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.
266^{th}
267^{th}
268^{th}
269^{th}
270^{th}
payment=100000*(9%/12)/(1-1/(1+9%/12)^360)=804.6226169
payment=principal+interest
principal>interest
payment-interest>interest
interest<payment/2
=>interest<804.6226169/2
=>interest<402.3113085
=>loan oustanding at the beginning of the
month<402.3113085*12/9%
=>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
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