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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