total dollar valuesduring the time period being analyzed.Test results also show that, for the time period considered, the TFRMdoes capture non-parallel interest rate movements. When we controlboth level and twist factors, we can replicate (or hedge) an Index moresuccessfully than with the other traditional method.
11.6
Adjusted Durations
The TFRM has taken a step further in quantitative description of interestrate risks; it opens up the door to many opportunities in risk managementand designing of profitable trading strategy. It is a descriptive model thatrelates the familiar semi-economic concepts of interest rate levels andyield curve shapes to interest rate exposure. The model's forecast powerlargely depends on the relative stability of interest rate correlation patterns that change relatively slowly as time elapses. According to marketexperience, interest rate volatilities are more stable than interest rate levels, and the impact of correlation pattern variations is relatively weakcompared with that caused by interest rate level movements.Experience reveals that yield curve shape and interest rate level areintimately correlated. Yield curve usually flattens as rates rise, andsteepens as rates fall. In the period from April 6 to July 6, 1995, the regression coefficient for the 25-year zero-coupon yield Y,, and the 2 9 5spread, Y,, - Y, , is p = -0.53 with r2 = 0.24 . Traditional scenarioanalyses do not contrive interest rate scenarios according to the observedcorrelation pattern, but rather create scenarios arbitrarily. On the otherhand, the TFF$M reconstructs the interest rate term structure as a functionof the level and twist factors supported by historically observed /3curves, as illustrated in Eq. (1 1.19). Under the framework of the TFRM,interest rate scenarios are generated by perturbing the current factor val-
428
Two-Factor Risk Model
Chapter 11
ues. For example, the scenarios can be “100 basis points up or down inthe level factor with the twist factor fixed”, or “50 basis points downin the level factor accompanied by 20 basis points up in the twist factor;”the yield curves derived based on these scenarios are consistent with thehistorical interest rate co-movement patterns. We want to emphasize thatthe introduction of the P -adjusted spread, defined by Eq. (1 1.18), as thetwist factor will not cause any ambiguity in designing scenarios. Investors who are reluctant to use our two-factor language can always translate their scenarios into the two-factor version.The current yield curve plus a scenario will always lead to an expected yield curve. By measuring the level and P -adjusted spread, wecan obtain the expected level and twist factors and their differences fromthe current values. This will translate the conventional scenario into thecorresponding two-factor scenario, and subsequently perform risk or return analyses.We claim that the TFRM approach leads to more realistic scenarios.Let us demonstrate this point by examining the most widely used “up ordown” scenario analyses. A pure “up 100 basis points” will translateinto a shift up of 100 basis points in the level factor f L , accompanied bya basis point change in the twist factor asAfT =l-loox
PLong,LevelPShort,Level*
Obviously, this is a rather unnatural constraint. By construction ofthe TFRM, the twist factor is independent of the level factor. When theinterest rate level changes by an amount, one should not have to assumethat the twist factor would also move by a certain amount. In the TFRMapproach, a scenario of level “up 100 basis points” means that the levelfactor will rise by 100 basis points, with the twist factor fixed, and thiswill lead to basis point change in the long term yield spread of(11.28)
Usually the short end moves more than the long end, namely,PShort,Lmel
’
&ng,Level
*
(11.29)
Therefore, the spread narrows as the level increases. This agrees withthe notion that yield curve often flattens or becomes inverted as rates
Sec. 11.6 Adjusted Durations
429
hike. 127One can see that a certain yield curve “form,” specified by the twistfactor in the context of the TFRM, refers to a family of yield curves withdifferent levels and naturally occurring shapes. The “form” is independent of the level factor, and, it changes if and only if the twist factorchanges. Notice that in the two-factor picture, the curve does not have toflatten as its level increases; the curve can assume any form. It can become steepened as the level hikes, if the twist factor happens to increasesignificantly. The basic assumption underlying the TFRM is that different parts of the yield curve move in proportion to historically observedco-movement pattern.The above discussion of level scenarios also applies to twist projections. The TFRM approach can design scenarios that correspond to pureyield curve “form” changes independent of level movements, while theconventional twist scenario (given in terms of spread widening or narrowing) unnaturally forces the level to be fixed, when in reality it is tiedto the spread. The two-factor model scenario such as “up or down 25basis points in the p -adjusted twist factor with the level fixed” maysound unfamiliar, but it will not lead to unrealistic yield curves.It is important to remember that the two-factor representation ofyield curves can cover a wide range of different interest rate term structures; any combination of the level factor fL and twist factor fT in Eq.