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)`
The present value of the franchise is approximately $105226.73