Answer:
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.
