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
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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
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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.
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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
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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%
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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
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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.
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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.
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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

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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

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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!
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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%
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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?
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