Pension contributions - Special cash values
Perpetual annuity
A perpetual (infinite) annuity is an infinite payment stream with constant amounts C. The present value of an infinite annuity is calculated as follows PV= C/r . Info: It is assumed that the first annuity payment (annuity payment in arrears) is made at the end of the first year.
An example to understand.
What amount would I have to invest today for the pension so that EUR 1 billion in interest income would be available each year from this at r = 10% (excluding taxes and others)? PV=1billion/0.1=10billion. I would therefore have to invest 10 billion today in order to receive 1 billion every year.
But what if I only had access to the money after four years? PV=1/(1.1)^3 -10bn=7.51. This would mean that you would have to invest 7.51bn today so that you can have 1bn paid out after 4 years.
Now a topic that flares up, but you can answer it with our discovery. For what reasons is it advisable to connect the statutory pension to the capital market (a yield product) as early as possible and preferably? Every year, around 100 billion additional dollars are paid into pensions. So what PV would you have to invest, assuming an r 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 in money if you have no lead time but what happens if we finally start but only use it later. PV=100/(1.08)^4 -1250=918 billion, PV=100/(1.08)^5 -1250=850 billion, PV=100/(1.08)^6 -1250=787 billion. That means with 7 years lead time we would get our current pension gap with only 787 billion annually (The example is only for understanding. It is very simplified and leaves out many real conditions). The investment of 787 billion would therefore be break even after just 11 years.
Finite pension
An annuity is a series of payments that provide equal payments C over a certain period of time. Examples are annuity loans or real estate loans. The annuity can be thought of as the difference between two infinite annuities in order to calculate the present value. The formula looks like this PV=C-[1/r - 1/r-(1+r)^t]. The value in the square bracket is also known as the annuity present value factor.
Our example: What is the present value if you have to pay 5,000 a year for a car for 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 . You would therefore pay an adjusted €20,501.
Another use case for the finite annuity.
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. 315.47 would be the annual payment including interest. The interest payments become smaller and smaller and the amortization increases.
Another use case for the finite annuity.
A real estate loan of 250,000 is to be repaid in full over the next 30 years in an annuity loan at an interest rate of 12% What is the constant (annual) annuity? Remember: Annuity = interest payment + repayment .
Amp=250000/[1\0.12 - 1/0.12-(1.12)^30]=31036 . So you would have to repay €31,036 per year.
Growing annuity
With the growing pension, we ask ourselves how our capital can continue to grow and still continue to pay us. To understand this, let's link our example to our pension gap: if the pension wants to pay out EUR 100 billion in the first year and these payments are then to increase by 4% annually, how much money must it invest today at an interest rate of 10%?
PV=100bn/0.1-0.04= 1,666 billions. So we would have to spend around 1666 billion in order to have a growing pension. But now we want to pay in first and pay out 7 years later. How much would we then have to invest for the pension? PV=100bn/0.1-0.04 / (1.1)^6= 940.78bn. To get 100bn for the pension, which grows by an additional 4% a year, we would only have to invest 940.78bn. The investment would break even after 18 years.
The lesson here is not that you should tie your pension to the stock market, but that a pension with returns would slowly but steadily pay for itself.
No quiz this time, but a question about your opinion. What do you think of the pension and how would you approach it?
Sneak Peak: The most important facts about bonds
Source: Brealey, R., S. Myers, F. Allen, A. Edmans (2022): Principles of Corporate Finance, 14th edition, McGraw Hill, ISBN 1260013901 and lecture slides
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