ment.In a stochastic interest rate environment, because the payoff of anEDF contract is a linear function of interest rate, but the price of a bond(or the price of the fixed side cashflows of a swap) is a convex functionof the interest rate, the expectation of price changes for an EDF positionis different from that for a bond position. The difference is called convexity value. In the simplest case, suppose the interest rate movementsare symmetrically distributed, then the expectation of the price movements is zero for an EDF position, whereas a positive value for a bond
Sec. 10.4
Swap Curve
409
position is positive.In order to estimate the convexity value, let us consider an EDF synthetic zero-coupon bond and the fixed side of a zero-coupon swap. Weassume that both the EDF rate and swap rate term structures are flat at acommon rate of S, the settlement date is exactly on the start date of thefirst EDF contract, and the maturities of both instruments are equal to thewhole number o f t years. Then the price of the fixed side of the zerocoupon swap is-2 r
Fiww=(l+;)
(10.32)
9
and the price of the EDF synthetic bond is(10.33)The corresponding Taylor expansions in small rate movements are-4 w a p - -tAS
OP
1+ -1( t 2 + -t)(AS)2+~ ( A S ) ~ ,
(10.34)
11+ -(t2--t)(AS)2 + 0 ( A S ) 3
(10.35 )
2
2
anddPEDs
OP
= -tAS
2
,
4
The difference isA f k w a p - 4 ~ ~ 3
OP
= -t(AS)2
8
+O(Aq3,
( 10.36)
which is always positive for small AS. The most interesting case is whenA S is normally distributed, then the expectation of the difference is
where 0 is the volatility of interest rates, and h is the holding period ofthe investment in these two instruments. We can see that, because ofconvexity, the investment in the swap instrument has a clear advantage,and investors will arbitrage the convexity value by selling the ED strip
Yield Curve Modeling
410
Chapter 10
bond and buy the swap bond, until the yield of the ED strip is driven upto a value such that the two investments will generate exactly the samereturn. This explains why the ED strip curve has to be higher than thezero-coupon swap curve, and the yield difference is the so called convexity adjustment given by the following equation:3x = -to28
(1 0.38)
In order to verify this, let’s compute the return from an investment ofa very short time period h in a ED strip with yield Y. It is given by
APPO
-=yh
(10.39)
It is clear now that if we adjust Y up by the value x given by Eq.(10.38), the extra return from the investment in the ED strip will matchthe advantage of the investment in the swap bond.
Chapter 11
Two-Factor RiskModel
In this chapter we deal with pricing and hedging problems with multifactors. We first introduce practical models, such as the principal components analyses (PCA) and the two-factor Risk Model (TFRM), andthen we provide and analyze their practical applications.
11.I PCA and TFRM MethodologiesThe PCA, is a methodology that tries to find the minimum number ofdegrees of freedom that is needed to describe the interest rates satisfactorily. PCA concentrates on analyzing the interest rate correlation patterns.The TFRM provides quantified level and curve risks for fixed income assets by analyzing the yield curve movements and quantifies theassociated interest rate risks in terms of a pair of level and twist durations, the TFRh4 provides hedging ratios and other hedging strategies.The resulting hedging strategies by means of two-factor risk matchingare simple and effective, avoiding the cumbersome task of subjectivescenario analyses. In comparison with PCA of the Treasury interest rateterm structure, we show that, for all practical purposes, a level and adjusted twist, two-factor model dynamically calibrated to the currentyield curve and recent history of interest rate correlation pattern can describe more than 93% of all interest rate movements; hedging strategiesso designed greatly outperform the traditional duration risk matching andad hoc scenario sensitivity neutralization.