(11.19) will produce a two-factor yield curve that is consistent with historical correlation pattern.Hedging strategy based on the TFRM automatically neutralizes therisks associated with these curve movements plus all other possible combinations. 12*The above prolonged discussion of scenario analyses is meant toclarify differences and similarities between the TFRM and traditionalapproaches. In fact one would never need to perform the cumbersomescenario analyses under the framework of the TFRM, unless one has adefinite and precise view on interest rates!~
This doesn’t imply that the shape of the yield curve can be determined by its level. Itonly suggests that sometimes the twist factor changes rather slowly in time with smallamounts, so that the part of shape change due to level movements dominates.The risk relationship given by Eq. (11.23) is a first order approximation. For largerfactor movements, e.g., q > 20 basis points, second order terms that involve convexities(second derivatives with respect to the factors) will become important. Then hedge ratiosneed to be refined.12’
’**
430
Two-FactorRisk Model
Chapter 11
The TFRM emphasizes the importance of using mutually independent factors in the representation of yield curves, so that it introduces themore abstract concept of p -adjusted spread twist factor. We have alsoobtained two-factor durations that can be positive or negative. With everimproving market efficiency and widespread use of information andquantitative technologies, the subject of finance is entering the stage ofscience. The task of quantifying all aspects of interest rate risks has longbeen pursued. Scientific concepts and methodologies are a necessity foraccurately describing, understanding, and controlling interest rate risks atoptimal efficiency. Sometimes it is necessary to introduce new concepts,re-examine certain existing ones, provide modifications and find newmeanings for them. The remainder of this section will discuss relatedtopics and future research directions.
77.6.7
PAdjusted Duration
The co-movement ratios of yield levels can be used to improve the traditional duration risk hedge. This methodology is similar to the treatmentof the level risks in the two-factor framework. You may have been familiar with the concept of /l-adjusted duration in computing hedge ratios.In the traditional treatment of price risks associated with the uncertaintiesin the yields to maturity of a pair of interest rate instruments, Bl and B2,the changes in prices are given by
MI = 4 DlAY,AP2 = P2D2AY2
(11.30)
If the objective is to hedge the price of 4 units of instrument B, byselling ;32 units of instrument B2 , one would apply the equationAM1 -AM2 =AqD,AY, -4P2D2AY2.
(11.31)
Combined with the assumption that the changes in yield to maturityis perfectly correlated with some benchmark yields, say the 10-year onthe run Treasury yield Ybenchmark, such that(1 1.32)
Then, we have
431
Sec. 11.6 Adjusted Durations
(11.33)By solving the above equation and Equation (1 1.31), we obtain
(44DlPI - 4P2D2P2 1AY2 = 0
(11.34)
3
which leads to the hedge ratio(1 1.35)If we defineD*=~D,
(11.36)
as the p -adjusted duration, then the hedge ratio equals the ratio of theproducts of D* and price P .When the volatilities associated with the yield to maturities are readily available, instead of using Equation (1 1.33, practitioners use(11.37)and define(11.38)
D’= DO
as the volatility adjusted duration. Let us examine the validity of thistreatment. If we assume that the interest rate term structure is entirelydriven by only one factor-the interest rate level f L , so that all rates areperfectly correlated, then the yields to maturity q and Y2 of two interestrate instruments are related by a deterministic function
andvar(AY,) = f
l2
var(AY2),
a, = f lo2,
-a’ - f’.02
(11.40)
432
Two-Factor Risk Model
Chapter 11
Therefore the ratio of the absolute volatilities is approximately equalto the ratio of the betas,/
PHedge = PHedge *
(1 1.41)
The approximation becomes exact if the yield to maturities is linearly related; it is rarely so. However the approximation is usually quitegood, especially for a period of small movements of interest rates. Thisreveals that the volatility adjusted duration hedging is a special case ofthe p -adjusted duration treatment, and it is an approximation. Conventionally, the volatility adjusted duration is also indiscriminately called thep -adjusted duration, due to their close relationships.As demonstrated in the previous sections, interest rates cannot beadequately described by a one-factor model; in other words the yields tomaturity of instruments are never perfectly correlated. One should always use the p ratio instead of the o ratio if the former is available. Inthe two-factor model context, this issue is entirely absent, because therisks are given in terms of changes in the well defined, application independent factors, not in terms of variations in the instrument specific yieldto maturity.
