Question

A 100,000 loan is being repaid in 360 monthly installments at a 9% nominal annual interest rate compounded monthly. The first payment is due at the end of the first month. Determine which payment is the first where the amount of principal repaid exceeds the amount of interest paid.

Answer 1

Answer:

The payment after 1 year will be

F=P(1+i)^n

n=1 year

P=100,000

F=100000(1+0.09)

F=109000 after 1 year

interest=9/100*100000=90000

exceeded payment=109000-90000=19000

Answer 2

Answer: The 269th

Explanation:

payment= 100000×(9%/12)/(1-1/(1+9%/12)^360)

=804.6226169

Note that,

payment equal to the sum of principal and interest.

Payment= Principal + interest.

principal>interest

payment - interest >interest

interest<payment/2

Therefore,

Interest <804.6226169/2

Interest<402.3113085

The oustanding loan at the beginning of the month<402.3113085 × 12/9%

The loan oustanding at the beginning of the month < 53641.5078

So, the first month will be month after the month where loan oustanding after the monthly payment is less than 53641.5078

Time taken for the balance to reach less than 53641.5078 is NPER(9%/12,-804.6226169,100000,-53641.5078)=267.234234 or 268 months

Hence, first month is 268+1=269