FNCE 4070: FINANCIAL MARKETS
AND INSTITUTIONS
Lecture 5: The Term Structure of
Interest Rates
Yield Curves
Spot and Forward Interest
Rates
The Pure Expectations
Model of Yield Curves
Where is this Financial Center?
Arab Spring Risk. Cairo stock
market closed from January
27, 2011 to March 23, 2011
Relationship of Yields to Maturity

In previous lectures we have discussed
various factors which can account for
differences in market interest rates:


For example, inflation, risk of default, business
cycles, flight to safety, and term to maturity.
In this lecture we will expand on term to
maturity as a factor and do so in the context
of yield curves (called the “term structure of
interest rates”).
Initial Observation Regarding
Market Rates and Maturity



“Normally,” longer term
interest rates are above
shorter term interest
rates.
Even when we adjust
for default risk (e.g.,
just looking at
Treasuries – see right
hand panel, this is the
case.
Why do you think this is
a normal relationship?
Why Might we Assume that the Long
Term Rate will be Above the Short Term?

Because of the risks associated with
committing one’s capital for longer periods of
time:




Uncertainty about inflation.
Uncertainty about economic activity.
Knowledge of the price (i.e., interest rate) risk
relationship regarding longer term issues.
And if dealing with debt which is not free from
default risk, uncertainty about future cash flows,
risk of default and credit ratings.
A Second Look at the Market Rate
and Maturity Relationship

On occasion, the long
term interest rate falls
below the short term
interest rate.


See right hand panel.
But this is not a typical
relationship. Most of
the time the yield curve
is upward sweeping.
see next slide.
Another View: 10-Year Treasury
Rate Minus 3-Month Treasury Rate
How Can We Illustrate the Relationship
Between Interest Rates and Maturity?

(1) We can look at interest rates over time.



Compare movements (or differentials) of long
term rates to short term rates over some historical
period.
OR
(2) We can look at interest rates at a point in
time, i.e., on a particular date.


What is the short term interest rate, the
intermediate term rate, and the long term interest
rate on a specific date?
This last approach is referred to as a yield curve.
Construction a Yield Curve

A yield curve is simply a graphic presentation of the
relationship of “term to maturity” and “yields to maturity”
(interest rate) on a given date for seasoned issues. To
construct it we plot:


Term to maturity on the X axis, with its
Corresponding interest rate on the Y axis, or:
Interest Rate (yield to maturity)
Term to maturity
Summary: Three Yield Curve Shapes

There are three basic shapes that yield curves can
take. These are:




Ascending Yield Curves (“Upward Sloping; Positive”) :
Long term interest rates higher than short interest term
Descending Yield Curves (“Downward Sloping; Inverted;
Negative”) : Short term interest rates higher than long term
interest rates.
(Relatively) Flat Yield Curves: Long term and short term
rates essentially the same.
Next slide illustrates these three basic shapes for
historical U.S. data.
Some Historical U.S. Yield Curves
Observations About Yield Curves
Over Long Periods of Time

More variation (i.e., basis point change)
associated with the shorter term segment of the
yield curve.
We can also
observe changes
in the yield curve
over time:
http://stockcharts.com/freecharts/yieldcurve.html

Theories to Explain the Yield Curve

There are three generally accepted theories or
explanations of the yield curve, these are:




These theories are potentially important because
they provide a framework for understanding the
shape of the yield curve, but perhaps more
importantly, we can use them to:




(Pure) Expectations Theory
Liquidity Premium Theory
Market Segmentations Theory
(1) Forecasting future interest rates.
(2) Forecasting future changes in economic activity (i.e.,
business cycle turning points).
(3) Forecasting future rates of inflation.
In this lecture, we will concentrate on the first
theory, the Pure Expectations Theory.
The (Pure) Expectations Theory



Financial market’s expectations regarding future
interest rates will shape the observed yield curve.
Model assumes that financial markets are efficient.
What does this mean for interest rates?




