Interest Rates
Empirical Properties
The Nominal Interest Rate

Suppose you take out a $1000 loan today. You
agree to repay the loan with a $1050 payment in
one year.






Interest = Payment (Face Value) – Principal (Price)
Interest = $1,050 - $1,000 = $50
Interest Rate = (Interest/Principal)
Interest Rate = ($50)/($1,000) = .05 (5%) Per Year
This is the one year spot rate
INTEREST RATES ALWAYS HAVE A TIME PERIOD
ASSOCIATED WITH THEM!!!
Annualizing

Suppose that you invest $1 at a quarterly
interest rate of 2%. What is your annual return?
$1
$1.02
X (1.02)
X (1.02)
$1.04
X (1.02)
$1.06
$1.082
X (1.02)
(1.02)(1.02)(1.02)(1.02) = 1.082 = 8.2%
Note: It is generally a safe approximation to multiply by 4
Annualizing

Suppose you earn a cumulative interest rate of 5% over
a 4 year period. What is your annualized return?
$1
$??
X (1+i)
$??
X (1+i)
$??
X (1+i)
$1.05
X (1+i)
(1+i)(1+i)(1+i)(1+i) = 1.05
(1+i) = (1.05)^(.25) = 1.012 = 1.2%
Note: Its generally a safe approximation to just divide by 4
The Yield Curve
6
4
2
0
1 yr


2 yr
5yr
10 yr
20yr
Spot Rates are interest rates charged for loans
contracted today: S(1), S(2), S(3), etc…
The Yield curve is a listing of current spot rates for
different maturities (on an annualized basis)
Forward Rates

Forward rates are interest rates for contracts to be
written in the future. (F)
 F(1,1)
= Interest rate on 1 year loans contracted 1
year from now
 F(1,2) = Interest rate on 2 yr loans contracted 1
year
 from now
 F(2,1) = interest rate on 1 year loans contracted 2
years from now
 S(1) = F(0,1)

Forward rates are not explicitly stated, but are implied
through observed spot rates
Calculating Forward Rates

The current annual yield on a 1 yr Treasury is 2.0% while
a 2 yr Treasury pays an annual rate of 2.6%

$1(1.02) = $1.02 ($1 invested for 1 year)
$1(1.026)(1.026) = $1.053 (invested for two years)

($1.02)(1+F(1,1)) = $1.053

Therefore, the implied return from the 1st year to the
second is
$1.053/$1.02 = 1.032 = F(1,1) = 3.2%

Calculating Forward Rates

The current annual yield on a 2 yr Treasury is 2.6% while
a 3 yr Treasury pays an annual rate of 2.9%

$1(1.026)(1.026) = $1.053 (invested for two years)
$1(1.029)(1.029)(1.029) = $1.09 (invested for 3 years)

($1.053)(1+F(2,1)) = $1.09

Therefore, the implied return from the 2nd year to the
third is
$1.09/$1.053 = 1.035 = F(2,1) = 3.5%

Spot Rates & Bond Prices

Zero Coupon (Discount) Bonds are convenient
because they only involve one payment.





Maturity date (Term)
Face Value (Assume $100)
A 90 Day T-Bill is currently selling for $99.70
Yield (Yield to Maturity) = ($100 - $99.70)/$99.70 = .003
(.3%)
Annualized YTM = (1.003)^(365/90) = 1.012 (1.2%)
Spot Rates & Bond Prices

STRIPS (Separately Traded Registered Interest
and Principal) were created by the Treasury
department in 1985.





Maturity date (Term)
Face Value (Assume $100)
A 10 Yr. STRIP is selling for $63.69
YTM = ($100 - $63.69)/$63.69 = .5701 (57.01%)
Annual YTM = (1.5701)^(.1) = 1.0461 (4.61%)
Forward Rates and Bond Prices

STRIP prices also imply forward rates…


An August 2015 STRIP is currently selling for $63.55
while an August 2014 STRIP is selling for $68.07.
F(9,1) = $68.07/$63.55 = 1.07 = 7%
Interest Rates & Bond Prices

Consider a 1 year, $100  Now, consider the same 1
discount bond with a price year, $100 discount bond
of $98.00
with a price of $94.00
i = ($100 – $98.00) *100 =2% i = ($100 – $94.00) *100 = 6.4%
$98.00
$94.00
Higher bond prices are associated with Lower
Returns!!
Interest Rates & Bond Prices
What’s the difference between a bond
price and an interest rate?
 They are both relative prices

Interest Rate = Price of a current $ in terms of
foregone future dollars.
 Bond Price = Price of a Future $ in terms of
foregone current dollars

