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, 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.


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 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
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103 103/52 500 500 *exp((.03)*1/52)) = 500.28
sum 52000 $53607.13


Generalizations : Suppose  the annual receipts are $R and that we consider the period between t  =  a  and t = b. Instead of  doing the estimation weekly ,  divided 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)).
Thus  FV = R dt*exp(.03*(b -t0)) + R dt*exp(.03*(b -t1)) + R dt*exp(.03*(b -t2)) + Rdt *exp(.03*(b -t3))+ ... + R*exp(.03*(b -tn-1)).
Now consider what happens to the estimate with n chosen very large. The estimate will also estimate a definite integral; Thus we have that
 
  Using R = $26000, a = 0 and b= 2 we have 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


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)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

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
 

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: