expectations theory

Chapter 5How Do The Risk and Term Structure Affect Interest RatesPart B

http://www.youtube.com/watch?v=b_cAxh44aNQhttp://www.youtube.com/watch?v=kFp6jk3zBhg

http://www.youtube.com/watch?v=BqKWcZTtxwg

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Term Structure Facts to Be Explained

Besides explaining the shape of the yield curve, a good theory must explain why:

• Interest rates for different maturities move together.

• Yield curves tend to have steep upward slope when short rates are low and downward slope when short rates are high.

• Yield curve is typically upward sloping.

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Three Theories of Term Structure

1. Expectations Theory – Pure Expectations Theory explains 1 and 2,

but not 3

2. Market Segmentation Theory– Market Segmentation Theory explains 3, but not 1

and 2

3. Liquidity Premium Theory– Solution: Combine features of both Pure

Expectations Theory and Market Segmentation Theory to get Liquidity Premium Theory and explain all facts

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Expectations Theory

• Key Assumption: Bonds of different maturities are perfect substitutes

• Implication: Re on bonds of different maturities are equal

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Expectations Theory

To illustrate what this means, consider two alternative investment strategies for a two-year time horizon.1. Buy $1 of one-year bond, and when it

matures, buy another one-year bond with your money.

2. Buy $1 of two-year bond and hold it.

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Expectations Theory

The important point of this theory is that if the Expectations Theory is correct, your expected wealth is the same (a the start) for both strategies. Of course, your actual wealth may differ, if rates change unexpectedly after a year.

We show the details of this in the next few slides.

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Expectations Theory - Notation

• We use zero-coupon bonds for this analysis to avoid the need to make reinvestment assumptions

• Zero coupon rates are “spot” rates• We will designate spot rates as S1, S2, S3 etc. where S1 = a

one-year bond• We will designate forward rates (rates for periods that start

in the future, not today) as 1f1, 1f2, 2f1 etc.• 2f1 = the two-year forward rate starting one year from now• 1f2 = the one-year forward rate starting two years from now

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(1+ it )(1+ it+1e ) −1= 1+ it + it+1

e + it (it+1e ) −1

Expectations Theory

• Expected return from strategy 1

Since it(iet+1) is also extremely small, expected return is approximately

it + iet+1

(1+S1)(1+ 1f1) = Ending Balance(End Balance/Beginning Balance)-1 = HPR(1+ HPR)^.5-1 = S2 = expected return

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Expectations Theory


(1+S1)(1+ 1f1) = Ending Balance(End Balance/Beginning Balance)-1 = HPR(1+ HPR)^.5-1 = S2 = expected return (annualized)

S1 1f1

S2

0 1 2

S2

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(1+ i2t )(1+ i2t ) −1= 1+ 2(i2t ) + (i2t )2 −1

Since (i2t)2 is extremely small, expected return is approximately 2(i2t)

Expectations Theory


(1+S2)(1+ S2) = Ending Balance(End Balance/Beginning Balance)-1 = HPR(1+ HPR)^.5-1 = S2 = expected return (annualized)

S2 S2

0 1 2

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i2t =it + it+1

e

2

Expectations Theory• From implication above expected returns of two

strategies are equal• Therefore

2 i2t( )= it + it+1e

Solving for i2t

(1)

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Expectations Theory• From implication above expected returns of two

strategies are equal• Therefore

Solving for 1f1

(1)

(1+S2)(1+ S2)-1 = (1+S1)(1+ 1f1)-1

1f1 = ((1+S2)(1+ S2))/(1+S1) – 1

1f1 = (End Balance/Beg Balance)-1Note that 1f1 is a HPR. We do not need to annualizeIt because it is exactly a one-year period

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Annualizing HPR

Remember that if we did need to annualize, we would use our same formula as always:

(1+HPR)^(# of HPs in one year) – 1

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Expectations Theory Example 1

• Assume S1 = 5% and S2 = 7%, solve for 1f1:

1f1 = ((1+S2)(1+ S2))/(1+S1) – 1

1f1 = (End Balance/Beginning Balance)-1

1f1 = ((1+.07)(1+ .07))/(1+.05) – 1

1f1 = 1.14490/1.05000 -1 = 1.0904 -1 = .0904

S1 = 5% 1f1 = ?

S2 = 7%

0 1 2

S2 = 7%

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• Assume S1 = 5% S3 = 10% 1f1 = 9.04%, solve for 1f2:

1f2 = ((1+S3)(1+ S3)(1+S3)/(1+S1)(1+1f1) – 1

1f2 = (End Balance/Beginning Balance)-1

1f2 = ((1+.10)(1+ .010)(1+.10))/(1+.05)(1.0904) – 1

1f2 = 1.3310/1.1449 -1 = 1.1625 -1 = .1625

S1 = 5% 1f1 = 9.04

S3 = 10%

0 1 2

S3 = 10%

1f2 = ?

S3 = 10%

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• Assume S1 = 4% S3 = 6%, solve for 2f1:

2f1 hpr = ((1+S3)(1+ S3)(1+S3)/(1+S1) – 1

2f1 hpr = (End Balance/Beginning Balance)-1

2f1 hpr = ((1+.06)(1+ .06)(1+.06))/(1+.04) – 1

2f1 hpr = 1.19102/1.04 -1 = 1.14521 -1 = .14521

2f1 = (1+.14521)^.5 -1 = .07014 annualized

S1 = 4% 2f1 = ?

