<p>Firstly, the formula you have used for FV is equivalent to that shown <a href="https://www.accountingtools.com/articles/what-is-the-formula-for-the-future-value-of-an-annuity-due.html" rel="nofollow noreferrer">here</a>. What it calculates, with your numbers, is the value after 7 years of investing 200,000 <em>annually</em> 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).</p>

<p>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 (<a href="https://financeformulas.net/Present_Value_of_Annuity.html" rel="nofollow noreferrer">source</a>):</p>

<p><span class="math-container">$$P\Bigg[\frac{1-(1+i)^{-n}}{i}\Bigg]$$</span></p>

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

CC BY-SA 4.0
2019-10-27T18:50:03.260Z
2019-10-28T10:00:06.457Z
2019-10-28T10:00:06.457Z
Adam Bailey
<p>I am supposed to solve the problem:</p>

<p>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?</p>

<p>What I did was that I used the formula for the future formula of pension:</p>

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

<p>But that is incorrect. The correct solution is 28331. </p>

<p>Can someone tell me, where I made a mistake?</p>

<mathematical-economics>
Regular annual pension
Peter F.
# Regular annual pension

(Asked by: username_0 on 2019-10-27)

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?

**Tags:** mathematical-economics

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## 1 answers

### Answer by username_1 on 2019-10-27. Score: 2

Firstly, the formula you have used for FV is equivalent to that shown [here](https://www.accountingtools.com/articles/what-is-the-formula-for-the-future-value-of-an-annuity-due.html). 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](https://financeformulas.net/Present_Value_of_Annuity.html)):

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

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


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