This one has me a bit confused

Question

An investor has up to $300,000 to invest in two types of investments. Type A pays 13% annually and type B pays 10% annually. To have a well-balanced portfolio, the investor imposes the following conditions. At least one-half of the total portfolio is to be allocated to type A investments and at least one-fourth is to be allocated to type B investments. What is the optimal amount that should be invested in each type of investment?

in type A

in type B

What is the optimal return?

Answer 1

x=amount invested

a=percent of A

b=percent of B 

y=return after one year

 

rules: .75>=a>=.5 ; b=1-a

 

equation: y=1.13(xa)+1.10(x(1-a))

               y=1.13(xa)+1.10(x-xa)

               y=1.13(xa)+1.10x-1.10xa

               y=1.10x+.03xa

 

you'll see that this creates a linear equation with a positive slope.  

 

substitute a dollar value for x.  I'll use 1.

 

              y=1.10(1)+.03(1)(a)

              y=1.10+.03a

 

when 'a' is between .5 and .75 the maximum value for 'y' is 1.123. This occurs when a=.75

 

so to sum up the answer

 

Optimal amount in type A 300,000*.75=225,000

Optimal amount in type B 300,000*.25=75,000

 

Optimal return in one year will be 300,000*1.123=336,900.