Question

A loan of $100,000 is made today. The borrower will make equal repayments of $898 per month with the first payment being exactly one month from today. The interest being charged on this loan is constant (but unknown). For the following two scenarios, calculate the interest rate being charged on this loan, expressed as a nominal annual rate compounding monthly. Give your answer as a percentage to 2 decimal places.

(a) The loan is fully repaid exactly after 180 monthly repayments, i.e., the loan outstanding immediately after 180 repayments is exactly 0.

(b) The term of the loan is unknown but it is known that the loan outstanding 2 years later equals to $86091.

please show working out. Thank you

Answer 1

a) Let the interest rate per month be i

898/(1+i)+ 898/(1+i)^2+...+898/(1+i)^180 = 100000

=> 898/i* (1-1/(1+i)^180) = 100000........................(1)

Using hit and trial method on the above equation

Putting i= 0.02 , the Left hand side of the equation becomes =43628.80

Putting i= 0.01 , the Left hand side of the equation becomes =74822.85

Putting i= 0.005 , the Left hand side of the equation becomes =106416.156

Putting i= 0.006 , the Left hand side of the equation becomes =98676.25

The actual value of i lies between 0.005 and 0.006

So, Using linear approximation method

i = 0.005+ (106416.156-100000)/(106416.156-98676.25)*(0.006-0.005) = 0.005829

Putting i= 0.005829 , the Left hand side of the equation becomes =99940.172

The actual value of i lies between 0.005 and 0.005829

So, Using linear approximation method

i = 0.005+ (106416.156-100000)/(106416.156-99940.172)*(0.005829-0.005) = 0.005821

Putting i= 0.005829 , the Left hand side of the equation becomes =99999.88

So, this is almost correct value of i

Annual Interest rate = i*12 = 0.06985 or 6.985% or 6.99% (rounded to two decimals)

b) Outstanding loan = Future value of loan - FV of 24 payments = 86091

=> 100000*(1+i)^24- 898/i*(1-1/(1+i)^24) = 86091.................(2)

Using hit and trial method on the above equation

Putting i= 0.01 , the Left hand side of the equation becomes =107896.9

Putting i= 0.005 , the Left hand side of the equation becomes =92454.52

Putting i= 0.003 , the Left hand side of the equation becomes =86689.56

Putting i= 0.0025 , the Left hand side of the equation becomes =85282.85

The actual value of i lies between 0.0025 and 0.003

So, Using linear approximation method

i = 0.0025+ (86091-85282.85)/(86689.56-85282.85)*(0.003-0.0025) = 0.002787

Putting i= 0.002787 , the Left hand side of the equation becomes =86088.64

The actual value of i lies between 0.002787 and 0.003

So, Using linear approximation method

i = 0.002787+ (86091-86088.64)/(86689.56-86088.64)*(0.003-0.002787) = 0.002788

Putting i= 0.00788 , the Left hand side of the equation becomes =86091.46

So, this is almost correct value of i

Annual Interest rate = i*12 = 0.033456 or 3.3456% or 3.35% (rounded to two decimals)