3. Generalization

The perpetuities so far made their payments either at the beginning or the end of the investment period. Obviously, it can well be that a perpetuity starts during the year (or investment period). For example, an investor could be interested to know at the beginning of the year the value of a stock that pays a constant annual dividend at the end of March (that is, 3 months or 1 quarter after the start of the "investment period").

 

How to handle such perpetuities?

  • If we use the formula for ordinary perpetuities, we assume that the first cash flow occurs in only one year. Hence, we would over-discount the cash flow stream by 0.75 years.
     
  • In contrast, if we use the formula for a perpetuity due, we assume that the first cash flow occurs today. Consequently, we would under-discount the cash flow stream by 0.25 years.

 

The solution is simple: We can use the formula for the ordinary perpetuity and then correct the resulting valuation by compounding it over 0.75 years at the cost of capital.

 

More specifically, let us use the letter n to denote how far out the first payment of the perpetuity is, expressed in investment periods: 

  • Ordinary perpetuity: The first cash flow occurs at the end of an investment period, hence n = 1
     
  • Perpetuity due: The first cash flow occurs at the beginning of the investment period, hence n = 0.
     
  • Example above: The first cash flow occurs in 3 months, hence n = 0.25.

  

Following the argument above, the standard formula for an ordinary perpetuity over-discounts the cash flow stream by (1-n) periods. In our example, that was 1 - 0.25 = 0.75 years. We can correct this by compounding the resulting cash flow over (1-n) periods at the cost of capital. The result is:

 

\( \bf{PV_0 = \frac{C}{R}\times (1+R)^{(1-n)}} \)

 

To complete our example, let us assume a perpetuity with an annual cash flow of 80 and a cost of capital of 10%. If the first cash flow occurs in 3 months, the resulting present value is 819.29:

 

\( PV_0 = \frac{80}{0.1}\times 1.1^{0.75} = 859.28 \)

 

Note that in the case of a perpetuity due, n equals 0 so that the equation above is:

 

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

 

This is the standard formula for a perpetuity due that we have derived on the previous page.

 

Example 4

It's the 1st of January. You are considering an investment that makes a constant annual payment of 200'000 at the end of September. Consequently, the first payment is 9 months or 0.75 years from now. Assuming a cost of capital of 6%, what's the present value of this cash flow stream?

 

Using the equation above, we can write (C = 200'000, R = 0.06, n = 0.75):

 

\( PV_0 = \frac{C}{R}\times (1+R)^{(1-n)} = \frac{200'000}{0.06} \times 1.06^{0.25} = 3'382'246\)

 

Consequently, the value of the cash flow stream is approximately 3.4 million.