Carlita invested $10,000 for 25 years. For the first 15 years she earned 15% compounded semi-annually. For the last 10 years she earned 9.6% compounded monthly. What was her effective rate of return over the 25 years?

A) 13.32%
B) 10.03%
C) 15.56%
D) 12.30%
E) 14.51%

The effective rate of return over the 25 years can be calculated by finding the total amount at the end of the period and then determining the rate that would result in this final amount if it were applied annually over the 25 years. First, calculate the amount after the first 15 years with a 15% interest rate compounded semi-annually: $$A=P(1+\frac{r}{n}{)}^{nt}$$$$A=10000(1+\frac{0.15}{2}{)}^{2\ast 15}$$$$A=10000(1+0.075{)}^{30}$$$$A=10000(1.075{)}^{30}$$$$A\approx 10000\ast 8.137=81370$$ Next, calculate the amount after the next 10 years with a 9.6% interest rate compounded monthly: $$A=81370(1+\frac{0.096}{12}{)}^{12\ast 10}$$$$A=81370(1+0.008{)}^{120}$$$$A=81370(1.008{)}^{120}$$$$A\approx 81370\ast 2.530=205,939.10$$ Now, find the effective annual rate (EAR) that would grow $10,000 to approximately $205,939.10 over 25 years: $$205939.10=10000(1+EAR{)}^{25}$$$$20.59391=(1+EAR{)}^{25}$$$$(1+EAR)=(20.59391{)}^{\frac{1}{25}}$$$$1+EAR=1.1332$$$$EAR=0.1332$$ Therefore, the effective rate of return over the 25 years is approximately 13.32%.