Market participants are constantly forming expectations
about future interest rates (these are referred to as forward
interest rates).
These forward rates are based upon the markets’ analysis
of all relevant events likely to affect interest rates in the
future (e.g., central bank actions, inflationary expectations,
business cycles, etc.).
These forward rates are incorporated into current
market interest rates (also referred to as spot
interest rates).
Finally, these spot interest rates are the interest
rates represented in observed yield curves.
Expectations Theory: What Determines the
Long Term Spot Interest Rate

This theory assumes that the current long term spot
interest rate which we can observe is comprised of 2
components



(1) A current shorter term spot interest rate and
(2) The expected, or forward, interest rate.
Determine the 2-year spot rate assuming:




(1) the current 1 year spot interest rate is 3% and
(2) the forward 1 year interest rate, 1 year from now is 5%.
Also, assume there is no risk of default and no required
premium for longer term financial assets.
Given the above, what will the market set as the current 2
year spot interest rate?
Solving for the Long Term Spot
Interest Rate


Recall, the one year rate is 3% and the expected
1 year rate one year from now (i.e., forward rate)
is 5%
Answer: The “equilibrium” 2 year spot rate is
4.0%

Why?
 Because at 4%, the financial markets are “indifferent”
to investing (or lending) for 1 year or 2 years.
 Note: A series of 2, one year instruments will offer an
average annual return or 4% (= 3+5/2) while the 2
year instrument will offer an annual return of 4%.
Illustrating the Expectations
Theory


Assume:
 (1) it, the current 1 year spot interest rate is 3% and
 (2) iet+1, the forward 1 year interest rate, 1 year from
now is 5%.
Then:
 (3) i2t, the current 2 year spot will equal 4%
More Examples

Assume the following:







Current (spot) one year rate is 5% and
The market’s forward one year rates over the next five
years (years 2, 3, 4, and 5) are:
Beginning of year 2: 6%,
Beginning of year 3: 7%,
Beginning of year 4: 8%, and
Beginning of year 5: 9%.
Given this data, calculate the following long term
spot rates:


Current (spot) two year bond rate
Current (spot) five year bond rate
Market’s Long Term Spot Rates


2 Year Interest Rate: The market’s current rate on a
two-year bond is calculated as follows:

Current (spot)1 year rate is 5% and forward 1 year rate, 1
year from now is 6%, then:

Current 2 year bond rate = (5% + 6%)/2 = 5.5%
5 year Interest Rate: The market’s current rate on a
five-year bond is calculated as follows:

Current (spot)1 year rate is 5% and forward 1 year rates, 1
year from now through five years from now are: 6%, 7%,
8%, and 9%, then:

Current 5 year bond rate = (5% + 6% + 7% + 8% + 9%)/5 =
7%
Expectations For Rising Interest Rates and
the Observed Yield Curve

If interest rates are expected to rise in future, forward
rates will be above today's observed spot rates.



And these higher forward rates will result in higher
observed spot rates as we move out the maturity.
Recall previous example:

The current (spot)1 year rate was 5% and forward 1 year
rate, 1 year from now was 6%.

As a result the 2 year spot rate is 5.5% and is higher than the 1
year spot rate (5%)
Yield Curve Question: Given the situation above where
forward rates are higher than current spot rates
(because the market expects higher rates in the future),
what will the current yield curve look like?
Yield Curve When Market Expects
Higher Interest Rates in the Future


If the current 1 year spot rate
(S1) is 5.0% and
Forward 1 year rates are:




if2 = 6%,
if3 = 7%,
If4 = 8%
If5 = 9%
Then observed long term spot
rates are:






Spot 2 year (S2) = 5.5%
Spot 3 year (S3) = 6.0%
Spot 4 year (S4) = 6.5%
Spot 5 year (S5) = 7.0%
Conclusion: The observed yield
curve is upward sloping
because the market expects
higher interest rates in the
future.
Spot interest rate
9.0%
oif5
8.5
8.0
oif4
7.5
7.0
oif3
os5
6.5
Os4
6.0
oif2Os3
5.5
os2
5.0 os1
1 2 3 4 5 year
Term to Maturity →
One More Example

Now assume the following:







Current (spot) one year rate is 9% and
The market’s forward one year rates over the next five
years (years 2, 3, 4, and 5) are:
Beginning of year 2: 8%,
Beginning of year 3: 7%,
Beginning of year 4: 6%, and
Beginning of year 5: 5%.
Given this data, calculate the following long term
spot rates:


Current (spot) two year bond rate
Current (spot) five year bond rate
Expectations For Falling Interest Rates and
the Observed Yield Curve


If interest rates are expected to fall in future, forward
rates will be below today's observed spot rates.

