risky investment is). To maximize your capital after your T successive periods, you need to choose your w so as to maximize g(w). (a) Assuming cash earns nothing (its return is 0), approximate g(w) by an expectation of a (suitably chosen) random variable. (b) Use an approximation Daniel and Moskowitz use in their derivation to derive an approximation for g(w) in terms of u and o (and w). (Hint: You may want to use the second order Taylor expansion for In(1 + x) around x = 0.] (c) Use your g(w) approximation from (b) to derive an optimal allocation w* that maximizes g(w). (Hint: Your optimal allocation should agree with the one Daniel and Moskowitz obtain.) (d) Use the optimal allocation w* from (c) to derive an expression for the optimal growth rate, g* = g(w*), of your capital. Can you recognize this expression as a familiar metric? (e) Repeat steps (a)-(d) with the assumption that cash earns a rate r per investment period. (f) Where did you use the independence of returns (over successive periods) in your derivation? Consider now the same setting but suppose that, instead of a single risky investment, you have at your disposal a number of risky assets with known return distributions. Now, you get to decide the allocation, w, of your capital to all the risky investments (together) as well as the allocation of your risky portfolio (how much of the capital you are putting at risk you allocate to each of the risky investments). As before, you choose these allocations once (before you begin investing) and you do not change them in-between successive investment periods. (g) Use your result from (d) (or (e)) above to argue that your allocation decisions reduce to maximizing the Sharpe ratio of your risky portfolio. [Note: For this part, your argument does not depend on the rate cash earns.)