Question

Maria has \(\$ 1\) she can invest in two assets, \(A\) and \(B\). A dollar invested in \(A\) has a \(50-50\) chance of returning 16 or nothing and in \(B\) has a \(50-50\) chance of returning 9 or nothing. Maria's utility over wealth is given by the function \(U(W)=\sqrt[0]{W}\) a. Suppose the assets' returns are independent. (1) Despite the fact that \(A\) has a much higher expected return than \(B\), show that Maria would prefer to invest half of her money in \(B\) rather than investing everything in \(A\) (2) Let \(a\) be the fraction of the dollar she invests in \(A\) What value would Maria choose if she could pick any \(a\) between 0 and \(1 ?\) Hint: Write down her expected utility as a function of \(a\) and then either graph this function and look for the peak or compute this function over the grid of values \(a=0,0.1,0.2,\) etc. b. Now suppose the assets' returns are perfectly negatively correlated: When \(A\) has a positive return, \(B\) returns nothing and vice versa. (1) Show that Maria is better off investing half her money in each asset now than when the assets' returns were independent. (2) If she can choose how much to invest in each, show that she would choose to invest a greater fraction in \(B\) than when assets' returns were independent.

Short answer

Answer: To find the optimal investment allocation for Maria in both scenarios, we need to calculate the expected utility for various allocations of her investments and then compare the expected utilities under both independent and negatively correlated scenarios. The optimal allocation will be the one that maximizes Maria's expected utility in each respective case.

Step-by-step solution

Calculate expected utility for different allocations

To show Maria's preference, let's compare expected utility of investing \(0.5\) in A and \(0.5\) in B to investing \(1\) in A. First, let's find the respective wealth allocation for each of the possible investment outcomes: - Maria invests \(0.5\) in A and \(0.5\) in B: - A returns 16, B returns 0: W = \((0.5\cdot16) + (0.5\cdot0)\) - A returns 0, B returns 9: W = \((0.5\cdot0) + (0.5\cdot9)\) - Maria invests \(1\) in A: - A returns 16: W = \(1\cdot16\) - A returns 0: W = \(1\cdot0\) Now, let's calculate utility for each wealth allocation and find the expected utility for each investment strategy by taking the average. - Investing \(0.5\) in A and \(0.5\) in B, expected utility: - \(U_A = \frac{\sqrt[0]{(0.5\cdot16) + (0.5\cdot0)} + \sqrt[0]{(0.5\cdot0) + (0.5\cdot9)}}{2}\) - Investing \(1\) in A, expected utility: - \(U_B = \frac{\sqrt[0]{1\cdot16} + \sqrt[0]{1\cdot0}}{2}\)

Compare expected utilities

Now we can compare the expected utilities to verify whether Maria would prefer to invest half of her money in B rather than investing everything in A. If \(U_A > U_B\), then Maria would prefer investing half of her money in B. #a.2. Choosing the optimal fraction a#

Write Maria's expected utility as a function of a

Let \(a\) be the fraction of the dollar Maria invests in A. Then \((1-a)\) is the fraction invested in B. Considering all possible outcomes, the expected utility function can be expressed as follows: \(E(a)= \frac{\sqrt[0]{a\cdot16 + (1-a)\cdot0} + \sqrt[0]{a\cdot0 + (1-a)\cdot9}}{2}\)

Determine optimal a

To find the optimal investment fraction \(a\), we can either graph the function and look for the peak or compute the expected utility for a range of values between \(0\) and \(1\). By calculating \(E(a)\) for a range of values for \(a\) between 0 and 1 (in increments of 0.1), we can find the optimal value for a that maximizes Maria's expected utility. #b.1. Negatively correlated returns# Now suppose the returns are perfectly negatively correlated.

Calculate expected utility for different allocations

Let's calculate the expected utility for the negatively correlated scenario: - Investing \(0.5\) in A and \(0.5\) in B, expected utility: - \(U_C = \frac{\sqrt[0]{(0.5\cdot16) + (0.5\cdot0)} + \sqrt[0]{(0.5\cdot0) + (0.5\cdot9)}}{2}\)

Compare expected utilities

Now we can compare \(U_C\) with \(U_A\) to determine if Maria would be better off investing half her money in each asset with negatively correlated returns than when the assets' returns were independent: If \(U_C > U_A\), then Maria would be better off investing half her money in each asset with negatively correlated returns. #b.2. Finding the optimal allocation under negatively correlated returns#

Write Maria's expected utility as a function of a

Let's express the function for negatively correlated returns: \(E'(a)= \frac{\sqrt[0]{a\cdot16 + (1-a)\cdot0} + \sqrt[0]{a\cdot0 + (1-a)\cdot9}}{2}\)

Determine optimal a

Similar to part a.2., we can find the optimal a that maximizes the expected utility under negatively correlated returns by calculating \(E'(a)\) for a range of values between 0 and 1 (in increments of 0.1). By comparing the optimal values of \(a\) in independent and negatively correlated scenarios, we can determine if Maria would choose to invest a greater fraction in \(B\) when assets' returns are negatively correlated.