02-Compounding

# Let r a the effective annual return set fv equal at

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• shi1hong
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Let r A = the effective annual return Set FV equal at the end of three years % 4 . 7 1 24 . 1 24 . 1 ) 1 ( 24 . 1 ) 1 ( 3 / 1 3 / 1 3 = - = = + = + A A A r r r

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Stated Rates Stated Rates do not take into account the effect of compound interest. Let m be the number of times an investment pays interest within a year Let r E be the effective interest rate every time interest is paid. Then the stated rate is simply r S = r E *m
Stated Rates Example: A bank pays 5% semi-annually. What is the stated rate? Stated rate=0.05*2 = 10% What is the effective annual rate? 1.05 2 -1=10.25%

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Future Value & Compounding ahead years of # year per periods g compoundin of # rate stated lue present va 1 = = = = + = N m r PV m r PV FV S mN S N
Effective Annual Rates What is EAR on account if stated rate is 12% with monthly compounding? % 67 . 12 1267 . 1 12 12 . 1 1 12 = = + = + EAR EAR

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Effective Monthly Rates The EAR is 15% What is effective 1-month rate? What is the stated rate if interest is paid with monthly compounding? % 04 . 14 12 * % 17 . 1 rate stated 12 / % 17 . 1 12 / 1 15 . 1 12 1 15 . 1 12 / 1 12 = = = + = + = S S S r r r
Compounding What is the EAR if account pays 10% With semi-annual compounding? (1.05)^2-1=10.25% With quarterly compounding? (1.025)^4-1=10.38% With monthly compounding? (1+.10/12)^12-1=10.47% With compounding of 10000 times/year? (1+.10/10000)^10000-1=10.517%

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Compounding
Continuous Compounding Future value with compounding an infinite number of times per year continuous compounding: Example: what is the EAR on an account that pays 10% with continuous compounding? 1+r A =e 0.10 r A =10.5171% 8 2.71828182 1 lim = + = e PVe m r PV FV S Nr mN S m

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Example If the EAR on an account is 9.1% what is the stated rate with continuous compounding? ? for solve to How 091 . 1 S S r r e =
Continuous Compounding Math review: Take natural log of both sides x e x = ) ln( ( 29 % 7 . 8 ) ln( 091 . 1 ln = = = r e S S r

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What we learned Given an effective return (9.1%), we find the equivalent return with continuous compounding by just taking the natural log of the gross effective return ln(1.091)=8.7%
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