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

7

Forward Rates

Example: Current YTM on 1-yr zero is 7%. Current YTM on 2-yr zero is 8.98%

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.

etYTM 11

13

Forward Rates

Expected gross return from investing $1 in

strategy 1 as of time t=1:

E[R1]=(1.07)(1+ )

The gross return from investing $1 in strategy 2 (known as of time t=1).

R2=(1.08)2

etYTM 11

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

21 )08.1()1)(07.1( etYTM

%9107.1

)08.1( 2

1 etYTM

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%

22 )08.1()1)(07.1( f

%9107.1

)08.1( 2

1 etYTM

18

Forward Rates

The n-period forward rate, , is found by solving

for

nf

ntnn

ntn YTMfYTM )1()1()1( ,

1,1

.nf

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Example

Suppose as of time t:– YTM on a 2-year zero is 10%– YTM on a 3-year zero is 12%– What is ? 3f

%11.16

)12.1()1()10.1(

3

33

2

f

f

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)=1.104 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= 793.83 in market value of 3-year

zeros– Fund the purchase by borrowing 793.83 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 793.83(1.104)=1162.25

This is 16.22% higher than the cash received

1162.25/1000 – 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:

211

22111 )1()1)(1(

fYTM

YTMYTMYTM

et

tett

is, That

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Side Note: Geometric Mean

The mean of N random variables is

The geometric mean is defined as

NNG XXXX /1

21 )...(

NXXXX N /)...( 21

25

Side Note: Geometric Mean

Example with gross returns: 1.1, 1.05, .95, 1.02 – Mean = 4.12/4 = 1.03– Geometric mean = (1.11.05.951.02)1/4 = 1.0285

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,

)1()]1)(1[(

)1()1)(1(

22/1

111

22111

tett

tett

YTMYTMYTM

YTMYTMYTM

so

)1(2/)]1()1[( 2111 tett YTMYTMYTM

27

Expectations Theory

But then

In other words, the two period rate is approximately an average of the two one-period rates

tett

tett

YTMYTMYTM

YTMYTMYTM

2111

2111

2/)(

12/)(1

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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.

tett YTMYTMYTM 2111 2/)(

ett YTMYTM 111

tt YTMYTM 21

ett YTMYTM 111

ett YTMYTM 111

tt YTMYTM 21

tt YTMYTM 21

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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.

t

et

ett

tet

ett

YTMYTMYTMYTM

YTMYTMYTMYTM

321111

3321111

3

)1()1)(1)(1(

implies ionapproximat an to which

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Example

Expected one-period spot rates

Then

A rising trend in expected short-term rates produces an upward sloping yield curve.

%7%,6%,5%,4 3121111 et

et

ett YTMYTMYTMYTM

%5.54/%)7%6%5%4(

%53/%)6%5%4(

%5.42/%)5%4(

4

3

2

t

t

t

YTM

YTM

YTM