412
Two-Factor Risk Model
Chapter 11
A wide range of applications, such as Level-Hedged BullemarbellTrades with Quantified Curve Risk, two-factor Portfolio Hedging Strategy, Bond Indices with level, and curve risk profiles, have been developed by applying the TFRM.Traditionally yield curve movements are described and expected interms of level shifts and curve twists. It is believed that a combination ofyield level changes together with long-term and short-term yield spreadwidening or narrowing can explain most (95%) of the yield curvemovements. The concept of duration, which is the percent change of theprice of an interest rate instrument per unit change of the yield to maturity, has been established and commonly used to quantify risks associatedwith interest rate level changes.Measurement of yield curve twist risk is usually performed in acrude manner, through a series of scenario analyses. One would assumea variety of “possible” yield curve scenarios, such as “up or down 50basis points,” “10s/30s spread steepened by 10 basis points”, and combinations of all, then compute the price sensitivity and design hedgingstrategies accordingly. The traditional treatment of curve risks is lackingquantifiability, and it contrives interest rate scenarios purely based onsubjective “market sense,” instead of explicitly incorporating the correlation patterns derivable from recent history of market movements. Theduration risk matching for hedging interest rate level risks is quite effective when the interest rate portfolio to be hedged has similar cashflows tothe hedging instrument, although this desirable condition is often absent.By the definition of duration, the price variations are driven by changesin the yield to maturity, which is specific to each interest rate instrumentor portfolio. The hedging methodology of duration risk matching for twodifferent instruments (portfolios) implicitly assumes that the change ofthe yield to maturity of one instrument is identical to the change of theyield to maturity of the hedging counterpart. Realizing this apparentlyproblematic assumption, financial experts find a remedy through meansof applying the “ ,8 -adjusted duration,” in essence they are assumingthat the yield to maturities move synchronously in proportion to the ratioof absolute yield volatilities of bonds of differing characteristics.In the following sections, we examine these traditional concepts andmethodologies, perform regression and PCA for the Treasury yieldcurve, and establish a level and p-adjusted spread two-factor model.Then we introduce the methodology of the TFRM, examine its performance in comparison with that of principal components analyses.
Sec. 11.2 Principal Components Analysis
11.2
413
Principal Components Analysis
Identifying, describing, and understanding the forces, or factors, thatdrive interest rates is one of the most intriguing tasks in fixed-incomeresearch. Economists approach this task by trying to relate interest ratesto economic/political situation, market supply and demand, inflation rate,mass psychology, as well as dozens of other economic indicators. Whatmakes things more complicated is that these factors themselves are ofteninterdependent, and the relationship between the factors and interest ratesas well as the correlation among the factors themselves also change overtime. A good knowledge of the underlying factors and the relationshipsis required in order to establish an economic interest rate model that correctly explains and describes interest rate movements.In this section we pursue a historical and statistical approach. Instead of trying to establish the relationship between economic factors andinterest rates, we focus on describing and understanding the interest ratemovements in terms of the yield curve itself. A convenient and widelyused description of the Treasury interest rate term structure is the zerocoupon yield curve with 120 yield points for maturities ranging from0.25 to 30 years in 0.25 year increments.These yield levels for different maturities constitute a numeric representation of the effects of the economic and market driving forces. Suchan interest rate term structure is driven by all the 120 yield levels, whichare intricately correlated. These yield levels are the factors that determine the interest rate term structure. The price of an interest rate instrument with cashflows at different times will move in accordance with thechanges of these factors. Our goal is to search for a parsimonious set ofindependent factors that can account for most of the observed variations.This task can be achieved by applying the well-established principalcomponents analysis (PCA).For any given time series, S' for t = l , 2 , . . . , T ,with historical mean(11.1)and variance(11.2)
414
Two-Factor Risk Model
Chapter 11
We define the corresponding zero-centered and normalized time series as(11.3)0
Now, let us consider the time series of a set of N zero-centered andnormalized yield curve points, y!" , where i = 1,2;..,N is the years-tomaturity index, and t=l,2,.-.,T is the time index. We proceed to construct a set of independent factors xi" in terms of linear combinations of
yi') ,that is,(11.4)i=l
or in matrix form,
XU)= BY'"
9
(11.5)
with the hope that we can find that only a small number (such as two orthree) of these factors significantly contribute to the variance of the yieldcurve movements. To find out this, let us multiply both sides of Equation (1 l .5) by B-' to obtain the representation of yield curve points interms of the factors:y(t)= B-'X(')
(11.6)
Since the factors are independent, we can write(11.7)t
or in matrix form,(11.8)t
where A is the diagonal matrix Ai, = 44.. . We can see that 4 is thevariance of the factor x i .The covariance matrix V of yield curve points is, by definition,
Sec. 11.2 Principal Components Analysis
415
t
(11.9)
or in matrix format,
v =B - ~ A ( B - ~ ) T ,
(11.10)
which can be re-written in the more familiar form
B V B ~= A .
