Question

For your risky investment, you choose a stock index fund. You're optimistic about stock returns going forward but you think their volatility is going to be higher than in the past. Specifically, you expect the stock index fund's return to be 15% and you think its standard deviation will be 30%.  The risk-free interest rate is 5%.

(1) How would you split your money between the stock fund and Treasuries if you wanted a return of 12%? (You're willing to settle for a lower return on your portfolio than the stock fund alone would provide in order to keep risk lower.)

(2) How would you split your money if you wanted to limit the risk of your portfolio so that its standard deviation would be 20%?

Answer 1

(1) Stock Fund Return = 15 % and Treasury Return = Risk-Free Interest Rate = 5 % (as Treasuries are considered to be risk-free investments)

Let the weight of the stock fund in the portfolio be Y, thereby making Treasury weight = (1-Y)

Target Portfolio Return = 12 %  

Therefore, Y x 15 + (1-Y) x 5 = 12

15Y + 5 - 5Y = 12

10Y = 12 - 5 = 7

Y = 0.7

Portfolio Weight's:

Treasury = (1-0.7) = 0.3 or 30 % and Stock Index Fund = 0.7 or 70 %

(2) Stock Standard Deviation = 30 %, Treasury Standard Deviation = 0 % (as it is risk-free), Coefficient of Correlation between stock fund and treasury = 0 (as treasury is risk-free and therefore has no relation to the returns of any other asset(s))

Target Pprtfolio Standard Deviation = 20 %

Let the weight of stock fund be K and weight of Treasury (1-K)

K x 30 + (1-K) x 0 = 20

K = (20/30) = 0.6667 or 66.67 % ~ 67 %

Stock Fund Weight = 67 % and Treasury Weight = 33 %