FIXED INCOME ATTRIBUTION

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1 Sotware Requirement Speciication FIXED INCOME ATTRIBUTION Authors Risto Lehtinen Version Date Comment /02/20 First Drat

2 Table o Contents 1 Introduction Purpose o Document Glossary, Deinitions, Acronyms and Abbreviations Reerences Business Speciication Overview Risk actors Implied actors Market movement decomposition Attribution o return Taylor approximation Sequential method Trading eect Attribution errors Scope Deinition Phased implementation User interace Attribution actor set deinition Attribution actor set selection Attribution key-igure display Calculations Risk models Taylor approximation Market movement decomposition Attribution calculation Example Setup Implied spread model (RS 1, RS 6) Risk actors (RS 2) Approximation method (RS 3, RS 4) Calculations Market movement decomposition (RS 8) Attribution calculation (RS 9, RS10) Open questions Appendix A: Calculation example WALLSTREET SYSTEMS PAGE 2 o 16

3 1 Introduction There are three dierent aspects to perormance attribution. First, we may look at the total result o a portolio, and examine which part is due to the market value change o a given instrument. Second, we can calculate which part o the result arises rom a given risk taken. For simple equity attribution (without derivative instruments) these two approaches are the same. That is, there is a one-to-one correspondence between the market value o the position and the risk taken. Similarly, i we consider a bond position against movements in bond prices, the changes in market value can be directly attributed to changes in the prices. However, i we want to consider zero-coupon pricing and risks taken against movements o the zero curve, such a simple connection between market variable movements and bond prices disappears. Similarly, i one wants to attribute part o the change in the market value o an option to a change in volatility, one needs more inormation than just the amount o change in the market value o the option. The irst type o perormance attribution already exists in TRM. This document describes how the second type o attribution can be eected. The third aspect o perormance attribution deals with perormance against benchmarks. Once the market value change has been attributed to dierent market movements, one can proceed to attribute the dierence between the portolio and benchmark perormances to dierent deviations rom the benchmark. This attribution mechanism already exists in Perormance Monitor and will not be discussed in this document. 1.1 Purpose o Document This document addresses the ollowing question: How to divide the change in the market value o an asset over a period into parts corresponding to changes in a given set o risk actors over the same period. The main ocus will be on a bond as the asset, and dierent movements o zero curve as the set o risk actors. However, the approach can be generalized or any asset and any set o risk actors. 1.2 Glossary, Deinitions, Acronyms and Abbreviations 1.3 Reerences WALLSTREET SYSTEMS PAGE 3 o 16

4 2 Business Speciication 2.1 Overview We are asking the ollowing question: given a portolio and a set o market variables at the beginning o a period, and the changes in the market quotes at the end o the period, how do we attribute the change in the market value o the portolio to dierent components o the market movement. Note that i the contents o the portolio have changed over the period, that is, i there has been trading, part o the change in the market value will be due to the timing o the trades. This eect is treated separately in section This section describes the process o ixed income attribution in broad terms without going into details. We can identiy the ollowing components: 1. Deine the risk actors. This is part o static setup. Typical risk actors are spread, dierent movements o zero curve, and time. 2. Decompose the daily market movement into movements o risk actors. For market variables such as FX rate this is trivial. More work is needed or example to recover the change in spread and dierent types o zero curve movements. 3. Decompose the daily market value change into parts corresponding to risk actor movements. There are two approaches to this: A. Use a Taylor approximation with risk sensitivities and actor movements (Taylor method) B. Apply the actor movements in sequence, and revaluate the position ater each movement (sequential method) 4. Link the daily attribution components over time. This part will not be addressed in this document, since it is a simple extension o what already exists in the system. Section 2.3 and Appendix A oer a numerical example o the above process, decomposing the change in the market value o a bond into movements o zero curve, spread, and time roll Risk actors By a set o risk actors we mean a decomposition o potential changes in the market into a set o market variables. The ex post counterpart o risk actors is a decomposition o the actual change o market variables into perormance actors. Subsequently, portolio or instrument returns can be attributed to changes in perormance actors. The identiication o actors begins with the set o observables in the market, which includes all market prices or which quotes exist. Let us call this set o actors Q. Another approach is to conjecture an underlying market model rom which all market prices can be derived. Let us call the set o variables in the market model M. and let q [m] be the model that ties the quoted prices (q) to the theoretical model variables (m). The number o variables in the model is smaller than the number o market quotes, so that the model cannot reproduce the exact market quotes, but rather an approximation. As an example, the quoted actors may consist o deposit and swap rates as well as bond quotes, all directly observable rom the market. The corresponding theoretical model could be the zero curve derived rom the deposit and swap rates and bond spreads derived rom the zero curve and bond prices. Once the undamental actors have been selected, the actual attribution actors can be chosen as combinations o the undamental actors. The idea here is that it would be impractical to analyze the eect o each market actor on the position result. Instead, one combines correlated undamental actors to create a smaller set o risk actors, which explain most o the market movements with ewer variables. For example, the movement o the zero curve may be approximated with just two risk actors, namely parallel shit and rotation. O course, there will remain a residual movement not explained by any risk actors. WALLSTREET SYSTEMS PAGE 4 o 16

