FI Performance Measurement
Content
Fixed Income Attribution Model
This paper proposes a flexible fixed income attribution model that can be adapted to a large range of investment processes. The StatPro return breakdown model can fit a very large range of fixed income investment processes. Pricing techniques are used to understand the impact bond-specific factors have on security returns. This allows us to give an accurate and detailed security performance decomposition. Bond specifics naturally lead to particular management methods and therefore require adapted ex-post performance attribution analysis. Mathieu Cubilié is a performance measurement consultant at StatPro Plc, where he is product manager for StatPro Fixed Income, the company bond performance measurement and attribution system. Mathieu holds a postgraduate degree in Market Finance and Economics from Toulouse University and began his career at Sinopia-Asset-Management, HSBC-group quantitative management subsidiary, in Paris, where he experienced bond portfolio management. lends itself to the use of derivative products. Unfortunately, futures contracts are replicating bond Once the various performance contributions are com- maturity behaviors rather than bond index returns. As a puted, they can be considered as elementary building result, bond managers practice benchmark sampling. blocks that can be rearranged according to the portfolio This technique consists of trying to track large indices with a limited number of assets. management decision process. INTRODUCTION The single currency return attribution model is flexible enough to permit concurrent elementary return decompositions. It is also able to attribute at different levels of detail corresponding to several available attribution effect aggregation capacities. To finish, we will see how this single currency model can easily be used ina multi-currency environment. FIXED INCOME MARKETS Bond and equity markets are clearly different; equity indices are intra-day quoting, while very few bond indices are. As a consequence, bond managers require market indicators during the day to run their investments. Actually, they’re using a sample of liquid bonds representing the various market liquid maturities. This set of bonds is known as the Yield Curve. The Yield Curves are continuously quoted and there information is accessable. They represent different markets: mainly local currency Treasury markets and also credit mar- THE PRICE EFFECT kets. A major difference with equity markets is that bond Index replication management is a difficult exercise; markets are traded “over-the-counter” while equity bond indices contain several thousands of securities, markets are organized markets. Prices are propriety to many with very large minimal trade size. This means brokers and index providers. This leads to a situation managers would require huge amounts of investments where price difference between a security held in a to closely replicate an index. This situation definitely portfolio and in the index can be very significant. This Winter 2005/2006 -49The Journal of Performance Measurement In a multi-currency universe, after making yield market and currency allocation decisions, managers have to define which bond in particular they want to invest in. They usually have duration objectives against the benchmark reflecting their market expectation. Market anticipations are mainly yield curve movements and credit spread variation. Bonds will therefore be selected for their characteristics: their duration, the spread they offer, and all other specificities. To achieve this goal, bond management is ruled more by risk control than the equity market. In the bond area, risk is first reflected by the Duration. MANGEMENT PROCESSES Despite the fact that there are many investment processes, bond managers generally face the same considerations.
phenomenon, known as “price difference effect,” can represent a large part of the excess return. This phenomenon can exist in an organized market where back office systems are not fed by index data. This is much less importance in equity markets, as volatility remains high compared to the price difference effect. For bonds, the price difference effect can be even more important than the excess return. In such case, a performance attribution exercise may be misleading.
The return formed by two consecutive intermediary prices is considered a security return contribution. With this approach, it is possible to go as far as required in the return breakdown and to adhere to the real management process. It is not sufficient to work at the security level only, as most of the management decisions are not made independently from one another, which makes their impact difficult to measure. For instance, selecting a long-term maturity bond on the yield curve also means, in the general case selecting a high duration bond. If this bond is a corporate one, this also means it is supposed to deliver a higher yield than the reference market. Of course, this also exposes the investment to spread variations that will impact the security return.
Whatever the reference account return source (back or front office), a first requirement is to clean the initial portfolio return from the price effect. This means that the first step is to reprice portfolios and benchmarks, using the same price base. For simplicity reasons, it is commonly accepted that the portfolio is repriced using the benchmark prices for common securities. Noncommon securities will keep the back office prices. What about transactions? Pricing techniques can only This is extended to common exchange rates for securi- explain price returns and therefore ignores the impact ties in multi-currency portfolios using benchmarks rates. of transactions made by the manager. For this reason, we also think an accurate attribution model must be transaction based and must clearly identify their Portfolio Reference impact.
Return
RETURN CONTRIBUTIONS For simplicity reasons we will consider plain vanilla bonds in this chapter.
Price Effect
Portfolio Adjusted Return
Let’s focus on the portfolio return, and let’s run a first In the following, we will concentrate on understanding decomposition of each instrument price return. In addithe portfolio recomputed return. tion, we will measure the trade impact by computing the return using the Modified Dietz and the all-in price rManagement = rreference + rPr ice return and consider their difference. SECURITY PRICING TECHNIQUES In order to understand the portfolio or benchmark rModified return, it makes sense first to analyze the performance of each instrument.
Dietz
=
MVe − MVs − ∑ CFt
t =1
n
MVe + ∑ CFt ×
t =1
n
s − t (See Appendix A) s−e
As Modified Dietz return captures trade return by defiEach security is associated with a pricing function. nition, we can derive Such a function establishes a link between the security rTrade = rModified Dietz − rPr ice price and its yield to maturity. Shocking the yield to maturity with elementary yield variations related to market conditions allows us to generate a set of inter- From now on, let’s consider the price return, . Pr ice mediary prices between the beginning and the end of period price. There are three major sources of return commonly identified when closely looking at a bond.
r
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rPr ice = rCarry + rYield Curve + rCredit 1
A first component of a bond return is the carry return2 reflecting the impact of time. Then, the Yield curve return reveals how the reference (Treasury) market reacted over the analysis period. This return is created by global yield curve movements. At last, the credit return is the part of the return, which is specific to the bond and not to the reference yield market. This corresponds to changes in the security spread with the reference yield curve. If we make the assumption a bond price depends on time and on its yield to maturity, then from a pricing point of view we can generate the following prices.
Carry return:
rCarry =
PCarry Pt −1
−1
Yield Curve return:
rYield
Credit return:
Curve
=
PYield
Curve
PCarry
−1
−1
rCredit =
Pt PYield Curve
This leads to the split of the account (or benchmark) return in terms of main sources of return. DETAILED RETURN CONTRIBUTIONS
sponding to date t-1 and to the yield to maturity at this Each of the three main sources of return can further be date. decomposed and give a more detailed vision on how returns were constructed. This can still be achieved Pt Carry = P(t , yt −1 ) , is the intermediary price taking thanks to the use of pricing functions. into account the time passing by. It is simply obtained by switching the date at the end of period one in the for- Carry Return: mula. The carry return can be split using two possible break, is the intermediary downs: price capturing the Yield Curve movement. It can be obtained by considering the security at date t and by • Coupon and convergence return adding the corresponding yield curve change to the start of period yield to maturity. In this breakdown, the return due to the bond yield to maturity can be decomposed into two distinct mechaPt Credit = P (t , yt −1 + YC[t −1,t ] + δ [t −1,t ]) , is the end of peri- nisms the coupon return component and the simple pasod price. It can alternatively be computed from the sage of time, denoted as roll down effect, reflecting the yield curve change and the spread change over the peri- bond convergence effect. od.
Pt −1 = P (t − 1, yt −1 ) , is the start of period price corre-
Pt
Yield Curve
= P (t , yt −1 + YC[t −1,t ])
Where δ [t −1,t ] allows Pt Credit = Pt and where
rCarry = rCoupon + rConvergence
y t = YCt + δ t
(See Appendix B for security spread calculation method) From this various intermediary prices, we can write: Price return:
The coupon return is simply the coupon rate multiplied by the time.
rCoupon = C.dt
The convergence effect reflects the fact that bonds are usually not traded at par, but the market price must converge toward par. The convergence effect is obtained by the difference between the yield to maturity and the coupon rate, multiplied by the time.
rPr ice =
Pt −1 Pt −1
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rRoll Down = (yt − C ) .dt
• Treasury and specific return
anticipate the various yield curve movements, known as Shift, Twist, and Butterfly movements and therefore make their investment choices. The Shift movement reflects the parallel movement of the yield curve. The Twist movement corresponds to the steepening or flattening shifts variation of the yield curve and the Butterfly movement is the curvature change of the yield curve. There are indeed various methods to compute the Shift, Twist, and Butterfly movements, and this article won’t describe how these various movements are measured. An additional movement is the natural aging process of the securities all along the Yield Curve. It corresponds to the fact that in a standard configuration, yields are decreasing with maturities. Shift, Twist and Butterfly movements are therefore calculated for constant security maturities (start of period maturities) while the rolldown captures the change of maturities due to the time passing by.
