chapter 7 the risk and term structure of interest ratesjneri/econ330/files/lecture... ·...

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Chapter Seven

Chapter 6

The Risk Structure and Term Structure

of Interest Rates

6-2

Bonds Are Risky!!!

Bonds are a promise to pay a certain amount in the future. How can that be risky?

1. Default risk - the chance the bond’s issuer may not make payment.

2. Inflation risk - investor cannot be aware of the real of the payments made.

3. Interest rate risk - a rise in interest rates before bond is sold could mean a capital loss.

• Investors’ concerns about risk affect their demand for U.S. Treasuries.

• 1998, Russia defaulted and people lost confidence in emerging market countries.

• Safest assets are U.S. Treasuries.

• Demand increased, price increased, and yields declined.

Default Case Study: What Happened Here?

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Default Risk

• Risk arises because an investment has many possible payoffs during the holding horizon.

• We need to look at the risk the bondholder faces - what are the possible payoffs, and how likely each is to occur.

• We will compare the risk on a bond’s return relative to the default risk-free rate – a US government bond.

Default Risk

• Corporate bond example:

• Assume the one-year risk-free interest rate is 5 percent.

• If the company is perceived as default risk-free, the company can issue a 5 percent coupon bond with a face value of $100.

Default Risk

• If this bond was considered default risk-free, the price of the bond would be the present value of the $105 payment. Price of risk free bond = ($100 + $5)/$1.05 = $100

• Suppose, there is now a 5% probability that the company will go bankrupt before paying back the loan. – Assume the outcome is either $105 or $0. – Expected value equals $94.50 Price of bond = $94.50/1.05 = $90

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Default Risk

Expected Value of bond payment

Possibilities Payoff Probability Payoff x

Probability

Full Payment $105 .95 $99.75 Default $0 .05 $0

Suppose 5% probability firm goes bankrupt – you get nothing

Expect to receive $99.75 one-year from now.

Discount at risk-free rate(5%) = 99.97

1.05 = $95

P = $95

Default Risk • If the price of the bond is $95, what is the

YTM? YTM = $105/$95 - 1 = 0.1052

• Default risk premium is the YTM minus the risk-free rate:

Risk Premium = 10.53% - 5%

= 5.53 percent.

• The higher the default risk, the higher the yield and risk premium.

Default Risk

Expected Value of bond payment

Possibilities Payoff Probability Payoff x

Probability

Full Payment $105 .90 $94.50 Default $0 .10 $0

90$05.1

50.94$

Suppose 10% probability firm goes bankrupt – you get nothing

Expect to receive $94.50 one-year from now.

Discount at risk-free rate =

P = $90; YTM = $105/$90 – 1 = 16.67%

Risk premium = 16.67 – 5 = 11.67%

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6-10

Default Risk Premium • We can calculate the probability of repayment

from the interest rates. Call it p.

• 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.

• At equilibrium: 1+i = p(1+k)

• The probability of repayment is

• the probability of default is 1 – p

• The probability of repayment:

1

1

ip

k

1.050.90

1.1667

6-11

3mo 6mo 9mo 1yr 2yr 3yr 5yr 10yr 20yr 30yr+

BONDS

U.S. Treasury 0.29% 0.51% 0.60% 0.68% 0.82% 0.94% 1.23% 1.67% 2.01% 2.40%

U.S. Treasury

Zeros -- -- -- 0.56% 0.73% 0.89% 1.25% 1.82% 2.32% --

Agency/GSE 0.71% 0.81% 0.96% 1.01% 1.13% 1.42% 1.62% 2.42% 3.13% 3.20%

Corporate

(Aaa/AAA) -- 0.39% -- 0.73% 1.05% 1.40% 1.59% 2.45% 3.35% 3.91%

Corporate (Aa/AA) 0.74% 1.15% 1.08% 1.22% 1.44% 1.62% 1.96% 2.75% 3.60% 4.49%

Corporate (A/A) 0.93% 1.25% 1.32% 1.42% 1.84% 1.89% 2.65% 3.44% 4.25% 4.32%

Corporate

(Baa/BBB) 1.21% 1.65% 2.15% 2.15% 2.54% 4.48% 4.88% 5.13% 6.08% 6.42%

Municipal

(Aaa/AAA) 0.60% 0.60% 0.77% 0.75% 0.97% 1.12% 1.43% 2.15% 2.85% 2.93%

Municipal (Aa/AA) 0.86% 0.75% 0.85% 0.88% 1.63% 1.61% 1.65% 2.59% 3.07% 3.28%

Municipal (A/A) 0.75% 0.80% 1.00% 0.99% 1.20% 1.68% 2.52% 3.45% 3.50% 3.45%

Taxable Municipal* 0.73% 1.11% 1.05% 1.10% 1.72% 2.05% 2.09% 4.30% 3.94% 3.36%

Yields on 9/20/16

Risk Structure of Interest Rates Risk Structure – hold time to maturity constant and compare yields.