5 A third alternative type o actors should be mentioned here: With (ex post) statistical analysis a lowdimensional actor model can be identiied and used as a basis o attribution analysis. This approach is not covered in this speciication Implied actors Zero coupon curve is an example o a market model where the model variables (zero coupon rates) are explicitly derived rom market quotes (deposit and swap rates or bond prices). Another example is a term structure interest rate model, where the model parameters (volatility, mean reversion, etc.) are derived rom quoted swaption or cap/loor volatilities. A special case is where the market value o an instrument is obtained rom a market quote, and there is a market model with one ree parameter. In this case the value o the parameter can be derived directly rom the market quote. For example, we can calculate the Black-Scholes implied volatility or an equity option, when the market value o the option is known Market movement decomposition Let us consider a movement in the market quotes rom q to p. As a consequence, the market value o the portolio moves rom V(q) to V(p). Alternatively, we might look at a change in the model variables rom m to n, the model variables corresponding to q and p, respectively. The ollowing discussion applies to either case. The only dierence is that in case the model variables don t completely explain the market prices, there will be an error term, the size o which may change rom the beginning o the observation period to its end. This change o error term will be included in the residual part o attribution. The task is to express the market movement as a combination o attribution actor movements plus a possible residual term. For attribution actors that are identical to market variables nothing needs to be done. For attribution actors that are aggregates o market variables, a little more is needed. For example, given an arbitrary yield curve movement, it is not evident how it can be expressed as the combination o, say, a parallel shit and a rotation. A simple method to decompose a given market movement into attribution actors is to express it as a linear combination o given actors so that the residual term is minimized. That is, given the shape o parallel shit and rotation, one chooses the multipliers A and B in: <total movement> = A * <parallel shit> + B * <rotation> + <residual> so that the residual term is as small as possible Attribution o return Once we know the attribution actor movements, there are two approaches to the decomposing o the return o the portolio into parts corresponding to actor movements. In what ollows we shall consider changes in market value ( V ). This will be converted into return according to the setup o Perormance Monitor Taylor approximation In Taylor approximation we consider the market value o our position a unction o our risk actors: V = V[ ] Where is the vector o risk actors. Now, i the market moves rom one set o risk actor values 0 to another set o values 0, the change in market value can be approximated by V[ 0 2 V 1 V ] V[ 0 ] u + 2 i i i, j i j i j +K WALLSTREET SYSTEMS PAGE 5 o 16

