Find the amount that should be invested now to accumulate the following amount?

Question

if the money is compounded as indicated: $7300 at 10% compounded quarterly for 5 years.

a) $4454.98
b) $11,961.90
c) $2845.02
d) $4532.73

Answer 1

10% compounded quarterly for 5 years is 20 quarters of 2.5% interest per quarter. The value of your investment would increase by a factor of (1.025)^20, which is about 1.6386.

If you want to end up with $7,300, you'd start with 7300/((1.025)^20), which is about $4,554.98. That matches your answer (a).

Answer (b) is what you'd end up with if you started with $7,300. The problem is a bit poorly worded (not your fault, the source's). "$7300 at 10% ..." sounds as if 7300 is the initial investment, however, the unknown is "the amount to be invested now."

I'd go with (a), though. (b) is the only number greater than $7300 and it's probably put in there to catch people who did the problem backwards.

Answer 2

The effective interest rate is found by
ieff = [1+r/m]^m -1 where r is the nominal (annual) interest rate and m is the number of compounding periods/year.
ieff = [1+ 0.10/4]^4 -1 = 0.1038
P = F(1+ ieff)^-n = 7300(1.1038)^-5 = $4455.24
Answer is a)