77.6.2 Hedging the Extremely Long EndSmall movements in the long term interest rates often result in relativelylarge price changes for long term instruments. What makes things worseis that the extremely long term ( > 2 6 years) interest rates are ratherweakly correlated with the rates of the rest of maturity terms; they move“on their own,” as financial analysts often say. This phenomenon isclearly demonstrated in Figure 11.1, the R-squared add up considerablyless than one beyond 26 years. In order to cover the long end, a thirdfactor is necessary. Since the poor correlation with the very long end,any short-term instruments cannot be used to adequately hedge a portfolio with a significant proportion of cash flows in the 26 to 30-year sector.The only logical solution to this problem seems to be the addition ofa zero-coupon bond with very long maturity time, such as a 29 year CSTRIP, to the basket of hedging instruments. For example, if the portfolio to be hedged includes all the Treasury bonds and notes with maturities longer than 5 years, one may use a 7-year note, a 10-year bond futures contract, a 20-year bond, and a 29-year C-STRIP to form the hedge,such that in addition to two-factor risk neutral and proceeds neutral, one
Sec. 11.7 Future Directions
433
also achieves cashflow match in the very long sector: the C-STRIP cashflow cancels the sum of cashflows in the 26 to 30 year region. In thisway one has effectively applied a three-factor model to capture interestrate risks more precisely.
11.7
Future Directions
An important future research direction following this two-factor approach is to study the stochastic process for the level and p-adjustedtwist factor, so that interest rate derivatives, such as bond futures contracts or interest rate spread options, can be treated within the sameframework. Without a uniform choice of factors in both static description of risks and dynamic valuation of interest rate ~ptionalities,’~’wewill not be able to handle a portfolio that includes interest rate underlyinginstruments and interest rate options. To construct an arbitrage free twofactor stochastic interest rate term structure model is also valuable forhandling interest rate options or derivatives in general.By applying the exponential spline model, we have systematicallyanalyzed some non-dollar government bond markets, such as French,Canadian, United Kingdom, Germany, Japanese, and Italian Treasurymarkets. Preliminary tests reveal that, with choices of suitable level andp -adjusted twist factors, the TFRM works well for each of these markets. In recent years, the international interest rate markets have becomemore and more efficient, accurate quantitative analyses are essential forinternational investors. The TFRM will find tremendous opportunitieswhen it is applied to the non-dollar markets where quantitative analysesare still under development. Most of the U.S. market version of application we have discussed can be readily applied to the international markets. Another important research project is to use the TFRM as an “equilibrium” reference of current interest rate term structure, so that one canidentify (abnormal) “rich” or “cheap” sectors of the yield curve, relativeto recent interest rate behavior. More specifically, one can compute theaverage 6 curve for the 3 month period preceding the analysis date, anduse Eq. (1 1.19) to back out the expected yield curve as it should be according to the two-factor model; a comparison of the current yield curvewith the expected one will reveal any abnormalities that may indicateprofitable investment opportunities.~
‘29 For example if one uses a different pair of factors in option pricing.
Chapter 12
The Holy Grail Two-FactorInterest RateArbitrage
This chapter is based on one of the so-called “million-dollar papers” theauthor wrote in 1994. The theory and methodology was put into practiceat major Wall Street trading houses and generated remarkable profits forthe proprietary trading businesses. Now many years later, we are readyto reveal it to the general public. In particular, we explain and emphasizeone crucial element, often ignored or neglected by academia, of interestrate trading: the financing costs associated with maintaining a tradingposition. Without the consideration of financing costs, all analyses oftrading strategies are impractical.This chapter presents the two-factor arbitrage method for identifyingstatistical arbitrage opportunities in the interest rate market and explainshow one can implement the trade by creating a three bond portfolio.
