Calculate amount of final payment.

Question

i have a question for my math practice but i do several ways but i still get the wrong answer, please help:

Loan payments of $700 due 3 months ago and $1000 due today are to be paid by a payment of $800 in two months and a final payment in five months. If 9% interest is allowed, and the focal date is five months from now, what is the amount of the final payment.

I calculate by using future value formula: S=P(1+r*t)

The first method i try is:

700(1+.0.09*8/12) + 1000(1+0.09*5/12) + 800(1+0.09*3/12)= 2597.5

2nd attemp:

700(1+0.09*8/12) + 1000(1+0.09*5/12)= 800(1+0.09*3/12) + X

==>X= 961.5

Can Anyone help me? ( this is simple interest)

Answer 1

I instead would try:

• \$700 to be paid in 5 months,
• \$100 to be paid in 2 months,
• \$900 to be paid in 5 months.

Assuming simple interest (without exponential formulas), a 9% anual, and liquidating first the oldest debt, and carrying the values into the focal date, we shall apply again the interest over every paid amount:

• 3 months from the first payment into the focal date
• no months from the second payment into the focal date

Hence the final paid value with the focal date correction should be: $\$700\cdot(1+0.09\cdot5/12)\cdot(1+0.09\cdot3/12)+\$100(1+0.09\cdot2/12)\cdot(1+0.09\cdot3/12)+\$900\cdot(1+0.09\cdot5/12)= \$1780.12$

From here the amount of final payment, at the focal date is: $\$900\cdot(1+0.09\cdot5/12)=\$933.75$

Note that making a payment at the 3rd past month involves a factor of $(1+0.09\cdot 3/12)$. This is the exact amount paid at that instant. Carrying this paid value with a focal date at the 5th next month involves a second factor of $(1+0.09\cdot 5/12)$, so the final paid quantity have a doubled factor of $(1+0.09\cdot 3/12)\cdot(1+0.09\cdot 3/12)$.
Of course, this depend on the system applied.

Answer 2

I use the present time as reference date. There are two methods with the corresponding equations:

$\color{blue}{\texttt{a) simple interest}}$

$$700\cdot (1+0.09\cdot 3/12)+1000=\frac{800}{1+0.09\cdot 2/12}+\frac{x}{1+0.09\cdot 5/12}$$

$\Rightarrow x=962.36$

The result is a little bit different from yours due the different reference dates.

---

$\color{blue}{\texttt{b) compound interest}}$

$$700\cdot (1+0.09/12)^3+1000=\frac{800}{(1+0.09/12)^2}+\frac{x}{(1+0.09/12)^5}$$

$\Rightarrow x=963.05$

Answer 3

The focal date is the date of the last payment. At this date, the amount of the debt is $$ FV_1=700\left(1+0.09\times\frac{8}{12}\right)+1000\left(1+0.09\times\frac{5}{12}\right)=1779.5 $$ and the amount of the repayments is $$ FV_2=800\left(1+0.09\times\frac{2}{12}\right)+P=818+P $$ We must have $FV_1=FV_2$ and then $$ P=1779.5-818=961.5 $$

If you use compound interest, then $$ 700\left(1+\frac{0.09}{12}\right)^8+1000\left(1+\frac{0.09}{12}\right)^5=800\left(1+\frac{0.09}{12}\right)^2+P $$ that is $$ P=1781.19-818.14=963.05 $$