1. 05-Expectations Hypothesis
Yield Curve
Expectations Theory

2. Yield Curve
All default-free bonds do not have the same growth rates, or YTM’s.
Bonds with different maturities have different YTM’s.
Yield Curve: a plot with
time-to-maturity on the x-axis
YTM on y-axis.

3. Yield Curves

4. Term Structure Theory The Big Picture:
Why don’t default-free bonds of different maturities grow at the same rate?
What defines the shape of the yield curve?
Why does the slope change?
Why does it usually slope upwards?

5. “Living” Yield Curve

6. Yield Curve and Recessions
The long and the short of it
Jan 5th 2006
From The Economist print edition
An inverted yield curve, then, suggests that short-term rates are higher today than they will be in
the future. But why should this necessarily spell recession? Normally, it is because the Federal
Reserve is in the midst of a campaign against inflation. To win this battle, short-term rates are
sometimes raised high enough to induce a recession, which squeezes inflation out of the system.
In due course, lower inflation will pave the way for lower short-term rates. But before this
happens, long-term bond yields fall in anticipation of the future victory. In this case, an inverted
yield curve is just a measure of the Fed's power.
Alternatively, inversions may be a measure of the Fed's ignorance. The bond market may know
something the central bankers don't. Long-term rates may be subdued, because the market
anticipates a recession that will eventually force the Fed to loosen monetary policy. But short-term
rates remain high, because the Fed has yet to act on what the bond market foresees.

7. Forward Rates Example: Current YTM on 1-yr zero is 7%.
Consider two strategies:
1: Buy 1-yr bond and roll over earnings at end
of year to another 1-yr zero-coupon bond
2: Buy a two year zero-coupon bond

8. Forward Rates Strategy 1: Invest in a one year bond and roll over ?7%
Strategy 2: Invest in a two year bond
8.98%
8.98%

9. Forward Rates
If we choose strategy 2, then we know for sure the growth rate of our money each year, provided we hold the bond to maturity.
If we chose strategy 1, we don’t know exactly what we will earn over the second year because we don’t know exactly what bond prices (yields) will be at the end of the year.

10. Notation YTMjt is the yield to maturity of a
Zero coupon bond that matures in j years from time t.
Is not completely observable until time t
Time t is today, t+1 is next year, etc.
The yield to maturity of a zero coupon bond is often referred to as the “spot” interest rate.

11. Notation Examples: YTM1t is the YTM on a ________that matures
________________
YTM2t is the YTM on a ________that matures
YTM1t+1 is the YTM on a _______that matures
YTM2t+2 is the YTM on a _______ that matures
1-yr zero
in one year from today.
2-yr zero
in two years from today.
1-yr zero
in two years from today.
2-yr zero
in four years from today.

12. Notation
Of course, we don’t know what yields (rates) will be when measured in the future.
We can develop some expectation
is the expected YTM on a 1-yr zero one year from now, that matures two years from now.

13. Forward Rates Expected gross return from investing $1 in
strategy 1 as of time t=1:
E[R1]=(1.07)( )
The gross return from investing $1 in strategy 2 (known as of time t=1).
R2=(1.08)2

14. Forward Rates The expected gross returns must be close
Suppose E[R1] is much larger than R2
No one buys the two-year bond
Two year bond price drops, yield increases
Everyone buys the one-year bond
Price of one-year bond jumps, yield decreases
Similar argument applies if R2 is much larger than E[R1]

15. Forward Rates Suppose the expected gross returns are identical
Then a forecast of one-year rate (YTM) in one year is

16. Forward Rates
In general, the gross returns of the two strategies should be close, but do not have to be equal.
Why might they differ? More on this later . . .
No matter what is true in reality, we can always set the gross returns equal to each other, and solve for the rate that would need to prevail for the two strategies to be equal.

17. Forward Rates
Forward rates: the inferred short term rate of interest for a future period that makes the expected gross return of a long-term bond equal to that of rolling over short term bonds
The two-year forward rate is 9%

18. Forward Rates
The n-period forward rate, , is found by solving
for

19. Example Suppose as of time t: YTM on a 2-year zero is 10%
What is ?

20. Forward Rates
You can always lock in a future loan at the forward rate.
Example:
YTM on 3 yr zero: 8%
YTM on 4 yr zero: 10%
What is 4-year forward rate?
1.083(1+f4)= or f4=16.22%

21. Forward Rates
Suppose you want to borrow $1000 three years from now to be paid back four years from now. How can you lock in at the 4th year forward rate?
Buy 1000/1.083= in market value of 3-year zeros
Fund the purchase by borrowing at 10% due in four years.
(Short-sell a bond)

22. Forward Rates
In three years from now, the three year zeros provide you with a cash flow of 1000.
In four years, your liability on the four year zeros has grown to (1.104)=
This is 16.22% higher than the cash received
/1000 – 1 = 16.22%

23. Expectations Theory
Expectations Theory: The gross return from investing in a long term bond is always equal to the expected return from rolling over short term bonds.
For example:

24. Side Note: Geometric Mean
The mean of N random variables is
The geometric mean is defined as

25. Side Note: Geometric Mean
Example with gross returns: 1.1, 1.05, .95, 1.02
Mean = 4.12/4 = 1.03
Geometric mean = (1.1´1.05´.95´1.02)1/4 =
For numbers close to 1, the geometric mean is approximately equal to the mean.

26. Expectations Theory According to the Expectations Theory
But the left side is a geometric mean, so to a close approximation,

27. Expectations Theory But then
In other words, the two period rate is approximately an average of the two one-period rates

28. Expectations Theory This implies that
if then the yield curve is upward sloping, i.e.
if then the yield curve is flat, i.e.
if then the yield curve is downward sloping, i.e.

29. Expectations Theory What about longer term bonds?
In general, the expectation theory says that the n-period spot rate equals the average of the one period rates expected to occur over the n-period life of the bond.

30. Example Expected one-period spot rates Then
A rising trend in expected short-term rates produces an upward sloping yield curve.