A concurrent possible decomposition insists more on the origin of the carry effect. Part of it comes from the local Treasury market represented by the local government yield curve. Another part comes from specific carry effect due to the purchase of bonds (governmental or others) not located on the reference yield curve. These bonds present a spread with the reference yield curve. (See Appendix A)
rCarry = rTreasury + rSpecific
With,
rTreasury = yt .dt rSpecific = δ t .dt
This breakdown requires systematically recovering the As a mathematical approach, we recommend the use of corresponding maturity treasury yield to maturity. the Nelson-Siegel formulation (1987). They give a parsimonious fit to a set of market forward rates. The fitYield Curve return: ting function they propose depends on four parameters: a scale parameter and three other parameters used to The yield curve return can also be broken down among describe the shape and the asymptotical behavior of the various components reflecting yield-curve manage- curve. A proper rearrangement of the Betas gives the ment. Shift, the Twist and the Butterfly figures. Once computed for each maturity, the three yield curve shifts can easily be used to shock the security yield to Most managers see their investments as bets they have- maturity in the pricing functions. This gives yield curve doing against the yield curve variations. They usually shift returns corresponding to yield curve management Figure 1: Return Breakdown
rYield Curve = rShift + rTwist + rButterfly + rRoll −down
Management Return
Carry return
Yield Curve return
Credit return
Trade Impact
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Figure 2: Shift, Twist, Butterfly and Roll-down Decomposition
yields
(%)
End of period yield Roll-down Start of period yield Shift + Twist + Butterly Reference yield curve t0 Reference yield curve t1
End of period maturity
Start of period maturity
maturities (years)
strategy.
Pt Shift = P (t , y t −1 + S [o t −1,t ])
1 2 1 Let’s consider S[0 Pt Twist = P (t , y t −1 + S [o t −1,t ], S [t −1,t ], S [t −1,t ] are respectively t −1,t ] + S [t −1,t ] ) yield changes due to Shift, Twist, and Butterfly moveButterfly 1 2 = P (t , y t −1 + S [o t −1,t ] + S [t −1,t ] + S [t −1,t ] ) ments over the interval [t-1, t] for a given security. Pt S[RD t −1,t ] is the yield curve change corresponding to rollPt Roll − down = P(t , y t−1 + S[0 + S[1t −1,t] + S[2 + S[ tRD ) t −1,t ] t− 1,t] − 1,t] down movement.
Figure 3: Comparison of the Reference Yield Curve at the Startand End of the Reference Period
yields (%) Reference Yield Curve t0
Reference Yield Curve t1
maturity (years)
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Figure 4 - Referenced yield curve at Start Date and Shifted Yield Curve
yields (%) Reference Yield Curve t0
Shifted Yield Curve
maturities (years)
This gives: Shift return:
Butterfly return:
rButterfly =
PButterfly PTwist
−1
rShift =
PShift PCarry
−1
Roll-down return: Credit Return:
rRoll −down =
PRoll −down −1 PButterfly
Twist return:
rTwist
P = Twist − 1 PShift
•
Credit shifts return
Figure 5 - Shifted and Twisted Yield Curve
yields (%)
Twisted Yield Curve
Shifted Yield Curve
maturities (years)
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4 5 Credit return can be seen the same way as yield curve Credit YC Sp[t −1,t ] = S[3 t −1,t ] + S [t −1,t ] + S [t −1,t ] return by considering the shift movements of the credit yield curves compared to the reference yield curve. This gives: This means the following breakdown can be achieved:
rCredit = rCredit Yield Curve + rBond Selection
The credit yield curve spread is the difference between the credit yield curve and the reference yield curve:
Pt Credit Shift = P (t , yt −1 + S[0 + S [1t −1,t ] + S [2 +S t −1,t ] t − 1,t ]
S [RD + S [t3− 1,t ]) t − 1,t ] Pt Credit Twist = P(t , yt −1 + S[0 + S[1t −1,t ] + S[2 + t −1,t ] t − 1,t ] S[RD + S[t3− 1,t ] + S [t4− 1,t ]) t − 1,t ]
Credit YC Spt = Credit YC t − YC t
As an example, let’s consider a credit yield curve formed by AA rating securities. The Credit yield curve Pt Credit Butterfly = P (t , yt −1 + S[0 + S[1t −1,t ] + S[2 + t −1,t ] t − 1,t ] return represents the impact of this credit yield curve RD 3 4 5 ) variation compared to the reference AAA yield curve. S[t − 1,t ] + S[t − 1,t ] + S[t − 1,t ] + S[t − 1, t] The bond selection return is therefore the part of the credit return due to the specific spread behavior of the The end of period price includes the bond specific security. This specific spread is measured with the spread. Credit yield curve so that:
yt = YCt + Credit YC Spt + σ t
3
1 Pt Selection = P (t , y t−1 + S [o + S[ + S [2t−1,1] + S [RD + t −1,1] t −1,1] t− 1,1]
Credit Yield Curve = Credit Shift + rCredit Twist + rCredit Butterfly
t 1,1 [ t− 1,1] [ t− 1,1 ] [− ] Let’s assume we also use Nelson-Siegel mathematical modeling for the credit yield curve. It is therefore possible to derive the various credit shift returns from the Where σ [t −1,t ] allows Pt Selection = Pt credit yield curve return. The credit shift return: r r
S4
+S 5
+σ
)
The credit shift return therefore represents the parallel variation of the credit spread (between AA credit curve and AAA reference yield curve). The credit twist return The credit twist return: represents the return due to slope variation of the credCredit Twist P it spread and credit butterfly return is the return due to rCredit Twist = t Credit Shift − 1 curvature change in the credit spread. Using pricing Pt functions, this could be written as followed: The credit butterfly return: 3 4 5 Let’s consider S[t −1,t ], S[t −1,t ], S [t −1,t ] are respectively Credit Butterfy credit spread changes due to shift, twist, and butterfly Pt = −1 r Credit Butterfly Credit Twist movements over the interval [t-1,t] and σ [t −1,1] the bond Pt specific spread variation over the same period. We will call selection return the return generated by the specif- The selection return: ic spread variation of the bond. This return reflects part P of the return due to the nature of the security independrSelection = Credit tButterfly − 1 ently from the reference market and the credit market. Pt (See appendix B for bond spread calculation method) The credit yield curve can be written as: • Swap spread and selection return
rCredit Shift
P = t Butterfly − 1 Pt
Credit Shift
A possible alternative breakdown is to consider the
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Figure 6: Twisted Yield Curve and the Final Fitting Yield Curve
yields (%)
Twisted Yield Curve
Final Fitting Yield Curve
maturities (years)
swap yield curve. This yield curve is formed of the Where σ [t −1,t ] allows Pt Selection = Pt rates offered by banks to initiate swap contracts. The Swap Spread return:
rCredit = rSwap Spread + rSelection
Let’s consider Swap YC Sp[t −1,t ], σ [t −1,t ] are respectiverSwap Spread = ly the variation of the swap yield curve spreads and the variation of the bond specific spread over the period we study. The swap yield curve spread corresponds to the The Selection return: difference between the Swap yield curve and the reference yield curve: Swap YC Spt = Swap YCt − YCt .