• Default risk—occurs when the issuer of the bond is unable or unwilling to make interest payments or pay off the face value

U.S. T-bonds are considered default free

Risk premium—the spread between the interest rates on bonds with default risk and the interest rates on T-bonds

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

• Income tax considerations

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Ratings and the Risk Structure of Interest Rates

• Default is an important risks a bond buyer faces.

• Independent companies (rating agencies) evaluate the creditworthiness of potential borrowers. – These companies estimate the likelihood that the

corporate or government borrower will make a bond’s promised payments.

Bond Ratings

• The best known bond rating services are

– Moody’s

– Standard & Poor’s

• A high rating suggests that a bond issuer will have little problem meeting a bond’s payment obligations.

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Bond Ratings • The highest-rated bonds, Triple A.

– Ex: ExxonMobil, Microsoft, and the government of Canada (also JNJ)

• The top four rating categories are considered investment-grade bonds. – These bonds have a very low risk of default.

– Reserved for most government issuers and corporations that are among the most financially sound.

7-17

7-18

Credit rating & historic default frequencies Moody’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%

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Bond Ratings

• The distinction between investment-grade and speculative, noninvestment-grade (less than BBB or Baa) is important.

– A number of regulated institutional investors are not allowed to invest in bonds rated below investment grade, which is Baa on Moody’s scale or BBB on Standard and Poor’s scale.

Bond Ratings

• Speculative grade bonds are bonds issued by companies and countries that may have difficulty meeting their bond payments but are not at risk of immediate default.

• Bonds with grades below investment grade are often referred to as junk bonds or high-yield bonds.

https://finance.yahoo.com/quote

/JNK?p=JNK

The Impact of Ratings on Yields

• Bond ratings are designed to reflect default risk.

• The higher the risk of default.

– The lower the rating

– The lower the bond price and the higher its yield.

• To understand quantitative ratings, it is easier to compare them to a benchmark.

https://finance.yahoo.com/quote/JNK?p=JNK



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The Impact of Ratings on Yields • U.S. Treasury issues are viewed as having

little default risk, so they are used as benchmark bonds.

• Yields on other bonds are measured in terms of the spread over Treasuries.

• Bond yield is the sum of two parts:

= U.S. Treasury yield + Default risk premium

7-23

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

The Impact of Ratings on Yields

• If bond ratings properly reflect risk, then the lower the rating of the issuer, the higher the default-risk premium.

• When Treasury yields move, all other yields move with them.

• We can see this from a plot of the risk structure of interest rates.

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Differences in Tax Status and Municipal Bonds

• Taxes affect the yield on a bond.

• Bondholders must pay income tax on the interest income they receive from owning privately issued bonds - taxable bonds.

• The coupon payments on bonds issued by state and local governments, municipal or tax-exempt bonds, are specifically exempt from taxation.

Differences in Tax Status and Municipal Bonds

• What are the tax implications for bond yields?

• Consider a one-year $100 face value taxable bond with a coupon rate of 6 percent.

– Par is $100, and yield to maturity is 6 percent.

– Government sees this 6 percent as taxable income.

– If tax rate is 30%, the tax is $1.80.

– Bond yields $104.20 after taxes, equivalent of 4.2 percent. This is the after-tax yield.

Taxes and Bond Prices

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

• For 30% tax bracket: • After tax yield = (taxable yield) x (1 – tax rate)

4.20% = 6% x (1 – 0.30)

• Tax equivalent yield = rate tax - 1

yieldexempt tax

http://www.bloomberg.com/markets/rates-

bonds/government-bonds/us/

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





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Interest Rates on Municipal and Treasury Bonds

Term Structure of Interest Rates

• Why do bonds with the same default rate and tax status but different maturity dates have different yields?

– Long-term bonds are like a composite of a series of short-term bonds.

– Their yield depends on what people expect to happen in the future.

• How do we think about future interest rates?

Term Structure of Interest Rates • The relationship among bonds with the same

risk characteristics but different maturities is called the term structure of interest rates.

