A G E N E R A L I Z E D H Y B R I D F I X E D I N C O M E AT T R I - B U T I O N M O D E L

A N D R E W C O L I N A N D K ATA L I N K I S S A G E N E R A L I Z E D H Y B R I D F I X E D I N C O M E AT T R I - B U T I O N M O D E L F L A M E T R E E T E C H N O L O G I E S P T Y LT D

Copyrght 2017 DST Global Solutons Lmted and Flametree Technologes Pty Lmted Revsed and corrected November 2017 All Rghts Reserved. No part of ths paper or any of ts contents may be reproduced, coped, modfed or adapted, wthout the pror wrtten consent of both DST Global Solutons Lmted and Flametree Technologes Pty Lmted.

a generalzed hybrd fxed ncome attrbuton model 3 Introducton The requrement for fxed ncome attrbuton contnues to rse dramatcally as the value t brngs to the nvestment process becomes more wdely apprecated. The attrbuton model descrbed n ths paper addresses both user demand and the lmtatons of exstng commercally avalable systems. It ncorporates extensve nput from ndustry practtoners, portfolo managers, software vendors and technology experts to provde a workable, cost-effectve soluton to many problems that have prevously affected ths area. The model has been mplemented by Flametree Technologes and s currently lcensed to several major ndustry vendors, wth further partnershps under dscusson. The FIA concept The Flametree approach arose as a response to the shortcomngs of commercally avalable attrbuton systems, ncludng requrements for large volumes of daly rsk data, long mplementaton tmes, hgh costs and nflexble modelng capabltes. Our am has been to provde a fxed ncome attrbuton engne that addresses all of these ssues. Ths approach has many advantages such as no requrement for rsk numbers, whch are calculated nternally use of user-suppled returns to ensure consstency between other performance data and attrbuton reports very hgh calculaton speed wdest possble range of attrbuton models, allowng an exact match between nvestment process and attrbuton reportng rapd deployment, snce only three data fles are requred to run attrbuton.

4 andrew coln and kataln kss Desgn consderatons The desgn of a successful fxed ncome attrbuton model s a dffcult problem that has defeated many practtoners. Perhaps the bggest - but least apprecated - obstacle s that attrbuton requres ntegraton of many apparently unrelated mathematcal, computng and fnancal factors such as the underlyng data model and workflow, depth and confgurablty of analyss, cost and data tradeoffs, and smplcty of deployment. The gudng prncples n the desgn of our model are: 1. The assumpton that weghts, base and local currency returns are avalable at the securty level. Vrtually any portfolo manager wth a requrement for attrbuton wll have performance systems n place that can provde ths nformaton, whle benchmark data s generally avalable at the same level of detal. We see the provson of accurate returns data as an entrely separate ssue from the provson of attrbuton analyss, especally snce performance calculaton packages are wdely avalable. The use of externally calculated returns ensures that attrbuton reports are always consstent wth other performance reports, so no reconclaton of results between systems s requred. 2. Low data requrements. Hstorcally, a major obstacle to the successful mplementaton of attrbuton models has been an over-relance on perturbatonal models whch requre large volumes of hghfrequency rsk numbers (yelds, modfed duratons, convextes) as proxes for exact prcng functons. The model descrbed here allows the perturbatonal approach to be replaced or supplemented by a frst-prncples approach to prcng, whch only requres yeld curves and lmted statc securty defntons. Instead of requrng daly rsk numbers, we allow securtes to be prced by dscountng ther cash flows under a range of nterest rate scenaros. Ths typcally reduces data volume requrements by orders of magntude. 3. Flexblty. Attrbuton reportng requrements vary greatly n complexty and scope, and t s crtcal for user acceptance that the attrbuton model be confgurable to match whatever nvestment process s n use. Many exstng fxed ncome attrbuton systems are constraned by desgn to a small number of attrbuton models that may not