Pt −1 Pt Butterfly
Swap Spread
rSelection =
The swap yield curve return reflects the return generated by the spread variation between the swap yield curve and the reference yield curve. The selection return • Sector and selection return reflects the returns offered by the specific bond spread existing between the bond yield to maturity and the Another way to decompose the Credit return is to conswap yield curve. sider managers have allocated their investments according to sector choices and inside each sector according to Where, yt = YCt + Swap YC Spt + σ t a bond selection process related to each issue characteristics. This requires using sector yield curves4 and conSwap Spread o 1 sidering the sector spread variation and the specific Pt = P(t , yt −1 + S [t −1,t ] + S [t −1,t ] + bond spread variation. With: 2
Pt
t Swap Spread
P
−1
S [t −1,t ] + Swap YC Sp[t −1,t ])
Sector YC Spt = Sector YCt − YCt rCredit = rSector + rSelection
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1 2 Pt Selection = P (t , y t −1 + S [o t −1,t ] + S [t −1,t ] + S [t −1,t ] + Swap YC Sp [t −1,t ] + σ [t −1,t ])
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1 2 Pt Sector = P (t , yt −1 + S [0 t −1,t ] + S [t −1,t ] + S [t −1,t ] +
•
Single currency attribution model
Sector YC Sp[t −1,t ])
1 2 Pt Selection = P (t , yt −1 + S[0 t −1,t ] + S [t −1,t ] + S [t −1,t ] +
The single currency version is based on return contribution differences.
Sector YC Sp[t −1,t ] + σ [t −1,t ])
Where σ [t −1,t ] allows Pt Selection = Pt The Sector return:
∆r = P − B
rSector =
Pt Sector −1 Pt Butterfly
+r ∑ ω (r ∆r = ∑ ( w r ⇔w r )+ ∑ (w
B B ,Carry
⇔ ∆r = ∑ ω P ( rP ,Carry + rP ,Curve + rP ,Credit )−
B ,Curve
P P ,Carry
− wB rB ,Carry )+ ∑ ( wP rP ,Curve −
+ rB ,Credit )
B B ,Curve
P P ,Credit
r
− wB rB ,Credit )
The Selection return:5
⇔ ∆r = Carry effect + Curve effect + Credit effect
An example of more detailed excess-return decomposition would give:6
rSelection =
Pt Pt Sector
−1
Below is a summary of the various main sources of return breakdowns that can be done. FIXED INCOME ATTRIBUTION MODELS Now that we have identified the various sources of return for a bond, let’s formulate a single currency fixed income attribution model.
rP ,Coupon + rP ,Convergence ∆r = ∑ ωP + rP, Shift + rP ,Twist + rP ,Butterfly + rP ,Roll− down − +r P ,Sector + rP ,Selection r B ,Coupon + r B ,Convergence + rB Butterfly + rB Roll ∑ ω B + r B ,Shift + r BTwist , , , − down +r B ,Sector + r B ,Selection
Figure 7: Return Decomposition
Carry return Yield Curve return Credit return
Treasury Yield return
Specific Yield return
Shift return
Twist return
Butterfly return
Credit Shift return
Credit Twist return
Credit Butterfly return
Or
Roll-down return
Or
Coupon return
Convergence return
Swap Spread return
Bond selection return
Or
Sector Effect
Bond selection return
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ωB r +∑( ω r ,Butterfly − ω + B ,Twist ) P P B rB , Butterfly ) ω r ∑( + ∑ (ω ∑ (ω r
P P , Roll − down P P , Sector
∑ (ω + ∑ (ω
⇔ ⇔ ∆r = ∑ ω P rP ,Coupon − ωB rB ,Coupon +
(
P P ,Convergence P P , Shift
r
r
− ωB rB ,Shift )+ ∑ ( ωP r P ,Twist − − ωB r B , Roll
− down
−ω B r B, Confergen
)
ce
)
Calculating the duration and the yield curve positioning effects requires the creation of synthetic or notional portfolios we will compare to the benchmark. Let’s define the Account D as having the same duration as the portfolio, but same positioning on the yield curve (weights all along the yield curve) as the benchmark. On any segment the Account D Yield Curve return can be measured as followed:
i i rD ,Yield curve = r B ,Yield curve
r
− ωB rB ,Sector )+
)
P P , Selection
− ωB r B ,Selection
)
MDP MDB
This Yield Curve return is obtained by using the bench+ Shift effect + Twist effect + Butterfly effect + R mark segment yield curve return and by adjusting it by ⇔ Rol l − down effect the ratio made of the total account and benchmark modified durations. + Sector effect + Selection effect Carry effect has been decomposed in the coupon carry effect and convergence carry effect. Yield curve effect has been decomposed in terms of Shift, Twist, Butterfly, and Roll-down effects. The credit effect is interpreted as a sector choice effect and a bond selection effect. So, for any segment i, we can split the yield curve effect into its duration and yield curve positioning component:
∆r = Coupon effect + Convergence effect
Yield curve effect i = Duration effect i + Yield curve positioning effect i
As described in Figure 5, other excess-return decompo- To understand these sources of return, we need to intrositions can be made according to the portfolio manage- duce synthetic portfolios. ment process. Let’s now introduce a slightly different way to split the yield curve effect. These two effects are computed as followed: • Duration and Yield Curve positioning effects
i i i i Duration effect i = w B rD ,Curve − w B rB ,Curve
The goal of this section is to make apparent how the duration management bet and how the impact of the This effect measures the impact of having a different asset allocation bet all along the yield curve, can be yield curve return due to a different modified duration measured. between the benchmark and the account.
i i i i i Let’s consider the following excess-return decomposi- Yield curve positioning = w Pr P ,Yield curve − w BrD ,Yield curve tion: This return gives the yield curve positioning manage∆r = Carry effect + Curve effect + Credit effect ment impact. This corresponds to the return related to different maturity allocation between the portfolio and Depending on the management process, many man- the benchmark. agers split the Yield Curve effect in terms of duration bets and Yield Curve positioning bets. Their analysis at As a summary, here is a template showing the various security level makes little sense; this is why we will excess-return breakdowns that can be done. drive our approach at segment level. For instance a typical way of decomposing main fixed Yield curve posi tioning effect + Credit effect income attribution effects could be to focus on the
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Figure 8: Excess-return Decomposition
Yield Effect Yield Curve Effect Credit Effect
Treasury Effect
Specific Yield Effect
Shift Effect
Twist Effect
Butterfly Effect
Credit Shift Effect
Credit Twist Effect
Credit Butterfly Effect
Or
Roll down Effect Or
Or
Coupon Effect
Convergence Effect
Duration Effect
Yield Curve Positioning Effect
Swap Spread Effect
Bond Selection Effect
Or
Sector Effect
Bond Selection Effect
Figure 9 Account and Benchmark Summary at Start of Period
Yield to maturity as of 2/28/2001: Modified Duration as of 2/28/2001: Account Benchmark 5.15% 4.76% 5.46 507.00%
Figure 10: Account and Benchmark Allocation and Reference Yield Curve Movement over the Reference Period
Benchmark Account Modified weights Durations 0.00 1.17 1.84 2.66 3.46 4.25 5.82 7.24 9.15 11.51 13.31 5.07 0.0% 7.0% 17.0% 12.0% 8.0% 7.0% 17.0% 12.0% 5.0% 7.0% 8.0% 100.0% Account Modified Duration 0.00 0.86 1.88 2.84 3.38 4.15 5.26 6.89 10.12 11.78 13.77 5.43
Maturities
Benchmark weights
Relative weights
Yield Curve Variation
[0;0.25] [0,25;1] [1;2] [2;3] [3;4] [4;5] [5;7] [7;10] [10;15] [15;20] [20;30]
0.0% 6.0% 15.7% 11.5% 9.7% 12.0% 20.20% 13.0% 2.0% 4.1% 6.0% 100.0%
0.0% 1.0% 1.3% 0.5% -1.7% -5.0% -3.2% -1.0% 3.0% 3.0% 2.0%
-0.17% -0.16% -0.17% -0.20% -0.17% -0.15% -0.11% -0.15% -0.12% -0.13% -0.07%
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Figure 11: Yield Curve Return and Intermediary Portfolio Return Contributions
Account Benchmark Yield Curve Yield Curve Maturities return return contributions contributions [0;0.25] [0;25;1] [1;2] [2;3] [3;4] [4;5] [5;7] [7;10] [10;15] [15;20] [20;30] 0.00% 0.01% 0.05% 0.07% 0.05% 0.04% 0.10% 0.12% 0.06% 0.11% 0.08% 0.00% 0.01% 0.05% 0.06% 0.06% 0.08% 0.13% 0.14% 0.02% 0.06% 0.06% Duration Portfolio Yield Curve return contributions 0.00% 0.01% 0.04% 0.06% 0.05% 0.07% 0.12% 0.13% 0.02% 0.06% 0.05%
Treasury carry effect and on the specific carry effect. This can be useful when part of the investments were made on corporate bonds. Also, the yield curve effect is seen in general as the combination of a duration bet and of a yield curve allocation strategy. To conclude, if we consider the same manager invested part of the account in corporate bonds, it may be of interest to monitor the sector choice effect as well as the specific bond selection effect inside each sector. This is of course meaningful when the corporate part of the portfolio has been managed by sectors. WORKING EXAMPLE The annual management fee for this account is 1.75%.
