Question

Mia Salto wishes to determine how long it will take to repay a ​$10,000 loan given that the lender requires her to make annual​ end-of-year installment payments of ​$2,142.

a.  If the interest rate on the loan is 14​%, how long will it take for her to repay the loan​ fully?

b.  How long will it take if the interest rate is 11​%?

c.  How long will it take if she has to pay 18​% annual​ interest?

d. Reviewing your answers in parts a​, b​, and c​, describe the general relationship between the interest rate and the amount of time it will take Mia to repay the loan fully.

Answer 1

a)

Interest rate on the loan is 14​%

The present value of the loan, the annual payment and the time required to repay the loan are related according to the following equation

In the above equation, solving for nt will let us know the time required to repay the loan

(1 + 0.14)^nt -1 = 0.65359 x (1 + 0.14)^nt

(1 + 0.14)^nt [ 1 - 0.6539 ] = 1

(1 + 0.14)^nt = 2.88679

Taking natural log on both sides of the equation

nt ln 1.14 = ln 2.88679

nt = 8.09 years

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b)

Interest rate on the loan is 11%

(1 + 0.11)^nt -1 = 0.5135 x (1 + 0.11)^nt

(1 + 0.11)^nt [ 1 - 0.5135 ] = 1

(1 + 0.11)^nt = 2.05566

Taking natural log on both sides of the equation

nt = ln 2.05566 ln 1.11

nt = 6.90 years

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c)

Interest rate on the loan is 18%

(1 + 0.18)^nt -1 = 0.84034 x (1 + 0.18)^nt

(1 + 0.18)^nt [ 1 - 0.84034 ] = 1

(1 + 0.18)^nt = 6.2632

nt = ln 6.2632 ln 1.18

nt = 11.08 years

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d)

As the interest rate decreases, the number of years required to repay the loan fully also decreases and when the interest rate increases, the number of years required to repay the loan in full also increases. Thus it can be said that interest rate and years required to repay the loan have a direct relationship.