Problem 5-23 Present Value of Multiple Annuities (LG5-4) A small business owner visits her bank to ask for a loan. The owner states that she can repay a loan at $2,800 per month for the next three years and then $5,600 per month for two years after that. If the bank is charging customers 9.50 percent APR, how much would it be willing to lend the business owner

The bank would be willing to lend the business owner $179,233.04.

Explanation:

To determine this, the formula for calculating the present value (PV) of an ordinary annuity as follows:

PV = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] …………………………………. (1)

The equation (1) is then applied to the two payment plans as follows:

(a) For a repayment of $2,800 per month for the next three years

PV today = Present value of 3 years payments today = ?

P = monthly payment = $2,800

r = monthly rate = 9.50% ÷ 12 = 0.791667%, or 0.00791667

n = number of months = 3 years * 12 months = 36

Substitute the values into equation (1), we have:

PV today = $2,800 × [{1 - [1 ÷ (1 + 0.00791667)]^36} ÷ 0.00791667]

PV today = $2,800 × 31.2178556171763

PV today = $87,410.00

(b) For a repayment of $5,600 per month for two years after that

PV in 3 years = Present value of 2 years payments in 3 years time = ?

P = monthly payment = $5,600

r = monthly rate = 9.50% ÷ 12 = 0.791667%, or 0.00791667

n = number of months = 2 years * 12 months = 24

Substitute the values into equation (1), we have:

PV in 3 years = $5,600 × [{1 - [1 ÷ (1 + 0.00791667)]^24} ÷ 0.00791667]

PV in 3 years = $5,600 × 21.7796154308224

PV in 3 years = $121,965.846412606

Note that this PV in 3 years we obtained is the present value of 2 years payments in 3 years time. Therefore, we need to convert it to the present value to as follows:

PV of PV in 3 years today = PV in 3 years ÷ (1 + r)^n (2)

Where;

PV in 3 years = $121,965.846412606

r = monthly rate = 9.50% ÷ 12 = 0.791667%, or 0.00791667

n = number of months = 3 years * 12 months = 36

Substitute the values into equation (2), we have:

PV of PV in 3 years today = $121,965.846412606 ÷ (1 + 0.00791667)^36

PV of PV in 3 years today = $121,965.846412606 ÷ 1.32827059801615

PV of PV in 3 years today = $91,823.04

(c) The amount the bank would be willing to lend the business owner

Amount the bank would lend = PV today + PV of PV in 3 years today

Since,

From part (a) above, PV today = $87,410.00

From part (b) above, PV of PV in 3 years today = $91,823.04

Therefore, we have:

Amount the bank would lend = $87,410.00 + $91,823.04 = $179,233.04

Therefore, the bank would be willing to lend the business owner $179,233.04.

Alternative Method:

Note that the amount of $179,233.04 the bank would be willing to lend the business owner can also be easily calculated in an excel sheet. Find attached the excel file to see how it is done.

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