First Simple Bank pays 8.6 percent simple interest on its investment accounts. If First Complex Bank pays interest on its accounts compounded annually, what rate should the bank set if it wants to match First Simple Bank over an investment horizon of 8 years? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e. g., 32.16.)

Answer with Explanations:

Given:

Simple bank pays 8.6 percent simple interest.

Compound bank pays x percent interest compounded annually.

Assume an equal amount deposited in an account of each bank.

Find x such that at the end of 8 year the future values are equal.

Solution:

Assume an initial deposit of P in either account.

We calculate the future value after 8 years.

Simple bank:

future value, Fs = P (1+8*0.086) = 1.688P (8.6% simple interest)

Compound bank:

future value, Fc = P (1+x) ^8 (x% compounded annually)

In order that both future values are the same, we equate Fc and Fs

Fc = Fs

P (1+x) ^8 = 1.688P

simplify

(1+x) ^8 = 1.688

Take 8th root on both sides

(1+x) ^ (8 * (1/8)) = 1.688^ (1/8)

simplify

(1+x) ^ (1) = 1.688^ (1/8)

take 8th root on right and simplify

1+x = 1.0676319

x = 1.0676319-1 = 0.0676319

Therefore the Compound Bank would offer an interest of 6.76%, rounded to 2 decimal places.