How do you compute compounded interest, when it changes half way through?

Question

You deposit $4000 in an account that pays 8% interest compounded semiannually. After 4 years, the interest rate is increased to 8.24% compounded quarterly. What will be the value of the account after 8 years?

Answer 1

If A is the original amout deposited, the account balanece, B, after 1 period is B = A(1 + i), where i is the interest for the period. After two periods, B = A(1 + i)(1 + i) or A(1 + i)^2. And carrying this out to n periods you will obtain B = A(1 + i)^n. This is the formula to use when the interest is compounded at intervals. Since your interst rate and compounding periods change you must apply this formula separately to each 4 year segment.

Answer 2

For the first term, the 8%/12month = 8%/yr  =  4%/6 months.  The number of intervals is 8, since interest is compounded twice/yr.

A = Pe^kt, where P is initial principal, e is euler's number, k is the percent rate/interval, and t is the number of intervals.

A = 4000e^(.04)(8) =  5508.51

For the second interval, 8.24%/year = 2.06%/3months, and t = (4 intervals/year)(4 yr) = 16 intervals

A = 5508.51e^(.0206)(16) = 7659.10