Continuous Income Streams
(draft based on Waner Costenoble 2nd Ed.)

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, T,  from time t = 0 to t = 2 years is T =  \$26000* ( 2-0) =  \$52000.

Notice that at this annual rate USH  receives \$26000 /year * 1 year/ 52 weeks =  \$500 / week.

Future Value of a Continuous Income Stream

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 103 weeks of interest +500 for 102 weeks of interest + 500 for 101 weeks of interest + ... +500 for 1 week of interest +500 for the last week ( no interest!)
= 500 *exp((.03)*103/52) +500 *exp((.03)*102/52)+500 *exp((.03)*101/52)+ ... +500 *exp((.03)*1/52 + 500)

 Step t = Part of Year Weekly Amount  Received Contribution to Future Value 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)) = 500.28 104 104/52 500 500 sum 52000 \$53576.22

Generalizations : Suppose  the annual receipts are \$R and that we consider the period between and t = b. Instead of  doing the estimation weekly, divide the time period  into n equal  time periods of length dt =  (b-a)/n so that the payment for that period is R* dt and for the kth time period [tk, tk+1] the final contribution to the future value will be  R*dt *  exp( (.03) (b - tk+1)).
Thus  FV = R dt*exp(.03*(b -t1)) + R dt*exp(.03*(b -t2)) + Rdt *exp(.03*(b -t3))+ ... + Rdt*exp(.03*(b -tn-1) + Rdt).
Now consider what happens to the estimate with n chosen very large. We can use the right hand endpoint estimates for a definite integral. The sums for estimating the FV will also approximate a definite integral; thus we have that
 Future value = FV = ∫ b a Re.03(b-t) dt = R/.03 [e.03(b-a) -1] .
Using R = \$26000, a = 0 and b= 2 we have
 Future value = FV = ∫ 2 0 26000e.03(2-t) dt = 26000/.03 [e.03*2 -1] = 53591.67.

Note: This Future Value is larger than the value computed in the table. This can be explained by the fact that in the continuous model for the computation, the deposits are decreasing continuously rather than discretely and therefore the table will give an underestimate for the continuous, or conversely the continuous model will give an overestimate compared to the discrete model.

If we now consider a rate of interest  of r% instead of 3%, we can see the general formula for the future value of a constant stream of  income is given by

 Future value = FV = ∫ b a Rer(b-t) dt = R/r [er(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 = ∫ b a R(t)er(b-t) dt.

Present Value of a Continuous Income Stream

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 = ∫ 2 0 26000e.03(2-t) dt.
Solving this equation for PV we find that
 PV = e -.03*2 ∫ 2 0 26000 e.03(2-t) dt.
and thus :
 PV =26000 ∫ 2   0 e -.03t dt = 26000/.03 [1-e-.03*2]=\$50470.73 .

Generalizations:

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 = ∫ b a Rer(a-t) dt = R/r [1-e-r*b].
If we consider the value for a long period of time from the starting time t = a, we can  consider this an infinite limit for b and we find the formula for the Present Value of an infinite stream of of a fixed income or R dollars per unit time  with interest at r  per unit time , compounded continuously, as PV = R/r.
If we consider the further generalization that the  revenues rate of receipt varies so that R  is a function  of t then we have the final general formula for present value:
 Present value = PV = ∫ b a R(t)er(a-t) dt.

Planning for retirement:

John Ush would like to retire in 30 years and thinks that this will be possible with ease if he has about \$500,000 in savings. So he has decided to start a regular savings plan in an IRA saving account that he expects to continue paying 4% per year compounded continuously. How much should John plan to put into his account on a monthly basis to achieve his goal?

Solution: Since John wants to accumulate \$500,000, we use this for the future value of the account and using the formula for FV we need to solve for R.

 Future value = FV = 500,000  = ∫ 30 0 Re.04(30-t) dt = R/.04 *[e.04(30) -1] .

Thus R = 500,000 (.04) /[e.04(30) -1]  = 20000/ 2.32 = 8620.25.

So to achieve his goal, John must plan to save about \$8620 annually or about \$718 per month.

Note that saving \$8620 without compounding interest for 30 years would give John only \$258,600. So compound interest makes a big difference over the 30 year period!

Discussion: If  John can find a better interest rate for his savings, perhaps 6%, and he plans to retire in 40 years (he's now 25 years old) then he would need to save only about R = 500,000 (.06) /[e.06(40) -1]  = 30000/10.02 = 2993.06 annually or about \$249.42 per month!