chapter 6 - yield curve analysis ii

8/7/2019 Chapter 6 - Yield Curve Analysis II

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Chapter 6Chapter 6

Yield Curve Analysis IIYield Curve Analysis II


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OverviewOverview

This chapter gets into more practical matters regardingThis chapter gets into more practical matters regardingyieldyield--curve analysis, although it also has somecurve analysis, although it also has sometheoretical issues as well. We will examine:theoretical issues as well. We will examine:

The differences between yield and price volatilities.The differences between yield and price volatilities.

How to calculate historical volatilities.How to calculate historical volatilities.

Basic TermBasic Term--Structure IssuesStructure Issues

Forward Rates of InterestForward Rates of Interest

Hypotheses of the Term StructureHypotheses of the Term Structure

Bootstrapping and the extraction of zero coupon prices.Bootstrapping and the extraction of zero coupon prices.


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Yield and Price VolatilityYield and Price Volatility

This is actually a pretty simple concept. Recall that modifiedThis is actually a pretty simple concept. Recall that modifiedduration tells us (in percentage terms) the relationship betweenduration tells us (in percentage terms) the relationship betweenprice and yield.price and yield.

Basically as he shows in page 206, the price volatility is aBasically as he shows in page 206, the price volatility is a

function of the yield volatility and duration:function of the yield volatility and duration:

* *P Y y MDW W!


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Yield and Price VolatilityYield and Price Volatility

To calculate price volatility from historical data:To calculate price volatility from historical data:

1.1. Compute the natural log of the price ratio:Compute the natural log of the price ratio:RRtt=ln(P=ln(Pt+1t+1/P/Ptt) for each date t.) for each date t.

2.2. Compute the mean (Compute the mean () of the natural log of the price ratio.) of the natural log of the price ratio.3.3. Compute the squared deviations for each t as xCompute the squared deviations for each t as xtt=(R=(Rtt--

))22..

4.4. Calculate the daily and annual volatility estimates:Calculate the daily and annual volatility estimates:

1 1

1 1, p, annual

*252and

1 1

N N

i ii i

P daily

x x

N NW W

! !! !


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Term Structure AnalysisTerm Structure Analysis

Generically, the phrase termGenerically, the phrase term--structure refers to thestructure refers to therelationship between the maturity date of defaultrelationship between the maturity date of default--freefreezero coupon bonds and their yield to maturity.zero coupon bonds and their yield to maturity.

The yield on a zero coupon bond between time 0 andThe yield on a zero coupon bond between time 0 andtime T (z=100/(1+ytime T (z=100/(1+yTT))

TT) is called the spot rate of) is called the spot rate ofinterest.interest.

Thus, there is a spot rate from 0 to 1, 0 to 2, 0 to 3, etc.Thus, there is a spot rate from 0 to 1, 0 to 2, 0 to 3, etc.

We also want to considerWe also want to consider forward rates.forward rates.

We are going to jump ahead of the book and cover themWe are going to jump ahead of the book and cover themnow, and then come back to extracting zeros.now, and then come back to extracting zeros.


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Spot and Forward RatesSpot and Forward Rates

In normal, everyday usage, when we say the phraseIn normal, everyday usage, when we say the phraseinterest rate we are talking about a spot rate.interest rate we are talking about a spot rate.

Somewhat formally, the nSomewhat formally, the n--period spot rate is theperiod spot rate is theinterest rate charged on money borrowed at time 0interest rate charged on money borrowed at time 0and repaid at the end of time n. We will denote this asand repaid at the end of time n. We will denote this asrrnn..

Note that when rNote that when rnn is a zero coupon yield.is a zero coupon yield.


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Graphically, this can be shown on a timeline.

For example, r4

the 4 year spot rate is the rate

extending from time 0, through the end of the

fourth year.

0 1 2 3 4 5

Spot rate covers this time.


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A forward rate, however, is the interest rate associatedA forward rate, however, is the interest rate associatedwith a loan that you contract to today, but which willwith a loan that you contract to today, but which willoccur at some future date.occur at some future date.

Note that once you sign the forward agreement, you areNote that once you sign the forward agreement, you arebound to it, that loan will occur at the specified terms.bound to it, that loan will occur at the specified terms.