a generalzed hybrd fxed ncome attrbuton model 5 match a manager s changng requrements. For nstance, an nvestor whose nvestment process nvolves extensve spread duraton allocaton wll fnd only lmted value n bottom-up curve analyss. 4. The ablty to mx top-down and bottom-up attrbuton models to generate hybrd models, thus blurrng the dstncton between equty, fxed ncome and balanced (or mxed) attrbuton requrements. 5. A plug and play approach. We beleve t should not be necessary to replace a manager s core performance system solely to ntroduce an attrbuton capablty. By deployng the attrbuton system as a stand-alone module wth clearly defned, self-contaned IT and data requrements, the user can contnue to use an exstng performance engne as a feed for ther new attrbuton capabltes. The performance software may then be replaced at a later date f need be wthout affectng the attrbuton workflow. 6. Recognton that FI attrbuton s sgnfcantly more complex than equty attrbuton, and that t requres entrely new sources of data, as well as new expertse. We see smplcty of deployment as crtcal to meetng ths requrement. Securty coverage Our model s desgned to be able to model all securty types traded n the marketplace, ether currently or n the future. Ths s acheved n three ways: 1. Use of a frst-prncples prcng lbrary. The Flametree engne uses a core lbrary of buldng block prcng routnes that allows the vast majorty of securtes to be accurately modeled. These nclude government and corporate bonds (both nvestment grade and hgh yeld), agences, securtzed debt such as ABS and MBS, CMBS, money market securtes, callable bonds, snkng securtes, varous types of FRNs, futures, forwards, FX optons, IRS, nflaton-lnked securtes, emergng market debt, and others. In addton, the program s nested portfolo and lookthrough capabltes allow the defnton of new securty classes that are portfolos of ndvdual sub-securtes. For nstance, a vanlla nterest-rate swap can be represented as a portfolo contanng +1 unt of a bond, and -1 unt of an FRN, where the bond and FRN are modeled usng the standard buldng blocks for these nstruments.

6 andrew coln and kataln kss 2. Mx-and-match frst-prncples and perturbatonal attrbuton. Our model allows both frst-prncples prcng and the use of rsk numbers as a prcng functon proxy. For cases where no prcng model s avalable (such as a complex CMBS), or where there are sources of accurate rsk numbers, t may be preferable to use perturbatonal attrbuton. Ths feature can be confgured from the securty level upwards, and can change over tme usng the program s effectve date functonalty. 3. Use of securty-level customzable output buckets. In some cases the ablty to drect return to a partcular rsk source may be needed. For nstance, a credt default swap may ether have ts return allocated to the generc Credt sector or to a custom sector such as CDS return. In practce, we fnd that ths combnaton of approaches allows flexble and complete coverage of all securty types wthout the requrement for complex workarounds. The mathematcs behnd frst-prncples and perturbatonal attrbuton s covered n more detal below. Attrbuton models No ndustry standards exst for fxed ncome attrbuton, and n our vew ths wll reman the case ndefntely, due to the wde range of nvestment approaches actve n the market. Rather than mposng a partcular attrbuton methodology on the user, our approach has been to acknowledge ths lack of standardzaton and to make the attrbuton model as flexble and confgurable as possble. It may then be adapted to measure the specfc returns of whchever nvestment process s n use. Although the Flametree engne has been desgned as a fxed ncome attrbuton system, the program s top-down attrbuton capabltes allow t to generate equty attrbuton reports as well, smply by allocatng all non-allocaton returns to a custom stock selecton category.

a generalzed hybrd fxed ncome attrbuton model 7 Top-down attrbuton Market allocaton Top-down attrbuton measures the effects of allocaton decsons by market weght, duraton contrbuton, or other measures such as duraton tmes spread. The well-known Brnson models are market weght allocaton algorthms, whch decompose actve return from each sector S nto return contrbutons from asset allocaton and stock selecton c AA S and c SS S, usng and c AA S = (w P S wb S ) (rb S rb ) (1) c SS S = (rp S rb S ) wp S (2) where ws P and wb S are the weght of the sector n portfolo and benchmark, rs P and rb S are the returns of the sector n the portfolo and benchmark, and r B s the overall return of the benchmark. 1 The sector weghts and returns are gven by 1 For smplcty, we have aggregated nteracton return wth stock selecton return. ws P = w P (3) S ws B = w B (4) S w rs P Pr = S S w P w rs B Br = S S w B (5) (6) and the benchmark return by r B = w B r (7) B where the sums are over ndvdual securtes n the gven sector or portfolo. Market allocaton attrbuton s a useful a tool to measure the effectveness of equty management. The reason t s not wdely used n