• Comparing 3-month and 10-year Treasury yields we can see: 1. Interest rates of different maturities tend to move

together.

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

3. Long-term yields tend to be higher than short-term yields – but NOT always

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7-31

Yield Curve 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.

https://www.treasury.gov/resource-center/data-chart-center/Pages/index.aspx

http://www.stockcharts.com/freecharts/yieldcurve.php

7-32

Term Structure of Interest Rates

Three Theories 1. Pure Expectations Theory/Hypothesis - explains

the first two facts but not the third

2. Segmented Markets Theory - explains fact three

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











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The Expectations Hypothesis • Assumes there is no uncertainty about the future.

• If there is no uncertainty, then an investor will be indifferent between holding a two-year bond or a series of 2 one-year bonds.

– Certainty means that the bonds of different maturities are perfect substitutes for each other.

• The expectations hypothesis implies that the current two-year interest rate should equal the average of current one-year rate and the one-year interest rate one year in the future.

The Expectations Hypothesis • If current interest rate is 5 percent and future

interest rate is 7 percent, then the current two-year interest rate will be (5+7)/2 = 6%.

• When interest rates are expected to rise, long-term interest rate will be higher than short-term interest rates.

– The yield curve will slope up.

• This also means: – If interest rates are expected to fall, the yield curve will

slope down.

– If expected to stay the same, the yield curve will be flat.

The Expectations Hypothesis

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The Expectations Hypothesis • If bonds of different maturities are perfect

substitutes for each other, then we can construct investment strategies that must have the same yields.

• Options:

1. Invest in a two-year bond and hold it to maturity

• i2t is interest rate on a 2-year bond bought today, t.

• One dollar yields (1 + i2t)(1 + i2t) two years later.


2. Invest in two one-year bonds, one today and one when the first matures.

– One-year bond today has interest i1t.

– One-year bond purchased in year 2 has interest ie1t+1, where e is expected.

– One dollar invested today returns

(1 + i1t)(1 + ie1t+1).


• The expectations hypothesis tells us investors will be indifferent between the two options.

• This means they must have the same return:

(1 + i2t)(1 + i2t) = (1 + i1t)(1 + ie1t+1)

• We can now write the two-year interest rate as the average of the current and future expected one-year interest rates:

i2t i1t i1t1

e

2

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7-40


• Returning to our example: • i1t = 5%

• ie1t+1 = 7%

• The interest rate on a 1-year bond is 5% and the interest rate on a 2-year bond is 6%.

%0.62

%7%5

2

1112

e

ttt

iii

7-41

A Note on Averages

• Geometric average of and =

• Arithmetic average =

1ti 1 1ti

1/ 2

1 1 1((1 )(1 )) 1t ti i

1 1 1

2

t ti i

7-42

Quiz: 2 year investment horizon – 2 options

• Option/strategy 1:

• Invest $1,000 for 2-years at 8%:

• Ending Balance = $1,166.40

• Option/strategy 2:

• Invest $1,000 1-year at 6% and expect 9% one year later:

• Ending Balance = $1,155.40

• Which is the better strategy and why?

• What happens to S and D?

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• We can generalize this: a bond with n years to maturity is the average of n expected future one-year interest rates:

int i1t i1t1

e i1t2e ... i1tn1

e

n

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

e

nt

e

t

e

tt

nt

1121111 ....

This is the only interest rate that is

known at time t

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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

e

nt

e

t

e

ttnt

1121111 ....

Side Note

n

iiiii

e

nt

e

t

e

ttnt

1121111 ....

n

rrrri

e

nt

e

nt

e

t

e

t

e

t

e

t

e

tt

nt

)(....)()()(11112121111111

Recall the Fisher Equation: i = r + πe

nn

rrrri

e

nt

e

t

e

t

e

t

e

nt

e

t

e

ttnt

)........ 11211111121111

This says long term interest rates = the average

real interest rate and the average rate of inflation

expected over the life of the bond

From 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.

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Expectations Theory: i = r + πe

i = r + πe

i

2

1

2

e

tt

t

iii

tt

e

t iii 21 2

3

21

3

e

t

e

tt

t

iiii

tt

e

t iii 232 23

tnnt

e

nt innii )1()1( )1(

In general:

)(3 132

e

ttt

e

t 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 Expectations Theory to Solve for

Expected 1-year (forward) Interest rates


Does this hypothesis explain the three observations we started with?

1. Interest rates of different maturities will move together. – Yes. Long term interest rates are averages of

expected future short-term interest rates.