i i i rD ,Yield Curve = − MDB × ∆YC[t ;t +1] ×
MDP MDB
Figure 12 Account and Benchmark Return Contributions
S tart date: 2/28/2005 End date: 3/30/2005 Total return Carry return contribution Management fees impact Yield curve return contribution Credit return contribution Account Benchmark 1.06% 1.08% 0.42% 0.39% -0.14% 0.69% 0.66% 0.09% 0.03%
The management fees impact is computed by using the The account yield curve return contribution can be annual management fees: obtained on any maturity band as follows:
r
i P ,Yield Curve
= − MD × w × ∆YC
i P i P
Management fees impact =
i [t ;t +1]
−1.75% ×
(3/ 30 / 2005 − 2 / 28 / 2005 )
365
In the same way, the benchmark yield curve return contribution can be obtained on any maturity band as follows:
Figure 13Performance Attribution: Main effects
Excess return Carry effect Management fees impact Yield curve effect Credit effect -0.02% 0.03% -0.14% 0.03% 0.06%
r
i B ,Yield Curve
= − MD × w × ∆YC
i B i B
i [t ;t +1]
The duration account yield curve return contribution can be obtained on each maturity band by doing:
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Effects are obtained by differentiating account and model only requires the addition of a market allocation effect reflecting the yield curve market choice and combenchmark contributions. puted as a standard asset allocation effect. Currency allocation effect also needs to be calculated. Figure 14: Performance Attribution Detailed: Detailed Attribution We can imagine more detailed versions of this template Excess return -0.02% using the various decomposition possibilities we pointCarry effect 0.03% ed out in the return contributions section. A part of this Management fees impact -0.14% attribution “tree” which has not been mentioned in this Duration effect -0.04% article is the residual effect, which is the portion of the Yield curves positioning effect 0.07% local currency excess return which is not explained by Credit effect 0.06% the attribution. The trade impact is the effect of transacHere, the Yield Curve effect has been replaced by the tions made at different prices than the valuation prices. Duration effect and the yield curve positioning effect. The Duration effect for any maturity band is calculated SUMMARY as the difference between the duration account yield curve return and the benchmark yield curve return for Fixed income attribution first requires using the same this segment. The yield curve positioning effect is cal- price and exchange rate reference in the portfolio and in culated as the difference between the account yield the benchmark so as to focus only on the excess return curve return and the duration account yield curve return part, which can be attributable to management decisions. Based on this, a flexible attribution framework is for a given segment. needed to decompose the excess return in a way reflecting the management process. A flexible calculation tool • Multi-currency attribution model is certainly the key driver for flexible reporting. With In a multi-currency environment, same effects can also such a described attribution framework, we can imagbe measured on single currency asset classes. The ine a first reporting line showing detailed attribution Figure 15 : A Final General Attribution Framework
Reference Excess Return
Price Effect
Management Managment Excess Return Excess Return
Market Allocation
Carry Effect
Yield Curve Effect
Credit Effect
Residual Residual
Trading Effect
Currency Allocation
Market Allocation Decision
Bond Selection Process
Currency Management Decision
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for bond managers and another reporting line summa- APPENDIX A rizing main attribution effects for clients having less knowledge in this area. Modified Dietz return: NOTATIONS
rModified Dietz =
y,
MVe − MVs − ∑ CFt
t =1
n
X [t −1,1]
Xt, MD, YC, C, δ ,
,
t, dt, S,
Credit YC Spt , Swap YC Spt , Sector YC Spt ,
σ ,
P, B,
rP , X ,
i, j,
∆YC ,
w ,
eCurrency ,
Yield to maturity Variation of quantity X between t-1 and t Quantity X at date t Modified duration Reference yield curve Coupon rate Given a bond, δ represents the spread that must be added to the yield curve rate to obtain the bond y Time Elementary time variation Designates a yield market shift Is the Credit Yield curve spread at date t Is the Swap Yield curve spread at date t Is the Swap Yield curve spread at date t Designates the security spread with the swap yield curve or with the credit yield curve, or with the sector yield curve. Is the portfolio return Is the benchmark return Is the return contribution of factor X to an asset class of the portfolio Asset class index Security index Yield curve change for over the reference period, also stated as Asset class weight Part of the excess return due to currency allocation effect
MVe + ∑ CFt ×
t =1
n
s−t s−e
MVe MVs CFt e s t
is the end of period market value is the start of period market value are cash flows occurring during the reference period is the end of period designates the start of period reflects the moment at which a transaction occurs
Modified Dietz returns take cash flows impact into account and gives an approximation of the internal rate of return. The shorter the reference period, the better the approximation. APPENDIX B
δ t 0 , the security spread with the reference yield curve
at t0 is given by:
pt 0 =
C1 C2 + + t1 (1 + YCt1 + δ t 0 ) (1 + YCt 2 + δ t 0 ) t 2 C3 C n + 100 + ...... + t3 (1 + YCt 3 + δ t 0 ) (1 + YCtn + δ t 0 ) tn
t
Where pt 0 is the price of the bond in t0, where C are the bond coupons for each following maturity and where YCt is the t maturity reference yield curve rate. (See G.R.A.P. document [2003]) APPENDIX C Let’s assume the price of a bond depends on time and on its yield to maturity.
p = f (t , y )
Let’s now differentiate this expression:
∂p ∂p ∂p = dt + dy p p∂t p ∂y
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Identification:
three different decompositions we have proposed.
∂p =y p∂t
∂p = perf p
∂p = − MD , , p∂y
6 Provided the different breakdown methods we provided in the return contribution chapter, the reader will understand we just focus on a few potential examples.
MD is the bond sensitivity to yield curve variation. We obtain:
perf = y.dt − MD.dy
By assuming the yield to maturity can be split among the yield curve yield and the security spread, we get:
perf = y.dt − MD .dYC − MD .dSpread
Which can be written as:
perf = rCarry + rYield Curve + rSpread
REFERENCES: G.R.A.P. (Groupe de Recherché en Attribution de Performance), research group in performance attribution, “Traitement des Produits de Taux,” Fall 2003. Nelson, Charles R. and Andrew F. Siegel, “Parsimonious Modeling of Yield Curve,” Journal of Business, October 1987, Vol. 60, Issue 4, pp. 473-489. ENDNOTES
The security return can be seen as an addition of various sub-returns by performing a Taylor development as shown in Appendix C.
1
Return and return contribution are regarded equivalent terminologies.
