IOE+201+Notes+2 - 1 M = Number of interest periods per year...

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Unformatted text preview: 1 M = Number of interest periods per year r = Nominal annual interest rate (or APR) r eff = Effective annual interest rate (or Effective APR or APY) Compound Interest: Formula for Effective Interest Rate Example: M = 12 interest periods per year r = 15% per year (0.15 per year) r eff = {1 + (0.15/12)} 12- 1 per year = {1 + 0.0125} 12- 1 = 16.08% per year 1 1- + = M eff M r r IOE 201 Lecture Notes 2 2 M = Number of interest periods per year r = Nominal annual interest rate (or APR) r eff = Effective annual interest rate (or Effective APR or APY) Compound Interest: Difference between Nominal and Effective Interest Rates 1 1- + = M eff M r r Difference between nominal and effective interest rates = r eff- r r M r M- - + = 1 1 3 Compounding Frequency Periods per Year M Effective Annual Interest Rate r eff Difference in Interest Rates r eff- r Annually 1 15% 0% Semi-annually 2 15.56% 0.56% Quarterly 4 15.87% 0.87% Monthly 12 16.08% 1.08% Weekly 52 16.16% 1.16% Daily 365 16.18% 1.18% Example: Nominal annual interest rate, r = 15% per year Difference Between Nominal and Effective Interest Rates 4 Compounding Frequency Periods per Year M Effective Annual Interest Rate r eff Difference in Interest Rates r eff- r Annually 1 15% 0% Semi-annually 2 15.56% 0.56% Quarterly 4 15.87% 0.87% Monthly 12 16.08% 1.08% Weekly 52 16.16% 1.16% Daily 365 16.180% 1.180% Hourly 8760 16.1833% 1.1833% Every minute 525600 16.1834% 1.1834% Example: Nominal annual interest rate, r = 15% per year Difference Between Nominal and Effective Interest Rates 5 Difference Between Nominal and Effective Interest Rates ( r eff- r ) Example: Nominal annual interest rate, r = 15% per year ( r eff- r ) 6 where r = Nominal annual interest rate M = Number of interest periods per year For interest compounded M times in 1 year: Compound-amount factor = (1 + i ) M = What happens to this factor as M becomes larger and larger? i.e., As M , ?? Compound Interest: Compound Amount Factor M M r + 1 Effective annual interest rate is M M r + 1 1 1- + = M eff M r r 7 r k r r M M k M r M r + = + = + 1 1 1 1 r M k = where +-- +- + + = + 3 2 1 ! 3 ) 2 )( 1 ( 1 ! 2 ) 1 ( 1 1 1 1 k k k k k k k k k k k +-- +- + + = ! 3 ) 1 )( 1 ( ! 2 ) 1 ( 1 1 2 1 1 k k k Take limit k : (1/ k 0 as k ) + + + + = + ! 3 1 ! 2 1 1 1 1 1 lim k k k e ( e = 2.71828......... ) Compound Interest: Compound Amount Factor = Use binomial expansion: 8 Compound Interest: Compound Amount Factor r M k = r r k k M M e k M r = + =...
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IOE+201+Notes+2 - 1 M = Number of interest periods per year...

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