6 The terms in the summation represent dierent attribution actors. For example, i our irst risk actor is a parallel shit o the yield curve, then 1 is the magnitude o the parallel shit, sensitivity (duration), and 2 V 2 1 can be called duration attribution and convexity attribution. V 1 is the interest rate is the convexity, and the corresponding terms in the Taylor approximation The choice o terms included in the approximation depends on what is deemed signiicant by the user o the system. In practice, adding terms beyond the second derivative will not be practical. The part o the market value change not captured by the Taylor approximation will orm a residual term, the size o which gives an indication o how well the set o derivatives was chosen Sequential method In sequential method the change in the market value is written as: V[ + ( V[ + K + + ( V[ Where ] V[ ] = ( V[ i th risk actor. 1 1 ] V[ 2 ] V[ i 1 ] V[ ]) + 1 ]) K + n ) is a vector where all the other elements are zero except element i, which gives the change in the In other words, we change one risk actor at the time, and evaluate the corresponding change in the market value, and that will be the attribution o the risk actor in question. The advantage o this approach is that there are ewer terms to consider, namely only one or each actor. Furthermore, there are no residual terms. On the other hand, the magnitudes o dierent attributions will depend on the order they are calculated. Also, a given attribution will depend on changes on other risk actors in addition to the changes in the actor in question. For example, we may get a dierent parallel shit attribution depending on the size o change in the spread. The two methods may also be combined. For example, we may calculate duration and convexity attributions using the Taylor approximation, and then a residual parallel shit attribution by revaluating the position with the parallel shit and taking the dierence o the total change due to parallel shit and the part explained by duration and convexity attributions Trading eect One commonly used attribution category, which cannot be handled with the approach described above, is trading eect. This means the part o perormance due to selection o under- or overvalued instruments. This eect is the dierence between the return over the period o a buy and hold strategy and the actual perormance o the instrument in the portolio. Trading eect diers rom the other attribution components in that it is (by deinition) not caused by market movements. Rather, it is the complement o the market value change contributable to (dirty) price change: r = r, t r p p where r is the return on the instrument within the position, and r = 1 p 1 is the return o holding the p instrument (with p 1 and p 0 the dirty prices o the instrument at the end and beginning, respectively, o the period Attribution errors It is useul to take a look at the dierent sources o inaccuracies over the course o the attribution process: 0 WALLSTREET SYSTEMS PAGE 6 o 16

7 1. Model error: I the model does not reproduce the quoted prices exactly, there will be a model error, which is not liable to stay constant. The changes in the model error could be shown as a separate category or included in the residual term. 2. Residual term: I the attribution actors do not sum up exactly to the total change in the market/model variables, there will be a residual term and the corresponding residual market value change not explained by the attribution actors. 3. Taylor approximation error: Changes due to nonlinearity o the market value unction V not captured by our Taylor approximation. This error type will not appear in the sequential method. 4. Order eect: I we use the sequential method, the order in which the dierent components o the market variable changes are introduced will aect the results. This is related to the Taylor approximation: i the position is linear, i.e., i all the higher order derivatives vanish, the order o revaluation will not change the results. 2.2 Scope Deinition This section lists the new unctionalities that have to be developed in order to carry out ixed income attribution as described in section Phased implementation Fixed income attribution can be implemented in phases. The minimal implementation includes the ollowing: 1. Risk actors comprise only direct market quotes and zero curve movements. 2. Only one method is implemented (Taylor or sequential). I Taylor approximation is chosen, only duration and convexity terms are included. 3. Only one set o risk actors is deined, and it is hard coded. In later phases one or more o these restrictions can be lited User interace Attribution actor set deinition Attribution actor deinition has two parts: First, the model used or valuation has to be chosen. Second, the possible movements o model variables have to be deined. Model is chosen at instrument level, while model variable movements have to be deined at model level Model variables The concept o model as deined in the context o this document is currently not explicit in the system, except or zero coupon curves. Other models, or example implied volatility and Hull-White term structure are embedded in the valuation modules. Most parts o the attribution approach taken here will apply to these implicit models as well. However, the deinition o attribution actors and the decomposition o market movement into actor movements would be dierent or implicit models. This document will concentrate on the case o zero coupon curves, where the model is explicit, and leave other cases or urther development. I Taylor approximation is used, one has to choose the order o the approximation and the cross derivatives taken into account as well as the actor movement decomposition. In sequential approach, the revaluation order has to be determined. RS 1. It shall be possible to choose the valuation model to be used or attribution calculations. WALLSTREET SYSTEMS PAGE 7 o 16

