Question

Suppose that every 4 months, you make a $595 payment toward a 10-year loan whose annual rate is 4.5%. How much was the loan for?

The answer is $14,289, but I'm not sure why. Can someone explain this to me?

Answer 1

Solution :

The amount of loan = PV of annuity payments made every four months for 10 years

The formula for calculating the present value of annuity is :

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

Where

PV = Present value of annuity   ;   A = Amount of payment /annuity per period    ;  

r = rate of interest per period ; n = no. of payment periods

As per the information given in the question we have

Each payment period is of 4 months ; Duration of the loan = 10 years

Thus total No. of payment periods = ( 10 years * 12 months) / 4 months

= 120 / 4 = 30 months

Thus n = 30

Annual Interest Rate = 4.5 %

Thus the Interest Rate for 4 months = ( 4.5 % / 12 ) * 4

= 1.5 % = 0.015

Thus r = 1.5 % = 0.015

A = Payment made every four months = $ 595

Applying the above information in the formula we have

= $ 595 * [ ( 1 - ( 1 / ( 1 + 0.015 ) ³⁰ )) / 0.015 ]

= $ 595 * [ ( 1 - ( 1 / ( 1.015 ) ³⁰ )) / 0.015 ]

= $ 595 * [ ( 1 - ( 1 / 1.563080 )) / 0.015 ]

= $ 595 * [ ( 1 – 0.639762 ) / 0.015 ]

= $ 595 * [ 0.360238 / 0.015 ]

= $ 595 * 24.015838

= $ 14,289.423614

= $ 14,289    ( When rounded off to the nearest whole number )

Thus the amount of loan = PV of annuity payments made every four months for 10 years

= $ 14,289

Note: The value of ( 1.015 ) ³⁰   is calculated using the Excel formula =POWER(Number,Power)

=POWER(1.015,30) = 1.563080