2. In this question we consider an unrealistic scenario, but it will help you better understand...

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2. In this question we consider an unrealistic scenario, but it will help you better understand returns and log-returns. Let R be the daily net return on some risky asset. Suppose that R as a discrete distribution: P(R = 2) 0.52 and P(R = − 1) = 0.48. For simplicity, assume that the risk-free rate is 0 and there are no transaction costs. = (a) What is the expected daily net return? [20] (b) What is the expected daily log return? Assume that log(0) limo log(x) rather than being undefined. [20] 1 = = = 0.5 and P(R (c) To simplify calculations, now assume that P(R = 2) -1) = 0.5. Suppose that we want to invest a fraction w of our current wealth in this asset. This means that at time t 1, .. , N we need to rebalance our position so that the fraction of our wealth invested in this asset is w. Since there are no transaction costs, this is equivalent to selling ".. = all stocks at time t and reinvesting the fraction w in the risky asset. The wealth not invested in the stock earns the risk-free rate of 0. Assume that we start with a wealth of $100. If our strategy is to maximise the expected wealth on day 500, we should set w = 1. But this is a good strategy? If we want to maximize the expected log wealth on day 500, what w should we use? [30] 2. In this question we consider an unrealistic scenario, but it will help you better understand returns and log-returns. Let R be the daily net return on some risky asset. Suppose that R as a discrete distribution: P(R = 2) 0.52 and P(R = − 1) = 0.48. For simplicity, assume that the risk-free rate is 0 and there are no transaction costs. = (a) What is the expected daily net return? [20] (b) What is the expected daily log return? Assume that log(0) limo log(x) rather than being undefined. [20] 1 = = = 0.5 and P(R (c) To simplify calculations, now assume that P(R = 2) -1) = 0.5. Suppose that we want to invest a fraction w of our current wealth in this asset. This means that at time t 1, .. , N we need to rebalance our position so that the fraction of our wealth invested in this asset is w. Since there are no transaction costs, this is equivalent to selling ".. = all stocks at time t and reinvesting the fraction w in the risky asset. The wealth not invested in the stock earns the risk-free rate of 0. Assume that we start with a wealth of $100. If our strategy is to maximise the expected wealth on day 500, we should set w = 1. But this is a good strategy? If we want to maximize the expected log wealth on day 500, what w should we use? [30]

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a To calculate the expected daily net return you can use the formula PR R Given PR 2 052 PR 1 048 05... View the full answer