8 In addition to the valuation method (par, zero coupon), one has to deine, or example, whether accrued interest is to be shown as a separate attribution category. Critical RS 2. It shall be possible to deine sets o risk actors or market or model variables. For zero curves, this will be similar to simulation scenario creation. The dierence is that there will be a wider variety o possible movements. Also, attribution actors are created as sets. Each attribution actor has to be given name, such as parallel shit, which will be used as the label o the corresponding attribution key igure. Optional: it is possible to have just one set o risk actors pre-deined or all clients. It is an open question whether this eature is important or not. RS 3. It shall be possible to choose the approximation method or each attribution actor. First, i sequential method is available, the order in which revaluation is carried out is determined. Second, or each attribution actor, one may choose to use irst and second order Taylor terms. I only Taylor method is used, then each actor has to have at least the irst order term. In this case also cross terms (o second order) are allowed. Critical, can be preset Attribution actor set selection RS 4. It shall be possible to select which attribution actor set is to be used or the calculation o key igures. From the implementation point o view, it will make a big dierence whether the attribution actor set is selected beore or ater perormance data is created. This is an open question. WALLSTREET SYSTEMS PAGE 8 o 16

9 Critical. However, i there is only one attribution actor set (either user-deined or preset), this step is not needed Attribution key-igure display Attribution actor key-igures. RS 5. Attribution actors should appear as new key-igures in the key-igure selection list. Note that i the number o attribution actors is not ixed, then the number o key-igures may change as well. Critical Calculations Risk models In the event that the existing valuation models do not provide suicient detail or perormance calculations, new models may have to be created. What kind o models are needed depends on the risk actors chosen and cannot be speciied within the scope o this document. To include implied spread attribution in bond perormance attribution, the ollowing is needed: Implied spread model. RS 6. The valuation model should calculate implied spread so that the market value calculated with zero coupon valuation with the added spread will be equal to the quoted market value. Optional Taylor approximation This is needed only or those attribution actors that are associated with Taylor approximation. For the purposes o urther processing, this is done at the lowest (instrument) level, so that we can later analyze attribution with dierent groupings in Perormance Monitor. For each instrument, calculate the required derivatives. RS 7. WALLSTREET SYSTEMS PAGE 9 o 16

10 I the market value o an instrument depends on all attribution actors associated with a term in the Taylor expansion, we need to calculate the corresponding dierential. This may be a number already available in TRM, or it may have to be calculated separately. Critical Market movement decomposition At this stage we assume that market quotes have been converted into model variables where necessary. Market/model variable changes are converted into attribution actor changes. RS 8. The change in market/model variables is presented as a linear combination o changes in risk actors plus residual terms. I there is an implied variable in the valuation model (spread, volatility), the change in it is calculated by solving its value rom the market price at the beginning and the end o the period. Critical Attribution calculation For each instrument calculate attribution terms. RS 9. This is done either by revaluating the instrument ater changing each attribution actor in turn (sequential method), or by multiplying the Taylor coeicients by the attribution actor changes (Taylor method). Critical RS 10. For each instrument calculate the trading eect. Critical WALLSTREET SYSTEMS PAGE 10 o 16

