Reading: Perpetuities

The third type of perpetuities we need to consider are so-called growing perpetuities. These are cash flow streams that grow at a constant rate of growth from one payment to the next and run forever.

An example could be a project which generates a certain initial cash flow C. In subsequent years, this cash flow then grows at the long-term expected rate of inflation. Suppose, for example, that the initial cash flow (due in 1 year) is 1 million and that the rate of growth equals 1%. Let us denote the constant rate of growth with the letter g.

The project's future cash flows are:

• \( C_1 = C_1 = 1'000'000 \)
   
• \( C_2 = C_1 \times (1 + g) = 1'000'000 \times 1.01 = 1'010'000 \)
   
• \( C_3 = C_1 \times (1+g)^2 = 1'000'000 \times 1.01^2 = 1'020'100 \)
   
• etc.

The valuation of the project takes the form of:

\( PV_0=\frac{C_1}{(1+R)}+\frac{C_1 \times (1+g)}{(1+R)^2} \) \(+\frac{C_1 \times (1+g)^2}{(1+R)^3} + ... \) 

It can be shown fairly easily that this valuation formula for growing ordinary perpetuities simplifies to:

\( \bf{PV_{0,\text{Growing Ordinary Perpetuity}} = \frac{C_1}{R-g}} \)

For the interested reader, the proof of this equation can be found at the bottom of this page.

In our example, with an initial cash flow (C1) of 1 million and a rate of growth of 1%, let us assume that the cost of capital is 10%. Consequently, the present value of the growing ordinary perpetuity is 1.11 million:

\( PV_{0,\text{Growing Ordinary Perpetuity}} = \frac{C_1}{R-g} = \frac{1'000'000}{0.10 - 0.01} = 11'111'111\)

Example 5

A stock is expected to pay a dividend of $2 in one year. Thereafter, the dividend payments are expected to grow at a rate of growth of 3% per year, forever. The relevant discount rate is 10%. Based on this information, what is the present value of all future expected dividend payments?

Using the formula of a growing ordinary perpetuity, the present value of all future dividend payments is 28.57:

\( PV_{0,\text{Growing Ordinary Perpetuity}} = \frac{C_1}{R-g} = \frac{2}{0.10 - 0.03} = 28.57 \)

Example 6

What if the stock from Example 4 pays the first dividend today, with a subsequent rate of growth of 3%?

As the first dividend payment arrives today (C₀), we are dealing with a growing perpetuity due rather than a growing ordinary perpetuity. The valuation formula for a growing perpetuity due is:

\( \bf{PV_{0, \text{Growing Perpetuity Due}} = C_0 + \frac{C_0 \times (1+g)}{(R-g)} = C_0 \times \frac{(1+R)}{(R-g)} } \)

In our example, with C₀ = 2, R = 0.1 and g = 0.03, the value of all dividends is 31.43:

\( PV_{0, \text{Growing Perpetuity Due}} = C_0 \times \frac{(1+R)}{(R-g)} = 2.00 \times \frac{1.1}{(0.1 - 0.03)} = 31.43 \)

Example 7

Assuming a cost of capital of 10%, which of the following investment proposals has the highest present value?

• Proposal A: A cash payment of 3 million today
• Proposal B: A constant annual payment of 250'000 that starts today
• Proposal C: A constant annual payment of 280'000 that starts in 1 year
• Proposal D: An annual payment of 250'000 that starts in one year and subsequently grows at a constant rate of growth of 2%
• Proposal E: An annual payment of 250'000 that starts today and subsequently grows at a constant rate of growth of 1%.

According to our computations, Proposal D, the growing perpetuity that starts in 1 year, has the highest valuation (3.125 million), followed by Proposal E (Growing Perpetuity Due, 3.056 million), and Proposal A (cash payment, 3 million).

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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 growing perpetuities.

The present value of a growing annuity that pays a first cash flow of C in 1 year, subsequently grows at an annual rate of g forever, and is exposed to a cost of capital of R, can be written as:
 

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

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

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

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: 

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

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

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

\( \longrightarrow PV_0\times\left(1-\frac{(1+g)}{(1+R)}\right) = \frac{C}{(1+R)} \)

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

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