8 andrew coln and kataln kss the fxed ncome world s that t gnores some mportant rsks. For nstance, f you choose to nvest 50% of your funds nto fxed ncome, the Brnson model wll gnore the dfference between a T-bll and a 30-year T-bond, despte the two havng radcally dfferent levels of nterest rate rsk. Ths s why duraton allocaton attrbuton s preferred by bond fund managers: t s a much closer match to the way that portfolo managers actually mplement nvestment strateges. For nstance, f a contracton n credt spreads n the 5-10 year regon of the curve s expected for US corporates, t makes sense to overweght duraton n that part of the portfolo, and duraton allocaton attrbuton wll measure the return made by ths decson. Duraton allocaton Duraton allocaton models measure return made by duraton msmatches between portfolo and benchmark at varous levels. Ths results n three categores of attrbuton return (market drecton return, market allocaton return, securty selecton return) rather than the two from the market allocaton model (asset allocaton, stock selecton). Duraton asset allocaton requres the followng data for each securty n portfolo and benchmark: w P, wb : weghts of securty n portfolo and benchmark; MD : modfed duraton; y : yeld to maturty; δy : total change n yeld to maturty; δt: nterval over whch the calculaton s to be performed, as a fracton of a year To perform duraton attrbuton, a yeld s frst calculated for each sector S n the benchmark: δy S = ( ) S w B MD δy ( ) (8) S w B MD Note that benchmark weghts are used for calculaton of all aggregated yeld changes. The market drecton return contrbuton s gven by c MD = (MD P MD B ) δy B (9)

a generalzed hybrd fxed ncome attrbuton model 9 where δy B s the overall change n yeld for the entre benchmark, and MD P and MD B are the modfed duratons of portfolo and benchmark respectvely. For each sector S, the contrbuton to market allocaton return s gven by c MA S = (w P S wb S ) MDB S (δyb S δyb ) (10) where δys B s the change n benchmark yeld for sector S. Lastly, ndvdual securty selecton returns for securty are gven by c SS = (w P w B ) MD (δy δy B S ) (11) where quanttes wth a suffx S refer to the sector of that name. The sum of returns over all rsks from equatons (9), (10) and (11) reduces to r = (w P w B) MD δy (12) whch s the securty-level actve return generated by each source of rsk, as expected. Note that t s possble to mx and match attrbuton models n the same analyss. For nstance, carry return may be analyzed usng market value attrbuton, but spread returns usng a duraton allocaton model. Ths pont s llustrated n the worked example below. Multple level allocaton attrbuton Allocaton decsons are frequently made at multple levels wthn a portfolo. For nstance, a manager mght frstly decde to overweght certan countres, and then to overweght partcular ndustral sectors wthn each country. Transparent attrbuton requres that the excess return generated by each type of decson be reported separately. To acheve ths result at the sector level requres that the benchmark be temporarly reweghted so that so that, at the country level, t becomes dentcal to the portfolo. Wth ths adjustment n place, there s no country overweghtng or underweghtng n play. Returns from country allocaton wll be zero, so any remanng asset allocaton returns wll be due to ndustry allocaton. Our model provdes ths functonalty as standard for both market weght and duraton allocaton attrbuton, for any number of nested allocaton decsons.

10 andrew coln and kataln kss Hybrd models A crtcal requrement for many users s the ablty to mplement a hybrd model, whch ncorporates both top-down and bottom-up returns. For nstance, we have seen credt desks that requre Brnson analyss for carry return, spread duraton allocaton attrbuton for credt spreads, and key rate duraton analyss for soveregn curve movements. Ths s easly acheved usng the buldng blocks descrbed above. A partcular example of a hybrd attrbuton model s a mxed attrbuton model. In ths case, asset allocaton returns are calculated usng asset type weghts. Where approprate, the remanng stock selecton weghts are then decomposed n terms of bottom-up effects. For nstance, a mxed portfolo mght contan both equtes and bonds, and n ths case a mxed attrbuton model s approprate. The frst decson made wll be whether to over- or under-weght entre countres or asset classes relatve to benchmark. The top level asset allocaton return measures the return made by ths decson. The next level of decson may have been at the level of ndustry or sector weghts wthn each asset type. Here, a second asset allocaton return provdes the return generated by these subsdary decsons. Lastly, stock selecton return measures ndvdual stock contrbutons to return. For fxed ncome securtes, ths return contrbuton may be decomposed further nto returns from carry, curve and credt effects. Mxed and hybrd attrbuton models can be combned f requred, to measure return from asset type weghtng, spread duraton allocaton decsons, and ndvdual fxed ncome returns. Bottom-up attrbuton Bottom-up attrbuton s the decomposton of a sngle securty s return n terms of ts sources of rsk. For equtes, the user must select a known set of rsks and run a complex statstcal study to measure the effects of these factors on the portfolo s return. For fxed ncome, the factors are usually assumed to be already known (passage of tme, movements n curves) and the decomposton s run n these terms. Here, fxed ncome attrbuton s n essence a specalzed form of rsk adjusted attrbuton.

