Question

A private investment club has $3000 earnmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high-risk, medium-risk, and low-risk. Management estimates that high-risk stocks will have a rate of return of 18%/year; medium-risk stocks, 9%/year; and low-risk stocks, 6%/year. The members have decided that the investment in low-risk stocks should be a quarter of the sun if the investments in the stocks of the other two categories. Assuming that all the money available for investments is invested, how much should the club invest in each type of stock if the investment goal is to have a return of $363/year on the total investment?

Answer 1

The club should invest $1450 in high-risk stocks, $950 in medium-risk stocks, and $600 in low-risk stocks to achieve the investment goal of a return of $363 per year.

How to determine how much to invest

Denote the amount invested in high-risk stocks as H, in medium-risk stocks as M, and in low-risk stocks as L.

Given:

Total investment amount: $3000

Rate of return for high-risk stocks: 18%

Rate of return for medium-risk stocks: 9%

Rate of return for low-risk stocks: 6%

Investment goal: Return of $363/year

Based on the given information, set up the following equations:

Equation 1: H + M + L = $3000 (Total investment amount)

Equation 2: 0.18H + 0.09M + 0.06L = $363 (Return on investment goal)

Additionally, we are given that the investment in low-risk stocks should be a quarter of the sum of the investments in the other two categories. This can be represented by the equation:

Equation 3: L = 0.25(H + M)

Now solve the system of equations to find the values of H, M, and L.

Substitute Equation 3 into Equation 1

H + M + 0.25(H + M) = $3000

1.25H + 1.25M = $3000

Divide both sides of the equation by 1.25, we have:

H + M = $2400

Next, substitute this value into Equation 2:

0.18H + 0.09M + 0.06L = $363

0.18H + 0.09M + 0.06(0.25(H + M)) = $363

0.18H + 0.09M + 0.015H + 0.015M = $363

0.195H + 0.105M = $363

Now we have a system of two linear equations:

H + M = $2400

0.195H + 0.105M = $363

Solving this system of equations will give us the values of H and M.

Multiply the first equation by -0.105 and the second equation by 1.25:

-0.105H - 0.105M = -$252

0.244H + 0.13125M = $453.75

Add the two equations together

0.139H = $201.75

Divide both sides by 0.139

H = $1450

Substitute this value back into the first equation, we can solve for M:

$1450 + M = $2400

M = $950

Now, find L by substituting the values of H and M into Equation 3:

L = 0.25(H + M)

L = 0.25($1450 + $950)

L = $600

Therefore, the club should invest $1450 in high-risk stocks, $950 in medium-risk stocks, and $600 in low-risk stocks to achieve the investment goal of a return of $363 per year.