12.1
Profit, Loss, and Financing Costs
The net profit or loss of a trade or a portfolio, in the industry’s terminology, is called the P&L. An interest rate proprietary trader seeks profitsby making trades: he buys or sells short bonds to create trading positions,then at a later time closes them out at a profit or loss. One of the keyelements of an interest rate portfolio is its “carry”- financing costs (orproceeds) one has to pay (or receive) in interest payments in order to put
Sec. 12.2 Two-factorArbitrage
435
on and hold the positions. In order to establish a portfolio and receive itsvalue appreciation, a proprietary trader pays interest to borrow cash tobuy bonds, andor sells bonds short and receives interest on credit balances from the short sales. For example, in order to buy a bond and holdon to it, one borrows money to buy the bond by using the bond itself ascollateral and pays interest for the holding period at the so-called rep0rate which is market determined and real-time quoted for each bond. Onthe other hand, in order to short a bond and hold on to the short position,one borrows the bond from the broker, sells it in the market, receivescash and earns interest on the balance at the rep0 rate available to him atthe time the trade was executed. A bond portfolio is just an aggregate ofsingle long and short positions, and its net financing costs is the sum ofthe financing costs of all the individual components. Therefore, the P&Lof the portfolio is equal to the price appreciation of the portfolio minusthe financing costs.
12.2
Two-factor Arbitrage
According to extensive empirical studies (see Chapter lo), for all themajor interest rate markets, the yield curve movements can be well described by a two-factor model. The two-factor description usually explains 90% to 99% of observed variances. In other words, the randomprice fluctuations of all the bonds in the interest rate market are mainlydriven by two independent sources of unpredictable noises (factors). Asa first order approximation, the random price fluctuation, over a timeinterval A t , is given by a linear function of the two factor movements:AB = RBAt + ABAf, + @BAfi(12.1)where R is the “local” yield; Afi and Af2 are the two-factor changeswhich are unpredictable random noises; A and 4 are the two-factor durations discussed in Chapter 10. R,A and @ are the risk properties ofthe particular bond and they can be readily computed by employing thetwo-factor model.Given three or more bonds, one can construct an overall position thatis independent of both factor movements. For example, let’s considerthe three bond portfolio:
436
The Holy Grail - Two-Factor Interest Rate Arbitrage
P = x B ~+ y B 2 - Bwith x and y satisfying the following conditions:AB = xAB, + y 4 B 2
Chapter 12
(12.2)
(12.3)4 B = XABI + Y A B ,The portfolio is riskless in the two-factor framework as the factorexposures are canceled out, with deterministic price increment given by(12.4)AP = (xR1Bl + yR2 B2 - RB)At .The financing cost associated with implementing this portfolio is(12.5)AF = (xq Bl + yr2B2 - rB)At ,where r , q and r, are the rep0 rates for the three bonds respectively. Inan arbitrage free market, such a position should satisfy AP-AF =0,namely,(12.6)XR,Bl + yR2B2 - RB = xq Bl + yr, B2 - rB .This equation is the two-factor arbitrage free condition for a threebond portfolio.For an intuitive explanation, let’s look at Figure 12.1. Curve (a)represents the price trajectory of a risk free investment, such as a moneymarket account. Curve ( c ) is the price movement over time of a riskyinvestment in a bond, which can be seen as a trend curve (b) plus randomfluctuations. Normally the trend curve should grow faster than the riskfree curve since investors should be compensated for taking risks.Price
A
Time
Figure 12.1. Price trajectories of risk free and risky investments.
A naive arbitrager would borrow at the risk free rate and buy thebond hoping to capture the higher growth rate corresponding to the trend
Sec. 12.3 Trading Strategy
437
curve.*3o But the outcome of the investment is uncertain. Subject toholding period constraints, the strategy is not viable. However, if he usesthree bonds to construct a portfolio such that the random components ofthe three bond price trajectories exactly cancel out, he will be left with ariskless position which grows at the combined trend curve rate which canbe higher or lower than the risk free rate. If equation (14.6) is violatedwith the left hand side greater than the right hand side, the combinedtrend rate is higher than the overall financing rate, and the self financedrisk free portfolio will make money over time, effectively creating anarbitrage ~ i t u a t i 0 n . l Such~ ~ a construction is possible because the threebond price trajectories have to “wiggle” in a certain way as required bythe two-factor nature of the yield curve movements.
12.3
Trading Strategy
In order to find profitable trading opportunities, one can construct allpossible triplets of bond positions and test them against Equation (14.6).If one finds a significant discrepancy, he can put on the trade to make aprofit.For example, let’s consider a proprietary trading operation in theU.S. government bond market. As discussed in Chapter 13, the short andlong maturity ends of the yield curve tend to move on their own and arenot well explained by the two-factor model. Therefore we shall limit ourselection of bonds to those with maturities between 2 to 27 years. Thereare over 200 such liquid bonds and STRIPS in the US. government market. For each instrument, the two-factor model in Chapter 10 provides alevel and twist duration(12.7)
In practice, in order to borrow money to buy the bond, the arbitrager needs to putdown a deposit to satisfy margin requirements and use the bond as collateral. He finances the bond by paying the rep0 rate which is roughly the risk free interest rate.13’ In case the right hand side is greater than the left hand side, he can do the oppositetrade: long B and short Bl and B2.