a generalzed hybrd fxed ncome attrbuton model 11 Sngle-securty return decomposton s typcally run n one of two ways: Prcng securtes from frst prncples The most drect way to prce a securty s to calculate ts ndvdual cash flows, to prce them usng the approprate dscount rate, and to add them together: p = C (1 + r ) t (13) where p s the securty s prce, C s the cashflow, r the nterest rate, and t the tme to maturty (n years) of the th cash flow. The securty s prced wth and wthout the effect of the current rsk (such as a parallel curve movement, or a change n spread due to a partcular credt factor), and the return due to that rsk s then gven by the dfference between the two prces, dvded by the startng prce. Calculatng return usng the perturbatonal equaton Assumng that the prce p of an arbtrary securty s a functon of tme t and yeld y, we can express δp n terms of a Taylor expanson, and wrte δp = p p δt + t t δy + 1 2 p 2 y 2 δy2 + O(δt 2, δy 3 ) (14) If we dvde throughout by p and wrte r = δp p (15) y = 1 p p t (16) MD = 1 p p y (17) C = 1 2 p 2p y 2 (18)

12 andrew coln and kataln kss equaton (14) becomes r y δt MD δy + 1 2 C δy2 (19) where r s the securty s local return, y ts yeld to maturty, MD ts modfed duraton, C ts convexty, δt the elapsed tme (n years), and δy the securty s change n yeld over the calculaton nterval. y, MD and C are often collectvely referred to as the securty s rsk numbers. It s temptng to vew equaton (19) as a one sze fts all approach to attrbuton, and several commercal systems have been bult on ths bass. Unfortunately, the assumpton s seldom vald. Many securtes have other sources of return, such as nflaton for TIPS and nflatonlnked glts; others (such as FRNs) have multple rsk senstvtes; the model s only exact for a securty wth a sngle cashflow; and some specalzed types of securtes, such as Australan and New Zealand bond futures, do not generate carry. Any system that offers perturbatonally-based attrbuton should therefore offer the ablty to customze the perturbaton equaton accordng to the type of securty. Unfortunately, supplyng daly rsk numbers can be a surprsngly dffcult (and expensve) problem. It can take many man-months to set up relable, robust feeds for rsk numbers. Even after ths pont, rsk numbers for some securty types such as OTC dervatves may stll need to be calculated n-house, and a sngle ncorrect value can dstort the entre analyss. Sources of fxed ncome return The man sources of fxed ncome return are carry, curve and credt, although there are many others dependng on the level of analyss requred and the securtes held. Carry return Carry return s the return generated by the passage of tme, due to the payment of coupons and the approach of maturty, when a fxed ncome securty must be redeemed at par. Carry return s closely approxmated by r carry = y δt (20) where y s the securty s yeld to maturty, and δt s the elapsed tme.

a generalzed hybrd fxed ncome attrbuton model 13 Carry return may be decomposed further n two ways: Pull-to-par and runnng yeld A manager who has purchased a bond at a prce below par wll show postve returns from pull-to-par effects as t approaches maturty. To vew carry return broken down n ths way, calculate the runnng yeld, whch s gven by r runnng yeld = C/P (21) where C s the securty s coupon, and P s the (clean) prce. Ths wll gve the nstantaneous return of the securty, gnorng any long-term captal gan effects. The pull-to-par yeld s then the yeld to maturty, mnus the runnng yeld. Rsk-free and credt carry Another way to break down yeld to maturty s to regard t as a sum of two yelds: a rsk-free yeld and a credt spread yeld. The carry return s then gven by the sum of the rsk-free carry and the credt carry: r rsk f ree = y rsk f ree δt (22) r credt spread = y credt spread δt = (y r rsk f ree ) δt (23) r carry = r rsk f ree + r credt spread (24) Ths type of decomposton s of partcular nterest to credt traders, who may add value by nvestng n hgh-yeld stocks wthout takng nterest rate rsk. If the strategy s successful, the credt carry of the portfolo wll exceed that of the benchmark and the value added wll be the dfference between the credt carry for the portfolo and the credt carry for the benchmark. Soveregn curve return The bulk of return n many portfolos s generated by parallel movements n the rsk-free curve. Ths s often referred to as dura-

