chapter 15 - university of colorado...

INVESTMENTS | BODIE, KANE, MARCUS

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin

CHAPTER 15

The Term Structure of Interest

Rates


Overview of Term Structure

• The yield curve is a graph that displays the relationship between yield and maturity.

• Information on expected future short term rates can be implied from the yield curve.

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Figure 15.1 Treasury Yield Curves

See

Treasury.gov Many other interesting links, for example:

stockcharts.com

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http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield

http://stockcharts.com/freecharts/yieldcurve.html


Figure 15.1 Treasury Yield Curves

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Bond Pricing

• Yields on different maturity bonds are not all equal – there is a term structure.

• We need to consider each bond cash flow as a stand-alone zero-coupon bond.

• The value of the bond should be the sum of the values of its parts.

• Bond stripping and bond reconstitution offer opportunities for arbitrage.

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Table 15.1 Prices and Yields to Maturities on Zero-Coupon Bonds ($1,000 Face Value)

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tt

ytm

CashFlow

1Price

These prices are in the form:

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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

• YTM = 6.88%

• 6.88% is less than the 3-year rate of 7%.

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32 07.1

1100$

06.1

100$

05.1

100$Price

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Two Types of Yield Curves

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Pure Yield Curve

• The pure yield curve

uses stripped or zero

coupon Treasuries.

• The pure yield curve

may differ

significantly from the

on-the-run yield

curve.

On-the-run Yield Curve

• The on-the-run yield

curve uses recently

issued 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 (or no arbitrage) requires that both strategies provide the same return.

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1+r1 1+r2

(1+y2)2

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INVESTMENTS | BODIE, KANE, MARCUS 15-10

Figure 15.2 Two 2-Year Investment Programs

1+r1 1+r2

(1+y2)2

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Yield Curve Under Certainty

• Buy and hold vs. rollover:

• Next year’s 1-year rate (r2) is just enough to make rolling over a series of 1-year bonds equal to investing in the 2-year bond.

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21

2

2 111 rry

21

212 111 rry

1+r1 1+r2

(1+y2)2

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Spot Rates vs. Short Rates

• Spot rate – the rate that prevails today for a given maturity

• Short rate – the 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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11...111

21 nnn rrry

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Short Rates and Yield Curve Slope

• When next year’s short

rate, r2 , is less than this

year’s short rate, r1, the

yield curve slopes down.

– May indicate

market expects

rates to fall.

• When next year’s short

rate, r2 , is greater than

this year’s short rate, r1,

the yield curve slopes

up.

– May indicate

market expects

rates to rise.

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Figure 15.3 Short Rates versus Spot Rates

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Forward Rates from Observed Rates

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1

1)1(

)1()1(

n

n

n

nn

y

yf

fn = one-year forward rate for period n

yn = yield for a security with a maturity of n n

nn

n

n yfy )1()1()1( 1

1

1+fn

(1+yn)n

(1+yn-1)n-1

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Example 15.4 Forward Rates

• The forward interest rate is a forecast of a future short rate implied by the market.

• Example: compute forward rate for year 4:

– rate for 4-year maturity = 8%

– rate for 3-year maturity = 7%

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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 today’s rate is 5% and the expected short 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:

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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.

or:

– Buy the 1-year bond today and hold to

maturity.

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• What if next year’s 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 a longer-term bond.

• This liquidity premium compensates short-term investors for the uncertainty about future prices.

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Theories of Term Structure

• Expectations

–Forward rates come from market

consensus

• Liquidity Preference

–Upward bias over expectations due to

premium the market requires

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Expectations Theory

• Observed long-term rate is a function of today’s short-term rate and expected future short-term rates.

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)1)(1()1( 21

2

2 fyy

• fn = E(rn) and liquidity premiums are

zero.

)1)(1()1( 21

2

2 rEyy

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Liquidity Premium Theory

• Long-term bonds carry more risk; therefore, fn generally exceeds E(rn)

• The excess of fn over E(rn) is the liquidity premium

• The yield curve has an upward bias built into the long-term rates because of the liquidity premium

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Figure 15.4 Yield Curves - A

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Figure 15.4 Yield Curves - B

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Figure 15.4 Yield Curves - C

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Figure 15.4 Yield Curves - D

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Yield Curves

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Yield Curves (cont.)

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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 the eventually 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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chapter 15 - university of colorado...

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