The Risk and Term Structure of Interest Rateseconweb.rutgers.edu/ctamayo/Teaching/lecture_7.pdf · The Risk and Term Structure of Interest Rates Money and Banking Cesar E. Tamayo

The Risk and Term Structure of Interest RatesMoney and Banking

Cesar E. TamayoDepartment of Economics, Rutgers University

July 20, 2011

C.E. Tamayo () Econ - 301 July 20, 2011 1 / 14

Revisiting risk premium


Term structure of interest rates: stylized facts

1 Interest rates on bonds of different maturities move together overtime:

2 When short-term interest rates are low, yield curves are more likely tohave an upward slope; when short-term rates are high, yield curvesare more likely to slope downward and be inverted

3 Yield curves almost always slope upward





3 Yield curves almost always slope upward





3 Yield curves almost always slope upwardC.E. Tamayo () Econ - 301 July 20, 2011 3 / 14

Term structure of interest rates: What may explain this?

Expectations theory (ET):

Assumes that bond holders consider bonds with different maturities tobe perfect substitutesThat is, buyers of bonds do not prefer bonds of one maturity overanother; they will not hold any quantity of a bond if its expected returnis less than that of another bond with a different maturityStatement of the ET: The interest rate on a long-term bond will equalan average of the short-term interest rates that people expect to occurover the life of the long-term bond

ExampleLet the current rate on one-year bond be 6%. You expect the interest rateon a one-year bond to be 8% next year. Then the expected return forbuying two one-year bonds averages (6% + 8%)/2 = 7%. The interestrate on a two-year bond must be 7% for you to be willing to purchase it.




Assumes that bond holders consider bonds with different maturities tobe perfect substitutes

That is, buyers of bonds do not prefer bonds of one maturity overanother; they will not hold any quantity of a bond if its expected returnis less than that of another bond with a different maturityStatement of the ET: The interest rate on a long-term bond will equalan average of the short-term interest rates that people expect to occurover the life of the long-term bond





Assumes that bond holders consider bonds with different maturities tobe perfect substitutesThat is, buyers of bonds do not prefer bonds of one maturity overanother; they will not hold any quantity of a bond if its expected returnis less than that of another bond with a different maturity

Statement of the ET: The interest rate on a long-term bond will equalan average of the short-term interest rates that people expect to occurover the life of the long-term bond













Term structure of interest rates: Expectations Theory

Let us be more general. Suppose that we compare two strategiesinvest a $1 as follows:

1 Buy a one period bond, hold it and when it matures, buy another one(roll-over strategy)

2 Buy a two period bond and hold it until maturity.(buy and hold)

So we have:

it = today’s interest rate on one period bonds

iEt+1 = next period expected interest rate on one period bonds

i2t = today’s interest rate on two period bonds






So we have:









So we have:









So we have:






Let us use our well known PV formula but in this case we know PVand want CF.

Note: in this case interest rate=rate of return=return (why?)

Expected return from the "buy and hold" strategy":

(1+ i2t ) (1+ i2t )− 1 = (1+ i2t )2 − 1

= 2i2t + (i2t )2

≈ 2i2t

Expected return from the "roll-over" strategy:

(1+ it ) (1+ iet+1)− 1 = 1+ it + iet+1 + it (iet+1)− 1

≈ it + iet+1






(1+ i2t ) (1+ i2t )− 1 = (1+ i2t )2 − 1

= 2i2t + (i2t )2

≈ 2i2t



≈ it + iet+1






(1+ i2t ) (1+ i2t )− 1 = (1+ i2t )2 − 1

= 2i2t + (i2t )2

≈ 2i2t



≈ it + iet+1






(1+ i2t ) (1+ i2t )− 1 = (1+ i2t )2 − 1

= 2i2t + (i2t )2

≈ 2i2t



≈ it + iet+1



The ET of the term structure tells us that both bonds will be heldonly if their returns are equal we equate these two results:

2i2t = it + iet+1 ⇒ i2t =it + iet+12

Or more generally:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2So the ET explains fact # 1, namely that interest rates of differentmaturities move together.It also explains fact # 2: if the short temr interest rate, it , is"abnormally" low, then people expect it to go back to a normal levelin the future; that is, they expect iet+1 to be higher than it . Therefore,the average of it and iet+1 would be higher than it and the two periodbond will have a higher interest rate.However, the ET fails to explain fact # 3.




2i2t = it + iet+1 ⇒ i2t =it + iet+12

Or more generally:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2

So the ET explains fact # 1, namely that interest rates of differentmaturities move together.It also explains fact # 2: if the short temr interest rate, it , is"abnormally" low, then people expect it to go back to a normal levelin the future; that is, they expect iet+1 to be higher than it . Therefore,the average of it and iet+1 would be higher than it and the two periodbond will have a higher interest rate.However, the ET fails to explain fact # 3.




2i2t = it + iet+1 ⇒ i2t =it + iet+12

Or more generally:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2So the ET explains fact # 1, namely that interest rates of differentmaturities move together.

