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FINC4101 Investment Analysis

Instructor: Dr. Leng Ling

Topic: Term Structure Analysis

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Learning objectives1. Define term structure of interest rates and yield curve.

2. Explain the purpose of theories of the term structure

3. Describe the expectations theory and discuss its implications.

4. Show how different expectations of future short-term interest rates can lead to different yield curves.

5. Use the expectations theory to infer future short-term interest rates (forward rates).

6. Describe the liquidity preference theory.

7. Define the liquidity premium.

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Term structure of interest rates Term structure of interest rates:

Relationship between yields to maturity and

term to maturity across bonds.

Yield curve: The graphical representation of the term structure.

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Term structure/ yield curve has 4 different ‘looks’

Term structure/ yield curve can take on one of the following shapes:

1. Rising/ upward sloping (most common)

2. Inverted/ downward sloping/ falling

3. Hump-shaped, rising and then falling

4. Flat

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Term structure/ yield curve has 4 different ‘looks’

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Expectations Theory (1) Yields to maturity are determined by

expectations of future short-term interest rates.

In other words, the shape of the term structure/ yield curve depends on the expected short-term interest rates in the future.

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Expectations Theory (2) Depending on what investors expect the

future short-term rate to be: Higher than current short-term rate, or Lower than current short-term rate, or Same as current short-term rate

We get different yield curves.

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Expectations Theory (3) market expecting increases in future short-

term interest rates leads to upward sloping yield curve.

market expecting decreases in future short-term interest rates leads to downward sloping yield curve.

market expecting no change in future short-term interest rates leads to flat yield curve.

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How does the expectations theory work?

Current 1-year interest rate is 8%. Suppose everyone in the market expects that

one year from now, the 1-year interest rate will rise to 10%.

How does this expectation determine the current 2-year interest rate?

All interest rates are quoted on an annual basis.

In this situation, investors can choose one of two possible strategies.

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Investment strategies For simplicity, assume

All bonds are zero-coupon bonds with $1000 face value. You can invest in fractional amount of bonds

Rollover: Buy 1-year bond now. When it matures, buy another 1-year bond next year and hold till

it matures. Buy-and-hold:

Buy a 2-year bond now and hold it till it matures in year 2.

The expectations theory says that the two strategies should produce the same expected total returns.

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Deriving the 2-year interest rate (1) Suppose you have $1 to invest. With the rollover strategy, at the end of year 1, you get:

$1 x (1 + current 1-year interest rate)

= 1 x (1.08) = 1.08 Suppose at the end of year 1, the 1-year interest rate is

10% as expected. You invest $1.08 (total proceeds) in a 1-year bond

promising 10%. After two years, you get: 1.08 x (1.10) = $1.188

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Deriving the 2-year interest rate (2)

Let y2 be the 2-year interest rate. According to the expectations theory,

investing $1 in a 2-year bond with an interest rate of y2 must also produce a total return of $1.188. That is,

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2

2

$1 (1 ) 1.188

1 1.188

1.188 1 0.08995 or 8.995%

y

y

y

´ + =

+ =

= - =

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Compare 1-year and 2-year rates 1-year interest rate: 8% 2-year interest rate: 8.995%

Conclusion: longer maturity bonds have higher interest rates/ yields to maturity.

The term structure is upward sloping!

* If we observe 1-year rate 8% and 2-year rate 8.995%, then market expects that, one year from now, the 1-year rate will increase from current 8% to 10%.

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Verify for yourself If the market expects future 1-year interest

rate to fall to 6%: 2-year interest rate is 6.995%. Yield curve is downward sloping.

If market expects future 1-year interest rate to stay at 8%: 2-year interest rate is 8%. Yield curve is flat.

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Inferring expected future rates (1) Expectations theory says that the expected total returns

from the buy-and-hold strategy and the rollover strategy are the same.

We can use this assumption to infer expected future interest rates, which are called “forward rates”.

Forward rate: The inferred short-term interest rate for a future period that makes the expected total return of a long-term bond equal to that of rolling over short-term bonds.

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Inferring expected future rates (2) In general, we obtain the forward rate by

equating the return on an n-period zero-coupon bond with that of an (n – 1)-period zero-coupon bond rolled over into a one-year bond in year n:

(1 + yn)n = (1 + yn-1)n-1(1 + fn)

YTM of n-period zero coupon bond

YTM of (n-1)-period zero coupon bond

Forward rate for period n.

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Inferring expected future rates (3) Thus, the forward rate formula is

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(1 )1

(1 )

nn

n nn

yf

y --

+= -

+

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Applying the forward rate formula Two-year maturity bonds offer yield-to-

maturity of 6%, and three-year bonds have yields of 7%. What is the forward rate for the third year?

Verify that the forward rate is 9.03%

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Problems (1) The current yield curve for default-free zero-

coupon bonds is as follows:

Assume that the zero-coupon bonds have a face value of $1000.

Maturity (Years) YTM

1 10%

2 11

3 12

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Problems (2)1. What are the implied one-year forward rates?

2. Assume that the pure expectations hypothesis is correct. If market expectations are accurate, one year from now what will be the yields to maturity on one- and two-year zero coupon bonds?

3. If you purchase a two-year zero-coupon bond now, what is the expected total rate of return over the first year? What if you purchase a three-year zero-coupon bond?

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Liquidity Preference Theory Investors prefer to hold short-term bonds because short-

term bonds have more ‘liquidity’ than long-term bonds. Short-term bonds offer greater price certainty and trade in more

active markets.

Investors like these attributes.

Because long-term bonds are less liquid by comparison,

investors demand a liquidity premium for holding long-

term bonds. Liquidity premium: extra expected return demanded by investors

as compensation for the lower liquidity of long-term bonds.

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Liquidity Preference Theory The theory says that investors demand a

liquidity premium on long-term bonds. Thus, long-term bonds have higher interest rates

than short-term bonds. This gives rise to an upward sloping yield curve. Even if future short-term interest rates are

expected to remain unchanged, yield curve will be upward sloping because of the liquidity premium.

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Liquidity premium The spread between the forward rate of

interest and the expected short-term rate: Liquidity premium= forward rate – expected short-term rate

Re-arranging the formula, we getForward rate = expected short-term rate + liquidity premiumSo, forward rate > expected short-term rate

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More problems (1) The yield to maturity on one-year zero

coupon bonds is 8%. The yield to maturity on two-year zero-coupon bonds is 9%.

a) What is the forward rate for the second year?

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More problems (2)

b) If you believe the expectations theory, what is your best guess as to the expected short-term interest rate next year?

c) If you believe the liquidity preference theory, is your best guess as to next year’s short-term interest rate higher or lower than in (b)?

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More problems (3) Which of the following statements is true?1. The expectations theory indicates a flat yield curve if

anticipated future short-term rates exceed current short-term rates.

2. The basic conclusion of the expectations theory is that the long-term rate is equal to the anticipated short-term rate.

3. The liquidity preference theory indicates that, all other things being equal, longer maturities will have higher yields.

4. The liquidity preference theory states that a rising yield curve implies that the market anticipates increases in interest rates.

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Summary1. Term structure of interest rates.

2. Theories of the term structure try to explain the shape of the term structure/ yield curve.

3. Pure Expectations theory and its implications.

4. Calculate forward rates using observed interest rates of different maturities.

5. Liquidity Preference theory and its implications.

6. Relationship between forward rate, expected short-term rate and liquidity premium.

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Practice 8 Chapter 10: 39 (a)(b), 41.