chapter 15leeds-faculty.colorado.edu/.../fnce4030-fall-2012-ch15-handout-part… · investments |...

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

INVESTMENTS | BODIE, KANE, MARCUS

15-2

• 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.

Overview of Term Structure

INVESTMENTS | BODIE, KANE, MARCUS

15-3

Figure 15.1 Treasury Yield Curves

See

Treasury.gov Many other interesting links, for example:

stockcharts.com

INVESTMENTS | BODIE, KANE, MARCUS

15-4

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.

INVESTMENTS | BODIE, KANE, MARCUS

15-5

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

tt

ytm

CashFlow

1Price

These prices are in the form:

INVESTMENTS | BODIE, KANE, MARCUS

15-6

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%.

32 07.1

1100$

06.1

100$

05.1

100$Price

INVESTMENTS | BODIE, KANE, MARCUS

15-7

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-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.

INVESTMENTS | BODIE, KANE, MARCUS

15-8

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.

1+r1 1+r2

(1+y2)2

INVESTMENTS | BODIE, KANE, MARCUS

15-9

Figure 15.2 Two 2-Year Investment Programs

1+r1 1+r2

(1+y2)2

INVESTMENTS | BODIE, KANE, MARCUS

15-10

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.

21

2

2 111 rry

21

212 111 rry

1+r1 1+r2

(1+y2)2

INVESTMENTS | BODIE, KANE, MARCUS

15-11

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.

11...111

21 nnn rrry

INVESTMENTS | BODIE, KANE, MARCUS

15-12

Short Rates and Yield Curve Slope

• 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.

• 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.

INVESTMENTS | BODIE, KANE, MARCUS

15-13

Figure 15.3 Short Rates versus Spot Rates

INVESTMENTS | BODIE, KANE, MARCUS

15-14

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

)1()1()1( 1

1 n

n

n

n

n fyy

Forward Rates from Observed Rates

(1+yn-1)n-1

1+fn

(1+yn)n

INVESTMENTS | BODIE, KANE, MARCUS

15-15

Example 15.4 Forward Rates

• The forward interest rate is a forecast of a

future short rate.

• Example: compute forward rate for year 4:

– 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