Total Value of an Constant Income Stream for a Fixed Time Interval
Suppose that USH Retail Incorporated receives income at a constant rate of 26000 dollars per year, then the total income received, TV, from time t = 0 to t = 2 years is TV = $26000* ( 2-0) = $52000.
Notice that at this rate USH receives $26000 /year
* 1 year/ 52 weeks = $500 / week.
During the two years USH plans to deposit its income as
it is received in a savings account paying 3% interest per annum
, compounded continuously. Following this plan, the amount of money in
the account at the end of the two years, the future value (FV) of this
income, when time t = 2 can be estimated by looking at how
the weekly receipts accumulate once deposited in the savings account.
FV = 500 for 104 weeks of interest +500 for 103 weeks
of interest + 500 for 102 weeks of interest + ... +500 for 1 week of interest
= 500 *exp((.03)*104/52) +500 *exp((.03)*103/52)+500
*exp((.03)*102/52)+500 *exp((.03)*101/52)+ ... +500 *exp((.03)*1/52)
Step | t = Part of Year | Weekly Amount Received | Contribution to Future Value |
0 | 0 | 500 | 500 *exp((.03)*104/52)) = 530.91 |
1 | 1/52 | 500 | 500 *exp((.03)*103/52)) = 530.61 |
2 | 2/52 | 500 | 500 *exp((.03)*102/52)) = 530.30 |
3 | 3/52 | 500 | 500 *exp((.03)*101/52)) = 530.00 |
.
. . |
.
. . |
.
. . |
.
. . |
103 | 103/52 | 500 | 500 *exp((.03)*1/52)) = 500.28 |
sum | 52000 | $53607.13 |
Future value = FV = | ò | b
a |
Re^{.03(b-t)} dt = R/.03 [e^{.03(b-a)} -1] . |
Future value = FV = | ò | 2
0 |
26000e^{.03(2-t)} dt = 26000/.03 [e^{.03*2} -1] = 53591.67. |
Future value = FV = | ò | b
a |
Re^{r(b-t)} dt = R/r [e^{r(b-a)} -1] . |
If we consider the further generalization that the
revenues rate of receipt varies so that R is a function
of t, or R = R(t), then we have the final general formula
for future value:
Future value = FV = | ò | b
a |
R(t)e^{r(b-t)} dt. |
Suppose Ush Retail is up for sale. And the present rate of receipt of income is expected to continue at $26000 per year for the next two years and the income could be deposited as it is received in an account paying interest 3% per annum, compounded continuously. How much would we have to deposit now in the bank to achieve the same bank balance at the end of two years?
Solution: Let PV represent the amount would we have to deposit now in the bank to achieve the same bank balance at the end of two years. Then PV*e^{.03*2} is the amount that the Present Value would yield after two years in the bank. But this is supposed to equal the future value of the USH receipts, so
PV e^{.03*2 }= FV = | ò | 2
0 |
26000e^{.03(2-t)} dt. |
PV = e ^{-.03*2} | ò | 2
0 |
26000 e^{.03(2-t)} dt. |
PV =26000 | ò | 2
0 |
e ^{-.03t} dt = 26000/.03 [1-e^{-.03*2}]=$50470.73 . |
If the rate of receipt of income from time t = a to t
= b is R dollars per unit of time and the income is deposited as it is
received in an account paying interest r per unit of time, compounded continuously,
then the present value of the income stream at time t = a is
Present value = PV = | ò | b
a |
Re^{r(a-t)} dt. |
Present value = PV = | ò | b
a |
R(t)e^{r(a-t)} dt. |