Introduction to Corporate Finance

PART 1: INTRODUCTION

3-6d FINDING THE PRESENT VALUE OF A PERPETUITY

A perpetuity is an annuity with an infinite life; it promises to pay the same amount at the end of every year

forever. One of the first, and certainly the most famous, perpetuities in modern history was the massive

‘consol’ bond issue sold by the British government after the Napoleonic Wars ended in 1815. This bond

issue got its name because it consolidated all the existing British war debts into a single issue that paid

a constant annual amount of interest into perpetuity. The issue itself never matured, meaning that the

principal was never to be repaid. These bonds are, of course, still traded.

Currently, not many corporations or governments issue perpetual bonds.^8 Perhaps the simplest

modern example of a perpetuity is preferred equity issued by corporations. Preferred shares promise

investors a constant annual (or quarterly) dividend payment forever. Therefore, we simply express the

lifetime (n) of this security as infinity (∞), and modify our basic valuation formulation for an annuity

accordingly. For example, we wish to determine the present value of an annuity (PV) that pays a constant

annual dividend amount (PMT) for a perpetual number of years (n = ∞) discounted at a rate r. Here, the

Greek summation notation is helpful in expressing the formula in Equation 3.9:

Eq. 3.9 PVPMT

t rt

=×

1

=1(1 +)

∞

∑

Fortunately, Equation 3.9 also comes in a simplified version, which says that the present value of

a perpetuity equals the annual, end-of-year payment divided by the discount rate. Equation 3.10 gives

this straightforward expression for the present value of a perpetuity (PV):

Eq. 3.10 (^) PVPMT
r
PMT
r
==×
1
It is important to make a subtle point here. Equations 3.9 and 3.10 calculate the present value of
a perpetuity that makes its first payment one year from today. We may need to make an adjustment to
these equations if we want to know the present value of a perpetuity that begins sooner or later than one
year from now.
In September 2008, following a series of tumultuous
events that included the bankruptcy of Lehman
Brothers and the bail-out of insurance giant AIG,
Warren Buffett expressed his faith in the US markets
by purchasing perpetual preferred shares from
Goldman Sachs. These shares had no maturity and
promised to pay $500 million annually in dividends.
Assuming that Buffett wanted a 10% annual return
on his investment, the purchase price would be:
PV = $500,000,000 ÷ 0.10 = $5 billion
A $5-billion purchase price makes sense, because
each year, Buffett would receive $500 million in
dividends, exactly the 10% return that he sought
example
Even though preferred equity usually offers the promise of paying dividends every year forever,
sometimes companies are unable to make those dividend payments. When a company has to suspend
8 Some long-term bonds are nearly perpetuities. In July 1993, the Walt Disney Company sold $300 million of bonds that matured in the year
2093, 100 years after they were issued. The market dubbed these ‘Sleeping Beauty bonds’ because their maturity matched the amount of time
that Sleeping Beauty slept before being kissed by Prince Charming in the classic story.
perpetuity
A level cash flow stream that
continues forever