Question

1. Calculate the amortization period of a $500,000 loan, with an annual interest rate of 18% requiring monthly payments of $7,716.56.

Answer 1

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time. Here cash flows are happened at the end of the period. PV of annuity is current value of cash flows to be received at regular intervals discounted at specified int rate or discount rate to current date.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

PV of annuity = Loan Amount

EMI = Cashflow

Particulars	Amount
PV Annuity	$    500,000.00
Int Rate	1.500%
Cash Flow	$         7,716.56

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
$ 500000 = $ 7716.56 * [ 1 - [ ( 1 + 0.015 ) ^ - n ] ] /0.015
0.97194 = [ 1 - [ ( 1.015 ) ^ - n ] ]
[ ( 1.015 ) ^ - n ] = 1 - 0.97194
[ ( 1.015 ) ^ - n ] = 0.02806

Take Log on both sides
Log [ ( 1.015 ) ^ - n ] = Log ( 0.02806)
Log( a^ b ) = b * Log (a )

-n * Log ( 1.015 ) = Log ( 0.02806)
-n * 0.00647 = -1.55191
-n = -1.55191 / 0.00647
n = 1.55191 / 0.00647
n = 239.86

I.e 240 Months or 20 Years

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