Chapter 6

The Risk Structure and Term Structure of Interest

Rates

Risk Structure of Interest Rates

• Bonds with the same maturity have different interest rates due to:

• Default risk• Liquidity • Tax considerations

Risk Structure of Interest Rates• Default risk: probability that the issuer of a

bond is unable or unwilling to make interest payments or pay off the face value

• U.S. Treasury bonds are considered default free (government can raise taxes).

• Risk premium: the spread between the interest rates on bonds with default risk and the interest rates on (same maturity) Treasury bonds

Response to an Increase in Default Risk on Corporate Bonds – Supply/Demand Application

Russian Default

Risk Structure of Interest Rates

• Liquidity: the relative ease with which an asset can be converted into cash

• Cost of selling a bond

• Number of buyers/sellers in a bond market

• Income tax considerations

• Interest payments on municipal bonds are exempt from federal income taxes.

Interest Rates on Municipal and Treasury Bonds

Taxes and Bond Prices

• Coupon payments on municipal bonds are exempt from federal Income taxes

• For 28% tax bracket:• After tax yield = (taxable yield) x (1 – tax rate)

3.60% = 5% x (1 – 0.28)

• Tax equivalent yield = rate tax - 1

yieldexempt tax

http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/

Risk Structure of Long-Term Bonds in the United States

Bond (credit) Ratings and Risk

Bond Ratings -

• Moody’s and Standard and Poor’s

Ratings Groups

• Investment Grade

• Non-Investment – Speculative Grade

• Highly Speculative

Bond (credit) ratingsS&P Moody’s What it means

AAA Aaa Highest quality and creditworthiness

AA Aa Slightly less likely to pay principal + interest

A A Strong capacity to make payments, upper medium grade

BBB Baa Medium grade, adequate capacity to make payments

BB Ba Moderate ability to pay, speculative element, vulnerable

B B Not desirable investment, long term payment doubtful

CCC Caa Poor standing, known vulnerabilities, doubtful payment

CC Ca Highly speculative, high default likelihood, known reasons

C C Lowest rated class, most unlikely to reach investment grade

D Already defaulted on payments

NR No public rating has been requested

+ or - & 1,2,3 Within-class refinement of AA to CCC ratings

Credit rating & historic default frequenciesMoody’s

Rating 1985 1990 1995 2000 2006 2008 2009 2010

Aaa 0% 0% 0% 0% 0% 0% 0% 0%

Aa 0% 0% 0% 0% 0% 0% 0% 0%

A 0% 0% 0% 0% 0% 1.201% 0% 0.36%

Baa1 0% 0% 0% 0.29% 0% 0.271% 1.144% 0%

Baa2 0% 0% 0% 0% 0% 0.794% 0.74% 0%

Baa3 0% 0% 0% 0.98% 0% 0.321% 0.70% 0%

Ba1 0% 2.67% 0% 0.91% 0% 0% 2.27% 0%

Ba2 1.63% 2.82% 0% 0.66% 0.51% 0% 0.60% 0%

Ba3 3.77% 3.92% 1.72% 0.99% 0% 2.715% 4.01% 0%

B1 4.38% 8.59% 4.35% 3.63% 0.66% 1.783% 4.10% 0.85%

B2 7.41% 22.09% 6.36% 3.84% 0.50% 0.825% 8.68% 0%

B3 13.86% 28.93% 4.10% 11.72% 1.93% 3.198% 8.52% 0.56%

Default Risk – Price and YTM

• Suppose risk-free rate is 4%• Suppose there is a company called FlimFlam that

issues one-year, 4% coupon bond, FV=$100.• If risk free, the price of the FlimFlam bond is

100$04.1

104$

04.1

100$4$

P

Default Risk

Expected Value of FlimFlam bond payment

Possibilities Payoff Probability Payoff x Probability

Full Payment $104 .95 $98.80Default $0 .05 $0

95$04.1

80.98$

Suppose 5% probability FlimFlam goes bankrupt – you get nothing

•Expect to receive $98.80 one-year from now.

