Question

Jeff is willing to invest $6,000 in buying shares and bonds of a company to gain maximum returns. From his past experience, he estimates the relationship between returns and investments made in this company to be:

R = –2S² – 9B² – 4SB + 20S + 30B.        

where,
R = total returns in thousands of dollars
S = thousands of dollars spent on Shares
B = thousands of dollars spent on Bonds
Jeff would like to develop a strategy that will lead to maximum return subject to the restriction provided on the amount available for investment.

a. What is the value of return if $4,000 is invested in shares and $2,000 is invested bonds of the company?
b. Formulate an optimization problem that can be solved to maximize the returns subject to investing no more than $6,000 on both shares and bonds.
c. Determine the optimal amount to invest in shares and bonds of the company. How much return will Jeff gain? Round all your answers to two decimal places.

Answer 1

Solution:

a. The returns is given by R = -2S^2 - 9B^2 - 4SB +20S + 30 B , so if S = $4000 and B = $ 2000, then the return will be = -2 * 4000 * 4000 - 9 * 2000 * 2000 - 4 * 4000 *2000 + 20 * 4000 + 30 * 2000

=   -99860000

b. The problem cab be solved through excel or through basic calculus , using excel we can solve it as shown below:-

Shares	0.00
Bonds	0.00
Total Invested	0.00
Max Investment Allowed	6000.00
Returns	0.00

Formula Sheet for the excel :-

Shares	0
Bonds	0
Total Invested	=SUM(B1:B2)
Max Investment Allowed	6000
Returns	= -2 * (B1^2) - 9 * (B2^2) - ( 4*B1*B2) + 20*B1 + 30*B2

Solving the problem on solver with the constraints as shown below:

Final Solution :-

Shares	$4.29
Bonds	$0.71
Total Invested	$5.00
Max Investment Allowed	$6,000.00
Returns	$53.57

C. So, in order to gain a profit Jeff must invest $ 4.29 in Shares and $0.71 in Bonds to make a net profit of $ 53.57.