(11.11)
This tells us that, the set of independent factors are the eigenvectorsof the covariance matrix with eigenvalues given by the diagonal elementsof A .According to the above discussion, the PCA of yield curve movements can be done in four steps:Compute the covariance matrix of the yield curve points;Compute the eigenvalues and eigenvectors;Rank the importance of the factors (eigenvectors) according to themagnitude of the corresponding eigenvalues;Write the yield curve point movements in terms of the significantfactors only, in this way we can usually explain the yield curve movements quite well by using two or three factors.For now we assume that only the first M factors are significant, thenwe can writeM
y:" =
CBiLixr) +Small Terms ,
(11.12)
k =I
so that, if M is two or three, we only need to use a two or three factormodel to describe the yield curve movements satisfactorily. The totalobserved variance is of course given by
416
Two-Factor Risk Model
Chapter 11
N
V = Tr(A) =
cak ,
(11.13)
k=l
and the variance explained by the M-factor model isN
(1 1.14)Therefore, the variance coverage ratio is- 'M
P M --.
(11.15)
V
Now let us apply the PCA to the study of the U.S. Treasury yieldcurve. The Exponential Spline Model provides us with a history of dailyyield curves based on the prices of Treasury Notes and Bonds. For agiven historical period of N days, say three months, we compute thecorrelation matrix H ,(11.16)where yi is the daily change of the yield level for the ithyear point, qj isthe linear correlation coeflcient between the two time series y,n and y sfor IZ = 1,2;. .,N . $ is often called R-Squared. It can be verified thatq; is a positive number between 0 and 1, and is a measure of the fractionof variations explained by the linear regression of one variable ( y i )against the other ( y j ). In order to quantify the extent to which a particular variable is correlated to all the rest of the variables, we define thecommunality as
(1 1.17)k=l
which is the sum of the R-squared of a variable against all other variables. Notice that, in case the variables are independent, all communalities are equal to one.
Sec. 11.2 Principal Components Analysis
417
We applied the PCA method to the correlation matrix for an 11yield-point yield curve that includes the 1, 2, 3, 4, 5, 7, 10, 15, 20, 25,and 30 year yield levels, which are taken from the splined C-STRIP yieldcurve for the period from April 6th to July 6th 1995. Figure 11.1 illustrates the PCA results for this 11-yield-point splined C-STRIP yieldcurve.We can see that the first principal component accounts for 10.0367of the observed total variance of 11, or 91.24%. The second principalcomponent covers another 6.95%, and the third one 0.8%. The third oneaccounts for less than 1 of the total variance, already less than any of theoriginal factors can cover. It is encouraging to see that all the variationsof the yield curve movement can be described to 98% by using only 2factors, if well chosen. This observation remains true when we performthe same analyses for the complete 120-point yield curve.Test results also show that all components have communalities of 1.
The objective of factor analyses is to identify a minimum number of
418
Two-Factor Risk Model
Chapter 11
the most communal factors (or combination of factors) that best describethe system in consideration. Very often it is desirable to find meaningfuland mutually independent variables so that the analyses can help us understand the causal relationship between the driving forces and the observed phenomena, as well as enable us to control the process by fixingor changing each of the independent variables. However, the principalcomponents we have computed for the 11-point or the 120-point termstructure are rather complicated. Normally, the first principal componentis a weighted average of yield levels, and the second principal component can be interpreted as some average long-term and short-term yieldspread. Such combinations are unacceptable or at least inconvenient forfinancial applications, such as hedging the risks associated with yieldcurve movements.