S3 = 6%

0 1 2

S3 = 6%

2f1 = ?

S3 = 6%

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• Assume S1 = 4% 1f1 = 4.5% 1f2 = 5.2% solve for S3

S3 hpr = ((1+S1)(1+ 1f1)(1+1f2)) – 1

S3 hpr = ((1.04)(1.045)(1.052)-1 = (1.14331) -1 =.14331S3 = (1.14331)^(1/3) – 1 = .04566 annualized

S1 = 4% 1f1 = 4.5%

S3 = ?

0 1 2

S3 = ?

1f2 = 5.2%

S3 = ?

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Expectations Theory and Term Structure Facts

• Explains why yield curve has different slopes1. When short rates are expected to rise in future,

average of future short rates = int is above today's short rate; therefore yield curve is upward sloping.

2. When short rates expected to stay same in future, average of future short rates same as today's, and yield curve is flat.

3. Only when short rates expected to fall will yield curve be downward sloping.

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• Pure expectations theory explains fact 1—that short and long rates move together1. Short rate rises are persistent

2. If it ↑ today, iet+1, iet+2 etc. ↑ ⇒average of future rates ↑ ⇒ int ↑

3. Therefore: it ↑ ⇒ int ↑(i.e., short and long rates move together)

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• Explains fact 2—that yield curves tend to have steep slope when short rates are low and downward slope when short rates are high1. When short rates are low, they are expected to rise

to normal level, and long rate = average of future short rates will be well above today's short rate; yield curve will have steep upward slope.

2. When short rates are high, they will be expected to fall in future, and long rate will be below current short rate; yield curve will have downward slope.

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• Doesn't explain fact 3—that yield curve usually has upward slope– Short rates are as likely to fall in future as rise,

so average of expected future short rates will not usually be higher than current short rate: therefore, yield curve will not usually slope upward.

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Market Segmentation Theory

• Key Assumption: Bonds of different maturities are not substitutes at all

• Implication: Markets are completely segmented;interest rate at each maturity aredetermined separately

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Market Segmentation Theory

• Explains fact 3—that yield curve is usually upward sloping– People typically prefer short holding periods and thus have

higher demand for short-term bonds, which have higher prices and lower interest rates than long bonds

• Does not explain fact 1or fact 2 because its assumes long-term and short-term rates are determined independently.

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Liquidity Premium Theory

• Key Assumption: Bonds of different maturities are substitutes, but are not perfect substitutes

• Implication: Modifies Pure Expectations Theory with features of Market Segmentation Theory

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• Investors prefer short-term rather than long-term bonds. This implies that investors must be paid positive liquidity premium, int, to hold long term bonds.

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• Modifies the expectations theory• Forward rates will be lower than calculated

by the expectations theory• Helps to explain the recurring upward slope

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Numerical Example

1. One-year interest rate over the next five years: 5%, 6%, 7%, 8%, and 9%

2. Investors' preferences for holding short-term bonds so liquidity premium for one-to five-year bonds: 0%, 0.25%, 0.5%, 0.75%, and 1.0%

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Numerical Example (Simplified)

• Interest rate on the two-year bond:0.25% + (5% + 6%)/2 = 5.75%

• Interest rate on the five-year bond:1.0% + (5% + 6% + 7% + 8% + 9%)/5 = 8%

• Interest rates on one to five-year bonds:5%, 5.75%, 6.5%, 7.25%, and 8%

• Comparing with those for the pure expectations theory, liquidity premium theory produces yield curves more steeply upward sloped

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Liquidity Premium Theory: Term Structure Facts

• Explains All 3 Facts– Explains fact 3—that usual upward sloped

yield curve by liquidity premium for long-term bonds

– Explains fact 1 and fact 2 using same explanations as pure expectations theory because it has average of future short rates as determinant of long rate

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• There is no easy way to observe the level of liquidity premium(s)

• One could statistically analyze past yield curve shapes

• In any case – from a trading standpoint, you may be able to “harvest” the premium

Market Predictions of Future Short Rates

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Evidence on the Term Structure

• Initial research (early 1980s) found little useful information in the yield curve for predicting future interest rates.

• Recently, more discriminating tests show that the yield curve has a lot of information about very short-term and long-term rates, but says little about medium-term rates.

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Case: Interpreting Yield Curves

• The picture on the next slide illustrates several yield curves that we have observed for U.S. Treasury securities in recent years.

• What do they tell us about the public’s expectations of future rates?

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Case: Interpreting Yield Curves, 1980–2008

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• The steep downward curve in 1981 suggested that short-term rates were expected to decline in the near future. This played-out, with rates dropping by 300 bps in 3 months.

• The upward curve in 1985 suggested a rate increase in the near future.

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• The slightly upward slopes in the remaining years can be explained by liquidity premiums. Short-term rates were stable, with longer-term rates including a liquidity premium (explaining the upward slope).

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Mini-case: The Yield Curve as a Forecasting Tool

• The yield curve does have information about future interest rates, and so it should also help forecast inflation and real output production.– Rising (falling) rates are associated with

economic booms (recessions) [chapter 4].– Rates are composed of both real rates and

inflation expectations [chapter 3].

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Chapter Summary

• Risk Structure of Interest Rates: We examine the key components of risk in debt: default, liquidity, and taxes.

• Term Structure of Interest Rates: We examined the various shapes the yield curve can take, theories to explain this, and predictions of future interest rates based on the theories.

expectations theory

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