Question

A moderately risk-averse investor has 50 percent of her portfolio invested in stocks and 50 percent in risk-free Treasury bills. Show how each of the following events will affect the investor's budget line and the proportion of stocks in her portfolio:

1. The standard deviation of the return on the stock market increases, but the expected return on the stock market remains the same.

2. The expected return on the stock market increases, but the standard deviation of the stock market remains the same.

3. The return on risk-free Treasury bills increases.

Short answer

1. The budget line will become flattered, and the proportion of stocks will fall.

2. The budget line becomes steeper, and the proportion of stocks will rise.

3. The budget line will shift upward and become flattered. The proportion of stocks can either increase or decrease.

Step-by-step solution

Explanation for part (a)

The budget line equation is $R_P=\left[\frac{R_m-R_f}{{\sigma }_m}\right]{\sigma }_p+R_f$; R_p is the expected return on a portfolio, Rm is the expected return from investing in the stock market, Rf is the risk-free return on treasury bills, σm and σp are the stock market and portfolio standard deviations.

With the increase in standard deviation, the slope of the budget line will fall, and it will become flattered. With the given return on the portfolio, as the standard deviation rises, the stocks become riskier. Thus, the proportion of stocks in the portfolio will decrease as the stocks become riskier, and the portfolio's expected return does not change.

Explanation for part (b)

With the increase in expected return, the slope of the budget line becomes steeper. The stocks are more attractive when the expected return increases with no change in risk. Thus, with the increase in expected return, the proportion of stocks in the portfolio will also increase.

Explanation for part (c)

As the risk-free return increases, the budget line will shift upwards, and the slope of the budget line also changes; the budget line becomes flatter. The return on treasury bills increases, making it more attractive; the investors could also hold fewer treasury bills and get the same level of return.Thus, the portion of the change in the portfolio depends on the preference of the investor.