11 2.3 Example In this section we consider a simple position consisting o holding one bond over one day, without trading, and see how the change in market value might be distributed between dierent risk actors. The numbers in parenthesis at section headings reer to requirement numbers Setup Implied spread model (RS 1, RS 6) We shall use a valuation model that does not exist in TRM at the moment: Zero coupon valuation with implied spread. This means that a constant spread is added to all (risk ree) zero rates used to discount the cash lows o the bond, in such a way that the total present value o all the cash lows is equal to the quoted market value o the bond. In this example we choose not to calculate accrued interest. This means that the nominal accrued interest will be embedded in time attribution Risk actors (RS 2) Choosing the implied spread model or valuation implies that risk actors consist o movements o the zero curve, time, and spread. I the currency o the bond were dierent rom the portolio currency, currency attribution would be automatically included. The movements chosen are parallel shit, rotation, and reshape. The exact shapes o these are shown in Appendix A. An additional risk actor will be ormed by the residual term, since the movement o the yield curve cannot be captured completely by the three movements deined here Approximation method (RS 3, RS 4) In this example we choose the sequential method, with the ollowing revaluation order: spread, parallel shit, rotation, reshape, time change. With sequential method, we do not have to speciy Taylor actors. (In Appendix A attribution is or comparison purposes calculated also or Taylor approximation and sequential method with revaluation order reversed.) Calculations Market movement decomposition (RS 8) Over the one day period we are examining there were the ollowing market movements: 1. Market price (dirty) o the bond changed rom to Yield curve shited. 3. Time moved orward Spread calculation Part o the change in the market price o the bond is due to the yield curve movement, part due to a change in spread. To recover the change in spread we calculate or both the start and end market prices what spread, when added to the zero curve, would price the bond at the market price. For the start o the period the answer is , or the end we get So the change in spread is Zero curve movement To decompose the zero curve movement into the predeined components we express the movement as a linear combination o the components plus a residual term: <zero coupon change> = A*<parallel shit> + B*<rotation> + C*<reshape> + <residual> WALLSTREET SYSTEMS PAGE 11 o 16

12 choosing the coeicients A, B, and C so as to minimize the residual term. The graph o the results is shown below. Note that the shape o each zero curve movement component is chosen at set up. Only the magnitudes are calculated to it the actual movement Parallel shit Rotation Reshape Residual Total Attribution calculation (RS 9, RS10) Here s a summary o the results o the sequential revaluation o the bond. In parenthesis are given results when the revaluation order is reversed. Detailed calculations are in Appendix A. Market value at start 102 Market value with changed spread Spread attribution = (3.4708) Market value with new spread and parallel shit Parallel shit attribution = Market value with new spread, parallel shit, and rotation ( ) Rotation attribution = Market value with new spread, parallel shit, rotation, and reshape. ( ) Reshape attribution = Market value with new spread and new zero curve. ( ) Residual zero curve movement attribution = Market value with new spread and new zero curve at the end o the period. (0.0354) WALLSTREET SYSTEMS PAGE 12 o 16

13 Time attribution = (0.0144) No deals were made over the period, so that no trading eect has to be calculated. 2.4 Open questions 1. Which is preerable, sequential or Taylor method, or should the choice be based on technical considerations? 2. Is it necessary to be able to mix market quote and model actors? For example, do we want to simultaneously have yield and zero curve movements. 3. Should attribution actor set be selected, say, at portolio level, or can it change dynamically rom Perormance Monitor. The answer will depend on/determine the system architecture. 4. Is it necessary to have a conigurable risk actor set, or is it suicient to deine a basic set (at least or the irst phase). WALLSTREET SYSTEMS PAGE 13 o 16

