05 the Risk and Term Strurcture of Interest Rates

7/30/2019 05 the Risk and Term Strurcture of Interest Rates

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Banking & Finance The Risk and Term Structure of

Interest Rates



1. Overview

• So far we have defined interest rates by the internal rate at

which the price of an asset today equals its present

discounted value today (yield to maturity) and further have

discussed the determinants of the interest rate on a

representative bond through asset demand and supply and

liquidity preference.

• In this discussion we have assumed that there is only one

bond and one interest rate. This oversimplifying assumption

will be relaxed next.



1. Overview

• The discussion of risk and term structure of bonds answers two

fundamental questions on how interest rates on different bonds

relate to each other

• Risk structure: Why do bonds with the same term to maturity

have different rates of return?

• Term structure: Why do bonds with different term to maturity

(and the same risk) have different rates of return?



2. Risk Structure

– Four factors are mainly responsible for differences in interest

rates if bonds have the same term to maturity

» Default risk

» Liquidity

» Income taxes

» Information cost



2. Risk Structure

• 1. Default risk

» Default occurs when a borrower is unable or unwilling tomake payments when scheduled.

» Default is likely to occur e.g. in the context of companies

suffering large losses, while it is less likely in the context

of government bonds (default free bonds)



2. Risk Structure

1. Default risk

» The difference in interest rates between bonds withdefault risk and default-free bonds is called the risk premium.

» Generally risky bonds pay off a higher interest rate since

lenders request a premium in order to accept an asset

with a higher risk. Otherwise they would be better off by

holding bonds with lower risk and the same interest rate.



2. Risk Structure

1. Default risk

» We can use the loanable funds framework to illustratethe effects of default risk.

» Let’s for simplicity assume that there are two bonds, a

government bond and a corporate bond which initially

have the same risk, maturity, payoff structure and

liquidity, so that they sell at the same price (see figure

below)



2. Risk Structure

LsG

Loanable funds

in $ bil.

Interest rate in%

LdG

E1

LsC

LdC

E1

Loanable funds

in $ bil.

Interest rate in%

Government Corporation

i*Gi*C



2. Risk Structure

• 1. Default risk

» Now suppose that the default risk of the corporate bondincreases. This changes the expected return of this bond

(the probability of receiving nothing increases).

» If the expected return of the corporate bond relative to

the government bond decreases, the demand for this bond (or the supply of funds for this bond) will decrease,

while the demand for the government bond will

increase.

» This is illustrated in the figure below.



2. Risk Structure

LsG

Loanable funds

in $ billion

Interest rate in%

LdG

E1

LsC

LdC

E1

Loanable funds

in $ billion

Interest rate in%

Government Corporation

i*G

i*CLsG’

E2

LsC’

E2

RiskPremium



2. Risk Structure

1. Default risk

» The higher the default risk of a bond, the higher will itsrisk premium be.

» Bonds are rated according to their default risk. The two

major rating institutions are Moody’s and Standard &

Poor’s

» The figure below overviews their classification schemes.



2. Risk Structure

• 1. Default risk

Moody’s Standard andPoor’s Descriptions

Aaa AAA Highest quality(lowest default risk)

Aa AA High quality

A A Upper mediumgrade

Baa BBB Medium grade

Ba BB Lower Mediumgrade

B B Speculative

Caa CCC,CC,C Poor

C D Lowest grade

Investment

grade

Junk bonds/

High yield bonds

Long term rating codes comparision

Ratings of Hungary

http://english.mnb.hu/Root/ENMNB/Statisztika/data-and-information/mnben_statisztikai_idosorok/mo_hitm/mnben_mo_hitm_kodok






2. Risk Structure

2. Liquidity

» Another factor driving interest rate spreads is the liquidityof bonds.

» Generally the more liquid a bond, the easier will it be toresell it and the interest rate will correspondingly belower.

» Thus, less liquid bonds sell at a liquidity premium.

» The most liquid bonds are Treasury bonds since they aremost widely traded.



2. Risk Structure

3. Income Tax Considerations

» In the U.S. traditionally municipal bonds, i.e. bondsissued by state and local government have had lower

interest rates even than U.S. Treasury bonds (see

Miskhin p.121).

» Since these bonds are not default risk free and not asliquid as T-bonds, this fact may seem puzzling.

» The reason for this difference is that municipal bonds are

exempt from federal income taxes, which increases

their expected return relative to T-bonds.



2. Risk Structure

4. Information cost

» Finally it matters how much effort has to be spent ondetermining the default risk of a bond.

» Generally this cost will be higher if a bond is less traded

and unrated.

» A bond which produces high information cost has a

relatively low expected return and, thus, will exhibit a

higher interest rate.



3. Term Structure

– If identical bonds have different terms to maturity, their

interest rates will differ even if they are otherwise identical.

– Since there are bonds which have the same risk, liquidity and

tax characteristics but different maturities, we can compare

the differences in their interest rates given their maturity

(e.g. 3-month T-Bills).

