Term structure of interest rates - e105.orge105.org/e266/download/ltermstructstep.pdf · 266: Financial Markets and Institutions Term structure of interest rates Jon Faust Faust,

266: Financial Markets and Institutions

Term structure of interest rates

Jon Faust

http://e105.org/e266

Faust, JHU e266, Spring 2015 –

Learning den

� John Schwarcz (excellent)

� Sessions: 6:30-8:00pm, Wednesdays


Daniel (also excellent)

� Adding: Thursday 4-5 pm at Greenhousein addition to Monday 4-5 pm in Greenhouse.

� Tutoring: The Econ dpt offers tutoring.Email Andy Gray agray18jhu.eduAndy will hook you up with a previous TA. Around$30/hour


Present value

� So far, we have said payment of s comingin h years has present val:

PV =s

(1 + i)n

� This assumes that the annualized rate ofinterest for borrowing for any length oftime, h, is the same: i.


. . .

In reality, a 5-year loan generally carries ahigher annual interest rate than a 1-year or2-year loan.


Yield curve

� On any given day, the market interestrates for loans of different maturities iscalled the term structure of interest rates

� When we plot these rates against the termor maturity we call this the yield curve.


WSJ, Feb. 10, 2016


WSJ, Feb. 10, 2016, as yield curve

.51

1.5

22.

5pe

rcen

t

0 5 10 15 20horizon (years)

source: Wall Street Journal

Treasury Yield Curve, Feb. 10, 2016


Up to now, we have assumed a flat yield curve


In practice,

� The yield curve is usually sloped upwardat least a bit and moves around a greatdeal

Faust, JHU e266, Spring 2015 – p

. . .

Let’s explore the history of the yield curve


. . .

� You can download this exciting moviefrom the course websitemp4: gohttp://e105.org/e266/download/yc.mp4


http://e105.org/e266/download/yc.mp4

A few things to notice

� Usually (but not always) a bit upwardsloping

� Negatively sloped when rates peakedaround 1980

� Term structure is very low right now


But not as low as, e.g., in Switzerland

l4/swissjyc20160210.eps


Bernanke

� In 2013, Chm. Bernanke gave a nicespeech about long-term rates and whythey are so lowWorth a read: gohttp://federalreserve.gov/newsevents/speech/bernanke20130301a.htm

� It’s ominous that 3 years later rates inmuch of the world are even lower


http://federalreserve.gov/newsevents/speech/bernanke20130301a.htm

. . .

A great deal of what goes on in financialmarkets depends on understanding the termstructure of interest rates.


Notation:

� Let’s define iht to be the annualizedinterest rate on day t on an h-year loan orzero coupon bondsingle payment coming in h years.


Present value

� With constant interest rates the streamst+1, . . . , st+M had

PVt =M∑

j=1

st+j

(1 + i)j

� When interest rates vary by how far thepayment is in the future:

PVt =M∑

j=1

st+j

(1 + ijt)j


Present value

� We have to use the j-period interest ratein discounting the payment j periods infuture.Still raised to jth power b/c ijt is stated at anannual rate.


. . .

How are rates of different maturities related?


Theories of the term structure

� We’ll discuss some economic theories ofterm structure behavior

� To do so, it is useful to step back anddiscuss the law of one price (LOOP)


Aside:: LOOP

� What is the law of one price (LOOP) ineconomics?


Aside:: LOOP

� What is the law of one price (LOOP) ineconomics?Identical items must sell for the same price underthe conditions of perfect competition

� Suppose A and B are identical, give theargument why the market will tend to pushthe prices to the same level.


Aside:: LOOP

� What is the law of one price (LOOP) ineconomics?Identical items must sell for the same price underthe conditions of perfect competition

� Suppose A and B are identical, give theargument why the market will tend to pushthe prices to the same level.Demand shifts from more expensive one tocheaper one, driving one price down and the otherup. Similarly, supply may shift as well, driving theprices together.


Aside:: LOOP and hedge fund strategies

� ‘LOOP thinking’ is at the center of manyhedge fund investment strategies

� Key intuition: you detect two ‘identical’ (ornearly identical) streams that sell forprices, you find some way to bet that thetwo prices will come together or converge

The idea is that violations of LOOP don’t tend tolast long

� Sometimes call ‘convergence strategies.’


