• Table of Contents • Introduction • 1-Time Value • 2-Risk/Return • 3-Accounting • 4-Securities • 5-Business • 6-Regulatory • Case Studies • Student Papers

## 1.3.5 Present Value of a Perpetuity

Sometimes annuities last forever -- or so we pretend. What is the value of payments that are received indefinitely, like proverbial AT&T dividends? On first reflection, you might think that a perpetual annuity would be infinite. But remember that \$1 paid in 10 years is worth only pennies today (assuming normal inflation and the inherent time value of money) and \$1 paid in 100 years is not worth picking up from the sidewalk.

The present value of a perpetual stream of future payments eventually reaches a limit. And, it turns out that the formula for an infinite series of equal payments, discounted by a constant discount rate, is simplicity itself:

 PV = Pymtn / i PV the present value (or initial principal) Pymtn the payment made at the end of each of an infinite number of periods i the discount rate for each period (assumed equal throughout)

You can convince yourself the formula is correct. Imagine you want to be paid \$2,000 a year forever and the discount rate is 8%. All you'll need is \$25,000 -- that is, \$2,000/.08. Each year, the \$25,000 will produce a \$2,000 return (assuming an interest rate of 8%), and this happy state of affairs will continue so long as the buffalo roam the plains and water gurgles in the brook and the sun rises in the east.

Alhough a perpetuity really exists only as a mathematical model, we can approximate the value of a long-term stream of equal payments by treating it as an indefinite perpetuity. Our computational method can be used even though we might have doubts about payments in the distant future -- given the effect of discounting, they have a miniscule effect on our final calculations.

What is a “Capitalization Rate”?

When converting a stream of income into a present value, valuators will often report on the “capitalization rate” or “capitalization multiplier” employed in their determination. A capitalization rate (or “cap” as it is often referred to) is simply an alternate method of performing the basic calculations we have already learned. The capitalization rate is nothing more than the reciprocal of the discount rate—reciprocal meaning one divided by a percentage. For example, a discount rate of 8% has a reciprocal (capitalization rate) of 12.5 (being 1/0.08). The amounts of the future cash flows are then multiplied by the capitalization rate to arrive at a present value.

The following table gives multipliers for various discount rates. Notice that as the discount rate (risk) increases, the multiplier decreases. As you can see, different discount rates (capitalization multipliers) result in quite different valuations.

 Discount rate Capitalization rate (multiplier) 4% 25 5% 20 6% 16.67 7% 14.29 8% 12.5 10% 10 12% 8.33 15% 6.67 20% 5 25% 4 50% 2

 Example: Basic Application A shopping center is expected to have returns of \$1.2 million next year. You determine that the discount rate (given the growth profile and risk of comparable center) is 15%. What is the center's value? Answer: The center is worth \$8 million (\$1.2 million times 6.67). Notice that if the business were considered less risky, so its discount rate was 8%, the multiplier would be 12.5 and the business would be worth \$15 million. If the business were considered more risky, so its discount rate was 25%, the multiplier would be 4 and the business would be worth \$4.8 million. Using a capitalization multiplier will sometimes be the easisest way to value a business with a steady earnings record.
 Example: Valuing a Closely Held Business Dr. and Mrs. Nehorayoff divorced. He owned a half-interest in a closely held medical practice, which interest earned at least \$50,000 annually, over and above a reasonable salary. In an equitable distribution proceeding, how would the court determine the value of the business by capitalizing earnings? Answer: The court considered, among other things, the earnings record and the risk involved -- each reflecting an assessment of the business -- to "capitalize" the earnings figure. Rev. Rul. 59-60 § 6. In making these judgment, expert testimony is essential, and the court should carefully consider the expert's rationale for adopting a particular earnings stream and choosing the multiplier. Based upon the nature of this enterprise, its history and prospects and "all the facts and circumstances of this case" -- judges fudge, too -- the court looked at actual earnings to impute future earnings and then said the appropriate capitalization factor would be in the range of 3 to 4 (a discount rate of 25% - 33%). From this the court concluded the value of Dr. Nehorayoff's interest in the business using capitalization of net earnings to be \$200,000. For an excellent discussion of this method, see Nehorayoff v. Nehorayoff, 108 Misc. 2d 311, 437 N.Y.S. 2d 584 (1981).
 1.3.4 Present Value of an Annuity 1.3.6 Constantly Growing Perpetuity