Question

Present Value of Multiple Annuities A small business owner visits his bank to ask for a loan. The owner states that he can repay a loan at $2,400 per month for the next three years and then $3,400 per month for the two years after that. If the bank is charging customers 7.50 percent APR, how much would it be willing to lend the business owner?

Answer 1

This is a question of two annuities, the first one from year 1 to year 3 and the second one from year 3 to year 5, on a monthly basis. The PV of the second annuity is the FV of the first annuity

The PV of the annuity can be found out using the following equation:

But we need to divide the PV in two parts as per the following equation:

The following part is an annuity for the first 3 years

Following is the second part of the equation. the numerator is the second annuity for the years 3 to 5 and the denominator is to discount the PV of the second annuity to the time 0. Heare as the compounding is monthly, therefore we are discounting it also monthly

Now coming back to the complete equation again:

So the loan amount that can be lent is $137,530.13