14 Appendix A: Calculation example WALLSTREET SYSTEMS PAGE 14 o 16

15 Comments Bond deinition These should not be changed Cashlow dates (do not change) Cashlow amounts Deine zero curve movements here. Parallel shit is Zero curve movement deinition obvious, others are quite arbitrary. The number o Yield curve gaps (do not change) movements (here it is ixed at 3) is limited by the Parallel shit shape number o gaps. Magnitudes don't matter - Rotation shape calculations are scaled. Reshape shape Initial situation Day 0 0 These should not be changed Yield curve gaps (do not change) Original yield curve (Risk-ree) Rates (continuous) Original bond price Bond price (dirty) 102 Maturity = cashlow date - day Cashlow maturities Interpolation actors are used both or interpolation Interpolation actors orward and key-rate duration. Interpolation actors backward Linear interpolation rom Rates Interest rates (interpolation) Theoretical price is calculated in order to isolate the Discounted coupon (no spread) zero curve movement and spread eects. Theoretical price (no spread) Interest rates (interpolation & spread) Spread is calculated implicitly by making zero Discounted coupon coupon (with spread) price equal to the direct quote. Theoretical price (with spread) 102 I Theoretical price!= Bond price (C25 is red), goal Dierence between market and theoretical seek to zero by changing spread (C26) Spread (implied) This should be compared to D32 Money value o spread IR sensitivities Key rate durations Sensitivities are dierentials with respect to the IR second derivatives variable in question. In case o yield curve Key rate convexities movements we irst have to split the IR sensitivity at Spread sensitivity each coupon date into sensitivities against rates at Time sensitivity (coupon) yield curve dates (Key-rate duration). Then we just Time sensitivity (total) multiply each key-rate sensitivity by the Key rate sensitivities corresponding movement deined or the yield curve Duration (parallel shit) shape change. Again, scaling is arbitrary and will be Rotation accounted or when movement magnitudes are Reshape calculated. Convexity Rotation convexity Reshape convexity Move time by changed market data Day These do not change. Yield curve gaps New yield curve New (Risk-ree) Rates (rom market) Yield curve change Bond price can be changed independently o zero rates. The spread will change accordingly. Bond price (dirty) Maturity dates diminish by one day Maturity dates Interpolation actors are used both or interpolation Interpolation actors orward and key-rate duration. Interpolation actors backward Linear interpolation rom Rates Interest rates (interpolation) Interest rates change (interpolation) Use day 0 maturities: we are only interested in yield Discounted coupon (old spread) curve movements and ignore time eect. Theoretical price (old spread) Spread calculation Interest rates (interpolation & spread) Discounted coupon Theoretical price I C60 is red, goal seek to zero changing C61 Dierence between market and theoretical 0.00 Spread (implied) Market movements Change in spread Market data changes that we are interested in are: implied spread, time (one day), yield curve movement (parallel shit, rotation, reshape). The magnitudes o the last three are meaningless - they depend on the scaling o the shape deinitions. Shape change magnitudes are calculated by decomposing the actual change into a linear combination o parallel shit, rotation, and reshape, minimising the residual curve movement. Change in maturity Decomposing yield curve movement Parallel shit magnitude Rotation magnitude Reshape magnitude Parallel shit Rotation Reshape Residual yield curve movement Interpolation eect Taylor attribution We decompose the change in market value Market value change Here are the components spread sensitivity * spread change spread attribution Keep spread and time constant - change ZC curve Total ZC attribution Total ZC is urther divided into shit sensitivity * shit magnitude Duration (parallel shit) attribution second order term in Taylor polynomial Convexity attribution rotation sensitivity * rotation magnitude Rotation attribution Rotation convexity attribution reshape sensitivity * reshape sensitivity Reshape attribution Reshape convexity attribution Residual due to unexplained part o ZC movement. Residual ZC linear approximation Residual ZC convexity approximation total ZC attribution - (shit + rotation + reshape + residual movement) Approximation error o ZC attribution time sensitivity * -1/365 (one day) Time attribution Market value change not explained by the actors above. Residual attribution Sequential attribution We decompose the change in market value by revaluating ater changing each risk actor in turn. Market value change Add new spread to the original zero curve Revaluation with spread change spread attribution ZC components have to be interpolated. We use interpolation actors o day 0. Add parallel shit Spread + parallel Revaluation with parallel shit added Duration (parallel shit) attribution Add rotation Spread + parallel + rotation Revaluation with rotation added Rotation attribution Add reshape E Spread + parallel + rotation + reshape Revaluation with reshape added Reshape attribution Add residual ZC change E E E E E E E-05 WALLSTREET SYSTEMS PAGE 15 o 16 In sequential method residual ZC change is due to

16 WALLSTREET SYSTEMS PAGE 16 o 16

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