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. |
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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).
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1.3.4 Present Value of an Annuity