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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

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

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

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

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

16 WALLSTREET SYSTEMS PAGE 16 o 16

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