LOOP and the term structure

� For a moment ignore all uncertainty.

� LOOP+certainty give strong implicationsfor the term structure


LOOP+certainty and the term structure

� Suppose you want to borrow or invest for10 years. In our notation, this interest rateat time t is denoted



� Suppose you want to borrow or invest for10 years. In our notation, this interest rateat time t is denoted i10,t

� If you invest $1, at the end of 10 years youwill have



� Suppose you want to borrow or invest for10 years. In our notation, this interest rateat time t is denoted i10,t

� If you invest $1, at the end of 10 years youwill have

(1 + i10,t)10


An equivalent alternative investment

� I could invest $1 for 3 years and then rollover the proceeds into a new 7 year loan.




� At the end of 3 years, I’ll have: (1 + i3,t)3

� At the end of 10 years, I’ll have




� At the end of 3 years, I’ll have: (1 + i3,t)3

� At the end of 10 years, I’ll have

(1 + i3,t)3(1 + i7,t+3)

7



� Without uncertainty, the LOOP says thatthese two ways to invest for 10 years mustreturn the same amount.

� Why



� Without uncertainty, the LOOP says thatthese two ways to invest for 10 years mustreturn the same amount.

� WhyIf, say the single 10-year loan returned more, folkswould shift funds out of the 3+7 alternative into the10. This drives up the price of the 10 (driving downit’s return) and drives down the price of thealternative (driving up its return). This ends whenthe two are equal.


Thus,

� Loop+certainty imply

(1 + i10,t)10 = (1 + i3,t)

3(1 + i7,t+3)7

� That is, market forces will drive these 3interest rates to a point where thisequation holds


Consider 10-year vs. rolling over 10 1-year bonds

� LOOP says:

(1+i10,t)10 = (1+i1,t)×(1+i1,t+1)×. . .×(1+i1,t+9)

� Take natural log:

10 ln(1 + i10,t) =9∑

j=0

ln(1 + i1,t+j)



� Apply ln(1 + z) ≈ z for small z:

i10,t ≈1

10

9∑

j=0

i1,t+j

� Under this story, the 10-year rate isapproximately equal to the average of the10, 1-year rates that will prevail over thenext 10 years.



� Setting aside uncertainty , LOOP gives usa simple way to relate long-term rates tothe future short-term rates that will prevail.


Question

� If the current 10-year rate is higher thanthe current 1-year rate, then in this worldwe know that the 1-year rate in the futurewill have to be


Question

� If the current 10-year rate is higher thanthe current 1-year rate, then in this worldwe know that the 1-year rate in the futurewill have to be higher.The long rate is the average of the future shortrates, so we have to pull the average up


Reality

� With no uncertainty, the LOOP plus a viewabout the short-term interest rate give acomplete characterization of the termstructure

� Tell me about the future path of shortrates, and I’ll tell you what long-term ratesare

� Tell me about current long-term rates, andI’ll tell you about the future path ofshort-term rates


Uncertainty.

� Once we bring uncertainty in, the theoryof the term structure gets much moresubtle.

� The first theory we’ll discuss simplypretends this complexity awayThis is the smallest possible modification of ourLOOP+certainty theory to account for uncertainty.


Expectations theory

� Under certainty, we said:Any two ways of investing for M periods must paythe same

� In the expectations theory we say that anytwo ways of investing for M periods mustbe expected to (that is, on average) paythe same returnAnd by ‘expected’ we mean the statistical sense ofexpectation.


Expectations theory

� Simple version. Take our LOOP theoryequation. Add uncertainty.Then, simply replace any unknown future-datedvalues with their expected value


Expectations theory

� LOOP+certainty:

(1 + i10,t)10 = (1 + i3,t)

3(1 + i7,t+3)7

� Expectations theory+uncertainty makesone change:

(1 + i10,t)10 = (1 + i3,t)

3(1 + ie7,t+3)7

And as always the e means the expected value ofthe item


Similarly,

�

(1+i10,t)10 = (1+i1,t)×(1+ie1,t+1)×. . . (1+ie1,t+9)

or doing our same approximation asbefore:

i10,t ≈1

10

(

i1,t +9∑

j=1

ie1,t+j

)

where now we have an e on all the ‘future’1-year rates.


