Question

You have just graduated and you owe $41,919 on your student loan that has an interest rate of 6% (compounded annually). How long will it take you to pay it off if you make one payment per of year of $5,000? Enter your answer in whole years to the closest full year. (ex. 4, 8 ,10)

Answer 1

Let n be the no. of years required

Now, the loan repayment installments form an annuity, the present value of which is given by

PV = A/r *(1- (1/(1+r)^n) )

where A is the annual installments (made at year end) , r is the interest rate per period and n is the no. of periods

Here r is 6% p.a. compounded annually

A =$5000

So, PV = 5000/0.06 * ( 1- (1/1.06)^n)

This present value is equal to the loan amount i.e. $41919

So, 5000/0.06 * [ 1- (1/1.06)^n] = 41919

=> [ 1- (1/1.06)^n] = 0.503028

=> (1/1.06)^n = 0.496972

=> 1.06^n = 1/0.496972=2.012186

Taking natural log of both sides

n *ln(1.06)= ln(2.012186)

=> n*0.058269 = 0.699222

=> n = 0.699222/ 00058269 = 11.99991 = 12 (rounded off)

So , It will take 12 years to finish the loan