And these lower forward rates will result in lower observed
spot rates as we move out the maturity.

Assume: The current (spot)1 year rate is 9% and forward 1
year rate, 1 year from now is 8%

Given this information, the 2 year equilibrium spot rate is
8.5% (9% + 8%)/2 = 8.5%

Note: The 2 year spot rate is lower than the 1 year spot rate.
Yield Curve Question: Given this situation where forward
rates are lower than current spot rates (because the
market expects lower rates in the future), what will the
current yield curve look like?
Yield Curve When Market Expects
Lower Interest Rates in the Future
If the current 1 year spot rate
(S1) is 9%, and :
 Forward 1 year rates are:






Then observed long term spot
rates are:





if2 = 8%,
if3 = 7%,
If4 = 6%
If5 = 5%
Spot 2 year (S2) = 8.5%
Spot 3 year (S3) = 8.0%
Spot 4 year (S4) = 7.5%
Spot 5 year (S5) = 7.0%
Conclusion: The observed yield
curve is downward sloping
because the market expects
lower interest rates in the future.
Spot interest rate
9.0% oS1
8.5
oS2
8.0
oif2 Os3
7.5
os4
7.0
oif3 os5
6.5
6.0
oif4
5.5
5.0
oif5
1 2 3 4 5 year
Term to Maturity →
Yield Curve When Market Expects No
Change in Interest Rates in the Future
If the current 1 year spot rate
(S1) is 7%, and :
 Forward 1 year rates are:






Then observed long term spot
rates are:





if2 = 7%,
if3 = 7%,
If4 = 7%
If5 = 7%
Spot 2 year (S2) = 7.0%
Spot 3 year (S3) = 7.0%
Spot 4 year (S4) = 7.0%
Spot 5 year (S5) = 7.0%
Conclusion: The observed yield
curve is flat because the market
expects no change in interest
rates in the future.
Spot interest rate
7.5%
7.0 os1
6.5
1
oei2 oie3 oie4 os5
2
3
4
5 year
Term to Maturity →
Summary of Expectations Regarding
Future Interest Rates

The shape and slope of the yield curve reflects the
markets’ expectations about future interest rates.
 Upward Sloping (Ascending, Positive) Yield Curves:
 Future (forward) interest rates are expected to
increase above existing spot rates.
 Downward Sloping (Descending, Inverted, Negative)
Yield Curves:
 Future (forward) interest rates are expected to
decrease below existing spot rates.
 Flat Yield Curves
 Future (forward) interest rates are expected to
remain the same as existing spot rates.
Appendix 1
Yield Curve web sites.
Yield Curves on Line


Visit the following sites:
(1) Bloomberg: U.S. Treasury and Selected Foreign
Country Yield Curves



(2) Dynamic U.S. Treasury Yield Curve
http://stockcharts.com/charts/yieldcurve.html


You can view yield curve changes over time.
(3)Yield Curves for Treasuries and Corporates


http://www.bloomberg.com/markets/rates/index.html
http://www.bondsonline.com/Todays_Market/Composit
e_Bond_Yields.php
(4) Nominal and Real U.S. Treasury Yield Curves

http://www.treasury.gov/resource-center/data-chartcenter/interest-rates/Pages/Historic-Yield-DataVisualization.aspx
Yield Curves for Foreign Countries

United Kingdom (and U.S.)

http://www.yieldcurve.com/marketyieldcurve.asp


Euro-Zone

http://www.ecb.int/stats/money/yc/html/index.en.html


This site lets you compare UK and US yield curves.
Note: Select “spot rate” curve for current Euro-zone yield curve
Other countries:


http://www.bloomberg.com/
Link to Markets, then to Government Bonds (current
yield curves for U.K., UK, Germany, Japan, Australia
and Brazil)