You deposit 5000 in an account earning 5 compounded semiannu

You deposit $5,000 in an account earning 5% compounded semi-annually for 2 years and 7%, compounded quarterly thereafter, what is the account value after 7 years?

Solution

5% compounded semi-annually = 2.5% per half-year

Future value (FV) after 2 years (4 half-years) = $5,000 x (1.025)⁴ = $5,000 x 1.1038 = $5,519.06

Next, 7% compounded quarterly = 1.75% per quarter

FV of $5,519.06 after another 5 years (20 quarters) = $5,519.06 x (1.0175)²⁰ = $5,519.06 x 1.4148 = $7,808.25

This is the account value after 7 years.

If equivalent annual rate be R, then

$7,808.25 = $5,000 x (1 + R)⁷

(1 + R)⁷ = 7,808.25 / 5,000 = 1.5616

Taking 7th root on each side:

1 + R = 1.0658

R = 1.0658 - 1 = 0.0658, or 6.58%