It also explains fact # 2: if the short temr interest rate, it , is"abnormally" low, then people expect it to go back to a normal levelin the future; that is, they expect iet+1 to be higher than it . Therefore,the average of it and iet+1 would be higher than it and the two periodbond will have a higher interest rate.However, the ET fails to explain fact # 3.




2i2t = it + iet+1 ⇒ i2t =it + iet+12

Or more generally:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2So the ET explains fact # 1, namely that interest rates of differentmaturities move together.It also explains fact # 2: if the short temr interest rate, it , is"abnormally" low, then people expect it to go back to a normal levelin the future; that is, they expect iet+1 to be higher than it . Therefore,the average of it and iet+1 would be higher than it and the two periodbond will have a higher interest rate.

However, the ET fails to explain fact # 3.




2i2t = it + iet+1 ⇒ i2t =it + iet+12

Or more generally:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2So the ET explains fact # 1, namely that interest rates of differentmaturities move together.It also explains fact # 2: if the short temr interest rate, it , is"abnormally" low, then people expect it to go back to a normal levelin the future; that is, they expect iet+1 to be higher than it . Therefore,the average of it and iet+1 would be higher than it and the two periodbond will have a higher interest rate.However, the ET fails to explain fact # 3.C.E. Tamayo () Econ - 301 July 20, 2011 7 / 14

Term structure of interest rates: Segmented markets theory

Radical departure: markets for different maturity bonds arecompletely separate and segmented.

Interest rates for each maturity are determined by D&S in eachmarket.

Investors have preferences for bonds of one maturity over another

If investors generally prefer bonds with shorter maturities that haveless interest-rate risk, then this explains why yield curves usually slopeupward (fact 3).




















Term structure of interest rates: Liquidity premium theory

The interest rate on a long-term bond will equal an average ofshort-term interest rates expected to occur over the life of thelong-term bond plus a liquidity premium that responds to supplyand demand conditions for that bond.

So we can add to our break-down of interest rates. If we arecomparing two bonds with the same risk structure:

Note: recall that if we are comparing bonds with different defaultrisk, we would add a risk premium.



The interest rate on a long-term bond will equal an average ofshort-term interest rates expected to occur over the life of thelong-term bond plus a liquidity premium that responds to supplyand demand conditions for that bond.So we can add to our break-down of interest rates. If we arecomparing two bonds with the same risk structure:

Note: recall that if we are comparing bonds with different defaultrisk, we would add a risk premium.



The interest rate on a long-term bond will equal an average ofshort-term interest rates expected to occur over the life of thelong-term bond plus a liquidity premium that responds to supplyand demand conditions for that bond.So we can add to our break-down of interest rates. If we arecomparing two bonds with the same risk structure:

Note: recall that if we are comparing bonds with different defaultrisk, we would add a risk premium.C.E. Tamayo () Econ - 301 July 20, 2011 9 / 14


Thus, we can adjust the resutl from the ET framework to account forthe liquidity premium:

int =it + iet+1 + i

et+2 + ...+ i

et+n−1

2︸︷︷︸facts #1 and #2

+ lnt︸︷︷︸fact #3

Note the subindex in lnt ; it has two components, n and t

Usually the longer the maturity, the higher the liquidity premium.

Also, the liquidity premium may change over time; in good times itmay be lower and in uncertain times it may be much higher.

A similar approach is that of the preferre habitat theory: Investorshave a preference for bonds of one maturity over another

They will be willing to buy bonds of different maturities only if theyearn a somewhat higher expected return




int =it + iet+1 + i

et+2 + ...+ i

et+n−1











int =it + iet+1 + i

et+2 + ...+ i

et+n−1











int =it + iet+1 + i

et+2 + ...+ i

et+n−1











int =it + iet+1 + i

et+2 + ...+ i

et+n−1











int =it + iet+1 + i

et+2 + ...+ i

et+n−1










Usually investor’s preferred habitat would be that of short term bondsin which case the analysis would be the same as the liquidity premiumtheory:

(The expectations yield curve is drawn flat because it is as likely to beupward sloping as it is to be downward sloping)However, the preferred habitat theory may allow for different investorsto prefer different "habitats".




(The expectations yield curve is drawn flat because it is as likely to beupward sloping as it is to be downward sloping)

However, the preferred habitat theory may allow for different investorsto prefer different "habitats".




(The expectations yield curve is drawn flat because it is as likely to beupward sloping as it is to be downward sloping)However, the preferred habitat theory may allow for different investorsto prefer different "habitats".C.E. Tamayo () Econ - 301 July 20, 2011 11 / 14

Term structure of interest rates: Testing the yield curve

Modern evidence suggests that yield curve data contain goodinformation about the liquidity premium between short and long termbonds but not so good information about intermediate term bonds.

Yield curve data are also helpful in forecasting future inflation.

Since rising interest rates are associated with economic booms andfalling interest rates are associated with recessions, the yield curvemay have predictive power over the business cycle.












Term structure of interest rates: interpreting the yieldcurve


Term structure of interest rates: interpreting the yieldcurve


Documents

The Risk and Term Structure of Interest Rateseconweb.rutgers.edu/ctamayo/Teaching/lecture_7.pdf · The Risk and Term Structure of Interest Rates Money and Banking Cesar E. Tamayo