2 3
This also means that: Credit YC Spt + σ t = δ t Or sector option adjusted spreads. The selection return will of course be different in the
4
5
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This paper proposes a flexible fixed income attribution model that can be adapted to a large range of investment processes. The StatPro return breakdown model can fit a very large range of fixed income investment processes. Pricing techniques are used to understand the impact bond-specific factors have on security returns. This allows us to give an accurate and detailed security performance decomposition. Bond specifics naturally lead to particular management methods and therefore require adapted ex-post performance attribution analysis. Mathieu Cubilié is a performance measurement consultant at StatPro Plc, where he is product manager for StatPro Fixed Income, the company bond performance measurement and attribution system. Mathieu holds a postgraduate degree in Market Finance and Economics from Toulouse University and began his career at Sinopia-Asset-Management, HSBC-group quantitative management subsidiary, in Paris, where he experienced bond portfolio management. lends itself to the use of derivative products. Unfortunately, futures contracts are replicating bond Once the various performance contributions are com- maturity behaviors rather than bond index returns. As a puted, they can be considered as elementary building result, bond managers practice benchmark sampling. blocks that can be rearranged according to the portfolio This technique consists of trying to track large indices with a limited number of assets. management decision process. INTRODUCTION The single currency return attribution model is flexible enough to permit concurrent elementary return decompositions. It is also able to attribute at different levels of detail corresponding to several available attribution effect aggregation capacities. To finish, we will see how this single currency model can easily be used ina multi-currency environment. FIXED INCOME MARKETS Bond and equity markets are clearly different; equity indices are intra-day quoting, while very few bond indices are. As a consequence, bond managers require market indicators during the day to run their investments. Actually, they’re using a sample of liquid bonds representing the various market liquid maturities. This set of bonds is known as the Yield Curve. The Yield Curves are continuously quoted and there information is accessable. They represent different markets: mainly local currency Treasury markets and also credit mar- THE PRICE EFFECT kets. A major difference with equity markets is that bond Index replication management is a difficult exercise; markets are traded “over-the-counter” while equity bond indices contain several thousands of securities, markets are organized markets. Prices are propriety to many with very large minimal trade size. This means brokers and index providers. This leads to a situation managers would require huge amounts of investments where price difference between a security held in a to closely replicate an index. This situation definitely portfolio and in the index can be very significant. This Winter 2005/2006 -49The Journal of Performance Measurement In a multi-currency universe, after making yield market and currency allocation decisions, managers have to define which bond in particular they want to invest in. They usually have duration objectives against the benchmark reflecting their market expectation. Market anticipations are mainly yield curve movements and credit spread variation. Bonds will therefore be selected for their characteristics: their duration, the spread they offer, and all other specificities. To achieve this goal, bond management is ruled more by risk control than the equity market. In the bond area, risk is first reflected by the Duration. MANGEMENT PROCESSES Despite the fact that there are many investment processes, bond managers generally face the same considerations.
phenomenon, known as “price difference effect,” can represent a large part of the excess return. This phenomenon can exist in an organized market where back office systems are not fed by index data. This is much less importance in equity markets, as volatility remains high compared to the price difference effect. For bonds, the price difference effect can be even more important than the excess return. In such case, a performance attribution exercise may be misleading.
The return formed by two consecutive intermediary prices is considered a security return contribution. With this approach, it is possible to go as far as required in the return breakdown and to adhere to the real management process. It is not sufficient to work at the security level only, as most of the management decisions are not made independently from one another, which makes their impact difficult to measure. For instance, selecting a long-term maturity bond on the yield curve also means, in the general case selecting a high duration bond. If this bond is a corporate one, this also means it is supposed to deliver a higher yield than the reference market. Of course, this also exposes the investment to spread variations that will impact the security return.
Whatever the reference account return source (back or front office), a first requirement is to clean the initial portfolio return from the price effect. This means that the first step is to reprice portfolios and benchmarks, using the same price base. For simplicity reasons, it is commonly accepted that the portfolio is repriced using the benchmark prices for common securities. Noncommon securities will keep the back office prices. What about transactions? Pricing techniques can only This is extended to common exchange rates for securi- explain price returns and therefore ignores the impact ties in multi-currency portfolios using benchmarks rates. of transactions made by the manager. For this reason, we also think an accurate attribution model must be transaction based and must clearly identify their Portfolio Reference impact.
Return
RETURN CONTRIBUTIONS For simplicity reasons we will consider plain vanilla bonds in this chapter.
Price Effect
Portfolio Adjusted Return
Let’s focus on the portfolio return, and let’s run a first In the following, we will concentrate on understanding decomposition of each instrument price return. In addithe portfolio recomputed return. tion, we will measure the trade impact by computing the return using the Modified Dietz and the all-in price rManagement = rreference + rPr ice return and consider their difference. SECURITY PRICING TECHNIQUES In order to understand the portfolio or benchmark rModified return, it makes sense first to analyze the performance of each instrument.
Dietz
=
MVe − MVs − ∑ CFt
t =1
n
MVe + ∑ CFt ×
t =1
n
s − t (See Appendix A) s−e
As Modified Dietz return captures trade return by defiEach security is associated with a pricing function. nition, we can derive Such a function establishes a link between the security rTrade = rModified Dietz − rPr ice price and its yield to maturity. Shocking the yield to maturity with elementary yield variations related to market conditions allows us to generate a set of inter- From now on, let’s consider the price return, . Pr ice mediary prices between the beginning and the end of period price. There are three major sources of return commonly identified when closely looking at a bond.
r
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rPr ice = rCarry + rYield Curve + rCredit 1
A first component of a bond return is the carry return2 reflecting the impact of time. Then, the Yield curve return reveals how the reference (Treasury) market reacted over the analysis period. This return is created by global yield curve movements. At last, the credit return is the part of the return, which is specific to the bond and not to the reference yield market. This corresponds to changes in the security spread with the reference yield curve. If we make the assumption a bond price depends on time and on its yield to maturity, then from a pricing point of view we can generate the following prices.
Carry return:
rCarry =
PCarry Pt −1
−1
Yield Curve return:
rYield
Credit return:
Curve
=
PYield
Curve
PCarry
−1
−1
rCredit =
Pt PYield Curve
This leads to the split of the account (or benchmark) return in terms of main sources of return. DETAILED RETURN CONTRIBUTIONS
sponding to date t-1 and to the yield to maturity at this Each of the three main sources of return can further be date. decomposed and give a more detailed vision on how returns were constructed. This can still be achieved Pt Carry = P(t , yt −1 ) , is the intermediary price taking thanks to the use of pricing functions. into account the time passing by. It is simply obtained by switching the date at the end of period one in the for- Carry Return: mula. The carry return can be split using two possible break, is the intermediary downs: price capturing the Yield Curve movement. It can be obtained by considering the security at date t and by • Coupon and convergence return adding the corresponding yield curve change to the start of period yield to maturity. In this breakdown, the return due to the bond yield to maturity can be decomposed into two distinct mechaPt Credit = P (t , yt −1 + YC[t −1,t ] + δ [t −1,t ]) , is the end of peri- nisms the coupon return component and the simple pasod price. It can alternatively be computed from the sage of time, denoted as roll down effect, reflecting the yield curve change and the spread change over the peri- bond convergence effect. od.
Pt −1 = P (t − 1, yt −1 ) , is the start of period price corre-
Pt
Yield Curve
= P (t , yt −1 + YC[t −1,t ])
Where δ [t −1,t ] allows Pt Credit = Pt and where
rCarry = rCoupon + rConvergence
y t = YCt + δ t
(See Appendix B for security spread calculation method) From this various intermediary prices, we can write: Price return:
The coupon return is simply the coupon rate multiplied by the time.
rCoupon = C.dt
The convergence effect reflects the fact that bonds are usually not traded at par, but the market price must converge toward par. The convergence effect is obtained by the difference between the yield to maturity and the coupon rate, multiplied by the time.
rPr ice =
Pt −1 Pt −1
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rRoll Down = (yt − C ) .dt
• Treasury and specific return
anticipate the various yield curve movements, known as Shift, Twist, and Butterfly movements and therefore make their investment choices. The Shift movement reflects the parallel movement of the yield curve. The Twist movement corresponds to the steepening or flattening shifts variation of the yield curve and the Butterfly movement is the curvature change of the yield curve. There are indeed various methods to compute the Shift, Twist, and Butterfly movements, and this article won’t describe how these various movements are measured. An additional movement is the natural aging process of the securities all along the Yield Curve. It corresponds to the fact that in a standard configuration, yields are decreasing with maturities. Shift, Twist and Butterfly movements are therefore calculated for constant security maturities (start of period maturities) while the rolldown captures the change of maturities due to the time passing by.
