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A loan of $12,000 is to be repaid within one year with level monthly payments, due at the beginning of each month. The 12 payments equal $1,000 each. A finance charge of $632 is also due with the first payment. Which of the following is closest to the effective annual interest rate on the loan?

a. 12.7%
b. 12.9%
c.13.1%
d. 13.3%
e. 13.5%

Sagot :

Solution :

It is given : loan amount = $12,000

Time to repay = 12 months

Finance charge = $ 632

AT the interest rate, outflow = inflow

The present value of the loan amounts = loan amount

[tex]$1000+632+[1000 \times (PVAF (r ,11))]=12000$[/tex]

[tex]$1000 \times PVAF(r,11)=12000-1632$[/tex]

[tex]$PVAF(r,11)=\frac{10368}{1000}$[/tex]

[tex]$PVAF(r,11)=10.368$[/tex]

Now using the annuity table we get

PVAF(1%, 11)=10.9676

This is equal to 10.368 (approximately)

∴ [tex]$r=1$[/tex] % per month of compounded monthly

So the annual interest rate is :

[tex]$=[(1+0.01)^{12}]-1$[/tex]

[tex]$r=[(1.01)^{12}]-1$[/tex]

[tex]$r = 12.68$[/tex] %

  = 12.70 %

Hence the correct option is (a).

The loan's effective yearly interest rate is 12.7 percent. As a result, option (a) is the proper response.

How do you compute the Annual Interest rate?

[tex]\text{It is given : loan amount} = $12,000\\\text{Time to repay} = 12 months\text{Finance charge} = $ 632\\\text{At the interest rate, outflow = inflow}\\\text{The present value of the loan amounts = loan amount}[/tex]

[tex]1000 + 632 + [ (P.V (r.11))] = 12,000\\\\1000 \text { x } P.V (r,11) = 12,000 - 1,632\\\\P.V (r.11) = \frac{10,368}{1000}\\\\P.V (r,11) = 10.368[/tex]

[tex]\text{Now using the annuity table we get} \\P.V (0.01, 11) =10.9676\\\text{This is equal to 10.368 (approximately)}[/tex]

[tex]r = 0.01 \text{ per month}\\\text{ Annual Interest rate}:\\r= [(1+0.01}^{12}] - 1\\r= [(1.01}^{12}] - 1\\r= 12.68\\[/tex]

Therefore, the closest option among the following choices is an option (a), i.e., 12.7%

For more information about the annual interest rate, refer below

https://brainly.com/question/16544946

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