Reading: Perpetuities

Suppose you win a lottery that guarantees a payment of 100'000 each year forever. The first payment is in one year. Moreover, "forever" means that the lottery payments will be passed on to your heirs, their heirs, etc. so that the annual inflow of 100'000 will never stop. The appropriate discount rate is 5%. What's the present value of this lottery jackpot?

The first reaction could be "infinite!" After all, if the cash flows run forever, the winner, over time, stands to receive an infinite amount of money!

The second reaction could be to argue that it is impossible to determine the present value, as the computation of the present values of a cash flow stream with an infinit number of cash flow never ends…:
 

\( PV = \frac{100'000}{1.05}+\frac{100'000}{1.05^1}+\frac{100'000}{1.05^2}+...+\frac{100'000}{1.05^{100}}+...+\frac{100'000}{1.05^\infty} \)

In fact, not even Excel can handle this calculation, as Excel "only" has 1'048'576 rows so that it would eventually run out of cells for payments in the VERY distant future…!

Both reactions are not really satisfying. And both turn out to be wrong. To understand why, remember what the present value factually means: The present value tells us how much money we have to invest today at the cost of capital to exactly replicate the cash flows of the investment proposal.

So the valuation problem can be rephrased as follows: How much money do we have to invest today at a rate of return of 5% so that we get annual interest payments of 100'000?

Now the answer is very simple: 2 million! If we invest 2 million today, we receive annual interest payments of 100'000 [= 2'000'000 × 0.05]. If we do not reinvest these annual proceeds, the balance of the account remains at 2 million, so that the following year also generates an interest payment of 100'000, etc.

More formally speaking, we just ran the following computation:

\( PV_0 \times R = C = 2'000'000 \times 0.05 = 100'000 \)

When we solve the above expression for PV₀, we obtain the valuation formula for cash flow streams that pay a constant cash flow C the end of each year forever, a so-called ordinary perpetuity: 
 

\(\bf{PV_{0,\text{Perpetuity}} = \frac{C}{R}} \)

(see at the bottom of this page for a mathematical proof of this equation)

Example 1

By law, the annual dividend of the stock of the Swiss National Bank SNB (the Swiss central bank) is limited to CHF 15 (it can't be any higher). Let's assume that SNB is expected to deliver these dividends each year forever and that the next dividend is due in exactly one year. Also, let us assume that investors expect a return of 5% to bear the risk that is associated with SNB's future dividend payments.

Based on this information, what is the theoretical stock price of SNB?

Given the assumptions, we can value the SNB stock as an ordinary perpetuity. Accordingly, the theoretical stock price is CHF 300:

\(PV_0 = \frac{C}{R} = \frac{15}{0.05}=300 \)

As it turns out, at the SNB stock was trading at a price of approximately CHF 5'500 in December 2018. So our initial computation was not very accurate. Let us assume that the model of the perpetuity is appropriate to value the SNB stock. What does the observed market price tell us about the return expectation of SNB shareholders?

To find out, we can solve the above expression for the discount rate R:

\( PV_0 = \frac{C}{R} \longrightarrow R = \frac{C}{PV_0} = \frac{15}{5'500} = 0.0027 = 0.27\% \)

Conditional on the model of an ordinary perpetuity, the observed valuation therefore implies that the SNB shareholders expect to earn an annual rate of return of 0.27%.

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Appendix: Proof of the formula for ordinary perpetuities

For the interested reader, we show below how to derive the simple valuation formula for ordinary perpetuities from the much more complex formula with which we started:

The original formula for the PV was:
 

\( PV_0 = \frac{C}{(1+R)}+\frac{C}{(1+R)^2}+\frac{C}{(1+R)^3}+... \)

When we factor out \( \frac{1}{(1+R)} \) from the second element on the right side of the equation onwards, we can rewrite the equation as:

\( PV_0 = \frac{C}{(1+R)}+\frac{1}{(1+R)}\times \left(\frac{C}{(1+R)}+\frac{C}{(1+R)^2}+\frac{C}{(1+R)^3}+... \right) \)

A closer look at the right side of the equation reveals that the expression inside the large parenthesis is identical to the original equation for PV₀. Therefore, we can write: 

\( PV_0 = \frac{C}{(1+R)}+\frac{1}{(1+R)}\times PV_0 \)

Now all we need to do is rearrange the resulting expression:

\( PV - \frac{PV_0}{(1+R)} = \frac{C}{(1+R)} \)

\( \longrightarrow \frac{PV_0\times (1+R) - PV}{(1+R)} = \frac{C}{(1+R)} \)

\( \longrightarrow PV_0 \times R = C \)

\( \longrightarrow PV_0 = \frac{C}{R} \)