Question

A small business owner visits his bank to ask for a loan. The owner states that he can repay a loan at $2,100 per month for the next three years and then $3,100 per month for the two years after that. If the bank is charging customers 6.75 percent APR, how much would it be willing to lend the business owner?

Answer 1

In order to calculate the amount that could be loaned to the business owner, we need to calculate the PV of the installments that would be made for 5 years at 6.75% APR.

Amount Loaned = PV of $2,100 monthly for 3 years + PV of $3,100 monthly for 2 years

r = 6.75% --> Monthly Rate = 6.75%/12 = 0.5626%

PV of annuity is calculated by mathematical representation:

First, calculating PV of $2,100 monthly at t = 0

Second, we need to calculate the PV of $3,100 monthly payment for 2 years today (calculating its PV at t = 3, and then discounting it further for 2 more years)

This PV₂ is as of year 3

Now we need to discount this further to year 0 (for 3 years or 36 months).

r = 0.5625% (monthly), n = 36 months

PV = FV/(1 + r)ⁿ

PV =69,414.39/(1 + 0.005625)³⁶

PV = $56,721.97 --> This is the PV at t = 0, for 3,100 monthly installment for 2 years (in year 4 and year 5)

Total Amount that can be loaned = 68,264 + 56,721.97 = $124,985.96 --> Answer