•Discount at risk-free rate =

•P = $95

Default Risk Premium

• We can calculate the probability of repayment from the interest rates.

• Let 1+k be the return on a one-year corporate debt and 1+ i be the return on a one-year default risk-free treasury.

• The probability of repayment is

• the probability of default is 1 – p

• The probability of repayment:

1

1

ip

k

1.040.95

1.0947

Default Risk

Expected Value of FlimFlam bond payment

Possibilities Payoff Probability Payoff x Probabilities

Full Payment $104 .90 $93.60Default $0 .10 $0

Suppose 10% probability FlimFlam goes bankrupt – you get nothing

•Expect to receive $93.60 one-year from now.

•Discount at risk-free rate =

•Yield = ($104 / $90) -1 = .1555 or 15.55%

•Default risk premium = 15.55% - 4% = 11.55%.

90$04.1

60.93$

Bond Ratings and Risk

• Increased risk reduces bond demand. • The resulting shift to the left causes a decline in

equilibrium price and an increase in the bond yield.

• Bond Yield = U.S. Treasury Yield + Default Risk Premium

• Risk spread or default risk premium =

Bond Yield - U.S. Treasury Yield

Information Content of Interest Rates:Risk Structure

• When the economy starts to slow, this puts a strain on private firms.

• A slower economy means a higher default probability

• Risk Spreads increase.

Information Content of Interest Rates: Risk Structure Risk spread = Baa Corporate minus 10-year Treasury

Term Structure of Interest RatesDefinition of the Term Structure:The relationship among bonds with the same risk, liquidity and tax characteristics but different maturities is called the term structure of interest rates.

Yield Curve: A plot of the term structure, with the yield to

maturity on the vertical axis and the time to maturity on the horizontal axis.

http://finance.yahoo.com/bonds/composite_bond_rates?desktop_view_default=true

Term Structure of Interest Rates

Term Structure of Interest Rates

http://stockcharts.com/index.html

Term Structure of Interest Rates:Facts to Explain

1. Interest rates (Yields) on different maturities tend to move together

2. Yields on short-term bond are more volatile than yields on long-term bonds

3. Long-term yields tend to be higher than short-term yields.

• Also want to explain the fact that yield curves can be inverted.

Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities

Sources: Federal Reserve: www.federalreserve.gov/releases/h15/data.htm.

Three Theories to Explain the Three Facts

1. Pure Expectations Theory explains the first two facts but not the third

2. Segmented Markets Theory explains fact three but not the first two

3. Liquidity Premium Theory combines the two theories to explain all three facts

Pure Expectations Theory

• The interest rate on a long-term bond will equal an average of the short-term interest rates that people expect to occur over the life of the long-term bond

• Key Assumption: Buyers of bonds do not prefer bonds of one maturity over another.

• Bonds of different maturities are considered to be perfect substitutes

Expectations Theory Notation

1ti interest rate on 1-year bond today (t).

2ti interest rate on 2-year bond today (t).

nti interest rate on n-year bond today (t).

1 1ti

1 1eti

interest rate on 1-year bond, 1-year from today (t+1).

Expected interest rate on 1-year bond, 1-year from today (t+1).

1et ni Expected interest rate on 1-year bond, n-years

from today (t+n).

A Note on Averages

• Geometric average of and =

• Arithmetic average =

1ti 1 1ti

1/ 21 1 1((1 )(1 )) 1t ti i

1 1 1

2t ti i

Expectations Theory:

• Let the current interest rate on one-year bond (i1t) be 6%.

• You expect the interest rate on a one-year bond next year ( ) to be 9%.

• Then the expected return from buying 2 one-year bonds averages (6% + 9%)/2 = 7.5%.

• Under the Expectations Theory the current interest rate on a two-year (i2t) bond must be 7.5% for you to be willing to purchase that bond.

