Question

Your uncle calls and asks for your investment advice. Currently, he has $100,000 invested in a risky portfolio. This portfolio has an expected return of 10.5% and a volatility of 8%. Suppose the risk-free rate is 5% and the market portfolio has an expected return of 18.5% and a volatility of 13%. To maximize your uncle’s expected return without increasing his volatility, what investment strategy would you recommend? If your uncle prefers to keep his expected return the same, but minimize the risk, which investment strategy would you recommend?

Answer 1

Given amount invested= $100,000

Portfolio expected return E[RxT] = 10.5%

Volatility of portfolio SD[RxT] = 8%

Risk free rate rf = 5%

Market portfolio return E[RT] = 18.5%

Volatility of market SD[RT]

= 13%

Note: volatility = standard deviation

In both the cases, the best portfolios are the combinations of risk free investment and market portfolio.

Let us take the amount to be invest be x

Now,

Expected return is calculated as

E[RxT] = rf + x*(E[RT] - rf)

Volatility is calculated as

SD[RxT] = x* SD[RT]

In case 1 volatility remains constant

8% = x*13%

x= 8%/13% = 61.5%

So, of total $100,000 amount,61.5% which is $61,500 is invested in market portfolio and remaining $38,500 is invested in risk free investment

Expected return = 5% + 61.5% *(18.5% -5%) = 13.3%

So, for the highest possible level of risk, keeping the volatility constant, the expected return = 13.3%

Now, in case 2

Expected return is constant, risk to be minimum

10.5% = 5% + x *(18.5% -5%)

5.5% = x* 13.5%

x= 5.5%/13.5%

x= 40.7%

So, of total $100,000 amount,40.7% which is $40,700 is invested in market portfolio and remaining $59,300 is invested in risk free investment

Volatility = 40.7%* 13% = 5.29%