Regular annual pension

Question

I am supposed to solve the problem:

A 24-year-old man decides to invest 200,000 euros at a 7% annual interest rate to bring him a regular annual pension from 31 to 50 years inclusive. What will be the pension?

What I did was that I used the formula for the future formula of pension:

$$FV=A\frac{\left ( 1+i \right )^{n}-1}{i}\left ( 1+i \right )=200,000\frac{\left ( 1+0,07 \right )^{7}-1}{0,07}\left ( 1+0,07 \right )=1851960.514$$ and then I divided it for twenty years which is 92598.02

But that is incorrect. The correct solution is 28331.

Can someone tell me, where I made a mistake?

Answer 1

Firstly, the formula you have used for FV is equivalent to that shown here. What it calculates, with your numbers, is the value after 7 years of investing 200,000 annually at a compound rate of 7% (as a rough confirmation, 1,851,960 is somewhat more than 7 x 200,000). Instead, you need the value after 7 years (or 6 years if the man is already at the end of his 24th year) of a one-off investment of 200,000 (call that V).

Secondly, the simple division by 20 takes no account of the interest earned during years 31-50. To allow for this, you need the present value of an annuity formula (source):

$$P\Bigg[\frac{1-(1+i)^{-n}}{i}\Bigg]$$

Set this formula equal to V and treat the annual (periodic) payment P as an unknown to be found.