. . .

Short hand for expectations theory of theterm structure: Long-term interest rates(approx.) equal the average of expectedfuture short-term rates.


Expectations theory: implications

� If today’s 10-year rate is above today’s1-year rate then, by the expectationstheory, the market expects the 1-year rateto


Expectations theory: implications

� If today’s 10-year rate is above today’s1-year rate then, by the expectationstheory, the market expects the 1-year rateto increaseby the same reasoning as in the LOOP+certaintycase.


. . .

Adding a realistic treatment of uncertainty


Economic content of the expectations theory

� The ‘expectations theory’ is essentially thetheory asserting that on average riskdoesn’t matterGoing from certainty to uncertainty, just put an e

superscript on unknown stuff

� Taken literally, this assertion goes againsteverything we’ve learned.Folks care about risk. . .



� More carefully: folks will pay a premiumfor ‘good’ risk, and expect to be paid tobear ‘bad’ risk



� In principle, the expectations theory couldbe close to right if on average risk doesn’tmatter much.

� It turns out that risk does matter, so weneed a richer theory.Let’s talk this through


Risk in the term structure

� Compare buying a 10-year bond to buyinga 5-year and rolling the proceeds into thenext 5 year bond.(assume there is no default risk)

� With the 10-year bond, the nominal returnis fixed . . .

� . . . but the real return is uncertainInflation could be higher or lower than expected



� If inflation risk is my main concern, buyingthe current 5-year bond and rolling intothe next one is better.

� If expected inflation has risen or fallenafter 5 years, this will be incorporated inthe yield I earn when I buy the second5-year bond

� If I buy the 10-year, I am locked in for thewhole period.



� On the other hand, suppose that the realinterest rate changes (say, it falls)

� Then I will earn a lower real rate over the10-years by rolling over 5-year bonds.

� I will wish that I’d locked in my return bybuying the 10-year.


. . .

Which should the investor prefer?[10-year, or 5-year rolled into 5-year]


. . .

The answer is the same answer that a goodeconomist gives to almost every question:


. . .

The answer is the same answer that a goodeconomist gives to almost every question: itall depends


Which earns a positive premium?

� For now, let me just assert that folks seemto demand a positive premium to hold the10-year versus rolling over shorter bonds.

� Thus, when expectated future short-terminterest rates are constant, the yield curveslopes upward (at least a bit)on average, most of the time.


Liquidity premium

� We will follow the text in calling the riskcompensation in the term structure a‘liquidity premium’

� And so we augment the standardequation implied by expectations theoryby adding a premium


Liquidity premium

� liquidity premium theory

i10,t =i1,t +

∑

9

j=1ie1,t+j

10+ ℓ10,t

Where ℓ10,t is the risk or liquidity premium themarket pays to compensate for risk in the 10-yearsecurity.

� In principle, ℓ could be either positive ornegative


Liquidity premium

� And in practice, it appears that usually inthe market ℓ is positive, but sometimes itis negative


Other theories

� The book lists segmented market theoryand in footnote: preferred habit theory.

� Just ignore these


. . .

� In reality, there seems to be somepremium: that is, the expectations theoryis does not hold in practice.

� Despite knowing all the stuff above, youwill often hear people (that is, pundits,market commentators, policymakers, etc.)reason based on the expectations theory.

� Thus, if long-term rates are aboveshort-term rates, folks regularly state that‘markets must expect short-term rates toincrease’


Macaulay (of duration fame)

� In the 1930s, Macaulay looked at suchpredictions and concluded‘Now experience is more nearly the opposite.’

� When an upward sloping yield curvepredicts rising short rates under theexpectations theory, short ratessubsequently tend to fall instead


Cite

� The Macaulay quote and more is in,Do Long-Term Interest Rates Overreact toShort-Term Interest Rates? N. Gregory Mankiw,Lawrence H. Summers and Laurence WeissBrookings Papers on Economic Activity, Vol. 1984,No. 1 (1984), pp. 223-247 gohttp://www.jstor.org/stable/2534279


http://www.jstor.org/stable/2534279

Up until the crisis, at least

� Up until the crisis, at least, experienceremained as in the 1930s

� With rates at many maturities pinned nearzero, we really don’t have muchexperience. So I won’t state generalitiesabout current circumstances.


