A $1000 loan is taken out today.The loan is repaid with a payment of $400 at the end of the 1^st year,$800 at the end of the 5^th year,and a payment of $X at the end of the 10^th year.If j₄ = 12%,what is X?

Question

Quizplus · Accepted Answer

To find $X$, we use the present value formula for each payment at the 12% annual interest rate. The present value $PV$ of a future payment $FV$ due in $n$ years at an interest rate $i$ is given by $PV = \frac{FV}{(1+i)^n}$. For the payments:1. The $400 payment in 1 year: $PV_1 = \frac{400}{(1+0.12)^1} = \frac{400}{1.12}$2. The $800 payment in 5 years: $PV_2 = \frac{800}{(1+0.12)^5}$3. The $X payment in 10 years: $PV_3 = \frac{X}{(1+0.12)^10}$The sum of these present values should equal the loan amount, $1000. So, we have:$$1000 = \frac{400}{1.12} + \frac{800}{(1.12)^5} + \frac{X}{(1.12)^10}$$Solving for $X$, we find:$$X = (1000 - \frac{400}{1.12} - \frac{800}{(1.12)^5}) \times (1.12)^10$$Plugging in the numbers:$$X = (1000 - 357.14 - 453.02) \times (1.12)^10$$$$X = 189.84 \times (1.12)^10$$$$X \approx 657.84$$Therefore, the correct answer is $X = 657.84$, which corresponds to choice B.

A $1000 Loan Is Taken Out Today