# The management of a national chain of fast-food outlets is selling a 12 year franchise in a certain city. Past experience in similar localities suggests that t years from now the franchise will be generating profit at the rate of f(t)=10,600 dollars per year. If the prevailing annual interest rate remains fixed at 3% compounded continuously, what is the present value of the franchise?

The management of the national fast food chain is selling a 12-year franchise. The profit generated is predicted to be \$10600 per year. The prevailing interest rate is 3% and is compounded continuously.

The present value of the franchise the sum of the future profit earned for the next 12 years discounted at 3%.

For continuous compounding the formula to be used is `P = A*e^(r*t)` where A in the initial amount, r is the rate of interest and P is the amount after t time periods.

To discount the future values use `P = A/e^(r*t)` . This gives the sum:

`PV = 10600/e^(1*0.03) + 10600/e^(2*0.03) + ... + 10600/e^(12*0.03)`

This is a geometric series with first term `10600/e^(1*0.03)` and common ratio `1/e^(1*0.03)`

= `10600*(e^-0.03 + e^-0.06 + ... e^-0.36)`

= `10600*e^-0.03*(1 - e^-0.36)/(1 - e^-0.03)`

`~~ 105226.73`

The present value of the franchise is approximately \$105226.73