– The common tool for this analysis are yield curves, which plot

the interest rates of same bond against their corresponding

different maturity terms. (See figure below)



3. Term Structure



3. Term Structure

• Yield curves can be classified as upward sloping (which is the

most commonly observed type of yield curve in developed

nations), downward sloping (sometimes called inverted yieldcurves) or flat.

• The relationship between an interest rate and the term to

maturity of a bond is usually not linear, so that yield curves can

often be classified as humped (such as in the figures above).



3. Term Structure

• If a yield curve is upward sloped short term interest rates are

below long term interest rates

• If a yield curve is flat, short term interest rates are equal to long

term interest rates.

• And finally, if a yield curve is downward sloped, long run

interest rates are below short term interest rates.



3. Term Structure

• What determines the shape of a yield curve?

» Any answer to this question has to incorporate three

empirical facts:

» 1. Interest rates on bonds of different maturities move

together over time

» 2. When short term interest rates are low, long run

interest rates tend to be high, such that yield curves are

upward sloped and vice versa.

» 3. Yield curves are generally almost always upward

sloped



3. Term Structure

Three theories have been brought forward to explain different yield

curves:

– 1. The expectations theory (which explains fact 1. and 2.)

– 2. The segmented market theory (which explains fact 3.)

– 3. The liquidity premium and preferred habitat theory

(which incorporates the first two theories and explains all

three facts)

3 h h



3.1. The Expectations Theory

– General Idea:

» The long-run interest rate is an average of expected

short term interest rates.

» Since interest rates are volatile (i.e. fluctuate), theshort and long term interest rates are different from

each other.

» For example: If you expect short run interest rate to

be equal to 10% on average for five years, the interest

rate on a five-year bond is going to be equal to 10%. If

you expect the interest rate to equal 12% on average

over the next ten years, the interest rate on ten year

bonds equals 12%, etc.

3 1 Th E i Th




• Key assumption: Identical bonds with different maturitydates are assumed to be perfect substitutes, i.e. investors donot prefer one maturity term over another.

» This means that individuals will only buy the bond withthe particular maturity term which offers the highest

rate of return.» Since this in turn drives up the price of this bond

today (and, thus, c.p. drives down the interest rateon this bond as well as potential future capitalgains), the rate of return on this bond starts to fall.

» The rate of return across different terms to maturity in equilibrium must be equal according to this theory.

3 1 Th E i Th




• To illustrate this process think of an easy example:

– An individual is given two options:

» 1. Either she/he buys a one-year bond and once itmatures she/he buys another one-year bond.

» 2. Or she/he buys a two-year bond instead and holds

it until maturity.

3 1 Th E t ti Th




• According to the expectation theory, the expected return onthese two strategies should be identical.

» If the interest rate today is equal to 5% and you expectthe interest rate next year to be equal to 10%, theinterest rate on the two year bond in equilibrium willequal 7.5%.

» If one strategy or the other exhibits a higher rate of return, individuals will be better off by only following this strategy. However, since this increases the

demand for the corresponding bond its price willincrease and the interest rate decrease until both of them are equal once more (arbitrage).

3 1 Th E t ti Th




•Generally:

» Let it be the interest rate on a one period bond today

» Let be the expected interest rate on a one period

bond one period from now

» And finally let i2t be the interest rate on a two periodbond today

e

t i

1

3 1 Th E t ti Th




– The expected return of investing one dollar in the two

period bond when held to maturity is given by:

– Given the fact that the square of i is generally very small,

we can reduce this expression to 2it.

22

22 21211)1)(1(t t t t t t iiiiii

Firstinterestpayment

Secondinterestpayment

Initial price of the bond(the investeddollar)

3 1 Th E t ti Th




– The expected return of investing one dollar in the twoone period bonds is given by:

– Again the product of the interest rates on the right handside of this equation is very small, s that this expression canbe reduced to:

e

t t

e

t t t

e

t

e

t t

e

t t iiiiiiiiii

11111111)1)(1(

Firstinterestpayment

Secondinterestpayment

Initial price of the bond(the invested

dollar)

e

t t ii 1

3 1 The E pectations Theor




– Since the expected return or both strategies in equilibrium

must be equal, it follows that:

– Thus, in equilibrium the long-term interest rate is equal to

the average of the short term interest rates (as expected).

e

t t t iii

122

2

1

2

e

t t

t

iii

3 1 The Expectations Theory




– This step can be extended to an arbitrary number of

periods n. In this case the long run interest rate can be

found by:

– Two numerical examples should further illustrate this point.

n

i

n

iiiii

n

i

e

it e

nt

ee

t t

nt

t

1

011 ...2





– 1. In the two period case, if the interest rate on a one period

bond is 5% today and expected to be 7% tomorrow, theinterest rate of a bond with a two period maturity is given

by:

– 2. Correspondingly if the short term interest rates over the

next 5 periods are expected to be 5%, 5.5%, 6%, 7% and 8%

correspondingly, the interest rate on a five period bond is

given by:

%6

2

%7%52

t

i

%3.65

%8%7%6%5.5%55

t i





• How can this theory be used to explain the shape of yield

curves?

