An individual has $40,000 to invest: $28,000 will be put into a low-risk mutual fund averaging 6.9% interest compounded monthly, and the remainder will be invested in a high-yield bound fund averaging 9.8% interest compounded continuously.
Required:
a. Write an equation for the total amount in the two investments.
b. Write the rate-of-change equation for the combined amount.
c. How rapidly is the combined amount of the investments growing after 6 months? after 15 months?
Answer:
a) F(x) = 28,000( 1.00575 )^12x + 12,000e^0.098x
b) F'(x ) = 28,000 ( In 1.071224 ) ( 1.071224 )^x + 1176 e^0.098x dollar per year
c) 3228.94 dollar/year, 3428.73 dollar/year
Step-by-step explanation:
Capital = $40,000
$28,000 = low-risk mutual fund
6.9% monthly compounded interest for the low risk mutual fund
$12,000 = high-risk yield bound fund
9.8% continuously compounded interest
A) Equation for total amount in two investments
F(x) = F1(x) + F2(x) —– ( 1 )
where :
F1(x) ( future value for monthly compounded interest)
= 28,000( 1 + 0.069/12 )^12x = 28,000 ( 1.00575 )^12x
F2(x) ( future value for continuously compounded interest )
= ( 40,000 – 28,000 )e^0.098x = 12,000 e^0.098x
back to equation 1
F(x) = 28,000( 1.00575 )^12x + 12,000e^0.098x
B Rate of change equation
f'(x) = d/dx (28,000( 1.00575 )^12x) + d/dx ( 12,000e^0.098x )
∴ f'(x) = 28,000 d/dx (1.00575^12)^x + 12,000 d/dx(b^x)
= 28,000 ( In 1.071224 ) ( 1.071224 )^x + 12,000 ( In b ) ( b^x )
f'(x ) = 28,000 ( In 1.071224 ) ( 1.071224 )^x + 1176 e^0.098x dollar per year
C) Determine how rapidly the combined amount will grow after 6 months and after 15 months
i.e. x = 0.5 , x = 1.25 years
after 6 months
28,000 ( In 1.071224 ) ( 1.071224 )^(0.5) + 1176*e^((0.098(0.50))
= 1993.88 + 1235.06 = 3228.94 dollar/year
after 15 months
28,000 ( In 1.071224 ) ( 1.071224 )^(1.25) + 1176*e^((0.098(1.25 ))
= 2099.47 + 1329.26 = 3428.73 dollar/year