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LO4 Effective Annual Rate (EAR) 6.3 • This is the actual rate paid (or received) after accounting for compounding that occurs during the year • If you want to compare two alternative investments with different compounding periods, you need to compute the EAR for both investments and then compare the EAR’s. © 2013 McGraw-Hill Ryerson Limited 6-0 LO4 Annual Percentage Rate • This is the annual rate that is quoted by law • By definition APR = period rate times the number of periods per year • Consequently, to get the period rate we rearrange the APR equation: • Period rate = APR / number of periods per year • You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate © 2013 McGraw-Hill Ryerson Limited 6-1 LO4 Computing APRs • What is the APR if the monthly rate is .5%? • .5(12) = 6% • What is the APR if the semiannual rate is .5%? • .5(2) = 1% • What is the monthly rate if the APR is 12% with monthly compounding? • 12 / 12 = 1% • Can you divide the above APR by 2 to get the semiannual rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate. © 2013 McGraw-Hill Ryerson Limited 6-2 LO4 Things to Remember • You ALWAYS need to make sure that the interest rate and the time period match. • If you are looking at annual periods, you need an annual rate. • If you are looking at monthly periods, you need a monthly rate. • If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly © 2013 McGraw-Hill Ryerson Limited 6-3 LO4 Computing EARs – Example 1 • Suppose you can earn 1% per month on $1 invested today. • What is the APR? 1(12) = 12% • How much are you effectively earning? • FV = 1(1.01)12 = 1.1268 • Rate = (1.1268 – 1) / 1 = .1268 = 12.68% • Suppose if you put it in another account, you earn 3% per quarter. • What is the APR? 3(4) = 12% • How much are you effectively earning? • FV = 1(1.03)4 = 1.1255 • Rate = (1.1255 – 1) / 1 = .1255 = 12.55% © 2013 McGraw-Hill Ryerson Limited 6-4 LO4 EAR - Formula m APR EAR 1 1 m Remember that the APR is the quoted rate m is the number of times the interest is compounded in a year © 2013 McGraw-Hill Ryerson Limited 6-5 LO4 Decisions, Decisions II • You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? • First account: • EAR = (1 + .0525/365)365 – 1 = 5.39% • Second account: • EAR = (1 + .053/2)2 – 1 = 5.37% • You should choose the first account (the account that compounds daily), because you are earning a higher effective interest rate. © 2013 McGraw-Hill Ryerson Limited 6-6 LO4 Decisions, Decisions II Continued • Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? • First Account: • Daily rate = .0525 / 365 = .00014383562 • FV = 100(1.00014383562)365 = 105.39, OR, • 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39 • Second Account: • Semiannual rate = .053 / 2 = .0265 • FV = 100(1.0265)2 = 105.37, OR, • 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37 • You have more money in the first account. © 2013 McGraw-Hill Ryerson Limited 6-7 LO4 Computing APRs from EARs • If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: APR m (1 EAR) 1 m -1 © 2013 McGraw-Hill Ryerson Limited 6-8 LO4 APR – Example 1 • Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? APR 12 (1 .12) or 11.39% 1 12 1 .1138655152 © 2013 McGraw-Hill Ryerson Limited 6-9 LO4 Mortgages • In Canada, financial institutions are required by law to quote mortgage rates with semi-annual compounding • Since most people pay their mortgage either monthly (12 payments per year), semi-monthly (24 payments) or bi-weekly (26 payments), you need to remember to convert the interest rate before calculating the mortgage payment! © 2013 McGraw-Hill Ryerson Limited 6-10 LO4 Continuous Compounding • Sometimes investments or loans are calculated based on continuous compounding • EAR = eq – 1 • The e is a special function on the calculator normally denoted by ex • Example: What is the effective annual rate of 7% compounded continuously? • EAR = e.07 – 1 = .0725 or 7.25% © 2013 McGraw-Hill Ryerson Limited 6-11 LO4 Quick Quiz – Part V • What is the definition of an APR? • What is the effective annual rate? • Which rate should you use to compare alternative investments or loans? • Which rate do you need to use in the time value of money calculations? © 2013 McGraw-Hill Ryerson Limited 6-12