Optimal Portfolio Allocation: Balancing Risk and Return in Investment”

QUESTION

Suppose an investor has to choose what fraction of her savings, b, she invests into the stock market (risky asset). The remaining fraction 1-b is invested in risk-free Treasury bills. Let U(Rp,𝞼p) = 4 + 2𝞼p 2. Let the expected return of the risk-free asset be Rf = 2, and the expected return of the risky asset be Rm = 4. Finally, let the standard deviation of the risky asset be 𝞼m = 2. a. Find the optimal fraction to invest in the stock market? b. Plot the indifference curves, budget line, and optimal portfolio bundle.

ANSWER

Optimal Portfolio Allocation: Balancing Risk and Return in Investment”

To find the optimal fraction to invest in the stock market (risky asset) and to plot the indifference curves, budget line, and optimal portfolio bundle, we can use the concepts of portfolio theory, specifically the Capital Market Line (CML) and the Investor’s Indifference Curve. These tools help investors make decisions about how to allocate their wealth between risky and risk-free assets to achieve the highest expected utility.

Finding the Optimal Fraction (b)

The investor’s goal is to maximize her utility, represented by the function U(Rp, 𝞼p) = 4 + 2𝞼p^2, where Rp is the expected return and 𝞼p is the standard deviation of her portfolio.

Calculate the utility of investing only in the risk-free asset:
U(Rf, 0) = 4 + 2(0)^2 = 4.

Calculate the utility of investing only in the risky asset

U(Rm, 𝞼m) = 4 + 2(𝞼m^2) = 4 + 2(2^2) = 4 + 8 = 12.

Calculate the slope of the indifference curve

The slope of the indifference curve represents the marginal rate of substitution (MRS), which is the rate at which the investor is willing to trade off expected return for lower risk. The MRS can be calculated as the derivative of the utility function:
MRS = dU/d𝞼p = 4𝞼p.

Set the MRS equal to the ratio of the risk-free asset’s expected return to its standard deviation (Rf/0) to find the optimal 𝞼p

4𝞼p = Rf/0 = 2/0 (since risk-free asset has zero standard deviation).
This simplifies to 4𝞼p = ∞, which means the investor is willing to accept infinite risk to invest in the risk-free asset.

As the investor prefers risk-free assets infinitely over the risky asset, the optimal fraction to invest in the stock market (risky asset) is b = 0. She should put all her savings into risk-free Treasury bills.

Plotting the Indifference Curves, Budget Line, and Optimal Portfolio Bundle:

Indifference Curves: The indifference curves represent different combinations of expected return (Rp) and standard deviation (𝞼p) that yield the same utility. In this case, they are upward-sloping curves because the investor prefers higher expected returns and lower risk.

Budget Line: The budget line represents the feasible combinations of Rp and 𝞼p that the investor can achieve given her wealth and the available risky and risk-free assets. The budget line equation is given by:
Rp = b * Rm + (1 – b) * Rf
𝞼p = b * 𝞼m

Optimal Portfolio Bundle: Since the investor decided to invest all her savings in the risk-free asset, her portfolio will have an expected return equal to the risk-free rate (Rp = Rf = 2) and zero standard deviation (𝞼p = 0). This point lies on the budget line where b = 0.

To create a graphical representation of these concepts, you can plot the indifference curves (upward-sloping), the budget line, and indicate the optimal portfolio bundle (where the budget line intersects with the indifference curve corresponding to maximum utility). This will visually demonstrate that the investor chooses to invest entirely in the risk-free asset due to her risk aversion, resulting in a portfolio with zero risk and the risk-free rate of return.

In summary, the optimal fraction to invest in the stock market for this risk-averse investor is zero (b = 0), meaning she should allocate all her savings to risk-free Treasury bills. The graphical representation of indifference curves, budget line, and optimal portfolio bundle will illustrate this decision.