• Why?

1 1eti

Example: 2 year investment horizon• Strategy 1:• Invest $1,000 for 2-years at 8%:• Ending Balance = (1+0.08)2($1,000) = $1,166.40• Strategy 2: • Invest $1,000 1-year at 6% and expect 9% one

year later:• Ending Balance = (1 +0.06)(1+0.09)($1,000) =

$1,155.40

• Come out $11 ahead with Strategy 1.

• What happens to S and D?

Expectations Theory ( Math)

1. Return from a 2-year bond over 2 years

1)1)(1( 2t2t ii

2. Return from a 1-yr bond and then another 1-yr bond

1-))(1(1 e11t1t ii

3. If one and two year bonds are perfect substitutes, then:

))(1(1))(1(1 e11t1t2t2t iiii

Term Structure of Interest Rates:Expectations Theory

From: ))(1i(1)i)(1i(1 e11t1t2t2t i

We can derive the following arithmetic approximation:

Which says the long-term interest rate = average of current and expected future short-term interest rates.

2

iii

e11t1t

2t

Here is how we get the approximation:

2 2

22 2

22 2

22

Expected return over the two periods from investing $1 in the

two-period bond and holding it for the two periods

(1 + )(1 + ) 1

1 2 ( ) 1

2 ( )

Since ( ) is very small

the expected re

t t

t t

t t

t

i i

i i

i i

i

2

turn for holding the two-period bond for two periods is

2 ti

Here is how we get the approximation:

1

1 1

1 1

1

1

If two one-period bonds are bought with the $1 investment

(1 )(1 ) 1

1 ( ) 1

( )

( ) is extremely small

Simplifying we get

et t

e et t t t

e et t t t

et t

et t

i i

i i i i

i i i i

i i

i i

Expectations Theory

2 1

12

Both bonds will be held only if the expected returns are equal

2

2The two-period rate must equal the average of the two one-period rates

For bonds with longer maturities

et t t

et t

t

t tnt

i i i

i ii

i ii

1 2 ( 1)...

The -period interest rate equals the average of the one-period

interest rates expected to occur over the -period life of the bond

e e et t ni i

nn

n

Actual math: No Approximation

)])(1i(1[)i(1 e11t1t

22t i

))(1i(1)i)(1i(1 e11t1t2t2t i

1)])(11([ 1/2e11t1t2 iii t

1/2e11t1t2t )])(1i(1[)i(1 i

This is a geometric average

Expectations Hypothesis - Arithmetic Average

In words: The interest rate on a bond with n years to maturity at time t is the average of the n expected future one-year rates.

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

Interest rate on a two-year bond:(5% + 6%)/2 = 5.5%

Interest rate for a five-year bond:(5% + 6% + 7% + 8% + 9%)/5 = 7%

Interest rate for one, two, three, four and five-year bonds are:5%, 5.5%, 6%, 6.5% and 7%.

n

iiiii

ent

et

ett

nt1121111 ....

This is the only interest rate that is known at time t

Expectations Hypothesis

Another example:One-year interest rate over the next five years 7%, 6%, 5%, 4% and 3%:

Interest rate on a two-year bond:(7% + 6%)/2 = 6.5%

Interest rate for a five-year bond:(7% + 6% + 5% + 4% + 3%)/5 = 5%

Interest rate for one, two, three, four and five-year bonds:7%, 6.5%, 6%, 5.5% and 5%.

n

iiiii

ent

et

ett

nt1121111 ....

Recall the Fisher Equation: i = r + πe

• Holding r constant:• If inflation is expected to rise in the future, expected

one-year interest rates will rise and the yield curve will slope upward.

• If inflation is expected to fall in the future, expected one-year interest rates will fall and the yield curve will slope downward.

• If inflation is expected to remain the same in the future, expected one-year interest rates will remain the same and the yield curve will be flat.