We have to denote the beginning and ending points ofWe have to denote the beginning and ending points ofthe forward rate. To do this we will use the notation fthe forward rate. To do this we will use the notation fm,nm,nwhere m is the beginning date for the loan and n is thewhere m is the beginning date for the loan and n is the

ending date. You enter into the forward contract at timeending date. You enter into the forward contract at time0.0.


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Graphically, this can also be shown on a

timeline. For example, there is a forward rate

for a 3 year loan which begins in exactly 1

year, f1,4.

0 1 2 3 4 5

f1,4 covers this time.Sign forward loanagreement at time

0


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Our goal is to understand the fundamental relationshipOur goal is to understand the fundamental relationshipbetween forward rates and spot rates, and how thebetween forward rates and spot rates, and how themarket enforces this relationship.market enforces this relationship.

Recognize that we define fRecognize that we define f0,10,1

rrmm. That is, the initial. That is, the initial

forward and future rates are the same.forward and future rates are the same.

We also have to realize that there is no fundamentalWe also have to realize that there is no fundamentaldifference between borrowing money on with a twodifference between borrowing money on with a twoperiod spot rate, or by contracting to two consecutiveperiod spot rate, or by contracting to two consecutive

forward rates.forward rates.


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Clearly all of the cash flows and risks will be the same. Thus

f0,1 and f1,2 must have an equivalence with r2.

0 1 2 3 4 5

Using 2 forward contracts (f0,1 and f1,2), you

lock in your borrowing rates at time 0 for

both periods 1 and 2.

0 1 2 3 4 5

Using 1 spot rate, you lock in your

borrowing rate for both periods 1 and 2.


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Perhaps the easiest way to see this is with an example.Perhaps the easiest way to see this is with an example.Lets say that you observe the following term structureLets say that you observe the following term structureof interest rates:of interest rates:

Spot RateSpot RateRRnn

Forward RateForward Rateffm,nm,n

rr11= 8= 8 ff0,,10,,1= 8= 8

rr22= 9= 9 ff1,21,2 10.000910.0009

rr33= 10= 10 f f2,32,3= 12.0276= 12.0276


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To keep things simple, we will assume an annualTo keep things simple, we will assume an annualcompounding frequency.compounding frequency.

Consider if you invested $1 at the two year spot rate. AtConsider if you invested $1 at the two year spot rate. Atthe end of the second year, you would have $1.188.the end of the second year, you would have $1.188.

2$1(1.09) $1.18FV ! !


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Similarly, if you were to lock in a series of two forwardSimilarly, if you were to lock in a series of two forwardrates, you would invest first at 8% for 1 year and thenrates, you would invest first at 8% for 1 year and thenat 10.0009 for the second year (but you lock in bothat 10.0009 for the second year (but you lock in bothrates at time 0!)rates at time 0!)

$1(1.08)(1.10009) $1.18FV ! !


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What if this were not the case? What if instead youWhat if this were not the case? What if instead youfound a bank that were willing to loan to you one yearfound a bank that were willing to loan to you one yearforward at 9%. How could you exploit this opportunityforward at 9%. How could you exploit this opportunityfor arbitrage?for arbitrage?

Clearly the bank is not charging enough in the secondClearly the bank is not charging enough in the secondyear, so you want to borrow from the bank and lend toyear, so you want to borrow from the bank and lend tothe market.the market.


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You do this in the following way. First, you contract withYou do this in the following way. First, you contract withthe bank at their (incorrect) forward rate of 9%.the bank at their (incorrect) forward rate of 9%.

You then simultaneously borrow $1 in the spot marketYou then simultaneously borrow $1 in the spot marketfor 1 year (at 8%) and lend in the spot market for 2for 1 year (at 8%) and lend in the spot market for 2years at 9%.years at 9%.


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You can see the timing of the events here.

Lock in the forward rate of 9% from the

bank for the second year.

0 1 2

Borrow $1 in the spot market at 8%, and

immediately lend it in the spot market at

9%.

