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

To know the amount that bank will be willing to lend the business owner can be calculated by the present value of such series of cash flows.

Formula of the present value (PV) of annuity

PV = PMT * (1- (1+i) ^-n / i)

Where,

Present value of first series of annuity payment (for first 3 years) PV1 =?

PMT is monthly cash flow = $2,100 per month

Interest rate rate i = 6.75%% per annum or 6.75%/12 = 0.56% per month

Time period of annuity n = 3 years or 3 *12 = 36 months

Therefore,

PV1= $2,100 * [1- (1+ 0.56%) ^-36 / 0.56%]

PV1 = $68,264.00

Now Present value of second series of annuity payment (for next 2 years) PV2 (after 3 years) =?

PMT is annual cash flow = $3,100 per month

And n = 24 months

Therefore,

PV2 (after 3 years) = $3,100 * [1- (1+ 0.56%) ^-24 / 0.56%]

PV2 (after 3 years) = $69,414.39

But this present value is after 3 years, to calculate today’s present value, we have to discount the above amount for 3 years at 6.75% per year discount rate

Todays present value PV2 =$69,414.39/ (1+6.75%) ^3 = $57,061.85

The amount you be willing to pay for these annuities = Present value of first series of annuity payment + Now Present value of second series of annuity payment

= PV1 + PV2 (today’s value)

= $68,264.00 + $57,061.85

= $125,325.85

The amount that bank will be willing to lend the business owner is $125,325.85