14 andrew coln and kataln kss ton return, snce ts magntude s equal to the (negatve) modfed duraton, tmes the sze of the parallel curve shft. The calculaton of ths return requres knowng the sze of any such parallel shft. As there s no ndustry-wde agreement on the defnton of ths quantty, we allow the average curve level to be defned usng one of Arthmetc averagng (smple but overweghts the short end of the curve, where samplng s more dense) Trapezodal ntegraton, whch approxmates the area under the curve and dvdes by the longest maturty The level of the curve at a maturty or modfed duraton pont equal to that of the benchmark. The parallel shft s then taken to be the dfference between the average curve level at successve dates. Other curve decomposton algorthms may be used as requred. For nstance, The shft/twst/butterfly (STB) model measures parallel shft as above, twst as the change n the slope of the curve between the 3 and 10 year ponts, and curvature as any remanng curve movement after parallel and twst movements have been removed. The prncpal component analyss (PCA) model calculates the egenfunctons of the curve and allocates movement to movements of order 0, 1 and 2. The key rate duraton (KRD) model perturbs the curve at usersuppled tenor ponts and uses ths curve to calculate the effect of a change n the curve at ths maturty only. Credt movements Credt effects are drven by changes n the spread between the soveregn curve and the sector curve for a partcular securty. Country curve allocaton s a partcular type of credt return, where (for nstance) alpha may be generated by contractng credt spreads between debt ssued by dfferent countres n the Euro-zone.

a generalzed hybrd fxed ncome attrbuton model 15 Our model allows famles of credt curves to be assocated wth partcular securtes, so that (for nstance) the return made by changes n the AAA-AA, AA-A, and A-B spreads may be measured. Alternatvely, a sector-specfc or ndustry curve may be assocated wth the securty, so that the return made by changes n the sector curve (sector return), and changes between the securty s market yeld and that curve (securty-specfc return) measured. Addtonal effects Dependng upon the type of analyss requred, other sources of return may be actve. For nstance, a portfolo wth many MBS may generate substantal return from convexty, and n ths case t makes sense to report convexty returns. In addton to the man three sources of return, any of the followng effects may be ncluded n the system s outputs: Rolldown return Convexty return Inflaton and break-even return (for nflaton-lnked securtes) FX return Paydown return (for snkng securtes such as amortzng bonds and MBS) Cash depost return Prce return Other effects can be defned by the user, dependng on the prcng model and the securty s confguraton parameters.

16 andrew coln and kataln kss Fgure 1: Sources of fxed ncome attrbuton returns Implementaton The mportance of mplementaton ssues s often overlooked when selectng an attrbuton platform, but they can form one of the largest (and costlest) barrers to successful provson of an attrbuton capablty. Ths rapd calculaton ablty also smplfes generaton of attrbuton reports over long tme perods. Fgure 2 shows the result of a contguous attrbuton calculaton on a corporate bond portfolo over a two-year perod. Ths capablty s partcularly useful when, for nstance, a new nvestment mandate requres the provson of hstorcal attrbuton reports n a partcular format. Fgure 2: Cumulatve attrbuton returns for an Australan corporate bond fund, 2010-2011

a generalzed hybrd fxed ncome attrbuton model 17 Future trends n fxed ncome attrbuton Strategy attrbuton A growng requrement for many managed funds s the ablty to provde strategy attrbuton capabltes, n whch holdngs wthn a portfolo are assgned to one or more nvestment strateges, such as duraton bet, curve steepenng, Latn Amercan credt spread play. Although ths s predomnantly a data management ssue, our model has the ablty to support such analyses n a natural way by assgnng returns from dfferent securtes to subportfolos. The full range of attrbuton and reportng capabltes s then avalable on each strategy. Lablty-drven nvestment (LDI) An LDI strategy s drven by the requrement to fund current and future lablty cash flows, rather than to beat a known benchmark. However, the requrement for attrbuton remans the same n both cases. Much of the responsblty of an LDI manager s to ensure that the portfolo s suffcently hedged aganst dfferent types of market movements, and an attrbuton analyss wll supply clear and unambguous feedback on whether ths am was acheved. The Flametree model can be appled to LDI portfolos n exactly the same way as conventonal managed portfolos, wth the lablty cashflows modeled n terms of conventonal and nflaton-lnked securtes. Reportng Although not strctly part of the attrbuton model, we regard reportng as an mportant part of the attrbuton process. The volume of data generated by an attrbuton analyss can easly overwhelm the user, and the provson of reportng tools and technques to generate nsght from ths frehose of data s a vtal part of the overall workflow. Sutable reportng technques range from the smple (rollng up performance contrbuton from benchmark stocks that are not held n the portfolo) to the wdely used (generaton of Excel reports usng drll-down and roll-up capabltes, allowng the user to dentfy areas of nterest) to the more exotc, such as nteractve treemaps to