Lend at 9% for 2 years

Borrow at 8% for 1 year

0 1 2

Borrow at 9%

in year 2


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We can thus look at the payments on all legs of thisWe can thus look at the payments on all legs of thistradetrade in particular note the net position:in particular note the net position:

TradeTrade Time 0Time 0 Time 1Time 1 Time 2Time 2

Borrow $1 at 8%Borrow $1 at 8%for 1 yearfor 1 year

+1.00+1.00 --1.081.08

Lend $1 at 9%Lend $1 at 9%for 2 yearsfor 2 years

--1.001.00 +1.188+1.188

Borrow (throughBorrow (throughforward) $1.08forward) $1.08for 1 year at timefor 1 year at time1 at 9%.1 at 9%.

+1.08+1.08 --1.1771.177

NetNet 00 00 +0.011+0.011


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Literally there is no risk to you, and you are guaranteedLiterally there is no risk to you, and you are guaranteedan positive cash flow lateran positive cash flow later hence this is an arbitragehence this is an arbitrage as the old Dire Straights song goes money for nothing.as the old Dire Straights song goes money for nothing.

Clearly arbitragers would quickly take advantage of thisClearly arbitragers would quickly take advantage of thismispricing in the market, and discipline the bank formispricing in the market, and discipline the bank forthis.this.


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To see a little more complicated example, lets look at aTo see a little more complicated example, lets look at arealreal--world case.world case.

This is taken from the Wall Street Journal of January 22,This is taken from the Wall Street Journal of January 22,2000.2000.


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On January 22, we observed the following zero couponOn January 22, we observed the following zero couponbond prices:bond prices:

Zero maturing in February, 2011: $58.6875Zero maturing in February, 2011: $58.6875

Zero maturing in February, 2012: $55.125Zero maturing in February, 2012: $55.125

The yields on these bonds are:The yields on these bonds are:

$58.6875 = 100/(1+r$58.6875 = 100/(1+r2020 /2)/2)2020: r: r2020 = 5.40= 5.40

$55.125 = 100/(1+r$55.125 = 100/(1+r2222/2)/2)2222: r: r2222 = 5.488= 5.488


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The one year spot rate beginning in year 10, i.e. inThe one year spot rate beginning in year 10, i.e. inperiod 20 (since we are now back to semiperiod 20 (since we are now back to semi--annualannualcompounding) is given by:compounding) is given by:

22

20,22 20

.054881

21 6.47152

.05401

2

! !


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Lets say that you now found a bank that was offeringLets say that you now found a bank that was offeringforward rates of 9% for year 10. How could you exploitforward rates of 9% for year 10. How could you exploitthis?this?

You would clearly want to borrow at the market rate ofYou would clearly want to borrow at the market rate of6.47% and lend to the bank at their incorrect rate of6.47% and lend to the bank at their incorrect rate of9%.9%.


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The trade you would put together would be as follows:The trade you would put together would be as follows: Borrow $1000 in the spot market for 11 years at 5.488%Borrow $1000 in the spot market for 11 years at 5.488%

Lend $1000 in the spot market for 10 years at 5.40%.Lend $1000 in the spot market for 10 years at 5.40%.

Use the forward rate to lock in to lend to the bank at time 10 forUse the forward rate to lock in to lend to the bank at time 10 for

1 year at a rate of 9%.1 year at a rate of 9%.


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The cashflows would be:The cashflows would be: Note that if you lend $1000 at 5.40 for 10 years, at the end of theNote that if you lend $1000 at 5.40 for 10 years, at the end of the

10 years you would receive 1000*(1+.054/2)10 years you would receive 1000*(1+.054/2)2020 = $1,703.76.= $1,703.76.

You would then lend this amount to the bank at 9% for the 11You would then lend this amount to the bank at 9% for the 11thth

year. At the end of the 11year. At the end of the 11thth year the bank would pay back to youyear the bank would pay back to you

1703.76*(1+.09/2)1703.76*(1+.09/2)22= 1,860.55= 1,860.55 You would then have to pay back the original $1000 you borrowedYou would then have to pay back the original $1000 you borrowed

at 5.488%, which would be: 1000*(1+.05488/2)at 5.488%, which would be: 1000*(1+.05488/2)2222 = $1,814.02.= $1,814.02.


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We wind up at the end of year 11, then, receivingWe wind up at the end of year 11, then, receiving$1860.55 and paying back to the market $1814.02$1860.55 and paying back to the market $1814.02 aanet gain of $46.52net gain of $46.52

So just like before we can see that we earn money forSo just like before we can see that we earn money fornothing.nothing.


