Portfolio Risk Decomposition (and Risk Budgeting)

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1 ortfolo Rsk Decomposton (and Rsk Budgetng) Jason MacQueen R-Squared Rsk Management

2 Introducton to Rsk Decomposton Actve managers take rsk n the expectaton of achevng outperformance of ther benchmark Mandates often stpulate the amount of Trackng Error (rsk relatve to the benchmark) allowed However, two other matters are at least as mportant - Havng the rght amounts of rsk (Rsk decomposton by holdngs) - Havng the rght knds of rsk (Rsk decomposton by factors) R-Squared 1

3 Rsk Decomposton by Factors The type of rsk beng taken should correspond to the way n whch the manager expects to outperform - A stock pcker should not be takng currency rsk - Market-tmers should be managng ther market beta - Sector rotators should be managng ther sector factor exposures (not ust ther holdng szes) Rsk Decomposton by factors clarfes what knds of rsk there are, and how bg the factor bets are relatve to each other, and to stock specfc rsk R-Squared 2

4 Rsk Attrbuton by Factors There s an mportant dstncton to be made between Rsk Decomposton usng the factors n a gven rsk model, and Rsk Attrbuton Rsk Attrbuton s used to determnng the exposure of a portfolo to factors that are not n the rsk model For example, many systems gve the Beta of the portfolo relatve to ts benchmark, whch s rarely a factor n the rsk model Statstcal rsk models use Rsk Attrbuton to relate the portfolo rsk to macro-economc and other factors R-Squared 3

5 Rsk Decomposton by Holdngs Ths talk focuses on Rsk Decomposton by holdngs We examne two cases :- - Decomposton of Absolute Rsk - Decomposton of Relatve Rsk or Trackng Error In each case, we want to answer two questons: - How much rsk s comng from each holdng? - How would the portfolo rsk be affected by a small change n each holdng (the margnal contrbuton to rsk)? R-Squared 4

6 A Sneak eek Ahead Ths knd of Rsk Decomposton forms the bass for Rsk Budgetng decsons We wll be lookng at the Actual, ercentage and Margnal Contrbutons to ortfolo Rsk, both from ndvdual holdngs and from groups of holdngs In the case of Absolute Rsk Decomposton, there s a unque answer to these questons Unfortunately for Rsk Budgetng, there sn t a unque answer n the case of Trackng Error Decomposton R-Squared 5

7 The Smplest ossble Example Before we get nto the fancy algebra, let s consder a very smple example Our benchmark s that old penson fund favourte : - 60% stocks, 40% bonds The manager s nervous about stocks and neutral on bonds, so our portfolo conssts of : - 50% stocks, 40% bonds, 10% cash The Rsk Decomposton should be smple R-Squared 6

8 ortfolo Holdngs data ortfolo Benchmark Dfference x() b() d() Totals = 100% 100% 0% Stocks = 50% 60% -10% Bonds = 40% 40% 0% Cash = 10% 0% 10% R-Squared 7

9 Absolute Volatltes & Correlatons Absolute Volatlty & Correlaton Matrx Stocks Bonds Cash Std. dev R-Squared 8

10 The Absolute Covarance Matrx! Absolute Covarance Matrx Stocks Bonds Cash R-Squared 9

11 The Algebra of Rsk Decomposton We begn by breakng down the total varance of a portfolo nto contrbutons from ndvdual holdngs We have V = ΣΣ x x C From whch we derve ndvdual contrbutons to varance as ACV = Σ x x C R-Squared 10

12 Actual Rsk Contrbuton Smplfed ACV Σ x = x C = x Σ x C ( ) = x Σ x cov R, R = x cov R, Σ x R ( R, R ) = x cov = x C where C p s the covarance of asset wth the portfolo. R-Squared 11

13 ercentage Rsk Contrbuton The converson of Actual Contrbutons to Rsk to ercentage Contrbutons s amazngly smple: CV ACV % = 100 % V We smply dvde the Actual Contrbuton by the Total Rsk and multply by 100 R-Squared 12

14 The tty Grtty Calculaton Detals Cov(,B) = Covarance(asset, portfolo) Stocks Bonds Cash Stocks = Bonds = Cash = Cov(,) = Corr(,) = R-Squared 13

