Pension contributions - Special cash values
Perpetual annuity
A perpetual (infinite) annuity is a payment stream that delivers constant amounts C for an infinite period of time. The present value of a perpetual annuity is calculated as follows:
PV= C/r
It is assumed that the first annuity payment (annuity payment in arrears) is made at the end of the first year.
An example to illustrate this:
What amount would have to be invested today so that, at an interest rate of r = 10%, EUR 1 billion in interest income would be available each year (excluding taxes and other items)?
PV=1billion/0.1=10billion
Therefore, you would have to invest 10 billion today in order to receive 1 billion every year.
But what would happen if you only accessed the money after four years?
PV=1/(1.1)^3 -10billion=7.51
This means that you would have to invest 7.51 billion today in order to be able to withdraw 1 billion in four years' time.
Another topic that is relevant:
Why does it make sense to connect the state pension to the capital market (a yield product) as early as possible? Every year, around 100 billion additional dollars are paid into the pension. What cash value would have to be invested, assuming an interest rate of 8% (assuming MSCI World) and a lead time of 5, 6 or 7 years?
PV=100 billion/0.08=1.25 trillion
It would cost 1.25 trillion if you have no lead time. But what happens if you want to use the money later?
PV=100/(1.08)^4 -1250=918 billion
PV=100/(1.08)^5 -1250=850 bn,
PV=100/(1.08)^6 -1250=787bn
With a lead time of 7 years, only 787 billion would be needed annually to close the current pension gap. (This example is for illustrative purposes only. It is very simplified and ignores many real conditions). This investment of 787 billion would be amortized after 11 years.
Finite annuity
An annuity, also known as an annuity, is a series of equal payments C over a certain period of time. Examples are annuity loans or real estate loans. The present value of a finite annuity can be calculated as the difference between two infinite annuities. The formula is
PV=C-[1/r - 1/r-(1+r)^t]
The expression in the square brackets is also known as the annuity present value factor.
Example:
What is the present value if you have to pay EUR 5,000 a year for a car over the next five years (in arrears) and a discount rate of 7% applies?
PV=5000-[1/0.07 - 1/0.07-(1+0.07)^5]=20,501
So you would pay EUR 20,501, adjusted for time.
Another use case:
A consumer loan of EUR 1,000 is to be repaid in full over four years (in arrears). How high is the annual repayment (annuity) if an interest rate of 10% has been agreed?
PV=Annual loan payment - (year annuity factor)=1000 =>
Annual loan payment=1000/(4 year annuity factor)=1000/ [1/0.1 - 1/0.1-(1.1)^4]=315.47
The annual payment, including interest, is EUR 315.47. The interest payments become smaller and smaller and the amortization increases.
Another use case:
A real estate loan of EUR 250,000 is to be repaid in full over the next 30 years in an annuity loan at an interest rate of 12%. How high is the constant (annual) annuity?
Amp=250000/[1\0.12 - 1/0.12-(1.12)^30]=31036
You would therefore have to repay EUR 31,036 per year.
Growing pension
With a growing pension, the question arises as to how the capital can continue to grow and still enable regular payouts. To illustrate this, let's take the following example: If the pension wants to pay out EUR 100 billion in the first year and these payouts are then to increase by 4% annually, how much money must be invested today at an interest rate of 10%?
PV=100bn/0.1-0.04= 1,666 billions
So you would have to invest EUR 1,666 billion in order to have a growing pension. However, if you only want to pay out in seven years' time, how much would you have to invest?
V=100bn/0.1-0.04 / (1.1)^6= 940.78bn
In order to receive EUR 100 billion for the pension, which grows by an additional 4% per year, you would only have to invest EUR 940.78 billion. This investment would pay for itself after 18 years.
Conclusion:
The key finding is not that you should link your retirement provision exclusively to the stock market. Rather, it shows that a high-yield pension can slowly but steadily pay for itself over time.
In this round, we will not be holding a quiz, but we would like to hear your opinions: What are your views on annuities, and what strategies would you pursue to make them successful?
Source: Brealey, R., S. Myers, F. Allen, A. Edmans (2022): *Principles of Corporate Finance*, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides