to pay off many smaller issues that had been floated in prior years to pay for the wars.
Since the purpose of the bonds was to consolidate past debts, the bonds were called
consols.Suppose each consol promised to pay $100 per year in perpetuity. (Actually,
interest was stated in pounds.) What would each bond be worth if the opportunity cost
rate, or discount rate, was 5 percent? The answer is $2,000:
Suppose the interest rate rose to 10 percent; what would happen to the consol’s value?
The value would drop to $1,000:
Thus, we see that the value of a perpetuity changes dramatically when interest rates
change.
What happens to the value of a perpetuity when interest rates increase?
What happens when interest rates decrease?
Uneven Cash Flow Streams
The definition of an annuity includes the words constant payment—in other words,
annuities involve payments that are the same in every period. Although many financial
decisions do involve constant payments, other important decisions involve uneven, or
nonconstant, cash flows. For example, common stocks typically pay an increasing
stream of dividends over time, and fixed asset investments such as new equipment nor-
mally do not generate constant cash flows. Consequently, it is necessary to extend our
time value discussion to include uneven cash flow streams.
Throughout the book, we will follow convention and reserve the term payment
(PMT)for annuity situations where the cash flows are equal amounts, and we will use
the term cash flow (CF)to denote uneven cash flows. Financial calculators are set up
to follow this convention, so if you are dealing with uneven cash flows, you will need
to use the “cash flow register.”
Present Value of an Uneven Cash Flow Stream
The PV of an uneven cash flow stream is found as the sum of the PVs of the individ-
ual cash flows of the stream. For example, suppose we must find the PV of the follow-
ing cash flow stream, discounted at 6 percent:
01 2 3 4 5 6 76%
PV ? 100 200 200 200 200 0 1,000
The PV will be found by applying this general present value equation:
(2-8)
a
n
t 1
CFta
1
1 i
b
t
a
n
t 1
CFt(PVIFi,t).
PVCF 1 a
1
1 i
b
1
CF 2 a
1
1 i
b
2
CFna
1
1 i
b
n
PV (Perpetuity)
$100
0.10
$1,000 at i10%.
PV(Perpetuity)
$100
0.05
$2,000 if i 5%.
Uneven Cash Flow Streams 79
Time Value of Money 77