A concurrent possible decomposition insists more on the origin of the carry effect. Part of it comes from the local Treasury market represented by the local government yield curve. Another part comes from specific carry effect due to the purchase of bonds (governmental or others) not located on the reference yield curve. These bonds present a spread with the reference yield curve. (See Appendix A)
rCarry = rTreasury + rSpecific
With,
rTreasury = yt .dt rSpecific = δ t .dt
This breakdown requires systematically recovering the As a mathematical approach, we recommend the use of corresponding maturity treasury yield to maturity. the Nelson-Siegel formulation (1987). They give a parsimonious fit to a set of market forward rates. The fitYield Curve return: ting function they propose depends on four parameters: a scale parameter and three other parameters used to The yield curve return can also be broken down among describe the shape and the asymptotical behavior of the various components reflecting yield-curve manage- curve. A proper rearrangement of the Betas gives the ment. Shift, the Twist and the Butterfly figures. Once computed for each maturity, the three yield curve shifts can easily be used to shock the security yield to Most managers see their investments as bets they have- maturity in the pricing functions. This gives yield curve doing against the yield curve variations. They usually shift returns corresponding to yield curve management Figure 1: Return Breakdown
rYield Curve = rShift + rTwist + rButterfly + rRoll −down
Management Return
Carry return
Yield Curve return
Credit return
Trade Impact
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Figure 2: Shift, Twist, Butterfly and Roll-down Decomposition
yields
(%)
End of period yield Roll-down Start of period yield Shift + Twist + Butterly Reference yield curve t0 Reference yield curve t1
End of period maturity
Start of period maturity
maturities (years)
strategy.
Pt Shift = P (t , y t −1 + S [o t −1,t ])
1 2 1 Let’s consider S[0 Pt Twist = P (t , y t −1 + S [o t −1,t ], S [t −1,t ], S [t −1,t ] are respectively t −1,t ] + S [t −1,t ] ) yield changes due to Shift, Twist, and Butterfly moveButterfly 1 2 = P (t , y t −1 + S [o t −1,t ] + S [t −1,t ] + S [t −1,t ] ) ments over the interval [t-1, t] for a given security. Pt S[RD t −1,t ] is the yield curve change corresponding to rollPt Roll − down = P(t , y t−1 + S[0 + S[1t −1,t] + S[2 + S[ tRD ) t −1,t ] t− 1,t] − 1,t] down movement.
Figure 3: Comparison of the Reference Yield Curve at the Startand End of the Reference Period
yields (%) Reference Yield Curve t0
Reference Yield Curve t1
maturity (years)
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Figure 4 - Referenced yield curve at Start Date and Shifted Yield Curve
yields (%) Reference Yield Curve t0
Shifted Yield Curve
maturities (years)
This gives: Shift return:
Butterfly return:
rButterfly =
PButterfly PTwist
−1
rShift =
PShift PCarry
−1
Roll-down return: Credit Return:
rRoll −down =
PRoll −down −1 PButterfly
Twist return:
rTwist
P = Twist − 1 PShift
•
Credit shifts return
Figure 5 - Shifted and Twisted Yield Curve
yields (%)
Twisted Yield Curve
Shifted Yield Curve
maturities (years)
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4 5 Credit return can be seen the same way as yield curve Credit YC Sp[t −1,t ] = S[3 t −1,t ] + S [t −1,t ] + S [t −1,t ] return by considering the shift movements of the credit yield curves compared to the reference yield curve. This gives: This means the following breakdown can be achieved:
rCredit = rCredit Yield Curve + rBond Selection
The credit yield curve spread is the difference between the credit yield curve and the reference yield curve:
Pt Credit Shift = P (t , yt −1 + S[0 + S [1t −1,t ] + S [2 +S t −1,t ] t − 1,t ]
S [RD + S [t3− 1,t ]) t − 1,t ] Pt Credit Twist = P(t , yt −1 + S[0 + S[1t −1,t ] + S[2 + t −1,t ] t − 1,t ] S[RD + S[t3− 1,t ] + S [t4− 1,t ]) t − 1,t ]
Credit YC Spt = Credit YC t − YC t
As an example, let’s consider a credit yield curve formed by AA rating securities. The Credit yield curve Pt Credit Butterfly = P (t , yt −1 + S[0 + S[1t −1,t ] + S[2 + t −1,t ] t − 1,t ] return represents the impact of this credit yield curve RD 3 4 5 ) variation compared to the reference AAA yield curve. S[t − 1,t ] + S[t − 1,t ] + S[t − 1,t ] + S[t − 1, t] The bond selection return is therefore the part of the credit return due to the specific spread behavior of the The end of period price includes the bond specific security. This specific spread is measured with the spread. Credit yield curve so that:
yt = YCt + Credit YC Spt + σ t
3
1 Pt Selection = P (t , y t−1 + S [o + S[ + S [2t−1,1] + S [RD + t −1,1] t −1,1] t− 1,1]
Credit Yield Curve = Credit Shift + rCredit Twist + rCredit Butterfly
t 1,1 [ t− 1,1] [ t− 1,1 ] [− ] Let’s assume we also use Nelson-Siegel mathematical modeling for the credit yield curve. It is therefore possible to derive the various credit shift returns from the Where σ [t −1,t ] allows Pt Selection = Pt credit yield curve return. The credit shift return: r r
S4
+S 5
+σ
)
The credit shift return therefore represents the parallel variation of the credit spread (between AA credit curve and AAA reference yield curve). The credit twist return The credit twist return: represents the return due to slope variation of the credCredit Twist P it spread and credit butterfly return is the return due to rCredit Twist = t Credit Shift − 1 curvature change in the credit spread. Using pricing Pt functions, this could be written as followed: The credit butterfly return: 3 4 5 Let’s consider S[t −1,t ], S[t −1,t ], S [t −1,t ] are respectively Credit Butterfy credit spread changes due to shift, twist, and butterfly Pt = −1 r Credit Butterfly Credit Twist movements over the interval [t-1,t] and σ [t −1,1] the bond Pt specific spread variation over the same period. We will call selection return the return generated by the specif- The selection return: ic spread variation of the bond. This return reflects part P of the return due to the nature of the security independrSelection = Credit tButterfly − 1 ently from the reference market and the credit market. Pt (See appendix B for bond spread calculation method) The credit yield curve can be written as: • Swap spread and selection return
rCredit Shift
P = t Butterfly − 1 Pt
Credit Shift
A possible alternative breakdown is to consider the
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Figure 6: Twisted Yield Curve and the Final Fitting Yield Curve
yields (%)
Twisted Yield Curve
Final Fitting Yield Curve
maturities (years)
swap yield curve. This yield curve is formed of the Where σ [t −1,t ] allows Pt Selection = Pt rates offered by banks to initiate swap contracts. The Swap Spread return:
rCredit = rSwap Spread + rSelection
Let’s consider Swap YC Sp[t −1,t ], σ [t −1,t ] are respectiverSwap Spread = ly the variation of the swap yield curve spreads and the variation of the bond specific spread over the period we study. The swap yield curve spread corresponds to the The Selection return: difference between the Swap yield curve and the reference yield curve: Swap YC Spt = Swap YCt − YCt .
