You want to buy a house that costs $500,000. You already have $100,000 and you decide to borrow the rest from your local bank, which charges an APR of 6% (semi-annually compounded) and requires you to make equal semi-annual payments of $25,500. How long will it take you to pay back the loan?
Loan Amount = $500,000 - $100,000 = $400,000
Annuity = [PVA x r] / [1 - (1 + r)-n]
$25,500 = [$400,000 x (0.06/2)] / [1 - {1 + (0.06 / 2)}-2n]
$25,500 = $12,000 / [1 - 1.03-2n]
1 - 1.03-2n = $12,000 / $25,500
1 - 0.4706 = 1.03-2n
ln(0.5294) = -2n[ln(1.03)]
-0.6360 = -2n[0.0296]
n = 0.6360 / [2 x 0.0296] = 10.76 , or 11 years
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