RISK-ADJUSTED PERFORMANCE ATTRIBUTION METHODOLOGIES. GiovanniZambruno. UniversityofMilano-Bicocca OUTLINE: ² performancemeasuresofmanagedfunds

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1 RISK-ADJUSTED PERFORMANCE ATTRIBUTION METHODOLOGIES GiovanniZambruno UniversityofMilano-Bicocca OUTLINE: ² performancemeasuresofmanagedfunds ² howtoadjustforrisk ² theaxiomaticsofperformanceattribution ² multi-periodperformanceattribution ² xed-incomeperformanceattribution

2 (RATEOF)RETURN ² ex{ante ² ex{post V(0) =) V(T) R[0;T] = V(T) V(0) V(0) This istheperiodicrateofreturn. Howto annualize? { simplerule { compoundrule r= 1 T R[0;T] r=(1+r[0;t]) 1=T 1

3 HOW TO MANAGE INTERMEDIATE CASH-FLOWS ² time{weightedrateofreturn R[T i 1 ;T i ] = 1+R[0;T] = Y i V(T i ) V(T i 1 )+F(T i ) 1 (1+R[T i 1 ;T i ]) (enhances the management skills) / consistent withcompoundrule! ² money{weightedrateofreturn W(T) = V(0)+ X T T i F(T i ) T R[0;T] = V(T) W(T) 1 rough approximation/ consistent with simplerule

4 WHATIS"RISK"??? when (and the extent to which) the outcome of an action di ersnegatively from what expected (hoped, forecasted,...) [Ref.: Elton,2000] Examples (internal measures): ² standarddeviation: ¾= ³ X pi (r i E(r)) 2 1=2 ² variance ¾ 2 = X p i (r i E(r)) 2 ² semivariance s= X p i (r i E(r)) +

5 Examples (external measures, wrt a target or benchmark) ² downsidedeviation 1 X DD= (bt r t ) + 1=2 T ² trackingerrorvolatility TEV = 1 T X (bt r t ) 2 ² shortfallprobability SFP = 1 T h #(bt r t ) +i

6 RISK{ADJUSTED PERFORMANCE MEASURES ² Sharpeindex SI= ¹ P r f ¾ P ² Modigliani RAPM=r f + ³ ¹ P r f "¾ B ¾ P # ² Sortino Sort= ¹ P ¹ B DD ² InformationRatio IR= ¹ TE TEV

7 ² Jensen'salpha =¹ P h r f + P ³ rm r f i ² Treynor TI= ¹ P r f P

8 what is Performance Attribution? "... themethodology whereby theexcessreturnis decomposedinto components, each referring to a particular aspect ofthestrategicprocess ofportfolio management." BENCHMARK + * ASSETALLOCATION + * MARKETTIMING + * STOCKSELECTION (top-down vs. bottom-up)

9 ArithmeticExcessReturn: AER = R B= X w i R i X v i B i = = X (w i v i )R i {z } Stock Selection + X v i (R i B i ) {z } Market Timing GeometricExcessReturn GER = 1+R 1+B = Y (1+R i ) w i : Y (1+B i ) v i = Y (1+R i ) w i v i {z } Stock Selection Y Ã 1+R i 1+B i! vi {z } Market Timing

10 MULTIPERIODCOMPOUNDING Geometric compounding is straightforward: GER 1jT = 1+R 1jT 1+B 1jT = Y t 1+R t 1+B t Arithmeticcompounding: AER 1jT = R 1jT B 1jT 6= X t (R t B t ) Equality can be recovered only if the one-period returnsareevaluatedonthebasisoftheinitialendowment: R t = V t V t 1 V 0

11 MAINSOURCESOFRISK (=POTENTIALEXCESSRETURN) ² Marketrisk(changesintheTermStructure) ² Defaultrisk ² Currencyrisk

12 ex post rateof return= V 1 V 0 V 0 (time-weightedvs. money-weighted) (tel-quel quotes) Ifmarket conditionsarestable, the value will follow a pattern such as to yield the same rate of return as established at time 0. Hence, part of the price variationis expected. Weareconcernedwiththe unexpected part(ifany), theonlyoneforwhichactivemanagementdeservesa reward. Usuallyattributedtoduration. Quoting vanbreukelen: LR=C+D ( y)

13 ReasonswhydurationisNOTacorrectmetric: ² theinitialtsis at=)allytmareequaltothe commonvalueofthespot rate. ² parallelshifts=)allytmremainequal. Only under these assumptions duration is a correct measureofsensitivitytochangesin"marketrate(s)". Neitheroftheseissatis edunderactualsituations: ² theinitialtsisusuallyupward-slopingandconcave; ² themostwidelyrecognizedmovementsare { shifts { twists { butter ies(humps)

14 "Isolate"thee ectofeverysinglecomponent ofthe TS. Hypotheses: 1. thetermstructureofinterestratesistheonlyrisk factorofthemarket; 2. all bonds are correctly priced according to the termstructure P 0 = 2 3 nx c 4 h Q ³ 5 hj=1 h=1 1+rj 3. the rational expectations theory holds (namely, one-period forward rates areexpected to become thespotratesnextperiod,andsoon).

15 P 1 = 2 3 º 1 X c 4 h Q ³ 5+ hj=2 h=2 1+rj {z } independent fromrº nx c 4 h Q ³ 5 (1+r º ) h=º hj=2;6=º 1+rj {z } independent from r º Takingthederivativewrtaspeci 1 = º (1+r º ) nx h=º 2 3 c 4 h Q ³ 5 hj=2 1+rj Aggregating thechangesinallthetsrates: P 1 ' = nx º=1 nx 1 r º º 8 < : 1 (1+r º ) nx h=º 2 39 c = 4 h Q ³ 5 hj=2 1+rj ; r º

16 Thisformulacanaccommodatevirtallyeveryreshape ofthets:forinstance ² shift: r º = ² twist: r º = (º º ) ² butter y: r º = + (º º ) 2 ² etc. These "rules" canbe tted into thegeneralformula toexhibitmorecompactexpressionsofthetotalprice variation.

17 Mainreferences: G. P.Brinson, L.R. Hood, G. L. Beebower, (1986), "Determinants ofportfolio Performance", Financial Analysts Journal,July-August; E. M. Ankrim, (1990), "Risk-Adjusted Performance Attribution",RussellResearchCommentaries,December G.vanBreukelen,(2000),"Fixed-IncomeAttribution", Journal of Performance Measurement