Problem 22 - A First Course in Probability

Question

Would the results of Example \(5 \mathrm{f}\) change if the investor were allowed to divide her money and invest the fraction \(\alpha, 0<\alpha<1,\) in the risky proposition and invest the remainder in the risk-free venture? Her return for such a split investment would be \(R=\alpha X+(1-\alpha) m\)

Step-by-Step Solution

Answer

In summary, when the investor divides her money between the risky and risk-free ventures with a fraction \(0 < \alpha < 1\), the expected return of the split investment \(E(R) = \alpha E(X) + (1 - \alpha)m\) will be less than the expected return of solely investing in the risky proposition. The standard deviation of her split investment, \(SD(R) = \sqrt{\alpha^2 * Var(X)}\), will also be reduced, indicating a reduction in risk.

Step 1: Calculate the Expected Return

To calculate the expected return, we will determine the expected value of R, which can be written as: E(R) = E(αX + (1 - α)m) Since m is a constant (the return of the risk-free venture), we can separate the expectation: E(R) = αE(X) + (1 - α)m

Step 2: Calculate the Standard Deviation of the Return

Next, we will calculate the standard deviation of R. The variance of R can be expressed as follows: Var(R) = Var(αX + (1 - α)m) Since m is a constant, its variance is 0. Thus, we can write: Var(R) = α^2 * Var(X) Now, to find the standard deviation, we take the square root of the variance: SD(R) = √(α^2 * Var(X))

Step 3: Compare the Results

Now that we have determined the expressions for the expected return and standard deviation of the split investment, we can compare these to the values from the case when the investor only invests in the risky proposition. In that case, the expected return is E(X) and the standard deviation is SD(X). If we compare the expected returns, we have: E(R) = αE(X) + (1 - α)m When α = 1, we get E(R) = E(X), which represents the case where the entire investment is in the risky proposition. For 0 < α < 1, E(R) will be a weighted average of the risky and risk-free returns. Since the risk-free return (m) is generally less than the expected return of the risky proposition (E(X)), E(R) will be less than E(X) for 0 < α < 1. In terms of the standard deviation, we found: SD(R) = √(α^2 * Var(X)) When α = 1, SD(R) = SD(X), representing the full investment in the risky proposition. As α decreases (0 < α < 1), the standard deviation of the split investment will decrease, indicating a reduction in risk. In summary, when the investor is allowed to divide her money between the risky and risk-free ventures, the expected return of her split investment will be less than the expected return of solely investing in the risky proposition, while the standard deviation (risk) will also be reduced.