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We can thus look at the payments on all legs of thisWe can thus look at the payments on all legs of thistradetrade in particular note the net position:in particular note the net position:

TradeTrade Time 0Time 0 Time 10Time 10 Time 11Time 11

Lend $1000 atLend $1000 at5.4% for 10 years5.4% for 10 years

--10001000 +1703.76+1703.76

Borrow $1000 forBorrow $1000 for11 years at11 years at5.488%5.488%

+1000+1000 1814.021814.02

Lend to bankLend to bankthrough forwardthrough forwardat 9%at 9%

--1703.761703.76 +1860.55+1860.55

NetNet 00 00 +46.52+46.52


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Theories of the Term StructureTheories of the Term Structure

Over the years various researchers have attempted toOver the years various researchers have attempted torelate spot rates to forward ratesrelate spot rates to forward rates that is one periodthat is one periodfuture spot rates to currently observed forward rates.future spot rates to currently observed forward rates.

Four primary hypotheses have emerged:Four primary hypotheses have emerged:1.1. The Expectations HypothesisThe Expectations Hypothesis

2.2. LiquidityLiquidity--Premium HypothesisPremium Hypothesis

3.3. Market Segmentation HypothesisMarket Segmentation Hypothesis

4.4. LocalLocal--Expectations HypothesisExpectations Hypothesis

Lets briefly examine each one.Lets briefly examine each one.


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Expectations HypothesisExpectations Hypothesis

There are multiple variants of this hypothesis. TheyThere are multiple variants of this hypothesis. Theyultimately all try to develop the notion that forward ratesultimately all try to develop the notion that forward ratesare some form of the markets estimate of future spotare some form of the markets estimate of future spotrates.rates.

TheThe unbiased expectations hypothesisunbiased expectations hypothesis formally statesformally statesjust that, i.e. fjust that, i.e. fk,k+1k,k+1= E= Ett[R[R

**kk].].

The difficulty is, that empirically forward rates areThe difficulty is, that empirically forward rates areterrible predictors of futures spot rates. If these are theterrible predictors of futures spot rates. If these are the

markets expectations, the market consistently sets itsmarkets expectations, the market consistently sets itsexpectations of future forward rates too high.expectations of future forward rates too high.


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LiquidityLiquidity--Preference TheoryPreference Theory

This says that lenders would, all things equal, prefer toThis says that lenders would, all things equal, prefer tolend for shorter periods of time. To induce them to lendlend for shorter periods of time. To induce them to lendlong term, therefore, borrowers must pay an additionallong term, therefore, borrowers must pay an additionalpremium, which shows up in the form of an upwardpremium, which shows up in the form of an upward

sloping yield curve, which generates forward rates thatsloping yield curve, which generates forward rates thatare greater than the markets actual expected spot rate,are greater than the markets actual expected spot rate,i.e. fi.e. fk,k+1k,k+1= E= Ett[R[R

**kk] +] + k,k+1.k,k+1.

The difficulty is that sometimes the yield curve isThe difficulty is that sometimes the yield curve isinverted, which would imply that investors preferencesinverted, which would imply that investors preferences

changed.changed. This is, in some ways, the most convincing of theseThis is, in some ways, the most convincing of these

arguments.arguments.


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MarketMarket--SegmentationSegmentation

This is not so much a theory as an antiThis is not so much a theory as an anti--theory: basicallytheory: basicallyit says that the longit says that the long--term and shortterm and short--term markets areterm markets aredifferent, and that only if they are grossly out of linedifferent, and that only if they are grossly out of linewith each other will participants in one market crosswith each other will participants in one market cross

over to participate in the other.over to participate in the other.

Basically it says that the fundamental premise of a yieldBasically it says that the fundamental premise of a yieldand/or forward rate coupon curve is flawed: that theand/or forward rate coupon curve is flawed: that thecurve tends to give the illusion that there is acurve tends to give the illusion that there is a

relationship that is not there.relationship that is not there. Generally this theory is not particularly useful, especiallyGenerally this theory is not particularly useful, especially

in the context of fixed income pricing.in the context of fixed income pricing.


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Local Expectations HypothesisLocal Expectations Hypothesis

This is a special case of the expectations hypothesis. ItThis is a special case of the expectations hypothesis. Itbasically says that all bonds (of the same credit quality)basically says that all bonds (of the same credit quality)have the same rate of return for a very short period ofhave the same rate of return for a very short period oftime. This is the only arbitragetime. This is the only arbitrage--free model!free model!