11.3
Two-factor Risk Model Specification
Investors and financial analysts often speak of interest rate risks in termsof level and cuwe exposure. However, a well defined and quantifiablecurve risk factor (or descriptor) has been lacking, while it has alwaysbeen arguable as to whether the short term (< 1 year), medium term (10year), or long term (>20 year) interest rates should be used as the maininterest rate level indicator. For a general description of interest rates,we are in favor of choosing the intermediate long term rates, usually the25-year region, as the interest rate level indicator for the following reason:The prices of interest rate instruments with evenly distributed cashflows are more sensitive to long term interest rates;Short-term interest rates are relatively weakly correlated with mostother parts of the yield curve;The extremely long term (29 - 30 years) rates are weakly correlatedwith the rest of the yield curve.The curve factor is usually a shape or slope related characteristic ofthe yield curve; a straight forward choice is the long tendshort term interest rate spread, with the drawback that the level and spread are correlated.The two-factor Risk Model (TFR) uses the 25-year zero-couponas the interest rate level factor fL, and the /3 -adjusted 25yield YLeve,
Sec. 11.3 Two-factor Risk Model Specification
419
year yield Yhng and 5-year yield YShort spread as the curve or twist factor fT . Model parameters, such as yield point to factor correlations andregression coefficients, are computed based on the time series of yieldcurves in the 3 months preceding the analysis date. The p-adjustedspread is defined by the following:(1 1.18)is the regression coefficient. This particular combination iswhere p(A,B)statistically independent of the level yield Ybvel. In this framework,since the two factors are meaningful and mutually independent, investorscan readily translate their views on interest rate movements into the twofactor inputs.The previous section has shown that, for the given time period, thefirst and second principal components together can describe about 98%of all observed yield variation. Under most circumstances, any arbitrarily chosen two factors would not perform as well as the first and secondprincipal components. However, most practitioners in the financial industry are tempted by market intuition that the level and spread description of the yield curve should not lead us too astray from the target; itshould work simply! Now it remains for us to justify this belief.For the same 11-point yield curve time series, we have computed thecommunality of the level and p-adjusted twist factors with all of theyield points. Amazingly the two factors can cover up to 97% of the observed variance (See B. Li, 1999).We can conclude that the level and p -adjusted twist model performsonly slightly less efficient than the rigorous PCA for the specific timeperiods. In practice, however, risk analyses and hedging must be doneprior to the realization of future interest rate movements; the modelworks only if the parameters derived based on the historical period in thepast can carry over to the future unknown period. Because of this subtlety, the distinction between the rigorous PCA and the approximate p adjusted two-factor model becomes blurred, if not completely vanished.For the unknown future, the approximate model may happen to performbetter or worse, depending on how the correlation pattern of interest rateschanges. In fact the PCA may be fitting excessively to the noise specific
420
Two-FactorRisk Model
Chapter 11
to the historical period being studied, while the fixed choice of level andtwist factors are capturing the average movements of the market,“squeezing” out much of the noise. The eigenvectors obtained for different periods of time are indeed quite different; if they are applied to otherperiods of time, the performance is in general poorer than that of thefixed choice of the level and ,8 -adjusted twist as the two factors. TheTFXM does have better forecast power than the PCA.With the choice of level and twist factors, f L and f T , we can express the yield curve by the two-factor representation:
(11.19)where t is the time to maturity; the two p curves PL(t) and P T (t ),arethe regression coefficients for every yield-point daily change time seriesAY(”)(t)for n = 1,2,...,N , against the time series of the daily change oflevel factor f L and twist factor f T for the same time period. The lastterm 8 ( t ) is only a function of t (years to maturity), independent of thefactors.Now we define the level duration DL , and the twist duration DT as
(11.20)
where P is the price of the instrument or portfolio, and the durations canbe positive or negative. The traditional duration is defined as:(1 1.21)Then for a zero-coupon bond with a single cashflow at time t , thetwo-factor durations are simply the minus of the product of P and thetraditional duration D , because
Sec. 11.4 Empirical Validation
421
A coupon carrying bond or any portfolio with fixed cashflows can betreated as a collection of zero-coupon bonds, so that the total level andtwist durations are the aggregates of the corresponding cashflow contributions. We can see that the two-factor durations are driven by the cashflow structure of the interest rate instrument, as well as by the p curves,or in other words, by the correlation patterns of the yield curve.An immediate application of the TFRM is the two-factor DurationTable that provides the indicative information and risk profile for all theTreasury notes, bonds and C-STRIPS. With the help of the duration table, we can easily compute the interest rate level and twist exposure forany interest rate instrument or portfolio. The price sensitivity to changesin the level or twist factor can be calculated according to theRiskDuration relationship:
AP = PDLAfL + PDTA$T .
(1 1.23)
With the changes in factors AfL and AfT given together for combined views, or given independently for individual factor exposure estimates.In summary, we have so far defined the TFRM and proven that, forthe historical periods studied, it performs almost as efficiently as thePCA. We also systematically provide quantified risk profile for interestrate instruments.