438
The Holy Grail - Two-Factor Interest Rate Arbitrage
Chapter 12
1 aBOn the other hand, the ‘‘local’’yield R = --,which is the partialB atderivative of the bond price with respect to time, can be numericallycomputed by taking the ratio of bond price changes and the corresponding time to maturity increment, while holding the yield curve fixed. Intuitively, this local yield is the instantaneous total rate of return which isequal to the bond yield plus the “roll effect” of the yield curve.The bond price B and all the risk parameters, R,A and 9 , as well asthe rep0 rate r are now available in real time. We can use a computerprogram to compute the units x and y for every triplet of bonds possibleand test it against equation (12.6) for arbitrage opportunities. Actual experience indicated that such profitable trading opportunities were plentiful back in 1997 when the method was put into practice.
Figure 12.2. A schematic chart of a 3-bond two-factor arbitrage trade.
A schematic chart of the P&L of a 3-bond two-factor arbitrage tradeis shown in Figure 12.2. In the trade, we buy x units of bond B,, y unitsof bond B,, short 1 unit of bond B, finance the position (given by Equation (14.2)) by paying interests at rep0 rate 5 for bond Bl , r2 for bondB, , and receiving interest payment at rep0 rate r for bond B. The trade isselected because Equation (14.6) is violated, such that8=(xR,B,+yR,B,-RB)-(x~B,+yr2B,-rB)>0.(12.8)The trading position (or P&L) should grow in time at the rate 8given by equation (14.8). In Figure 12.2, the upward slope of the position trend line is equal to 8.Since the two-factor model does not explain 100% of the yield curve
Sec. 12.3
Trading Strategy
439
movements, as illustrated in the figure, the actual trading positionevolves around the trend line. The difference between the actual positioncurve and the trend line is the residual noise not explained by the twofactor model. In practice, we enter the trades when the position line is atleast one standard deviation below the trend line and expect it to revertback to the trend and move further up. Actual experiences showed thatsuch trades have high success rates.One caveat is that the two-factor model, the usefulness of which depends on the persistence of the correlation patterns of the yield curve, isnot stable over time, and the noise residual can grow so large that thetrading position moves permanently astray from the expected trend line,leading to an unexpectedly large loss or gain. Therefore this tradingstrategy is not strictly a risk-free arbitrage trading strategy - it should becategorized as a stat arb (statistical arbitrage) method.Today’s global interest rate markets are very efficient, and thusstrictly risk-free arbitrage opportunities are rare if not non-existent. Statistical arbitrage methods become one of the main tools of professionalproprietary traders. The methodology presented above can be readilygeneralized and applied to multi-bond trades as well as trades that involve bonds, swaps, futures and credit instruments. The central idea is tohedge away risks that can be well explained by correlation patterns,search for an edge - such as the growing trend - and take on risks thatone can understand and control.
Chapter 13
YieldDecompositionModel
This chapter presents a yield decomposition model (YDM) that studiesthe US.Treasury spot curve by decomposing it into three different components:The expectation curve;The risk premium curve;The volatility/convexity contribution.The model reveals the market expected FED funds rates and provides a phenomenological theory of yield curve movements.In today's market environment, what is the risk premium one can getfor extending an extra year of duration? What is the dollar value of convexity? What is the market expected FED funds rate? Why the impliedforward rates always seem to be biased higher? The yield decompositiontheory can answer these questions.The objective of this chapter is to provide a phenomenological theory of the yield curve and its driving forces. We try to provide the methodology for quantifying such important but ambiguous concepts as risk,risk premium, convexity value, and market expectations, and explaintheir relationship with the yield curve. Our goal is to formulate a practical and useful theory, a model that reveals the nature-financial phenomena-in simple and elegant forms.