Short is normally defined to be whatever is theShort is normally defined to be whatever is thesmallest time period in the model being used, i.e. itssmallest time period in the model being used, i.e. itsinstantaneous for continuous time models, but maybeinstantaneous for continuous time models, but maybeone month for discreteone month for discrete--time models.time models.

Some of the discrete time models really push this to theSome of the discrete time models really push this to theabsolute limit.absolute limit.

Formally it is: E[ZFormally it is: E[ZTT--(t+1)(t+1)]/Z]/ZTT--tt= (1+r= (1+rtt))


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Coupons, Zeros and Strips (oh my!)Coupons, Zeros and Strips (oh my!)

In the market we primarily observe coupon bearingIn the market we primarily observe coupon bearingbonds, although many times we wish to work with zerobonds, although many times we wish to work with zerocoupon bondscoupon bonds this is especially true in the context ofthis is especially true in the context ofbuilding arbitragebuilding arbitrage--free term structure models.free term structure models.

We really have two ways of getting zero information:We really have two ways of getting zero information:extracting them from coupon bearing bonds in a processextracting them from coupon bearing bonds in a processknown asknown as bootstrappingbootstrapping, and directly observing them in, and directly observing them inthe strips market.the strips market.

Why ever use the bootstrapping method? PrimarilyWhy ever use the bootstrapping method? Primarily

because the coupon market is still larger and more liquidbecause the coupon market is still larger and more liquidthan the strips market, and as a result traders havethan the strips market, and as a result traders havemore confidence in the quotes from that market.more confidence in the quotes from that market.


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BootstrappingBootstrapping

Bootstrapping is simply a procedure for extracting zeroBootstrapping is simply a procedure for extracting zerocoupon bond rates from coupon bearing bonds. It iscoupon bond rates from coupon bearing bonds. It ismost commonly used in the Treasuries market.most commonly used in the Treasuries market.

The basic idea is that you start with an initial short termThe basic idea is that you start with an initial short term

bondbond typically a 1 month or 3 month bond, which istypically a 1 month or 3 month bond, which istruly a zero coupon bond, and you calculate its yield.truly a zero coupon bond, and you calculate its yield.

You continue calculating the zero coupon bonds until youYou continue calculating the zero coupon bonds until yourun into your first couponrun into your first coupon--bearing bond. For anbearing bond. For anexamples sake, lets say that you observe a six monthexamples sake, lets say that you observe a six month

and a twelve month zero (which is typical), and thatand a twelve month zero (which is typical), and thatyour first coupon bearing bond matures at month 18.your first coupon bearing bond matures at month 18.Denote the zero coupon yields as ZDenote the zero coupon yields as Z66 and Zand Z12.12.


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Typically you use aTypically you use a parpar--valuevalue coupon bearing bond, that means onecoupon bearing bond, that means onefor which you know the price is 100. Given that you know the valuesfor which you know the price is 100. Given that you know the valuesof Zof Z66 and Zand Z1212, what you do is find the value of Z, what you do is find the value of Z1818 that makes thisthat makes thisstatement true:statement true:

For example, let us say that ZFor example, let us say that Z66=10%, Z=10%, Z1212 = 12%, and that the= 12%, and that thecoupon on a parcoupon on a par--priced coupon bond is 13%. The 18 month zeropriced coupon bond is 13%. The 18 month zerowould be:would be:

2 3

6 12 18

/ 2 / 2 100 / 21001 / 2 1 / 2 1 / 2

C C C

Z Z Z

!

2 3

18

183

18

13/ 2 13/ 2 100 13/ 21001.05 1.06 1 / 2

106.588.02 ; so Z 13.114

1

Z

or

Z

!

! !


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One major problem with bootstrapping is that you dontOne major problem with bootstrapping is that you dontreally get bonds spaced evenly every six months, and ifreally get bonds spaced evenly every six months, and ifyou are working with a model where you need monthlyyou are working with a model where you need monthlyzeros, you really have trouble finding it.zeros, you really have trouble finding it.

