If a loan of $10,000 is paid back in equal annual end-of-year payments of $2,570.69 during the next five years, what is the annual interest rate on the loan?
A) 2%
B) 5%
C) 9%
D) 12%

Question

Quizplus · Accepted Answer

This problem can be solved using the present value formula for an ordinary annuity:

PV = PMT x [1 - (1 + r)^(-n)] / r

Where PV is the present value (in this case, the loan amount of $10,000), PMT is the annual payment of $2,570.69, r is the annual interest rate, and n is the number of payments (in this case, 5).

Substituting these values gives:

10,000 = 2,570.69 x [1 - (1 + r)^(-5)] / r

To solve for r, we can use trial and error or an online calculator. When we plug in answer choice C (9%), we get:

10,000 = 2,570.69 x [1 - (1 + 0.09)^(-5)] / 0.09
10,000 = 2,570.69 x 4.3295
10,000 = 11,116.79

This is greater than the loan amount, which means the interest rate is too high. If we try answer choice B (5%):

10,000 = 2,570.69 x [1 - (1 + 0.05)^(-5)] / 0.05
10,000 = 2,570.69 x 4.8683
10,000 = 12,499.94

This is too low, meaning the interest rate is too low. Answer choice A (2%) can be quickly eliminated because it is too low. Finally, when we try answer choice D (12%):

10,000 = 2,570.69 x [1 - (1 + 0.12)^(-5)] / 0.12
10,000 = 2,570.69 x 3.0374
10,000 = 7,804.38

This is also too low, meaning the interest rate is too high. Therefore, the only possible answer is C (9%).

If a Loan of $10,000 Is Paid Back in Equal