18 andrew coln and kataln kss summarze the sources of actve return. The reportng requrements of portfolo managers and clent reportng staff wll usually dffer wdely. For a flagshp fund, a smple statement of carry, curve and credt returns may be all that s needed by a marketng team, but the front offce may requre much more detaled analyss. The model descrbed handles both cases easly by usng dfferent confguraton sets. For nstance, predefned templates are provded for duraton/curve reshapng, Camps, Tm Lord, key rate duraton and top-down attrbuton models. Summary Test deployments and feedback strongly ndcate that ths model meets the vast majorty of fxed ncome managers attrbuton requrements, and that t can be mplemented quckly wth mnmal busness rsk and at reasonable cost.

a generalzed hybrd fxed ncome attrbuton model 19 Worked example All performance n ths example s presented as performance contrbuton, whch s the product of a securty or a sector s weght and ts return. Performance contrbutons can then be aggregated to sector or portfolo level. Consder the followng portfolo and benchmark. Each securty les n one of two sectors {S 1, S 2 }. Sector Securty w P w B MD y δy S1 A 13% 5% 1.97 3.30% -0.70% S1 B 13% 0% 2.33 3.40% -0.60% S1 C 22% 44% 2.89 3.25% -0.40% S1 D 6% 8% 3.05 4.40% -0.20% S2 E 8% 13% 3.43 4.40% -0.10% S2 F 10% 5% 4.80 4.90% 0.00% S2 G 11% 10% 5.20 5.10% 0.10% S2 H 17% 15% 5.80 5.10% 0.20% Table 1: Weghts, rsks and yelds for sample portfolo and benchmark For securty, w P and w B are securty weghts n portfolo and benchmark, respectvely; MD s modfed duraton; y s yeld to maturty; δy s aggregate change n yeld. The portfolo and benchmark have vrtually dentcal modfed duratons (3.6902 vs 3.6900 years respectvely). The perod over whch the returns are measured s 0.25 of a year. The changes n yeld may be decomposed further by source of rsk. Assumng that yeld changes are due to parallel, non-parallel and credt shfts, Table 1 can be supplemented by the data n Table 2. Bottom-up attrbuton One way to nterpret these results s to take a bottom-up vew, and to regard all return contrbutons as havng been made at the securty level. The dsplayed contrbutons are actve (portfolo performance contrbuton mnus benchmark performance contrbuton):

20 andrew coln and kataln kss Securty Securty δy Parallel δy Non parallel δy Credt A -0.20% -0.50% 0.00% B -0.20% -0.40% 0.00% C -0.20% -0.30% 0.10% D -0.20% -0.20% 0.20% E -0.20% -0.10% 0.20% F -0.20% 0.00% 0.20% G -0.20% 0.10% 0.20% H -0.20% 0.20% 0.20% c Carry c Parallel c Non parallel c Credt A 0.0660% 0.0315% 0.0788% 0.0000% B 0.1105% 0.0606% 0.1212% 0.0000% C -0.1788% -0.1272% -0.1907% 0.0636% D -0.0220% -0.0122% -0.0122% 0.0122% E -0.0550% -0.0343% -0.0172% 0.0343% F 0.0613% 0.0480% 0.0000% -0.0480% G 0.0128% 0.0104% -0.0052% -0.0104% H 0.0255% 0.0232% -0.0232% -0.0232% Subtotal 0.0203% 0.0000% -0.0485% 0.0285% Total 0.0002% Table 2: Changes n yeld, decomposed by rsk Table 3: Securty-level performance contrbutons from bottom-up rsks Actve carry contrbuton s calculated as c Carry = (w P w B ) y δτ (25) and the actve return contrbuton due to ndvdual sources of rsk s calculated as c rsk = (w P w B ) MD δy rsk (26) For nstance, actve carry return from securty A was gven by (13% 5%) 3.69% 0.25 = 0.0660%, whle actve parallel shft return was (13% 5%) 1.97 0.20% = 0.0315%. At the aggregate level, carry generated an actve return of 2.03 bp. The modfed duraton of portfolo and benchmark were vrtually dentcal, so overall return due to parallel curve movements was zero, as expected. In addton, the portfolo lost 4.85 bp from non-parallel movements n the curve, but made back 2.85 bp from credt shfts. Overall, the portfolo s actve return was very close to zero.

