Annual Percentage Rate vs. Effective
Annual Rate
• APR = r m * m ,
where m is the number of compounding
periods per year
• EAR = (1+ APR / m) m - 1
5-1
Compound Interest
EAR = (1+ APR / m) m - 1
APR = 6%
i
Periods
per
year
ii
Interest
per
period
iii
iv
Value
APR after
(i x ii) one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
6.000%
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
5-2
Continuous compounding
Remember that FV = PV [ 1 + r / m ] m
When m ->

FV = PV * e r t
Also, EAR = e r - 1
5-3
AER vs. APR example
• It is just the middle of the month, but you already ran out of
money. You go to Soprano’s Check Cashing, who loan
you $500 now. In exchange, you will repay $570 at the
end of the month. What are the Annual Percentage Rate
(APR) and the Effective Annual Rate (EAR) that they
charge you?
5-4
Computing the Outstanding Loan Balance
5-5
Example – monthly compounding
• You just graduated from
the Neeley school and
bought a new car for
$40,000. You finance
the entire amount at 6%
APR with monthly
payments over 4 years.
What is your monthly
payment?
5-6
Inflation and Discount Rates
• Key issues:
– What is the difference between a real and a
nominal return?
– How can we convert from one to the other?
• Example:
Suppose we have $1,000, and Diet Coke costs $2.00 per
six pack. We can buy 500 six packs. Now suppose the
rate of inflation is 5%, so that the price rises to $2.10 in
one year. We invest the $1,000 and it grows to $1,100 in
one year. What’s our return in dollars? In six packs?
5-7
Calculating the Real Interest Rate
5-8
Calculating the Real Interest Rate
5-9
The Determinants of Interest Rates
•
•
•
•
Inflation
Maturity and interest rate risk
Default risk
Taxability
5-10
U.S. Interest Rates and Inflation Rates,
1955–2005
Interest rates are average three-month Treasury bill rates and inflation
rates are based on annual increases in the U.S. Bureau of Labor
Statistics’ consumer price index. Note that interest rates tend to be high
when inflation is high.
5-11
Term Structure of Risk-Free U.S. Interest Rates,
January 2004, 2005, and 2006
The figure shows the interest rate available from investing in risk-free U.S.
Treasury securities with different investment terms. In each case, the interest
rates differ depending on the horizon. (Data from U.S. Treasury STRIPS.)
5-12
Short-Term Versus Long-Term U.S.
Interest Rates and Recessions
One-year and ten-year U.S. Treasury rates are plotted, with the spread between
them shaded in blue if the shape of the yield curve is increasing (the one-year
rate is below the ten-year rate) and in red if the yield curve is inverted (the oneyear rate exceeds the ten-year rate). Gray bars show the dates of U.S.
recessions. Note that inverted yield curves tend to precede recessions as
determined by the National Bureau of Economic Research. In recessions, interest
rates tend to fall, with short-term rates dropping further. As a result, the yield
curve tends to be steep coming out of a recession.
5-13
Interest Rates on Five-Year Loans for
Various Borrowers, June 2006
5-14