441
Sec. 13.1 Volatility Adjusted Duration
13.1
Volatility Adjusted Duration
The spot curve (zero-coupon curve) and the corresponding volatilitycurve constitute the interest rate term structure. In the study we apply theorthogonal exponential spline model to build the yield curve for the U.S.Treasury market. The volatility curve is derived by using the time seriesof the yield. Each point on the spot curve represents the yield of a synthetic zero-coupon bond. We gain knowledge of the interest rate marketby studying these synthetic zero-coupon bonds.The modified duration is a measure of price sensitivity to yieldmovements. For the purpose of measuring the relative price risk on different days, we define the volatility adjusted duration (VAD) as theproduct of the modified duration and the relative volatility, i.e.,
(13.1)where d is the VAD, D the modified duration, 0 the normalized volatility (product of the yield level and the percent volatility), and Ore, thelong term reference volatility. In the subsequent discussions, we fixare,= 80 basis points.IIOL,,
3040
0
, ,I , , I , , ,, , , I I , , ,
, , , I , , ,
I
5
LO
, , , , I , , , g
, , , II5
, , , , I , , , ,
, , , , , , , , , ,,
l , , , , l , , , ,20
, , , ,
25
, , , ,
30
Years u, maarity
Figure 13.1. Volatility adjusted duration as a measure of risk.
The VADs for all the points on the spot curve form a VAD curve, orrisk curve. Figure 13.1 illustrates the normalized volatility curves andVAD curves for 4 different trade dates. We can see that, in order to
Yield Decomposition Model
442
Chapter 13
compare risks for different dates and different maturities, it is indeed important to use the VAD. A 10-year zero-coupon bond on August 26th,1997 has a VAD of 9.87. On the other hand, the VAD of a 10-year zerocoupon bond on August 26th, 1996 is 12.82. The difference is as largeas 23%. Without the VAD, one would have naively thought that therisks are about the same for the 10-year zero-coupon bonds on both days.The VAD provides a uniform risk measure across different times andmaturities. It facilitates the comparison of risks for different instrumentson different days.
13.2
Dollar Value of Convexity
Given today's spot curve, volatility structure, and the investment horizon,we can compute the dollar value of convexity for an investment in aTreasury instrument. Let us define 7 the horizon yield implied by thespot curve, y the horizon yield, z the maturity of the zero-coupon bondon the horizon date, o the horizon yield volatility and h the investmenthorizon. According to the stochastic nature of interest rates and arbitrage-free pricing arguments, the horizon yield y is normally distributedwith mean of j7 and volatility of o&For convenience, we will use the continuous compounding convention in the discussion for the rest of this chapter. On the horizon date, theprice of the bond is e-yr . The expectation of the horizon price is then(13.2)
P=E[e-yr]=exp
Now let us assume that we are instead in a still interest rate environment, that means o = 0 , then the expected horizon price is(13.3)po= E[e-Yr 1 = e-7' .From the above two equations we can see that the expected horizonprice is higher in a stochastic interest rate environment. The differencebetween P and Po is said to be the dollar value of convexity, that is,1
-r2c2h(13.4)Pc=P-P,=Po(e2-1).For a short horizon period h = 0 , the dollar value of convexity be13* It is a good approximation to assume that y is normally distributed for a short horizonperiod h and small interest rate movements.
Sec. 13.3 Expected Total Rate of Return
443
comes:(13.5)c-2From a total rate of return (TRR) point of view, we can re-write Eq.(13.4) as
P --r1 2o2hP,.
1
-r2a2h
P = p0 + Pc = &e2
(13.6)9
which tells that the convexity/volatilitycontribution to the TRR is(13.7)c-2where C = r2/2 is the convexity of the zero-coupon bond of maturity r .This result has been obtained by using more elaborate stochastic analyses.A coupon baring bond is equivalent to a portfolio of zero-couponbonds. The convexity/volatility contribution to the dollar return of acoupon bond is just the sum of the convexity/volatility contributions tothe dollar returns of the zero-coupon bonds.
R - -1r o2 2 = o2 C ,
13.3
Expected Total Rate of Return
In a stochastic interest rate environment, the yield of a zero-coupon bondcan be assumed to follows the geometric Brownian motiondy = pydt + OydW ,(13.8)where y is the yield to maturity, p the “drift” rate (percent change ofyield per year), o the normalized volatility, and W the Brownian motion. The expected total rate of return (TRR) of an investment in thezero-coupon bond is given by four different contributions:The yield;The roll return;The convexity/volatility contribution;The drift.Therefore the expected TRR can be written as:(13.9)E[TRR] = y + kD + 0 2 C- p y D ,where k = dydr is the slope of the spot curve, and D the duration. The
444
Chapter 13
Yield Decomposition Model
second term, kD , is the roll return.’33The last term, p y