Bottom line, expectations theory

� It has a large grain of truth.Expected future short rates must (in anyreasonable theory) be reflected in current longrates

� But liquidity premia also seem to be largeand variable in practice.


Bottom line, expectations theory

� Despite knowing this, people often usethe expectations theory to derivepredictions of future interest rates.


Alan Blinder

� Famous economist, former vice chairmainof Fed,Yet everyone—and here I mean analysts, marketparticipants and central bankers alike—continues[despite the evidence] to “read” the market’sexpectations of future short rates from the yieldcurve, as if doing so made sense. I find it hard toexplain why everyone is doing what everyoneknows to be wrong.. . . (1997, p.16)


Alan Blinder

� CiteBlinder, Alan, Distinguished Lecture on Economicsin Government: What Central Bankers could learnfrom Academics—and Vice Versa, Journal ofEconomic Perspectives, vol. 11, no. 2, Spring1997.


As Vonnegut would say,

and so it goes. . .


What you should know

� The expectations theory is an importantbaseline caseIt would hold if there were no uncertainty (no risk)or if people did not demand compensation forbearing risk

� It provides a baseline interpretation ofexpectations of future short ratesThis baseline interpretation is influential, inpractice, and often quite wrong.


Case study: Recent long-term rates in U.S.

� From mid-November 2014 to mid-January2015, the yield on 10 year U.S.government bonds fell from about 2.4percent to 1.7 percent.

� Q:If long rates fall, what does theexpectations theory say has happened toexpected future short rates?

� A: Expected future short rates must havefallen


Case study: Recent long-term rates in U.S.

� But over this period the Fed had beencommunicating a shift up in the likelihoodthat short-term rates would soon rise fromzero.

� This suggests a problem for theexpectations theory:long rates down, expected future short rates up.


Case study: Two possibilities

� 1. Folks don’t believe the FedPerhaps they think that the bad economic news wehear from abroad will soon come to the U.S. andthat the Fed will not raise rates.

� 2. Premia in the 10-year rates are fallingSay, risks abroad have risen, shifting out thedemand for U.S. bonds and decreasing thepremium demanded for holding them


Case study: Two possibilities

� Note: the effect that government bondrates fall in the U.S. when the world getsscarier in some way is called a ‘flight tosafety’ effect.

� Many folks actually probably believe amixture of these two stories


Case study: general lesson

� When long-term rates change, we canusually tell a story driven by changingexpectations of future short rates and astory of changing premia.

� You should get used to thinking up bothkinds of story.


Finally some terminology

� Financial markets folks often speak offorward interest rates.

� These are closely related to ourexpectations theory discussion.


Forward rates

� If I know the current 10-year rate and thecurrent 3 year rate, I can ask,Q: What 7-year rate 3 years from now would makethese two investments pay off the same?

� A: This rate will be f in:

(1 + i10,t)10 = (1 + i3,t)

3(1 + f)7


Forward rates

� Solving, we get

1 + f =

(

(1 + i10,t)10

(1 + i3,t)3

)1/7


Forward rates

� f in the above equation is known at the7-year forward rate, 3 years hence.


Generally,

� The M -year forward rate S years in thefuture is

1 + f =

(

(1 + iM+S,t)M+S

(1 + iS,t)S

)1/M


Another way of describing the expectations theory

Expectations theory of term structure saysthat forward rates are the market’s expecationof future spot interest rates


Note: You can look at forward rates on Bloomberg

� There are computers around that you canuse to access Bloomberg servicesEssentially a proprietary ‘web-like’ services, filledwith a wealth of financial data


Forward rates on Bloomberg

� To see U.S. forward rates at any point usethe command:FWCM <go>


FWCM screenshot, Feb. 10, 2014


Documents

Term structure of interest rates - e105.orge105.org/e266/download/ltermstructstep.pdf · 266: Financial Markets and Institutions Term structure of interest rates Jon Faust Faust,