Interest Rates in the US (1984 – 2004)
14
12
10
8
6
4
2
0
1/1/84
1/1/89
1/1/94
1 YR TBILL
1/1/99
1/1/04
1 Year Treasury Rate
18
16
14
12
10
8
6
4
2
0
1/1/59
1/1/64
1/1/69
1/1/74
1/1/79
1/1/84
1/1/89
1/1/94
1/1/99
1/1/04
Interest Rates in the US
Term
Federal
Funds
Mean
1Yr TBill 5 Yr.
TBill
5.88
Std. Dev.
2.98
Corr (+1)
.988
Corr (+2)
.968
Corr (+3)
.949
Corr (+4)
.934
10 Yr.
TBill
Interest Rates in the US
16
14
12
10
8
6
4
2
0
1/1/84
1/1/89
1 YR
1/1/94
5 YR
10 YR
1/1/99
Fed Funds
1/1/04
Interest Rates in the US
Term
1Yr
5 Yr.
10 Yr.
Mean
Federal
Funds
5.80
5.88
6.49
6.69
Std. Dev.
3.39
2.98
2.75
2.68
Corr (+1)
.986
.988
.992
.994
Corr (+2)
.961
.968
.979
.985
Corr (+3)
.937
.949
.968
.976
Corr (+4)
.915
.934
.957
.969
Correlations
1YRTB
5YRTB
1YRTB
5YRTB
10YRTB
1
0.966104
0.934983
1
0.993211
FF
0.973375
0.914724
10YRTB
1
0.879391
FF
1
Interest Rates
Mean reverting (stationary)
 Long term rates are less volatile than short
term rates
 Long term rates show more persistence
than short term rates
 High degree of persistence
 Highly correlated with one another (long
rates less correlated with shorter rates)

Interest Rates & Inflation
14
12
10
8
6
4
2
0
1/1/84
1/1/89
1/1/94
1/1/99
1/1/04
Interest Rates & Inflation
15
10
5
0
1/1/84
1/1/89
1/1/94
1/1/99
-5
-10
1 YR TBILL
INFLATION
1/1/04
Interest Rates & Inflation
MEAN (Inflation Rate)
3.90
STDEV (Inflation Rate)
3.6746435
Corr(FF)
0.5899089
Corr(1YRTB)
0.5552795
Corr(5YRTB)
0.4879992
Corr(10YRTB)
0.4666077

Inflation rates are highly correlated with interest
rates (less so for longer term rates)
Characteristics of Business
Cycles

All recessions/expansions “look similar”, that is, there
seems to be consistent statistical relationships between
GDP and the behavior of other economic variables.
 Correlation (procyclical, countercyclical)
 Timing (leading, coincident, lagging)
 Relative Volatility

4
3
18
2
1
14
0
-1
10
8
-3
-4
6
-5
-6
2
Annual Yield
1/
1/
00
1/
1/
95
12
1/
1/
90
-2
16
1/
1/
85
1/
1/
80
Annual Growth
Interest Rates vs. GDP
GDP
1YR TBILL
4
0
Nominal Interest Rates tend to be Procyclical and
lagging
Interest Rates vs. Money
5
14
4
12
1/1/2004
1/1/2002
1/1/2000
1/1/1998
1/1/1996
1/1/1994
-1
1/1/1992
6
1/1/1990
0
1/1/1988
8
1/1/1986
1
Annual Yield
10
2
1/1/1984
Annual Growth
3
M1
1YR TBILL
4
-2
-3
-4

2
0
Interest rates tend to be negatively correlated with
changes in money (in the short run)
Nominal vs. Real Interest Rates

A $1000 investment at a 10% annual interest rate will
pay out $1100 in one year.

Nominal Return (i) = ($1100 - $1000)/$1000 = .10 (10%)
or
(1+i) = $1100/$1000 = 1.10
Nominal vs. Real Interest Rates

A $1000 investment at a 10% annual interest rate will
pay out $1100 in one year. To get a real (inflation
adjusted) returns, we must divide by the price level
(current and future)

Real Return (r) = (($1100/P’) – ($1000/P))/($1000/P)
or
(1+r) = ($1100/$1000)/(P’/P)
(1+r) = (1+i) / (1+ inflation rate)
Nominal vs. Real Interest Rates

A $1000 investment at a 10% annual interest rate will
pay out $1100 in one year. To get a real (inflation
adjusted), we must divide by the price level (current and
future).
Suppose that the inflation rate is equal to 5% annually

Real Return (1+r ) = (1.10) / (1.05) = 1.048%

An Easy Approximation
 We
have the following:
(1+i) = (1+r)(1+inflation)
(1+i) = 1 + r + inflation + r*inflation
i = r + inflation. + r*inflation ( usually r*inf is small)
Ex) r = 10% - 5% = 5%
Real Interest Rates: 1975-1985
20
15
10
1YR
5YR
10YR
5
0
1/1/1975
1/1/1977
1/1/1979
1/1/1981
1/1/1983
1/1/1985
-5
-10

Why would anyone accept a negative real rate of return?
Ex Ante. Vs. Ex Post
Ex Ante real interest rates are the rates
investors expect based on anticipated
inflation rates
 Ex Post real interest rates are the rates
investors actually receive after the fact.
 The difference between the two depends
on the accuracy of inflationary
expectations

1/
1/
-5
-10
1/
1/
7/
1/
1/
1/
7/
1/
1/
1/
7/
1/
1/
1/
7/
1/
1/
1/
7/
1/
1/
1/
7/
1/
1/
1/
7/
1/
19
85
19
84
19
84
19
83
19
83
19
82
19
82
19
81
19
81
19
80
19
80
19
79
19
79
19
78
19
78
Inflation Expectations
20
15
10
5
Expected
Actual
Real Rate
0
Inflation Expectations and Real
Returns

Inflation expectation tend to be quite
persistent (i.e. investors don’t seem to
update to new information). Therefore,
real interest rates also have a high degree
of persistence.