5. Generalization

As already discussed for the case of level perpetuities, it can well be that the first cash flow occurs within the investment period rather than at its beginning or end. The same logic applies to growing perpetuities.

 

Using the notation from before, where n denotes how many investment periods it takes for the first cash flow to arrive, we can adjust the formula for growing perpetuities to allow for any starting point:

 

\( \bf{PV = \frac{C_{n}}{(R-g)} \times (1+R)^{(1-n)}} \)

 

In the case of a perpetuity due, n = 0 so that the above equation becomes:

 

\( PV = \frac{C_0}{(R-g)} \times (1+R)^{1} \)

 

This is the same expression as we have derived on the previous page for growing perpetuities due:

 

Example 8

Consider a stock that makes an annual payment of 12, starting in 3 months (n = 0.25). Thereafter, the payment is expected to grow at an annual rate of 3% forever. The cost of capital is 8%. What's the present value of the cash flow stream?

 

\( PV = \frac{C_{n}}{(R-g)} \times (1+R)^{(1-n)} = \frac{12}{(0.08-0.03)} \times 1.08^{0.75} = 254.26 \)

 

The present value of the cash flow stream is 254.26.

 

Note that the same logic applies for perpetuities that start much later. The following example illustrates this:

  

Example 9

Let's consider a project that will not make any payments over the next 5.75 years because, for example, all funds are required for internal developments. Only 5.75 years from now, the first annual payment will arrive. For the sake of illustration, let us assume that this payment (C5.75) is 1 million and that it will grow at an annual rate of 4% forever. The cost of capital is 12%. What's the present value of the cash flow stream?

 

In this case, Cn = 1, n = 5.75, g = 4%, and R = 12%. We can plug these value into the generalized equation for growing perpetuities:

 

\( PV = \frac{C_{n}}{(R-g)} \times (1+R)^{(1-n)} = \frac{1}{(0.12-0.04)} \times 1.12^{-4.75} = 7.30 \)

 

In this case, the standard formula for a growing perpetuity under-discounts the cash flow stream by 4.75 years (it assumes the first payments arrives in 1 year, whereas in fact it only arrives in 5.75 years). We therefore discount the result of the standard growing perpetuity over an additional 4.75 years, which yields a present value of approximately 7.30 million.