Mod 6 - Ir - Apr and Ear

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Michael R. Roberts

William H. Lawrence Professor of Finance

The Wharton School, University of Pennsylvania

Interest Rates:APR and EAR

Copyright © Michael R. Roberts


Last TimeTime Value of Money• Intuition, tools and discounting• Compounding• Useful shortcuts• Taxes• Inflation

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This TimeInterest Rates• Interest rate quotes• Non-annual cash flows and

compounding

APR & EAR


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Current 5-Year Jumbo CD Rates

*Bankrate.com as of 12/16/2014




Difference between “Rate” and “APY”?


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Rate = APR or Annual Percentage Rate





Measures amount of simple interest earned in a year


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Simple interest = interest earned without compounding





Many bank quotes are in terms of APR


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APR typically not what we earn or pay




APY = Annual Percentage Yield


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APY = EAR or Effective Annual Rate




APY = EAR or Effective Annual Rate

EAR measures actual amount of interest earned/paid in year


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HOW ARE DIFFERENT RATES RELATED?


Lesson: EAR is a discount rate

EAR is what matters for computinginterest and discounting cash flows


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Lesson: APR is not a discount rate.

APR is a means to an end. We use itto get a discount rate (e.g., EAR)


How do we get from an APR to an EAR (and vice versa)?


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Lesson: The relation between APRand EAR is:


k is the number of compounding periods

per year



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i is the periodic interest rate, orperiodic discount rate



Example

• Invest $100 in CD offering 5% APR with semi-annual compounding. How much money will you have in one year?


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Example


0 ?

?100

Period


Example


0

?100

2Period

k = 2


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Example


0

?100

2Period

Period 1 (n = 1)

1

Period 2 (n = 2)


Example


0

?100

2Period

1


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Example


0

?100

2Period

1

Earn 2.5% interest Earn 2.5% interest


Example


0

?100

2Period

1

100 x (1+0.025)1102.5


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Example


0

100

2Period

1

100 x (1+0.025)1102.5

102.5 x (1+0.025)1105.0625


Example


0

100

2Period

1

100 x (1+0.025)1102.5

102.5 x (1+0.025)1105.0625


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Example


0

100

1Year

?


Example


0

100

1Year

0.5

?


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Example


0

100

1Year

0.5

?


Example


0

100

1Year

0.5

105.0625

Earn 5.0625% interest


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Example


0

100

1Year

0.5

105.0625

Earn 5.0625% interest


Lesson: If you discount cash flowsusing EAR, then measure time inyears. If you discount cash flows usingperiodic interest rate, then measuretime in periods.


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Proof

where N = kT = # of periods


Periods vs Years

0 2

105.0625100

1

100 x (1+0.025)102.5

102.5 x (1+0.025)

Periods


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Periods vs Years

0 1

105.0625100

0.5

100 x (1+0.050625)1/2102.5

102.5 x (1+0.050625)1/2

0 2

105.0625100

1

100 x (1+0.025)102.5

102.5 x (1+0.025)

Periods

Years




APR = 2.37%


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APR = 2.37%k = 365 (or 360, 252)




APR = 2.37%k = 365 (or 360, 252)

i = 2.37% / 365= 0.006714%


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APR = 2.37%k = 365 (or 360, 252)

i = 2.37% / 365= 0.006714%

EAR = (1+0.006714%)365-1= 2.398%


Summary


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Lessons

• EAR is a discount rate– Measures cash flows in years

• Period interest rate, i, is a discount rate– Measures cash flows in periods

• APR is not a discount rate


Lessons

• Moving between EAR and APR

where

i = APR / k and k = # of periods per year


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Coming up next

• Interest Rates– Term Structure of interest rates and the

yield curve


Documents

Mod 6 - Ir - Apr and Ear