Pt −1 Pt Butterfly
Swap Spread
rSelection =
The swap yield curve return reflects the return generated by the spread variation between the swap yield curve and the reference yield curve. The selection return • Sector and selection return reflects the returns offered by the specific bond spread existing between the bond yield to maturity and the Another way to decompose the Credit return is to conswap yield curve. sider managers have allocated their investments according to sector choices and inside each sector according to Where, yt = YCt + Swap YC Spt + σ t a bond selection process related to each issue characteristics. This requires using sector yield curves4 and conSwap Spread o 1 sidering the sector spread variation and the specific Pt = P(t , yt −1 + S [t −1,t ] + S [t −1,t ] + bond spread variation. With: 2
Pt
t Swap Spread
P
−1
S [t −1,t ] + Swap YC Sp[t −1,t ])
Sector YC Spt = Sector YCt − YCt rCredit = rSector + rSelection
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1 2 Pt Selection = P (t , y t −1 + S [o t −1,t ] + S [t −1,t ] + S [t −1,t ] + Swap YC Sp [t −1,t ] + σ [t −1,t ])
The Journal of Performance Measurement
1 2 Pt Sector = P (t , yt −1 + S [0 t −1,t ] + S [t −1,t ] + S [t −1,t ] +
•
Single currency attribution model
Sector YC Sp[t −1,t ])
1 2 Pt Selection = P (t , yt −1 + S[0 t −1,t ] + S [t −1,t ] + S [t −1,t ] +
The single currency version is based on return contribution differences.
Sector YC Sp[t −1,t ] + σ [t −1,t ])
Where σ [t −1,t ] allows Pt Selection = Pt The Sector return:
∆r = P − B
rSector =
Pt Sector −1 Pt Butterfly
+r ∑ ω (r ∆r = ∑ ( w r ⇔w r )+ ∑ (w
B B ,Carry
⇔ ∆r = ∑ ω P ( rP ,Carry + rP ,Curve + rP ,Credit )−
B ,Curve
P P ,Carry
− wB rB ,Carry )+ ∑ ( wP rP ,Curve −
+ rB ,Credit )
B B ,Curve
P P ,Credit
r
− wB rB ,Credit )
The Selection return:5
⇔ ∆r = Carry effect + Curve effect + Credit effect
An example of more detailed excess-return decomposition would give:6
rSelection =
Pt Pt Sector
−1
Below is a summary of the various main sources of return breakdowns that can be done. FIXED INCOME ATTRIBUTION MODELS Now that we have identified the various sources of return for a bond, let’s formulate a single currency fixed income attribution model.
rP ,Coupon + rP ,Convergence ∆r = ∑ ωP + rP, Shift + rP ,Twist + rP ,Butterfly + rP ,Roll− down − +r P ,Sector + rP ,Selection r B ,Coupon + r B ,Convergence + rB Butterfly + rB Roll ∑ ω B + r B ,Shift + r BTwist , , , − down +r B ,Sector + r B ,Selection
Figure 7: Return Decomposition
Carry return Yield Curve return Credit return
Treasury Yield return
Specific Yield return
Shift return
Twist return
Butterfly return
Credit Shift return
Credit Twist return
Credit Butterfly return
Or
Roll-down return
Or
Coupon return
Convergence return
Swap Spread return
Bond selection return
Or
Sector Effect
Bond selection return
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ωB r +∑( ω r ,Butterfly − ω + B ,Twist ) P P B rB , Butterfly ) ω r ∑( + ∑ (ω ∑ (ω r
P P , Roll − down P P , Sector
∑ (ω + ∑ (ω
⇔ ⇔ ∆r = ∑ ω P rP ,Coupon − ωB rB ,Coupon +
(
P P ,Convergence P P , Shift
r
r
− ωB rB ,Shift )+ ∑ ( ωP r P ,Twist − − ωB r B , Roll
− down
−ω B r B, Confergen
)
ce
)
Calculating the duration and the yield curve positioning effects requires the creation of synthetic or notional portfolios we will compare to the benchmark. Let’s define the Account D as having the same duration as the portfolio, but same positioning on the yield curve (weights all along the yield curve) as the benchmark. On any segment the Account D Yield Curve return can be measured as followed:
i i rD ,Yield curve = r B ,Yield curve
r
− ωB rB ,Sector )+
)
P P , Selection
− ωB r B ,Selection
)
MDP MDB
This Yield Curve return is obtained by using the bench+ Shift effect + Twist effect + Butterfly effect + R mark segment yield curve return and by adjusting it by ⇔ Rol l − down effect the ratio made of the total account and benchmark modified durations. + Sector effect + Selection effect Carry effect has been decomposed in the coupon carry effect and convergence carry effect. Yield curve effect has been decomposed in terms of Shift, Twist, Butterfly, and Roll-down effects. The credit effect is interpreted as a sector choice effect and a bond selection effect. So, for any segment i, we can split the yield curve effect into its duration and yield curve positioning component:
∆r = Coupon effect + Convergence effect
Yield curve effect i = Duration effect i + Yield curve positioning effect i
As described in Figure 5, other excess-return decompo- To understand these sources of return, we need to intrositions can be made according to the portfolio manage- duce synthetic portfolios. ment process. Let’s now introduce a slightly different way to split the yield curve effect. These two effects are computed as followed: • Duration and Yield Curve positioning effects
i i i i Duration effect i = w B rD ,Curve − w B rB ,Curve
The goal of this section is to make apparent how the duration management bet and how the impact of the This effect measures the impact of having a different asset allocation bet all along the yield curve, can be yield curve return due to a different modified duration measured. between the benchmark and the account.
i i i i i Let’s consider the following excess-return decomposi- Yield curve positioning = w Pr P ,Yield curve − w BrD ,Yield curve tion: This return gives the yield curve positioning manage∆r = Carry effect + Curve effect + Credit effect ment impact. This corresponds to the return related to different maturity allocation between the portfolio and Depending on the management process, many man- the benchmark. agers split the Yield Curve effect in terms of duration bets and Yield Curve positioning bets. Their analysis at As a summary, here is a template showing the various security level makes little sense; this is why we will excess-return breakdowns that can be done. drive our approach at segment level. For instance a typical way of decomposing main fixed Yield curve posi tioning effect + Credit effect income attribution effects could be to focus on the
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Figure 8: Excess-return Decomposition
Yield Effect Yield Curve Effect Credit Effect
Treasury Effect
Specific Yield Effect
Shift Effect
Twist Effect
Butterfly Effect
Credit Shift Effect
Credit Twist Effect
Credit Butterfly Effect
Or
Roll down Effect Or
Or
Coupon Effect
Convergence Effect
Duration Effect
Yield Curve Positioning Effect
Swap Spread Effect
Bond Selection Effect
Or
Sector Effect
Bond Selection Effect
Figure 9 Account and Benchmark Summary at Start of Period
Yield to maturity as of 2/28/2001: Modified Duration as of 2/28/2001: Account Benchmark 5.15% 4.76% 5.46 507.00%
Figure 10: Account and Benchmark Allocation and Reference Yield Curve Movement over the Reference Period
Benchmark Account Modified weights Durations 0.00 1.17 1.84 2.66 3.46 4.25 5.82 7.24 9.15 11.51 13.31 5.07 0.0% 7.0% 17.0% 12.0% 8.0% 7.0% 17.0% 12.0% 5.0% 7.0% 8.0% 100.0% Account Modified Duration 0.00 0.86 1.88 2.84 3.38 4.15 5.26 6.89 10.12 11.78 13.77 5.43
Maturities
Benchmark weights
Relative weights
Yield Curve Variation
[0;0.25] [0,25;1] [1;2] [2;3] [3;4] [4;5] [5;7] [7;10] [10;15] [15;20] [20;30]
0.0% 6.0% 15.7% 11.5% 9.7% 12.0% 20.20% 13.0% 2.0% 4.1% 6.0% 100.0%
0.0% 1.0% 1.3% 0.5% -1.7% -5.0% -3.2% -1.0% 3.0% 3.0% 2.0%
-0.17% -0.16% -0.17% -0.20% -0.17% -0.15% -0.11% -0.15% -0.12% -0.13% -0.07%
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Figure 11: Yield Curve Return and Intermediary Portfolio Return Contributions
Account Benchmark Yield Curve Yield Curve Maturities return return contributions contributions [0;0.25] [0;25;1] [1;2] [2;3] [3;4] [4;5] [5;7] [7;10] [10;15] [15;20] [20;30] 0.00% 0.01% 0.05% 0.07% 0.05% 0.04% 0.10% 0.12% 0.06% 0.11% 0.08% 0.00% 0.01% 0.05% 0.06% 0.06% 0.08% 0.13% 0.14% 0.02% 0.06% 0.06% Duration Portfolio Yield Curve return contributions 0.00% 0.01% 0.04% 0.06% 0.05% 0.07% 0.12% 0.13% 0.02% 0.06% 0.05%
Treasury carry effect and on the specific carry effect. This can be useful when part of the investments were made on corporate bonds. Also, the yield curve effect is seen in general as the combination of a duration bet and of a yield curve allocation strategy. To conclude, if we consider the same manager invested part of the account in corporate bonds, it may be of interest to monitor the sector choice effect as well as the specific bond selection effect inside each sector. This is of course meaningful when the corporate part of the portfolio has been managed by sectors. WORKING EXAMPLE The annual management fee for this account is 1.75%.
