Question

You borrow 10,000 from a loan company. The company wants you to pay it back in equal monthly installments in 10 years under a monthly interest rate of i%.

If the equivalent future amount of your payments at the end of year 10 is 33,000, what is the nominal annual rate?

Answer 1

First of all we are required to calculate the nominal interest rate. Then, we can determine the monthly installment using the calculated interest rate.

Present value = $ 10,000

Future value = $ 33,000

Time = 10 years

Let us assume the nominal rate of return = i% per year

$\large FV = PV(F/P,i\%,n)$

FV (F/P,1%, n) = PV

$\large \implies (1 +i)^n= \frac{FV}{PV}$

$\large \implies 1 +i= [\frac{FV}{PV}]^{\frac{1}{n}}$

$\large \implies i= [\frac{FV}{PV}]^{\frac{1}{n}} - 1$

Now plug in all the values in the above equation we get

$\large \implies i= [\frac{33,000}{10,000}]^{\frac{1}{10}} - 1$

$\large \implies i= 1.126811 - 1$

$\large \implies i= 0.126811$

$\large \implies i \approx 12.68\% \ per \ year$

Nominal interest rate = 12.68% per year

Now calculating the monthly installment using nominal interest rate = 12.68% per year

Borrowed amount = $ 10,000

Time = 10 years = 120 months

$\large A = 10,000(A/P,\frac{12.68}{12}\%,120)$

$\large \implies A = 10,000 \times \frac{\frac{0.1268}{12}}{1-(1+\frac{0.1268}{12})^{-120}}$

$\large \implies A = 10,000 \times 0.0147429$

$\large \implies A \approx \$ \ 147.43$

Monthly installment = $ 147.43

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