Interest Rates: APR and EAR

1.

Interest Rates:
APR and EAR
Michael R. Roberts
William H. Lawrence Professor of Finance
The Wharton School, University of Pennsylvania
Copyright © Michael R. Roberts

2.

Last Time
Time Value of Money
• Intuition, tools and discounting
• Compounding
• Useful shortcuts
• Taxes
• Inflation
Copyright © Michael R. Roberts

3.

This Time
Interest Rates
• Interest rate quotes
• Non-annual cash flows and
compounding
Copyright © Michael R. Roberts

4.

APR & EAR
Copyright © Michael R. Roberts

5.

Current 5-Year Jumbo CD Rates
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

6.

Current 5-Year Jumbo CD Rates
Difference between
“Rate” and “APY”?
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

7.

Current 5-Year Jumbo CD Rates
Rate = APR or Annual
Percentage Rate
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

8.

Current 5-Year Jumbo CD Rates
Rate = APR or Annual
Percentage Rate
Measures amount of
simple interest earned
in a year
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

9.

Current 5-Year Jumbo CD Rates
Rate = APR or Annual
Percentage Rate
Simple interest =
interest earned
without compounding
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

10.

Current 5-Year Jumbo CD Rates
Rate = APR or Annual
Percentage Rate
Many bank quotes are
in terms of APR
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

11.

Current 5-Year Jumbo CD Rates
Rate = APR or Annual
Percentage Rate
APR typically not
what we earn or pay
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

12.

Current 5-Year Jumbo CD Rates
APY = Annual
Percentage Yield
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

13.

Current 5-Year Jumbo CD Rates
APY = EAR or
Effective Annual Rate
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

14.

Current 5-Year Jumbo CD Rates
APY = EAR or
Effective Annual Rate
EAR measures actual
amount of interest
earned/paid in year
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

15.

HOW ARE DIFFERENT
RATES RELATED?
Copyright © Michael R. Roberts

16.

Lesson: EAR is a discount rate
EAR is what matters for computing
interest and discounting cash flows
Copyright © Michael R. Roberts

17.

Lesson: APR is not a discount rate.
APR is a means to an end. We use it
to get a discount rate (e.g., EAR)
Copyright © Michael R. Roberts

18.

How do we get from an APR to an
EAR (and vice versa)?
Copyright © Michael R. Roberts

19.

Lesson: The relation between APR
and EAR is:
Copyright © Michael R. Roberts

20.

Lesson: The relation between APR
and EAR is:
k is the number of
compounding periods
per year
Copyright © Michael R. Roberts

21.

Lesson: The relation between APR
and EAR is:
i is the periodic interest rate, or
periodic discount rate
Copyright © Michael R. Roberts

22.

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

23.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
?
100
?
Copyright © Michael R. Roberts

24.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
k=2
Period
0
2
100
?
Copyright © Michael R. Roberts

25.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year? Period 1
Period 2
Period
0
100
Copyright © Michael R. Roberts
(n = 1)
(n = 2)
1
2
?

26.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
100
Copyright © Michael R. Roberts
1
2
?

27.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
1
2
100
?
Earn 2.5% interest
Copyright © Michael R. Roberts
Earn 2.5% interest

28.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
100
100 x (1+0.025)1
Copyright © Michael R. Roberts
1
2
102.5
?

29.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
100
1
2
105.0625
102.5
102.5 x (1+0.025)1
100 x (1+0.025)1
Copyright © Michael R. Roberts

30.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Period
0
100
1
2
105.0625
102.5
102.5 x (1+0.025)1
100 x (1+0.025)1
Copyright © Michael R. Roberts

31.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Year
0
1
100
?
Copyright © Michael R. Roberts

32.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Year
0
100
Copyright © Michael R. Roberts
0.5
1
?

33.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Year
0
100
Copyright © Michael R. Roberts
0.5
1
?

34.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Year
0
0.5
100
105.0625
Earn 5.0625% interest
Copyright © Michael R. Roberts
1

35.

Example
• Invest $100 in CD offering 5% APR
with semi-annual compounding. How
much money will you have in one
year?
Year
0
0.5
100
105.0625
Earn 5.0625% interest
Copyright © Michael R. Roberts
1

36.

Lesson: If you discount cash flows
using EAR, then measure time in
years. If you discount cash flows using
periodic interest rate, then measure
time in periods.
Copyright © Michael R. Roberts

37.

Proof
where N = kT = # of periods
Copyright © Michael R. Roberts

38.

Periods vs Years
Periods
0
1
2
100
102.5
105.0625
100 x (1+0.025)
Copyright © Michael R. Roberts
102.5 x (1+0.025)

39.

Periods vs Years
Periods
0
1
2
100
102.5
105.0625
100 x (1+0.025)
102.5 x (1+0.025)
Years
0
100
0.5
1
102.5
105.0625
100 x (1+0.050625)1/2 102.5 x (1+0.050625)1/2
Copyright © Michael R. Roberts

40.

Current 5-Year Jumbo CD Rates
APR = 2.37%
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

41.

Current 5-Year Jumbo CD Rates
APR = 2.37%
k = 365 (or 360, 252)
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

42.

Current 5-Year Jumbo CD Rates
APR = 2.37%
k = 365 (or 360, 252)
i = 2.37% / 365
= 0.006714%
*Bankrate.com as of 12/16/2014
Copyright © Michael R. Roberts

43.

Current 5-Year Jumbo CD Rates
APR = 2.37%
k = 365 (or 360, 252)
i = 2.37% / 365
= 0.006714%
EAR =
(1+0.006714%)365-1
*Bankrate.com as of 12/16/2014
= 2.398%
Copyright © Michael R. Roberts

44.

Summary
Copyright © Michael R. Roberts

45.

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
Copyright © Michael R. Roberts

46.

Lessons
• Moving between EAR and APR
where
i = APR / k and k = # of periods per year
Copyright © Michael R. Roberts

47.

Coming up next
• Interest Rates
– Term Structure of interest rates and the
yield curve
Copyright © Michael R. Roberts