Question

We have $1000 to invest, and three possible investments - a stock index fund, a bond index fund, and a money market.. Average returns on the stock index fund, the bond index fund, and the money market fund are 8%, 5%, and 2% respectively.

Total investment in the stock index fund and the bond index fund should not exceed 80% of available funds. Total investments in the bond index fund and the money market fund should be at least 30% of the available funds. And investment in stock index fund should not exceed the sum of investments in the bond index fund and the money market fund.

To maximize our average return, how much should we invest in the stock market fund?

(Hint: decision variables are how much to invest in each of the three funds)

Answer 1

We need to calculate the weight of investment in stock market fund to maximize our average return.

1st condition is the investment in stock index fund should not exceed the sum of investments in the bond index fund and the money market fund. Total weight of investment in all 3 funds is 100%.

So, maximum we can invest in stock index fund is 50% to fulfill the above condition and remaining 50% equally in bond index fund and money market fund.

50% stock index fund = 25% bond index fund + 25% money market fund

now 2nd condition is Total investments in the bond index fund and the money market fund should be at least 30% of the available funds.

Let's check whether we fulfill this condition or not.

Total investments $1,000*30% at least in bond index fund and the money market fund = $300

our investments in bond index fund and the money market fund is 50% which is $500 and higher than minimum amount of $300. So this condition is also fulfilled.

now 3rd condition is Total investment in the stock index fund and the bond index fund should not exceed 80% of available funds.

80% of $1,000 = $800

so, Total investment in the stock index fund and the bond index fund should not exceed $800. our investment in stock index fund and the bond index fund is 50%+25% = 75% or $1,000*75% = $750 which is lower than $800. 3rd condition is also fulfilled.

Maximum return as per 50% investment in stock index fund and 25% each in bond index fund and the money market fund.

average return = weight of stock index fund*return of stock index fund + weight of bond index fund*return of bond index fund + weight of money market fund*return of money market fund

average return = 0.50*8% + 0.25*5% + 0.25*2% = 4% + 1.25% + 0.5% = 5.75%

If we invest less than 50% in stock index fund than average return will be lower than our maximum average return of 5.75%. Stock index fund has highest return among the 3 funds. so to maximize average return it should have maximum permissible investment which is 50%.