35 Present and Future Value of a Continuous Income Stream

Arkansas Tech University
MATH 2243: Business Calculus
Dr. Marcel B. Finan
35
Present and Future Value of a Continuous
Income Stream
When an income stream ?ows into an investment, the investment grows be-
cause of the continuous ?ows of money and the interest compounded on the
money invested. Thus, two functions are required: a function de?ning the
?ow of money, and a function de?ning a function multiplier.
Discrete Income Stream
Many business deals involve payments in the future. For example, when a
car or a home is bought on credits, payments are made over a period of time.
The future value, F V, of a payment P is the amount to which P would have
grown if deposited today in an interest bearing bank account. The present
value, P V, of a future payment F V, is the amount that would have to be
deposited in a bank account today to produce exactly F V in the account at
the relevant time future.
If interest is compounded n times a year at an annual rate r for t years, then
the relationship between F V and P V is given by the formula
r
F V = P V (1 +
)nt.
n
In the case of continuous compound interest, the formula is given by
F V = P V ert.
Example 35.1
You need $10,000 in your account 3 years from now and the interest rate is
8% per year, compounded continuously. How much should you deposit now?
Solution.
We have F V = $10, 000, r = 0.08, t = 3 and we want to ?nd P V. Solving the
formula F V = P V ert for P V we ?nd P V = F V e?rt. Substituting to obtain,
P V = 10, 000e?0.24 ? $7, 866.28.
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Continuous Income Stream
In the above discussion we introduced the present and future value of a sin-
gle payment. Next, we want to calculate the present and future value of a
continuous stream of payments.
The revenues earned by a huge corporation (such as an electricity company),
for example, come in essentially all the time, and therefore they can be rep-
resented by a continuous income stream.
Since the rate at which revenue is earned may vary from time to time, the
income stream is described by a function S(t) which represents the ?ow rate
in dollars per year. Note that the rate depends on time , t, usually measured
in years from the present.
To ?nd the present value of a continuous income stream over a period of M
years we divide the interval [0, M ] into n equal subintervals each of length
?t = M and with division points 0 = t
n
0 < t1 < · · · < tn = M. That is,
over each time interval we are assuming a single payment is made. Assuming
interest r is compounded continuously, the present value of the total money
deposited is approximated by the following Riemann sum:
n
P V ? S(t1)e?rt1?t + S(t2)e?rt2?t + · · · S(tn)e?rtn?t =
S(ti)e?rti?t.
i=1
Letting ?t ? 0, i.e. n ? ?, we obtain
M
P V =
S(t)e?rtdt.
0
The future value is given by
M
F V = erM
S(t)e?rtdt.
0
Example 35.2
An investor is investing $3.3 million a year in an account returning 9.4%
APR. Assuming a continuous income stream and continuous compounding
of interest, how much will these investments be worth 10 years from now?
Solution.
Using the formula for the future value de?ned above we ?nd
10
F V = e.94
3.3e?0.094tdt ? $54.8 million.
0
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Example 35.3
At what constant, continuous rate must money be deposited into an account
if the account contain $20,000 in 5 years? The account earns 6% interest
compounded continuously.
Solution.
Given F V = $20, 000, M = 5, r = 0.06. Since S is assumed to be constant
then we have
5
20, 000 = S
e?0.06tdt.
0
Solving for S we ?nd
20, 000
S =
? $4, 630 per year.
5 e?0.06tdt
0
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