2. Yields on short-term bonds will be more volatile than yields on long-term bonds. – Yes. Long-term rates are averages of short-term

rates, so changing one short-term rate has little effect on the long term rate.

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3. This hypothesis cannot explain why long-term yields are normally higher than short term yields.

– It implies that the yield curve slopes upward only when interest rates are expected to rise.

– This hypothesis would suggest that interest rates are normally expected to rise.

Segmented Market Theory – not in the

book

• Bonds of different maturities are not perfect

substitutes for each other.

• Key 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.

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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


• 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


• 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.

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To decode charts and graphs, use these three strategies:

1. Read the title of the chart.

2. Read the label on the horizontal axis.

3. Read the label on the vertical axis.

• What are the ranges of the axis? • Are the measurements evenly spaced? • Without noting these things, charts can be

misleading.

The Liquidity Premium Theory • In order to explain all 3 facts we need to extend

the expectations hypothesis to include risk.

• Bondholders face both inflation and interest-rate risk.

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

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The Liquidity Premium Theory Inflation Risk

• Real return is what matters and computing real return from nominal return requires a forecast of expected future inflation.

– The further into the future we look, the greater the uncertainty.

– A bond’s inflation risk increases with its time to maturity.

The Liquidity Premium Theory

Interest-rate risk

• arises from the mismatch between the investor’s investment horizon and a bond’s time to maturity.

– If a bondholder plans to sell a bond prior to maturity, changes in the interest rate generate capital gains or losses.

– The longer the term of the bond, the greater the price change for a given change in interest rates and the larger the potential for capital losses.

The Liquidity Premium Theory

• Investors require compensation for the increase in risk they take for buying longer term bonds.

• We can think about bond yields as having two parts:

– One that is risk free - explained by the expectations hypothesis.

– One that is a risk premium - explained by inflation and interest-rate risk.

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The Liquidity Premium Theory • Together this forms the liquidity premium theory

of the term structure of interest rates.

• We can add the risk premium (rpn) to our previous equation to get:

• The liquidity premium theory explains all three of our observations about the term structure of interest rates.

n

iiiirpi

e

nt

e

t

e

ttnnt

1121111 ....

Numerical Example

Term in years (n)

1 2 3 4 5

One year interest rate

expectations 5% 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 are

expected to remain constant)

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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

Market

Predictions

of Future

Short

Rates

The Information Content of Interest Rates

• Risk spreads provide one type of information, the term structure another.

• We can apply what we have just learned to recent U.S. economic history to show how forecasters use these tools.

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Information in the Risk Structure of Interest Rates

• The immediate impact of a pending recession is to raise the risk premium on privately issued bonds.

– Note that an economic slowdown or recession does not affect the risk of holding government bonds.

– The impact of a recession on companies with high bond ratings is also usually quite small.

• The lower the initial grade of the bond, the more the default-risk premium rises as general economic conditions deteriorate.

Information in the Risk Structure of Interest Rates

• Panel A of Figure 7.8 shows the annual GDP growth over four decades superimposed on shading that shows the dates of recessions.

– During shaded periods growth is negative.

• Panel B of figure 7.8 shows GDP growth against the spread between yields on Baa-rated bonds and U.S. Treasury bonds.

– risk spread rises during recessions.

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• During financial crises, people take cover.

• They sell risky investments & buy safe ones.

• An increase in the demand for government bonds coupled with a decrease in the demand for virtually everything else is called a flight to quality.

– This leads to an increase in the risk spread.

• The 1998 Russian default on its bonds led to a serious flight to quality causing the financial markets to cease to function properly.

Information in the Term Structure of Interest Rates

• Information on the term structure, particularly the slope of the yield curve - helps to forecast general economic conditions. – The yield curve usually slopes upward.

– On rare occasions, short-term interest rates exceed long-term yields leading to an inverted yield curve.

• This is a valuable forecasting tool because it predicts a general economic slowdown. – Indicates policy is tight because policymakers are

attempting to slow economic growth and inflation.


• Figure 7.9 shows GDP growth and the slope of the yield curve, measured as the difference between the 10-year and 3 month yields: term spread.

• Panel A shows GDP growth together with the term spread at the same time.

• Panel B shows GDP growth in the current year against the slope of the yield curve one year earlier. – The two lines clearly move together.

2/24/2018

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• When the term spread falls, GDP growth tends to fall one year later.

• This shows that the yield curve is a valuable forecasting tool.

chapter 7 the risk and term structure of interest ratesjneri/econ330/files/lecture... ·...

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