This is a short note covering pensions as done in LC HL Project Maths. I do not know how pensions are calculated “in the real world”.

## Fixed Number of Payouts

Suppose you want a pension that will pay you €20,000 per year for 25 years after retirement. How much should you have in your pension fund on retirement in order to have this? Suppose further that money can be invested at 3% per annum.

### Method 1

Suppose we need the pension fund to contain €X on retirement. Let $P(t)$ be the amount of money in the pension fund after $t$ years and suppose the pension fund is invested at 3%. Well, in the first year we need: $p(0)=X$,

but then take out €20,000: $p(1)=X-20,000$

We then accrue interest on this (capital + interest = $(1+i)$) — but then withdraw €20,000 at the end of the first year so: $p(2)=X(1+i)-20,000(1+i)-20,000$.

Now this accrues interest but €20,000 is withdrawn: $p(3)=X(1+i)^2-20,000(1+i)^2-20,000(1+i)-20,000$ $\vdots$ $p(25)=X(1+i)^{24}-20,000[(1+i)^{24}+(1+i)^{23}+\cdots+(1+i)+1]$.

Now at this point we will have exhausted the pension fund so that $p(25)=0$: $X(1+i)-20,000[(1+i)^{24}+\cdots+(1+i)+1]=0$, $\Rightarrow X=20,000\frac{(1+i)^{24}+\cdots+(1+i)+1}{(1+i)^{24}}$.

Now $(1+i)^{23}+\cdots+(1+i)+1$ is simply a geometric series with 25terms, a first term of $1$ and a common ratio of $(1+i)$. Hence using the formula for the sum of geometric series from the tables: $(1+i)^{24}+\cdots+1=\frac{1-(1+i)^{24}}{1-(1+i)}=\frac{(1+i)^{25}-1}{i}$.

Now of course in this example, $1+i=0.3$ so using a calculator: $(1+i)^{24}+\cdots+1=36.4593$,

so that $X=358,711$.

### Method 2

This is the method that is done in the textbooks. The above method is equivalent but we need this method to analyse the case of an infinite number of payouts. The reason they didn’t tally with each other last Saturday is because I never took out €20,000 in the first year.

As far as I can see ye are just told that the amount ye will need is: $20,000+\frac{20,000}{(1+i)}+\frac{20,000}{(1+i)^2}+\cdots+\frac{20,000}{(1+i)^{24}}$.

The idea being that, for example, $20,000/(1+i)$ is the present value of $20,000$ in one year’s time. However this doesn’t seem to take into account the fact that the pension fund should accrue interest. However all of these questions will be set up so that the interest and inflation are balanced. Hence, we have that, in the notation above, $p(0)=20,000+\frac{20,000}{1+i}+\cdots+\frac{20,000}{(1+i)^{24}}$.

But now we withdraw that €20,000 (at the front): $p(1)=\frac{20,000}{(1+i)}+\cdots+\frac{20,000}{(1+i)^{24}}$.

However the fund now accrues interest at a rate of $i$ ( $=3\%$), followed by a withdrawel of €20,000: $p(2)=\left(\frac{20,000}{(1+i)}+\cdots+\frac{20,000}{(1+i)^{24}}\right)(1+i)-20,000$ $=\left(20,000+\frac{20,000}{(1+i)}+\cdots+\frac{20,000}{(1+i)^{23}}\right)-20,000=\frac{20,000}{1+i}+\cdots+\frac{20,000}{(1+i)^{23}}$.

Hence we kind of get this telescoping such that after $t$ years the pension fund is worth: $p(t)=\frac{20,000}{1+i}+\cdots+\frac{20,000}{(1+i)^{25-t}}$,

so that in the 25th year we have just €20,000 left for the final payment.

It should be clear that this matches the above method exactly. To calculate what $p(0)$ actually is note: $p(0)=20,000+\frac{20,000}{(1+i)}+\cdots+\frac{20,000}{(1+i)^{24}}$ $=20,000+20,000\left(\frac{1}{1+i}\right)+20,000\left(\frac{1}{1+i}\right)^2+\cdots+20,000\left(\frac{1}{1+i}\right)^{24}$,

that is $p(0)$ is a geometric series with first term $a=20,000$ and common ratio $1/(1+i)$. Now there are $25$ terms and using the sum of a geometric series formula in the tables. Let $r=1/(1+i)$ ( $=1/1.03$ in this example): $p(0)=20,000\frac{1-r^{25}}{1-r}=358,711$,

using a calculator (the same as above).

## Infinite Number of Payouts

Suppose you want a pension that pays you € $A$ indefinitely until you die. How much money do you need in the fund. We suppose that inflation is running at a rate of $i$ and that money can also be invested at this rate. By the formula we need: $X= A+\frac{A}{1+i}+\frac{A}{(1+i)^2}+\cdots+\frac{A}{(1+i)^t}+\cdots$, $=A+A\left(\frac{1}{1+i}\right)+A\left(\frac{1}{1+i}\right)^2+\cdots+A\left(\frac{1}{1+i}\right)^t+\cdots$.

That is an infinite geometric series with first term and common ratio $1/(1+i)$ (note that for an infinite geometric series to converge we require that $|r|<1$ — this is certainly the case here). The formula for the sum of an infinite geometric series is in the tables and is given by: $S_\infty=\frac{a}{1-r}$, $X=\frac{A}{1-\left(\frac{1}{1+i}\right)}$.

Multiplying above and below by $1+i$: $X=\frac{A(1+i)}{1+i-1}=\frac{A(1+i)}{i}$.

So far example if you want €20,000 per year this sums to €686,667.

### Remark

If on the other hand you have a pension pot of $X$ and what to find out what payment $A$ you can get per time period (with interest $i$) you end up with (again) $\displaystyle X=\frac{A(1+i)}{i}$

and this is an equation to solve for $A$ in terms of $X$ and $i$: $\underset{\times i}{\Rightarrow} Xi=A(1+i)$ $\displaystyle \underset{\div(1+ i)}{\Rightarrow} A=\frac{Xi}{1+i}$

### Conclusion

You should use the formula as it makes the case of infinite payments easier but realise that the actual mechanics of the pension fund are more like Method 1 above.