Term Structure of Interest Rates:Expectations Theory

i

21

2

ett

t

iii

ttet iii 21 2

321

3

et

ett

t

iiii

ttet iii 232 23

tnnte

nt innii )1()1( )1(

In general:

)(3 132ettt

et iiii

From the formula for the yield on a 2-year bond:

From the formula for the yield on a 3-year bond:

Using the Pure Expectations Theory to Solve for Expected 1-year (forward) Interest rates

Actual math: No Approximation

)1)(1()1)(1( 11122etttt iiii

t

t

i

ii

1

22e

11t 1

)1()(1

11

)1(

1

22e

11t

t

t

i

ii

Term Structure Facts and the Expectations TheoryExpectations Theory Explains:1. Interest Rates of different maturities tend to move

together - long term interest rates are averages of expected future

short-term interest rates.

2. Yields on short-term bond are more volatile than yields on long-term bonds –

- long term interest rates are averages of expected future short-term interest rates.

But Expectations Theory does not explain:

3. Long-term yields tend to be higher than short-term yields.

Segmented Market Theory

• Bonds of different maturities are not perfect substitutes for each other.

Segmented Markets Hypothesis

• Assumptions:• Investors have specific preferences about

the maturity or term of a security.• Investors do not stray from their preferred

maturity.

Segmented Markets Hypothesis

• The slope of the yield curve is explained by different demand and supply conditions for bonds of different maturities.

• If the yield curve slopes up, it does so because the demand for short term bonds is relatively greater than the demand for long term bonds.

• Short term bonds have a higher price and a lower yield as a result of the relatively greater demand. So the yield curve slopes upward.

Segmented Markets Hypothesis

Price Price

0 0

S S

P2s

P1s P1

l

P2l

D1s

D2s

D1l

D2l

Quantity of Short-term Bonds Quantity of Long-term Bonds

Upward Sloping Yield Curve

Segmented Markets Hypothesis

• The segmented markets hypothesis explains why….• Yield curves typically slope upward.

• On average, investors prefer bonds with shorter maturities that have less interest rate risk.

• Therefore, the demand for short term bonds is relatively greater than the demand for long-term bonds

Segmented Markets Hypothesis

• But, the segmented markets hypothesis does not explain why…• Interest rates on different maturities move

together.• The segmented markets hypothesis assumes that

short and long markets are completely segmented.

Liquidity Premium Theory of the Term Structure of Interest Rates

• Yield curve upward slope is explained by the fact that long-term bonds are riskier than short-term bonds.

• Bondholders face both inflation risk and interest rate risk.

• The longer the term of the bond, the greater both types of risk.

• Investors need to be compensated for the greater risk.

Term Structure of Interest Rates

Liquidity Premium Theory

n

ent

et

ett

nt RPn

iiiii

1121111 ....

Liquidity or Risk Premium

(explains fact 3)

Pure Expectations Theory: average of expected future short-term rates

(explains facts 1&2)

Numerical Example

Term in years (n)

1 2 3 4 5One year interest rate

expectations5% 6% 7% 8% 9%

Liquidity premium 0% 0.25% 0.5% 0.75% 1.0%

Pure expectations predicted n-year bond

interest rates

5% 5.5% 6% 6.5% 7%

Actual n-year bond interest rates,

accounting for liquidity preference

5% 5.75% 6.5% 7.25% 8%

5% 6%

2

5% 6% 7%

3

5 6 7 8%

4

5 6 7 8 9%

5

Relationship Between the Liquidity Premium and Expectations Theories

(if short term interest rates areexpected to remain constant)

Information Content of Interest Rates:Term Structure

• When the yield curve slopes down, it is called inverted

• An inverted yield curve is a very valuable forecasting tool

• It signals an economic downturn

Information Content of Interest Rates:10-year T-bond compared to 3-month T- bill

Market Predictions of Future

Short Rates

The actual math is a lot more interesting. Refer to the note on

“Term Structure and Forward Interest Rates.”