Retirement at 120?

If people could live forever, then would they still be able to retire or would pension schemes become bankrupt from having to pay out never ending streams of money to eternally youthful pensioners?
 
This might sound like an overly academic question, however I've noticed a glut of articles recently about the prospects of substantially extending the human lifespan, for example:
 
http://gizmodo.com/peter-thiel-is-right-about-one-thing-1785104345
 
And a feature in this week's economist:
 
http://www.economist.com/news/leaders/21704791-science-getting-grips-ways-slow-ageing-rejoice-long-side-effects-can-be
 
Many of these articles raise the point that extending the human lifespan would have an effect on the affordability of retirement. Claiming that retirement as we know it would come to an end.
 
Is this correct though? Most of the articles that I found simply state that retirement would now be impossible without explaining precisely why. I thought I'd do some calculations and see how much of an effect increased lifespans would have on pension scheme funding, It turns out that the situation is not as bad as it seems.
 
First we need to introduce an actuarial technique that allows us to calculate the cost of paying someone a pension into the future.

Perpetuity

​Suppose that we have a payment stream of amount $£1$, payable annually, and paid forever. Graphically we are talking about the following situation::

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​Can we place a value on this series of payments?
 
You might suppose that since we are paying out an infinite amount of money that the value of the perpetuity (which is the name of a payment stream that continues indefinitely) should be infinite.
 
However finance uses a concept called ‘net present value’ or NPV, to assign a value to this stream, and this value turns out to be finite.
 
Time value of money

First we need some extra information, let's assume that we have access to a risk-free bank account that pays out an interest rate of $i$% per annum. (For example, $i$ might be $5$). So that if we invest $1$ at time $t=0$, it will be worth $1*(1+i)$ at time $t=1$, and worth $1*(1+i)*(1+i)$ at time $t=2$.
 
Let's first solve a simplified problem and just consider the value of the first payment of $1$ at time $1$. If we wish to invest an amount of money now, so that we can pay the $1$ due at time $t = 1$ with the money we have invested, then if we put $1*(1+i)^{-1}$ in the bank account now, this will be worth $1$ at time $1$. Similarly to invest an amount now so that we will be able to pay the amount $1$ at time $n$, we need to invest ${1/(1+i)}^{-n}.$
 
Going back to the original problem, we can use this result to calculate the amount we should invest now so that we can pay all the future payments. It will be:
 
$\sum_{k=1}^{\infty} {1/(1+i)}^{-k} $ 

This is just a geometric series, which sums to $1/ (1- (1+i)) = 1/i.$
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Therefore the amount we need to invest is $1/i$. So, if $i=5%$ as we had earlier, then we  actually only need to invest $20$ now in order to pay able to pay someone $1$ every year, for ever!
 
Checking the Result
 
Let's check that this makes sense. Suppose that we do invest this amount at $t=0$, then at $t=1$ we will have $(20)(1.05) = 1 + 20$.
 
Which gives us $1$ to pay the perpetuity, and leaves enough to invest the original amount again. We can see that if nothing else changes, this system can continue indefinitely.
 
Returning to our previous question, would it be possible to pay a pension to someone who will live forever? The answer is yes. We can even calculate the amount that we would need to invest now to pay the pension.
 
Increasing pensions
 
This system we have derived is not very realistic though. Most pensions increase over time, for example the state pension in the UK increases by the minimum of CPI, $2.5%$ and the average annual wage growth. Given that a pension that increases over time will not just pay out an infinite amount, but will also grow to be infinitely big. Would such a pension still work in a scenario where pensioners live forever?
 
It turns out in fact that yes, such a system is still sustainable under certain conditions.
 
Let’s suppose that our perpetuity increases at a rate of $g%$ per year, so here we might assume that $g=2.5$. Then if we derive the amount that we will need to invest now (at an investment rate of $i%$) we find that we will need $((1+g)/(1+i))^n$. Summing all of these values gives the following initial value:
 
$\sum_{k=1}^{\infty} {(1+g)/(1+i)}^{-k} $this is another geometric sum, however now we need to consider convergence, this sum will converge iff $((1+g)*(1+i)) <1$. That is, the sum will converge iff we can find an investment that grows faster than the perpetuity we have promised to pay.
 
If the sum converges then it will converge to $(1/(i-g))$.
 
So using the example we had earlier, where $i=5%$, $g=2.5%$ we would need to invest $ 1/0.025 = 40$. Which is substantially more than the non-increasing annuity but still finite.
 
So we see that even when the pension is increasing, we can still afford to set it up.

Mortality Premium
 
Can’t we still make the argument that pensions will become unaffordable due to the fact that these perpetuities will still cost a lot more.
 
Let's compare the value of an annuity assuming the pensioner will live to the average life expectancy with one where the pensioner is assumed to live forever.

Looking in my orange tables (the name for the Formula and Tables for Actuarial Exams) I see that the cost of a non-increasing pension of $1$ per year, paid to a $65$ year old male at a discount rate of $4%$ is $12.66$. Compare this to the value of $25$ to pay the pension forever, we see that the cost of the pension is roughly double. This amount is much higher, but it's interesting to note that the increase in cost is similar as that between increasing and non-increasing pensions!

Can we actually live forever?

We should also consider the fact that mortality will never actually be $0$ even if ageing is eliminated. Presumably accidents, and possibly even illness unrelated to age would still exist and for this reason we would not expect to have to pay these pensions forever. If you live long enough, then the probability of all non-impossible events occurring at some point should eventually reach $1$. This means that eventually even the most unlikely accidents would eventually happen if you lived long enough. 
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