a generalzed hybrd fxed ncome attrbuton model 21 Top-down attrbuton Suppose that the portfolo had nstead been managed usng a topdown duraton allocaton strategy, n whch rsk s apportoned to portfolo and benchmark sectors. In ths case, allocaton decsons wll have been made wth a vew to generatng both excess carry return and excess return from duraton. The attrbuton analyss wll allow the effects of both decsons to be compared. To see the effects of the allocaton decson on carry return, break down the carry contrbuton as follows. Note that we are usng a Brnson analyss for carry, snce carry returns are drven by absolute yeld rather than changes n yeld over an attrbuton nterval. Allocaton return for carry The sector-level contrbuton to allocaton return s gven by c AA S = (w P S wb S ) (rb S rb ) (27) Sector w P S wb S r B S r B c AA S Sector 1-3% 0.8539% 1.0098% 0.0047% Sector 2 3% 1.2163% 1.0098% 0.0062% 0.0109% Table 4: Asset allocaton performance contrbutons for carry Asset allocaton return s a sector-level effect, so we only show returns at the sector level. Stock selecton return for carry Stock selecton s a securty-level effect. Includng nteracton returns, t s gven by c SS = (w P w B ) (r r B S ) (28) Overall, the weghtng decsons made by the manager generated outperformance from carry, wth a roughly equal contrbuton from allocaton to sectors (1.09 bp) and from ndvdual stock carry contrbutons (0.94 bp).

22 andrew coln and kataln kss Securty w P w B r carry rs B c SS A 8% 0.8250% 0.8539% -0.0023% B 13% 0.8500% 0.8539% -0.0005% C -22% 0.8125% 0.8539% 0.0091% D -2% 1.1000% 0.8539% -0.0049% E -5% 1.1000% 1.2163% 0.0058% F 5% 1.2250% 1.2163% 0.0004% G 1% 1.2750% 1.2163% 0.0006% H 2% 1.2750% 1.2163% 0.0012% 0.0094% Table 5: Stock selecton performance contrbutons for carry Duraton allocaton return Carry returns have been analyzed usng a Brnson analyss, snce these are drven by market weghts. By contrast, returns due to duraton allocaton allocaton should be analyzed usng a non-brnson approach, snce actve returns from curve effects are drven by duraton contrbutons. For a duraton allocaton analyss, recall that there are three sources of actve return rather that the two from Brnson attrbuton: market drecton, duraton allocaton, and duraton selecton return. Market drecton return Performance contrbuton from market duraton effects c MD s a global source of return, and s calculated at the portfolo level: c MD = (MD P MD B ) δy B (29) MD P MD B δy B c MD 3.6902 3.6900-0.2000% 0.0000% Table 6: Market drecton performance contrbuton Ths zero value for market return reflects the vrtually equvalent modfed duratons of portfolo and benchmark, reflectng the manager s decson to be neutral nterest rate rsk at the portfolo level. In fact, the portfolo had several actve duraton decsons n play, but at the sector level rather than the overall portfolo level. The return made from these lower level duraton allocaton decsons are reflected n the next source of return.

a generalzed hybrd fxed ncome attrbuton model 23 Duraton allocaton return Just as for carry allocaton return, duraton allocaton return contrbuton cs DA s calculated by sector, usng c DA S = (w P S wb S ) MDB S (δyb S δyb ) (30) where δy B S s the change n benchmark yeld for sector S. Sector w P S MD S w B S MD S δy B S δy B c DA S Sector 1 1.3778 1.6141-0.3982% -0.2000% -0.0468% Sector 2 2.3124 2.0759 0.0628% -0.2000% -0.0622% Total 3.6902 3.6900-0.1090% Table 7: Duraton allocaton performance contrbutons These results show that the manager made poor duraton allocaton decsons n both sectors. For nstance, n Sector 1 the fund was short duraton by 0.2363 of a year, makng the fund less senstve to yeld changes than the benchmark. Yelds n ths sector fell by 39 bp compared to the benchmark s decrease of 20 bp. Ths decrease n yelds generated a rse n prces, but because the portfolo was shorter duraton than the benchmark, t underperformed, generatng a net actve return of (1.3778 1.6141) ( 0.3982% 0.20000%) = 4.68 bp. Conversely, n Sector 2 the fund was 0.2365 years long, makng t more senstve to yeld changes than the benchmark. The yeld of ths sector rose whle that of the benchmark fell, drvng down prces and hence returns. The net result was agan negatve for the fund, generatng a performance contrbuton of -6.22 bp. Duraton selecton return Duraton selecton return s calculated on a per-securty bass, usng c SS = (w P w B ) MD (δy δy B S ) (31)

