Future Value of Annuity

Question

Textbook: If you invest \$2000 a year (at 9%) from ages 31 to 65, these funds will grow to $470,249 by age 65.

***the textbook did not say how they got this number, I just assumed it used FVA because it is in the same section

My calculation:

$$FVA =2000 \frac{(1.09)^{35}-1}{0.09}$$

$$FVA = 431,421.5093$$

Not sure if $35$ is the correct amount of years, but I tried $34$ and $36$ and did not get the answer from the book. What am I missing?

Answer 1

You found the future sum of ordinary annuity. The book expects the future sum of annuity due: $$S=1.09\cdot 2000\cdot \frac{(1.09)^{35}-1}{0.09}=470,249.45.$$

Answer 2

Assume the periodic payment of $2,000$ is compounded continuously. Thus the formula is: $FV = \dfrac{P}{k}\left(e^{kT} - 1\right)= \dfrac{2,000}{0.09}\left(e^{0.09\cdot 34}-1\right)=451,723.49$ dollars.