7.0Term Structure of Interest Rates

8/12/2019 7.0Term Structure of Interest Rates

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CHAPTER 15

The Term Structure of Interest

Rates


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The yield curve is a graph that displays therelationship between yield and maturity.

Information on expected future short termrates can be implied from the yield curve.

Overview of Term Structure


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Figure 15.1 Treasury Yield Curves


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Bond Pricing

Yields on different maturity bonds are not allequal.

We need to consider each bond cash flow

as a stand-alone zero-coupon bond. Bond stripping and bond reconstitution

offer opportunities for arbitrage.

The value of the bond should be the sumof the values of its parts.


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Table 15.1 Prices and Yields to Maturities on

Zero-Coupon Bonds ($1,000 Face Value)


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Example 15.1 Valuing Coupon Bonds

Value a 3 year, 10% coupon bond using

discount rates from Table 15.1:

Price = $1082.17 and YTM = 6.88% 6.88% is less than the 3-year rate of 7%.

32 07.11100$

06.1100$

05.1100$Price


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Two Types of Yield Curves

Pure Yield Curve

The pure yield curve

uses stripped or zero

coupon Treasuries.

The pure yield curve

may differ significantly

from the on-the-runyield curve.

On-the-run Yield Curve

The on-the-run yield

curve uses recentlyissued coupon bonds

selling at or near par.

The financial press

typically publishes on-the-run yield curves.


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Yield Curve Under Certainty

Suppose you want to invest for 2 years. Buy and hold a 2-year zero

-or-

Rollover a series of 1-year bonds

Equilibrium requires that both strategies

provide the same return.


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Figure 15.2 Two 2-Year Investment Programs


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Yield Curve Under Certainty

Buy and hold vs. rollover:

Next years 1-year rate (r2) is just enough

to make rolling over a series of 1-year

bonds equal to investing in the 2-yearbond.

22 1 2

1

2

2 1 2

(1 ) (1 ) (1 )

1 (1 ) (1 )

y r x r

y r x r


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Spot Rates vs. Short Rates

Spot ratethe rate that prevails today for a

given maturity

Short ratethe rate for a given maturity (e.g.

one year) at different points in time.

A spot rate is the geometric average of its

component short rates.


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Short Rates and

Yield Curve Slope

When next years short

rate, r2, is greater than

this years short rate, r1,

the yield curve slopesup.

May indicate rates

are expected to rise.

When next years short

rate, r2, is less than this

years short rate, r1, the

yield curve slopesdown.

May indicate rates

are expected to fall.


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Figure 15.3 Short Rates versus Spot Rates


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11)1(

)1()1(

nn

nn

ny

yf

fn= one-year forward rate for period n

yn= yield for a security with a maturity of n

)1()1()1( 1

1 n

n

n

n

n fyy

Forward Rates from Observed Rates


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Example 15.4 Forward Rates

The forward interest rate is a forecast of a

future short rate.

Rate for 4-year maturity = 8%, rate for 3-year

maturity = 7%.

1106.107.1

08.1

1

11

3

4

3

3

4

44

y

yf

%.f 06114


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Interest Rate Uncertainty

Suppose that todays rate is 5% and the

expectedshort rate for the following year is

E(r2)= 6%. The value of a 2-year zero is:

The value of a 1-year zero is:

47.898$

06.105.1

1000$

38.952$05.1

1000$


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The investor wants to invest for 1 year.

Buy the 2-year bond today and plan to sell

it at the end of the first year for $1000/1.06

=$943.40.

0r-

Buy the 1-year bond today and hold to

maturity.


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What if next years interest rate is more (or

less) than 6%?

The actual return on the 2-year bond is

uncertain!


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Investors require a risk premium to hold alonger-term bond.

This liquidity premium compensates short-term investors for the uncertainty about

future prices.


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Expectations

Liquidity Preference

Upward bias over expectations

Theories of Term Structure


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Expectations Theory

Observed long-term rate is a function of

todays short-term rate and expected

future short-term rates.

fn= E(rn) and liquidity premiums are zero.


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Long-term bonds are more risky; therefore,

fngenerally exceeds E(rn)

The excess offnover E(rn) is the liquiditypremium.

The yield curve has an upward bias builtinto the long-term rates because of the

liquidity premium.

Liquidity Premium Theory


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Figure 15.4 Yield Curves


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Figure 15.4 Yield Curves


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Interpreting the Term Structure

The yield curve reflects expectations of future

interest rates.

The forecasts of future rates are clouded by

other factors, such as liquidity premiums. An upward sloping curve could indicate:

Rates are expected to rise

And/or Investors require large liquidity premiums to

hold long term bonds.


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Interpreting the Term Structure

The yield curve is a good predictor of the

business cycle.

Long term rates tend to rise in anticipation of

economic expansion.

Inverted yield curve may indicate that interest

rates are expected to fall and signal a

recession.


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Figure 15.6 Term Spread: Yields on 10-year vs.

90-day Treasury Securities


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Forward Rates as Forward Contracts

In general, forward rates will not equal theeventually realized short rate

Still an important consideration when

trying to make decisions : Locking in loan rates


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Figure 15.7 Engineering a Synthetic Forward

Loan

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7.0Term Structure of Interest Rates