11.4
Empirical Validation
When we apply the TFRM to quantify interest rate risks or design hedging strategies for an unknown future period, the effectiveness of ourtreatment will mainly depend on the stability of the interest rate correlation pattern. The other source of ineffectiveness is the incomplete description - 97% coverage - of yield curve movements by simply usingtwo factors. We claim that this effect is rather small compared with thatdue to the time evolution of interest rate correlation patterns. This point
422
Two-Factor Risk Model
Chapter 11
of view argues against the attempt to use a third factor to improve hedging interest rate risks, since the model risk (due to uncertainties with regards to the third factor correlation pattern) will diminish whatever gainsthat may result from inclusion of a third d i m e n ~ i 0 n . However,I~~it is important to stay with as few hedging variables as possible. In order tovalidate the TFRM, we study the historical correlation patterns and examine performance of hedging strategies for various time periods.Our TFRM results show that the yield curve usually flattens withinterest rate hikes, and steepens when interest rate declines. However,values may change by 10 to 20% over a threethe medium termmonth period. These results are consistent with the observation andanalysis of the historical data, which shows that the shape of the P and
PL
r2 curves are quite stable over time periods of a few months, and theinterest rate levels for all maturity terms are positively correlated withthe front end usually moving more than the long term end; the short endyield levels are negatively correlated.Another way to examine the model is to test the hedging strategiesbased upon it. At no extra effort, by simply adjusting the par holdingratios according to the two-factor durations, we generally can obtainabout a 30% reduction in price risk. The two-factor strategy exactlyachieves its goal of eliminating curve risks. Since traditional durationmatching is similar to the level risk hedging in our two-factor approach,we would expect little gain in hedging efficiency in periods of time wheninterest rate curve moves up or down without changing its shape,whereas we should observe significant improvements when the shapechanges.The actual performance of any hedging strategy usually depends onmany influencing market conditions, in addition to interest rate movements. For example, each participating bond may trade according to itsown line of fortune; a bond may become special, and an “on the run”issue may cheapen as it becomes “off the run.”The TFRM performs exceptionally well when applied to trackingbond indices by using a small number of instruments, usually 4 to 10bonds and strips.In essence the TFRM uses average historical co-movements of interFor a better descrhtion of interest rate term structure. a third factor that focuses on theextremely long region will improve performance; this will be further discussed in Section11.6.2.
See. 11.5 Applications
423
est rates to determine hedge ratios, so that it aims to hedge risks in a relatively long time horizon, usually one or two months rather than a fewdays. Beyond a few months, if not rebalanced, the correlation patternmay have changed too far for the hedging to be still effective.
11.5
Applications
Effectiveness and simplicity empower the TFRM to find a wide range ofapplications. The merit of the methodology mainly lies in its ability tocapture most of the interest rate risks by using two factors with clear financial interpretation. An investor with a definite view on interest ratescan readily estimate the expected performance, while one with a partialview can specifically hedge the uncertain part of the risks and designinvestment or trading strategies in accordance with the view.
7 7.5.7
Level-hedged BulletlBarbell Trades
In reality, interest yield curve shapes are just as unpredictable as interestrate levels. However, the risk associated with shape uncertainties is usually much smaller than the level related parts. One of financial servicefirms' main businesses is to provide clients with level-hedged but shapesensitive trade recommendations, so that the remaining exposure maymeet the investor's specific requirements. Traditionally this task is doneby various kinds of scenario analyses; often the process is complicatedand time consuming as it is unsystematic.The Level-Hedged BulletlBarbell Trades application is designed toserve these clients. By using the two-factor duration table, we constructproceeds and level risk neutral bulletharbell trades and compute theirremaining sensitivity to unit change of the /?-adjusted twist factor.When combined with the ricwcheap analyses application based on theexponential spline model, we produce a daily report of favorable bulletharbell trades with quantified curve risk.Some investors may be interested in executing proceeds neutral andtwo-factor risk neutral trades. In this case it is necessary to include afourth instrument.7 7.5.2
Two-factor Portfolio Hedging Strategy
Quantitative hedging in general involves the following three steps:
424
Two-Factor Risk Model
Chapter 11
Define and analyze the properties to be hedged.’25Identify the independent factors (or variables) that are driving theproperties. Usually it is necessary to quantify the movements of theseunderling factors.’26Choose hedging instruments that only depend on the same underlingfactors to hedge the portfolio. Hedge ratios are computed and maybe rebalanced dynamically as necessary.The TFRM provides a reliable and convenient framework for takingthese three steps towards practical hedging strategies. For most interestrate portfolios, the property to be hedged is usually the price, or the return performance relative to a benchmark or index. As demonstrated inprevious sections, under most circumstances the two independent variables - level and twist factors - together with the current interest rate termstructure are sufficient to describe all kinds of interest rate movements,so that the interest rate risks for the entire portfolio can be characterizedby using the aggregate two-factor durationsand.Suppose we have a portfolio P of N interest rate instruments, Bifor i = 1,2,-..,N , with corresponding two-factor durations 0;)and0;’ . Our objective is to hedge the price of the portfolio by using three, levelhedging instruments X ( k ) for k = 1,2, and 3 , with pricesdurations DE),and twist durations 0%).The hedging methodologyshould provide us with the position of each of the three instruments, A(k)for k = 1,2,...,3,so that we achieve a proceeds and two-factor risk neutral hedge.In order to achieve level and twist risk hedge, we should have
0y)
Pik’
12’ Sometimes movement of the property in one direction is desirable so that we onlyneed to hedge it against possible changes in the opposite direction. For example, an investor would like to hedge his portfolio against price depreciation, but would rather leaveit free to appreciate in value. However, this would imply the necessity of using optionsor adding convexities.In a realistic situation, the number of factors may be too many to be handled reliably.We must settle with using only a few (usually one or two) variables, as in the case ofhedging interest rate risks.
