Question

The management of a national chain of fast-food outlets is selling a 10-year franchise in Cleveland, Ohio. Past experience in similar localities suggests that t years from now the franchise will be generating profit at the rate of f(t)=31700 dollars per year. If the prevailing annual interest rate remains fixed at 7 percent compounded continuously, what is the present value of the franchise?

Answer 1

In the time interval t to t + dt, the present value of the profit is [e^(-0.09 t)](37,200 dt) dollars.

The present value of the profit, in dollars, in the interval 0 <= t <= 10 is

integral 0 to 10 of [e^(-0.09 t)](37,200 dt)
= [e^(-0.09 t)](37,200) / (-0.09), evaluated from t = 0 to t = 10
= [e^(-0.09(10)) - e^(-0.09(0))](37,200) / (-0.09)
= 245,284.54

The present value of the franchise is $245,284.54 .