7. Extensions

For the interested reader, this section provides some extensions and additional examples for the valuation of perpetuities. In particular, we discuss how to handle situations where the payment frequency and the compounding frequency are not the same. For example, this could be a perpetuity that makes quarterly payments but the discount rate is expressed on an effective annual basis.

 

Example 10

A stock pays a constant quarterly dividend of EUR 2.00 forever. The first dividend is paid in one quarter. The cost of capital (effective annual rate) is 10%. What's the present value of all future dividend payments?

 

In this example, the payment frequency (quarterly) and the compounding frequency (annual) do not match. In order to value the perpetuity, we therefore have to either convert the quarterly payment into an equivalent annual payment or we have to convert the annual cost of capital into a quarterly cost of capital. Let's look at these two approaches:

 

Annual Cash Flow Equivalent

The first approach is to figure out what annual cash flow is equivalent to a quarterly cash flow of EUR 2.00. The point is that the quarterly dividends can be invested at the time they occur so that they earn interest until the end of the year. A quarterly dividend of EUR 2.00 will therefore be more valuable than an annual dividend of EUR 8 [= 2×4], payable at year end.

 

The first quarter's dividend arrives after three months and can therefore be invested for 9 month (0.75 years) until the end of the year. Similarly, the second and third quarterly dividend arrive after 6 and 9 months, respectively, so that the remaining investment period is 0.5 years and 0.25 years, respectively. Finally, the last quarter's dividend arrives at year end, so there is no reinvestment until the end of the year. The future value of the quarterly dividends, therefore, is:

 

\( FV_1 = C \times (1+R)^{0.75} +C \times (1+R)^{0.50} +C \times (1+R)^{0.25} + C \)

 

\( FV_1 = 2 \times 1.1^{0.75}+2 \times 1.1^{0.50} +2 \times 1.1^{0.25} + 2 = 8.294 \)

 

Put differently, a quarterly dividend of EUR 2, payable at the end of the quarter, is equivalent to an annual dividend of EUR 8.294, payable at year end, assuming an interest rate of 10%.

 

Now that we have the annual equivalent of the dividend, compounding frequency and payout frequency are consistent, so that we can compute the present value of the perpetuity using the standard formula of an ordinary perpetuity:

 

\( PV_0 = \frac{C_{Annual \ equivalent}}{R} = \frac{8.294}{0.1} = 82.94 \)

 

Based on these computations, the present value of all future dividend payments is EUR 82.94.

 

Quarterly Cost of Capital

Alternatively, we can express the annual interest rate of 10% on a quarterly basis. More formally, we are looking for a quarterly interest rate (Rm) that generates an annual return of R when compounded over 4 quarters. In keeping with the section Compounding Frequency and Effective Annual Rate, we use the letter m to denote the compounding frequency.

 

We are looking for Rm such that: 

 

\( (1+R_m)^4 = (1+ R) \)

 

When solving this expression for Rm, we find:

 

\( R_m = (1+R)^{\frac{1}{m}} - 1 \)

 

In our example: 
 

\( R_m = (1+R)^{\frac{1}{m}} - 1 = 1.1^{\frac{1}{4}}-1 = 0.02411 = 2.411\% \)

  

Put differently, an interest rate of 10% with annual compounding (R) is equivalent to a quarterly interest rate of 2.411%. Now we can use this quarterly interest rate to value the quarterly dividend stream:

 

\( PV = \frac{C}{R_m} = \frac{2}{0.02411} = 82.94 \)

 

The result is a present value of EUR 82.94, which is identical to the solution above using equivalent annual cash flows.

   

Example 11

A project pays a cash flow of $1 million every other year, forever. The first cash flow is due today, the second cash flow will be due in 2 years, etc. What's the present value of the cash flow stream if the cost of capital (effective annual rate) is 6%?

 

In this example, the compounding frequency (annual) is shorter than the payment frequency (every other year). To value the perpetuity, we therefore either have to express the cash flow on an annual basis or convert the interest rate into a biennial rate (m = 0.5). For simplicity, we only show the latter approach below:

 

An annual interest rate of 6% corresponds to a biennial rate of 12.36%:

 

\( R_m = (1+R)^{\frac{1}{m}} - 1 = 1.06^{\frac{1}{0.5}}-1 = 0.1236 = 12.36\% \)

 

The perpetuity in question is a perpetuity due, as the first cash flow already arrives today. Consequently, the value of the cash flow stream is $ 8.09 million:

 

\( PV = \frac{C}{R_m} = \frac{1}{0.1236} = 8.09 \)

  

Example 12

The same logic applies to growing perpetuities. Consider, for example, a stock that pays a quarterly dividend of $1 that grows at 1% from each quarter to the next. The first payment arrives in exactly 1 quarter and the cost of capital (effective annual rate) is 12%.

 

Using the logic from above, we can convert the effective annual rate of 12% into a quarterly rate of 2.874%:

 

\( R_m = (1+R)^{\frac{1}{m}} - 1 = 1.12^{\frac{1}{4}}-1 = 0.02874 = 2.874\% \)

 

With this quarterly rate, we can then use the equation of a growing ordinary perpetuity to value the cash flow stream in question:

 

\( PV = \frac{C}{(R_m-g)} = \frac{1}{0.02874-0.01} = 53.4 \)

 

The result is a value of $53.4.