425
Sec. 11.5 Applications3
APL = PDy’AfL =
CAkP f ) D g A f Lk=l3
APT = PDY’ AfT =
CA k P f ’D g )AfT
(1 1.24)
k=l
The hedging should cost nothing and add no value to the portfolio(In this discussion we ignore transaction costs), so that we impose(1 1.25)k=l
In Equation (1 1.24), the aggregate durations for the portfolio aregiven in terms of those of its constituents by the following relationship:
(11.26)
where si is the number of shares of the i* instrument.Solving the above three simultaneous equations will give us thenumber of shares A,, 4, and 4 for the three hedging instruments X , ,X , ,and X , .For most investors, the objective of hedging is not to fix the price ofthe portfolio, but rather to achieve some target risk levels. For example,the investor may buy a selection of cheap instruments according to his orher judgment, while trying to match the level and twist durations with anindex in the hope that his portfolio will perform better. By using threehedging instruments (one of them can be cash, with zero duration), theinvestor can easily achieve this goal with the help of the TFRM. Let usassume that the index has level duration D P d e x,) and twist dura, the portfolio has level duration O F ) , twist duration D y ’ ,tionand price P “ ) , then by solving the following three simultaneous equations
LIPdex)
426
Two-Factor Risk Model
Chapter 11
AP(1) + ;12P(2)+ h P ( 3 ) = 0P (1)DL(P) +ADy+;12Dy+/23DL(3) = P (P)Dt(Index) ,
(1 1.27)
A$) + ;12,0p+ 4 4 3 ) = p ( P ) D p d e x )
p ( 1 ) ~+p
we will obtain the number of shares 4,4, and 4 of the hedging instruments.In the above discussion, the types of instruments that can be used ashedging instruments or as constituents of the portfolio are rather general,as long as they are mainly driven by the level and twist factors and thecontract. The derivation of the DL and DT for futures contracts willinvolve treatment of delivery options and other complications. The corresponding durations can also be calculated. For example, each of thethree hedging instruments can be a bond, a C-STRIP, cash, or a bondfuture.
11.5.3 Bond Indices with Level and Curve Risk ProfileFixed income bond indexing is a very popular investment vehicle amonginstitutional investors. To construct an indexed bond portfolio, it is unrealistic for investors to buy all instruments in a chosen index because ofliquidity and transaction cost. A more practical approach is to buy asmall basket of instruments that will replicate the characteristic or riskprofile of a selected index. A successful index bond fund should be ableto benchmark against an index when it is exposed to both level and twistyield curve movement. It is common knowledge that one can track anindex by matching its duration when yield curve movement is of parallelfashion, but it is often problematic to match an index when the yieldcurve shift is non-parallel or twists. By applying TFRM,we can construct a portfolio to track an index more closely whether yield curve ismoving in a parallel or non-parallel mode. To illustrate this application,we structure two Treasury portfolios to match a Government TreasuryIndex.The first portfolio takes TFRM duration into consideration while thesecond portfolio tracks traditional duration only. In both cases, linearprogramming techniques are employed to search for the optimal portfoliorequired to match the index characteristic and other prescribed constraints. We define the objective function to be maximizing marketweighted yield. We also constrain the portfolio to have approximately
Sec. 11.6 Adjusted Durations
427
equal average coupon and maturity versus the selected index. For thetraditional portfolio the duration is set to equal the index. For the twofactor tracking portfolio the level and twist duration are set to equal tothe index. We tracked the performance of the Treasury Index and bothportfolios from 6/30/94 to 6/30/95. For the purpose of comparison, therewill be no rebalancing during the period. Results show that both portfolios tracked the Treasury Index quite closely in total return, but the twofactor Tracking Portfolio is less volatile than the Traditional Portfoliowhen they are compared with the Treasury Index in