PDF Compound Interest and Mortgage Payments
Compound Interest and Mortgage Payments
Yuan Gao
Applied Mathematics University of Washington
yuangao@uw.edu
Spring 2015
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Annual Compounding
P(t): Principal at time t. P(0): Initial principal. r : Annual interest rate. If we compound annually,
P(1yr ) = P(0)(1 + r ? 1yr ) P(2yr ) = P(1yr )(1 + r ? 1yr ) = P(0)(1 + r ? 1yr )2
... P(nyr ) = P(0)(1 + r ? 1yr )n
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Varying the Compounding Interval
t: Compounding interval. Annual Compounding: t = 1yr Quarterly Compounding: t = 1/4yr Monthly Compounding: t = 1/12yr Daily Compounding: t = 1/365yr After the first compounding interval,
P(t) = P(0)(1 + r t)
After the second compounding interval, P(2t) = P(t)(1 + r t) = P(0)(1 + r t)2 ... P(nt) = P(0)(1 + r t)n
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Comparison of Different Compounding Intervals
Let t = nt, we have
P(t) = P(0)(1 + r t)t/t
Take t = 1yr ,
P(1) = P(0)(1 + r t)1/t
Compounding Frequency Year-End Balance Annual Yield
Annually Semi-annually Quarterly Monthly Daily
$1,060.00 $1,060.90 $1,061.36 $1,061.68 $1,061.83
6% 6.090% 6.136% 6.168% 6.183%
Table: Interest rate at r = 6% with initial principal P(0) = $1, 000
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Continuous Compounding
Let t 0,
P(t) = P(0)(1 + r t)t/t
P(t) = lim P(0)(1 + r t)t/t
t 0
In honor of the Swiss mathematician Euler (1707-1783):
lim (1 + 1 )n = e 2.71828
n
n
Taking n = 1/r t we get
P(t) = P(0)ert
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