» If a yield curve is upward sloped, it indicates that

short term interest rates are below long run interest

rates.

» Since long run interest rates in turn are an average

of current and future expected short run interest

rates an upward sloped yield curve suggests that

future short run interest rates are expected to

increase (et vice versa). If they are flat, future

expected interest rates are assumed to remain about

the same.





• How can this theory be used to explain our empirical facts?

» Fact 1. Interest rates of bonds with different

maturities move together over time.

» Historically a rise in short run interest rates led to

higher short run interest rates in the successive

periods, such that individual expectations also tended

to increase. As a consequence long term interest

rates –being the average of these short run interest

rates – increased as well.





•How can this theory be used to explain our empirical facts?

» Fact 2. Yield curves are upward sloped if short term

interest rates are low et vice versa.

» If short term interest rates are very low, people willexpect them to increase to some normal level in the

future and thus, long term interest rates tend to be

high. Conversely if short term interest rates are very

high, people will expect them to decrease and long

term interest rates will be low. In this case, the yieldcurve will be inverted.





• Shortcomings:

» While the expectations theory explains the general

shape of a yield curve as well as facts 1. and 2., it

lacks support form the third empirical observation

that yield curves -at large- are upward sloped:

» Since interest rates increase and decrease all the time,

inverted yield curves would have to be observed

much more often if the expectations theory was true.

3 2 Segmented Market Theory



3.2. Segmented Market Theory

– General idea: Bonds with different maturities are traded

in distinct markets, i.e. are no substitutes for each other.

» The justification for this assumption is that investors

have strong preferences for a bond of a certain maturity and care only about the return to that bond.

» Different yield curves are then determined by different

supply and demand situations for each bond. Since in

general the demand for long run bonds is smaller than for short term bonds, short term bonds will

always have higher prices and lower interest rates

(corresponding to fact 3.)

3 2 Segmented Market Theory



3.2. Segmented Market Theory

– Shortcomings:

» While the segmented market theory is able to explain

the third empirical observation, it fails to explain fact

1. and 2.

» There is no apparent reason why interest rates for

bonds of different maturity should be moving together

if these markets are completely independent.

Analogously, there is no apparent reason, why lowshort term interest rates should correspond to high

long term interest rates et vice versa.

3 3 Liquidity Premium and Preferred Habitat Theories



3.3. Liquidity Premium and Preferred Habitat Theories

The Liquidity Premium theory is a combination of theexpectation theory and the segmented market theory.

» It states that individuals do not only care about the

expected return of identical bonds with the different

maturity terms, but in addition to that also request aliquidity premium corresponding to the supply and

demand conditions for bonds of this specific

maturity term.

» While bonds in this setup are substitutes, they are notperfect since investors’ timing preferences are

incorporated as well.





• Generally investors prefer short run bonds, since their interest

rate risk is lower. Thus, in order to incentive investors to hold a

long term bond, the liquidity premium an issuer has to pay on

this bond will have to be higher.

• Modifying the expectations theory, this idea can be stated as:

nt

n

i

e

it

nt

e

nt

ee

t t

nt l

n

i

ln

iiiiit

1

011 ...2





•The preferred habitat theory follows a similar method andcomes to basically the same conclusion.

» In a nutshell it states that normally investors prefer a

bond with a certain maturity (preferred habitat).

» They will only be willing to hold a bond of a different

maturity it the expected return on this bond is higher.

» Since short run habitats are usually preferred, issuers

of long-run bonds have to offer higher interest rates inorder to incentive individuals to buy their bonds

instead.





• The graph below indicates the difference between theexpectations theory and the liquidity premium / preferred

habitat theory if expected short term interest rates are

assumed to be constant

» In this scenario the expectations theory predicts a flatyield curve (I).

» Liquidity premium and preferred habitat, however,

still predict an upward sloped yield curve (II) since

the liquidity premium increases with the maturityterm.





int

Periods to maturity n510 15 20 25 30

II

I

LiquidityPremium l nt





• A numerical example: Consider again the case, in which

expected short-term interest rates for five years are given by:

5%, 5.5%, 6%, 7%, 8%, respectively.

» Assume that the liquidity premia for each of these fiveyears is given by 0%, 0.25%, 0.5%, 0.75% and 1%.

» The interest rate on a two year bond then is given by:

%5.525.02

%5.5%52 t

i





– The interest rate on a five year bond in turn would be equal

to:

– We could in principle repeat this exercise for all five periods

and compare it to the expectations theory. The predictions

by the liquidity premium/ preferred habitat theory are

always going to be higher – increasing the slope of thepredicted yield curves.

%3.71

5

%8%7%6%5.5%5

5

t

i





• How does this theory fit empirical facts?

» Since the general idea is following the expectations

theory, fact 1 and 2 can easily be explained through

the liquidity premium and preferred habitat theory.

» However, also for fact 3 predictions are more accurate.

Even if short term interest rates are expected to remain

constant or fall slightly, the predicted yield curves will

be flat to upward sloped.

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05 the Risk and Term Strurcture of Interest Rates