Determine the effective annual interest rate on the loan

Question

A borrower owes \$5000 today and has promised to pay \$1900 at the end of the next three years to repay the loan. Determine the effective annual interest rate on the loan. What is the outstanding balance immediately after the second payment?

For the effective annual interest rate I get a negative interest. I am not sure where I have gone wrong. And I am not sure on how to find the second part of the question( outstanding balance). Any help will be appreciated.

a) Effective annual interest rate:

$ PV= FV/(1+i)^t $

$ 5,000=(1900)/(1+i)^3 $

$ i=-0.275 $

Answer 1

Your PV formula is right, but it applies to each amount paid so your second equation should be $$ 5,000= \frac{1900}{(1+i)^1} + \frac{1900}{(1+i)^2} + \frac{1900}{(1+i)^3} $$

Once you have calculated $i$ (by trial and error), then the outstanding balance after, say , 1 year just before the \$1900 repayment is $5000 (1 + i)$ so the outstanding balance just after the first repayment instalment is $$5000(1+i) - 1900.$$ You can repeat that process to get the outstanding balance after the second repayment instalment. You can check you answer by repeating it again and checking that the outstanding balance after the third repayment is zero.

See also related question repayment of loan with compound interest

Answer 2

Since the 1900 are paid for 3 periods, you must use the annuity formula:

$5000=\frac{1900}{i}\left[1-\frac{1}{(1+i)^3}\right]$

and solve for $i$, this will be the root of a cubic polynomial. It is approximately equal to 6.85% (obtained from Wolfram Alpha).

After the second payment, there must be an outstanding debt such that after interests, it is exactly equal to 1900, so $\frac{1900}{1+i}\approx 1778.21$.