Question

A company must pay a ​$308, 000 settlement in 3 years. ​(a) What amount must be deposited now at 8​% compounded semiannually to have enough money for the​ settlement? ​(b) How much interest will be​ earned? ​(c) Suppose the company can deposit only ​$100, 000 now. How much more will be needed in 3 ​years? ​(d) Suppose the company can deposit ​$100 ,000 now in an account that pays interest continuously. What interest rate would they need to accumulate the entire ​$308,000 in 3 ​years?

Answer 1

Amount after t years is given by

A (t) = P (1 + r/n)^nt

In this case,

A = $308000

t = 3 years

a) r = 8% = 0.08

n = 2 [Since compounded semiannually]

Therefore,

308000 = P(1 + 0.08/2)^{2 * 3}

=> 308000 = P(1.04)⁶

=> P = $ 243416.87

Therefore,

Amount that must be deposited now = $ 243416.87

b) Interest earned = $308000 - $243416. 87 = $ 64583.13

c) P = $100,000

Amount after 3 years = 100000(1 + 0.08/2)^{2 *3} = $ 126531.90

Therefore,

Amount that will be needed more in 3 years  = $308000 - $ 126531.90 = $ 181468.10

d) Amount after t years when compounded continuously = Pe^rt

Let the rate of interest be r%

Therefore,

100000 = 308000e^r(3)

=> e^3r = 308000 / 100000 = 3.08

=> ln e^3r = ln 3.08

=> 3r = ln 3.08

=> r = 0.37497

=> r = 37.5%

Therefore,

Rate of interest needed = 37.5%