i i i rD ,Yield Curve = − MDB × ∆YC[t ;t +1] ×
MDP MDB
Figure 12 Account and Benchmark Return Contributions
S tart date: 2/28/2005 End date: 3/30/2005 Total return Carry return contribution Management fees impact Yield curve return contribution Credit return contribution Account Benchmark 1.06% 1.08% 0.42% 0.39% -0.14% 0.69% 0.66% 0.09% 0.03%
The management fees impact is computed by using the The account yield curve return contribution can be annual management fees: obtained on any maturity band as follows:
r
i P ,Yield Curve
= − MD × w × ∆YC
i P i P
Management fees impact =
i [t ;t +1]
−1.75% ×
(3/ 30 / 2005 − 2 / 28 / 2005 )
365
In the same way, the benchmark yield curve return contribution can be obtained on any maturity band as follows:
Figure 13Performance Attribution: Main effects
Excess return Carry effect Management fees impact Yield curve effect Credit effect -0.02% 0.03% -0.14% 0.03% 0.06%
r
i B ,Yield Curve
= − MD × w × ∆YC
i B i B
i [t ;t +1]
The duration account yield curve return contribution can be obtained on each maturity band by doing:
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Effects are obtained by differentiating account and model only requires the addition of a market allocation effect reflecting the yield curve market choice and combenchmark contributions. puted as a standard asset allocation effect. Currency allocation effect also needs to be calculated. Figure 14: Performance Attribution Detailed: Detailed Attribution We can imagine more detailed versions of this template Excess return -0.02% using the various decomposition possibilities we pointCarry effect 0.03% ed out in the return contributions section. A part of this Management fees impact -0.14% attribution “tree” which has not been mentioned in this Duration effect -0.04% article is the residual effect, which is the portion of the Yield curves positioning effect 0.07% local currency excess return which is not explained by Credit effect 0.06% the attribution. The trade impact is the effect of transacHere, the Yield Curve effect has been replaced by the tions made at different prices than the valuation prices. Duration effect and the yield curve positioning effect. The Duration effect for any maturity band is calculated SUMMARY as the difference between the duration account yield curve return and the benchmark yield curve return for Fixed income attribution first requires using the same this segment. The yield curve positioning effect is cal- price and exchange rate reference in the portfolio and in culated as the difference between the account yield the benchmark so as to focus only on the excess return curve return and the duration account yield curve return part, which can be attributable to management decisions. Based on this, a flexible attribution framework is for a given segment. needed to decompose the excess return in a way reflecting the management process. A flexible calculation tool • Multi-currency attribution model is certainly the key driver for flexible reporting. With In a multi-currency environment, same effects can also such a described attribution framework, we can imagbe measured on single currency asset classes. The ine a first reporting line showing detailed attribution Figure 15 : A Final General Attribution Framework
Reference Excess Return
Price Effect
Management Managment Excess Return Excess Return
Market Allocation
Carry Effect
Yield Curve Effect
Credit Effect
Residual Residual
Trading Effect
Currency Allocation
Market Allocation Decision
Bond Selection Process
Currency Management Decision
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for bond managers and another reporting line summa- APPENDIX A rizing main attribution effects for clients having less knowledge in this area. Modified Dietz return: NOTATIONS
rModified Dietz =
y,
MVe − MVs − ∑ CFt
t =1
n
X [t −1,1]
Xt, MD, YC, C, δ ,
,
t, dt, S,
Credit YC Spt , Swap YC Spt , Sector YC Spt ,
σ ,
P, B,
rP , X ,
i, j,
∆YC ,
w ,
eCurrency ,
Yield to maturity Variation of quantity X between t-1 and t Quantity X at date t Modified duration Reference yield curve Coupon rate Given a bond, δ represents the spread that must be added to the yield curve rate to obtain the bond y Time Elementary time variation Designates a yield market shift Is the Credit Yield curve spread at date t Is the Swap Yield curve spread at date t Is the Swap Yield curve spread at date t Designates the security spread with the swap yield curve or with the credit yield curve, or with the sector yield curve. Is the portfolio return Is the benchmark return Is the return contribution of factor X to an asset class of the portfolio Asset class index Security index Yield curve change for over the reference period, also stated as Asset class weight Part of the excess return due to currency allocation effect
MVe + ∑ CFt ×
t =1
n
s−t s−e
MVe MVs CFt e s t
is the end of period market value is the start of period market value are cash flows occurring during the reference period is the end of period designates the start of period reflects the moment at which a transaction occurs
Modified Dietz returns take cash flows impact into account and gives an approximation of the internal rate of return. The shorter the reference period, the better the approximation. APPENDIX B
δ t 0 , the security spread with the reference yield curve
at t0 is given by:
pt 0 =
C1 C2 + + t1 (1 + YCt1 + δ t 0 ) (1 + YCt 2 + δ t 0 ) t 2 C3 C n + 100 + ...... + t3 (1 + YCt 3 + δ t 0 ) (1 + YCtn + δ t 0 ) tn
t
Where pt 0 is the price of the bond in t0, where C are the bond coupons for each following maturity and where YCt is the t maturity reference yield curve rate. (See G.R.A.P. document [2003]) APPENDIX C Let’s assume the price of a bond depends on time and on its yield to maturity.
p = f (t , y )
Let’s now differentiate this expression:
∂p ∂p ∂p = dt + dy p p∂t p ∂y
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Identification:
three different decompositions we have proposed.
∂p =y p∂t
∂p = perf p
∂p = − MD , , p∂y
6 Provided the different breakdown methods we provided in the return contribution chapter, the reader will understand we just focus on a few potential examples.
MD is the bond sensitivity to yield curve variation. We obtain:
perf = y.dt − MD.dy
By assuming the yield to maturity can be split among the yield curve yield and the security spread, we get:
perf = y.dt − MD .dYC − MD .dSpread
Which can be written as:
perf = rCarry + rYield Curve + rSpread
REFERENCES: G.R.A.P. (Groupe de Recherché en Attribution de Performance), research group in performance attribution, “Traitement des Produits de Taux,” Fall 2003. Nelson, Charles R. and Andrew F. Siegel, “Parsimonious Modeling of Yield Curve,” Journal of Business, October 1987, Vol. 60, Issue 4, pp. 473-489. ENDNOTES
The security return can be seen as an addition of various sub-returns by performing a Taylor development as shown in Appendix C.
1
Return and return contribution are regarded equivalent terminologies.
2 3
This also means that: Credit YC Spt + σ t = δ t Or sector option adjusted spreads. The selection return will of course be different in the
4
5
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