Answer:
The the largest loan this buyer can afford is 14,533.75.
Step-by-step explanation:
This can be determined using the formula for calculating the present value of an ordinary annuity as follows:
Step 1: Calculations of the present value or the loan the buyer can afford for a 30 year loan at 5 1/2%
PV30 = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)
Where;
PV30 = Present value or the loan the buyer can afford for a 30 year loan at 5 1/2% =?
P = monthly payment = 1000
r = interest rate = 5 1/2% = 5.50% = 0.055
n = number of years = 30
Substitute the values into equation (1) to have:
PV30 = 1000 * ((1 - (1 / (1 + 0.055))^30) / 0.055)
PV30 = 1000 * 14.5337451711221
PV30 = 14,533.75
Step 2: Calculation of the present value or the loan the buyer can afford for a 20 year loan at 4 1/2%
PV20 = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (2)
Where;
PV30 = Present value or the loan the buyer can afford for a 20 year loan at 4 1/2% =?
P = monthly payment = 1000
r = interest rate = 4 1/2% = 4.50% = 0.045
n = number of years = 20
Substitute the values into equation (1) to have:
PV20 = 1000 * ((1 - (1 / (1 + 0.045))^20) / 0.045)
PV20 = 1000 * 13.0079364514537
PV20 = 13,007.94
Conclusion
Since 14,533.75 which is the present value or the loan the buyer can afford for a 30 year loan at 5 1/2% is greater than the 13,007.94 which is the present value or the loan the buyer can afford for a 20 year loan at 4 1/2%, it therefore implies that the the largest loan this buyer can afford is 14,533.75.