A commercial real estate developer plans to borrow money to finance an upscale mall in an exclusive area of the city. The developer plans to get a loan that will be repaid with uniform payments of $400,000 over a 15-year period beginning in year 2 and ending in year 15. How much will a bank be willing to loan at an interest rate of 10% per year

The amount a bank will be willing to loan at an interest rate of 10% per year is $2,678,795.44.

Explanation:

This can be calculated using the following 2 steps:

Step 1: Calculation of the present value of the loan in year 2

This can be calculated using the formula for calculating the present value of an ordinary annuity as follows:

PV2 = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (1)

Where;

PV2 = Present value of the loan in year 2 =?

P = Annual payment = $400,000

r = Interest rate = 10%, or 0.10

n = number of years to make payments beginning in year 2 and ending in year 15 = 14

Substitute the values into equation (1) to have:

PV2 = $400,000 * ((1 - (1 / (1 + 0.10))^14) / 0.10)

PV2 = $400,000 * 7.3666874569392

PV2 = $2,946,674.98

Step 2: Calculation of the present value of the loan in year 1 or the amount a bank will be willing to loan at an interest rate of 10% per year

This can be calculated using the simple present value formula as follows:

PV1 = PV2 / (1 + r)^n (2)

Where;

PV1 = Present value of the loan in year 1 = ?

PV2 = Present value of the loan in year 2 = $2,946,674.98

r = Interest rate = 10%, or 0.10

n = number of year = 1

Substitute the values into equation (2) to have:

PV1 = $2,946,674.98 / (1 + 0.10)^1

PV1 = $2,946,674.98 / 1.10

PV1 = $2,678,795.44

Therefore, the amount a bank will be willing to loan at an interest rate of 10% per year is $2,678,795.44.