Constant Maturity Treasuries (CMT) help this a lot. TheyConstant Maturity Treasuries (CMT) help this a lot. Theyprovide you on a daily basis with rates for bonds thatprovide you on a daily basis with rates for bonds thathave maturities of 1 year, 2 year, 3 years, 5 years, etc.have maturities of 1 year, 2 year, 3 years, 5 years, etc.

You still have to make assumptions about what happensYou still have to make assumptions about what happens

inin--between those bonds. One approach is to get morebetween those bonds. One approach is to get moredatadata i.e. find interest rate derivatives that do maturei.e. find interest rate derivatives that do maturebetween the bonds.between the bonds.


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The second approach is to make some assumption aboutThe second approach is to make some assumption aboutwhat is going on inwhat is going on in--between the observed rates. Inbetween the observed rates. Inessence you simply assume how the rates will behave.essence you simply assume how the rates will behave.

A common (but not particularly good) assumption is toA common (but not particularly good) assumption is to

assume that the par bonds are linear in coupon betweenassume that the par bonds are linear in coupon betweenobserved points. That is, if Cobserved points. That is, if Cnn and Cand Cn+2n+2 are the couponsare the couponsof par bonds maturity at times n and n+2, and you needof par bonds maturity at times n and n+2, and you needa coupon for a bond which matures at time n+1 buta coupon for a bond which matures at time n+1 butthere is no such bond trading, simply use linearthere is no such bond trading, simply use linear

interpolation to estimate Cinterpolation to estimate Cn+1n+1, i.e. C, i.e. Cn+1n+1 = (C= (Cnn+C+Cn+2n+2)/2)/2 The problem with this approach is that it can lead toThe problem with this approach is that it can lead to

some rather bizarresome rather bizarre--looking forward rate structures.looking forward rate structures.


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Other commonly used methods include various curveOther commonly used methods include various curvefitting algorithms such as piecewise cubic splines, cubicfitting algorithms such as piecewise cubic splines, cubicdiscount rate generating functions, and statisticaldiscount rate generating functions, and statisticalmethods such as Diaments method outlined inmethods such as Diaments method outlined in

Sundaresans book.Sundaresans book. While cubicWhile cubic--spline methods are pretty common inspline methods are pretty common in

commercial software, there is still a lot of variation incommercial software, there is still a lot of variation inhow various firms elect to fit their curves. All methodshow various firms elect to fit their curves. All methodsinvolve tradeoffs, such as being willing to accept a moreinvolve tradeoffs, such as being willing to accept a more

jagged curve in some regions in exchange for ajagged curve in some regions in exchange for asmoother curve in other regions.smoother curve in other regions.

There is not one universally accepted method.There is not one universally accepted method.


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A Simple CubicA Simple Cubic

One method that is sometimes used is to assume thatOne method that is sometimes used is to assume thatthere is a specific functional form for the discountthere is a specific functional form for the discountfunction, d(t).function, d(t).

Realize that the discount function is just thatRealize that the discount function is just that it tellsit tellsyou how much a dollar to be received at time t is worthyou how much a dollar to be received at time t is worthtodaytoday and from that you can obviously calculate theand from that you can obviously calculate thezero coupon rate.zero coupon rate.

A potential method is to estimate the parameters of thisA potential method is to estimate the parameters of this

formula using regression analysis:formula using regression analysis:d(t) = 1 + at + btd(t) = 1 + at + bt22 + ct+ ct33


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A Simple CubicA Simple Cubic

Then, when you need the zero coupon rate forThen, when you need the zero coupon rate for anyany timetimet, you simply plug in the value of t, and it spits out thet, you simply plug in the value of t, and it spits out thediscount price, and you can then solve for the zerodiscount price, and you can then solve for the zerocoupon yield.coupon yield.

Unfortunately, there is no guarantee that it will exactlyUnfortunately, there is no guarantee that it will exactlyfit the current yield curve, and there are number of yieldfit the current yield curve, and there are number of yieldcurves that we do occasionally see, such as invertedcurves that we do occasionally see, such as invertedcurves, which is it very difficult or impossible for thecurves, which is it very difficult or impossible for the

simple cubic function to match.simple cubic function to match. A cubic spline really just takes this a step further. PleaseA cubic spline really just takes this a step further. Please

see the handout for an example.see the handout for an example.

chapter 6 - yield curve analysis ii

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