24 andrew coln and kataln kss Securty w P S MD S w B S MD S δy B δy B S c SS A 0.2561 0.0985-0.7000% -0.3982% 0.0476% B 0.3029 0.0000-0.6000% -0.3982% 0.0611% C 0.6358 1.2716-0.4000% -0.3982% -0.0011% D 0.1830 0.2440-0.2000% -0.3982% 0.0121% E 0.2744 0.4459-0.1000% 0.0628% -0.0279% F 0.4800 0.2400 0.0000% 0.0628% 0.0151% G 0.5720 0.5200 0.1000% 0.0628% -0.0019% H 0.9860 0.8700 0.2000% 0.0628% -0.0159% Total 0.0889% Table 8: Duraton selecton performance contrbutons For nstance, securty A was overweght n the portfolo, and contrbuted 0.1576 years to the portfolo s actve duraton poston. Its yeld fell by 70 bp compared to the 39.82 bp decrease wthn Sector 1. The securty s net contrbuton to performance (as dstnct to the contrbuton of the sector, or the portfolo as a whole) was therefore 0.1576 ( 0.7000% 0.3982%) = 4.76 bp. Summarzng the duraton allocaton analyss, we have Rsk Return Carry allocaton 0.0109% Carry selecton 0.0094% Market drecton 0.0000% Duraton allocaton -0.1090% Duraton selecton 0.0889% TOTAL 0.0002% Table 9: Summary results for duraton allocaton attrbuton Comparng the top-down and bottom-up vews In both cases, the carry return aggregates to the same total, although n the top-down model carry can be decomposed n terms of allocaton to partcular sectors. The negatve curve return can ether be assgned to adverse twst movements n the curve (bottom-up attrbuton) or to poor allocaton to market sectors. The two rsks are closely related, snce Sector 1 held securtes wth shorter duratons, whle Sector 2 held securtes wth longer duratons. In other words, ths s exactly the type of behavour we would have expected f the curve had flattened or steepened. However, we dd not need a securty-level breakdown to supply ths nformaton. Whch analyss should be used depends on how the portfolo was

a generalzed hybrd fxed ncome attrbuton model 25 managed. For a bottom-up manager, a duraton allocaton analyss would be napproprate, as t would decompose returns n terms of rsks that had never been examned or managed. Spread duraton allocaton attrbuton Many portfolos, partcularly for emergng-market debt (EMD) are managed n terms of spread duraton allocaton rather than modfed duraton. The technques descrbed n ths paper are equally applcable to portfolos managed n ths way. Hybrd attrbuton To construct a hybrd analyss, the duraton selecton term can be decomposed further. Table 8 shows the duraton selecton return due to the aggregated return for each securty. If we use the data n Table 3 and construct smlar tables, one for each source of rsk, the result s a report that shows return from both top-down sources (market drecton, duraton allocaton) and bottom-up sources, whch replace the duraton selecton term. The result s shown n Table 10: Rsk Return Carry allocaton 0.0109% Carry selecton 0.0094% Market drecton 0.00004% Duraton allocaton -0.1090% Parallel curve return 0.0000% Twst curve return 0.0380% Credt return 0.0509% TOTAL 0.0002% Table 10: Summary results for duraton allocaton attrbuton Note that the sum of the last three terms equals the duraton selecton return, and that the parallel curve return s zero. Ths s expected, as any parallel curve return wll be appear n the market drecton term. We have left the term n place as other curve breakdowns may not have a parallel curve shft term, such as a prncpal component or a key rate duraton analyss. In ths case, the term wll not be zero.

26 andrew coln and kataln kss Bographes Andrew Coln s founder of Flametree Technologes, a company that provdes nnovatve performance and attrbuton software to fund managers of all szes. He was prevously Head of Fxed Income Research at StatPro Ltd, and has held postons n fnance, academa and defence n the UK and Australa. Andrew holds a PhD n appled mathematcs from the Unversty of St Andrews. He s a Fellow of the Insttute of Mathematcs and ts Applcatons, and holds Chartered Mathematcan (C.Math) accredtaton. He s also Adjunct Professor n the School of Busness at the Unversty of Tasmana. Andrew may be contacted at andrew.coln@flametree.global. Kataln Kss s Global Solutons Manager - Performance at DST Global Solutons. Pror to jonng DST, Kate was UK Head of Performance at UBS Global Asset Management where she was responsble for performance reportng and analyss. Kate also led the strategc mplementaton of a new global performance tool across the organsaton. Pror to UBS Kate began her career as a performance analyst at Russell Mellon CAPS n 2002. Kataln may be contacted at kate.kss@dstgs.com.