15 Absolute Rsk Decomposton Absolute ortfolo Varance V(p) = SD(p) = 9.26 Stocks = Bonds = Cash = ACV() = CV() = 77% 23% 0% R-Squared 14

16 Contrbutons from Groups of Holdngs We can generalse these expressons from ndvdual holdngs to groups of holdngs as follows :- ACV = Σ ACV Energy Energy CV Energy CV % = Σ % Energy R-Squared 15

17 Margnal Contrbuton to Varance For the total portfolo rsk we have :- V = The Margnal Contrbuton to Varance s defned as :- whch really couldn t be smpler! Σ Σ x x C V MCV = = 2 Σ x C = 2C x R-Squared 16

18 Margnal Contrbuton to Rsk (S.D.) Bearng n mnd that : 2 V p = S p we have : V = S 2S And so, trvally, MCR = S x = S V V x = V x V S = MCV 2S = 2C 2S = C S R-Squared 17

19 Margnal Contrbutons to Rsk Margnal Contrbutons to ortfolo Varance Stocks Bonds Cash MCV() = MCV() = MCV() = Margnal Contrbutons to ortfolo Rsk Stocks Bonds Cash MCR() = MCR() = MCR() = R-Squared 18

20 Summary of Absolute Decomposton Absolute Rsk Decomposton by Holdngs - ortfolo Formulae = x() * C(,) 100*ACV() / V(p) 2 * C(,) / 100 (C(,)/100) / (S(p) C(,) / (S()*S(p)) CV() / x() C(,) / V(p) Holdng x() S() C(,) = ACV() = CV() = MCV() = MCR() = Corr(,) = Beta(,) Stocks 50% % Bonds 40% % Cash 10% % ortfolo 100% % ortfolo varance = V(p) = ortfolo rsk (s.d.) = S(p) = 9.26 R-Squared 19

21 Trackng Error Decomposton Trackng error s the varance of relatve returns Relatve returns are defned as follows :- Rˆ = R R B where Rˆ s the relatve return on the portfolo, R R B s the absolute return on the portfolo, and s the absolute return on the benchmark R-Squared 20

22 R-Squared 21 Here s the Important Bt!! ortfolo and benchmark returns are defned as: So relatve returns can be defned as :- = = = p R x R Σ = B R R Σ b = Rˆ ( ) R x b Σ ( ) B R R x Σ ( )( ) B R R b x Σ

23 R-Squared 22 Three Defntons of Trackng Error Correspondng to each of these formulatons, we get three dfferent expressons for Trackng Varance :- TV ( )( ) C b x b x = ΣΣ Ĉ x x = Σ Σ ( )( ) Ĉ b x b x = Σ Σ

24 Three Versons of Trackng Error These three sets of equatons for relatve return and rsk (T.E.) may be charactersed as follows: - Frst uses relatve holdngs and absolute returns - Second uses absolute holdngs and relatve returns - Thrd uses relatve holdngs and relatve returns It s very easy to demonstrate that these expressons are equvalent at the aggregate level However, they lead to dfferent decompostons R-Squared 23

25 Relatve Volatltes & Correlatons Relatve Volatlty & Correlaton Matrx Stocks Bonds Cash Std. dev (1.00) (0.67) 8.34 (1.00) (0.67) R-Squared 24

26 The Relatve Covarance Matrx! Relatve Covarance Matrx Stocks Bonds Cash (46.32) (39.92) (46.32) (39.92) R-Squared 25

27 Trackng Error Decomposton - 1 Relatve weghts & Absolute covarances TV(p) = 2.25 TE(p) = 1.50 Stocks = Bonds = Cash = ACV() = CV() = 100.0% 0.0% 0.0% R-Squared 26

28 Trackng Error Decomposton - 2 Absolute weghts & Relatve covarances TV(p) = 2.25 TE(p) = 1.50 Stocks = 7.72 (9.26) (2.00) Bonds = (9.26) Cash = (2.00) ACV() = (3.54) CV() = % 188.8% 68.5% R-Squared 27

29 Trackng Error Decomposton - 3 Relatve weghts & Relatve covarances TV(p) = 2.25 TE(p) = 1.50 Stocks = Bonds = Cash = ACV() = CV() = 31.5% 0.0% 68.5% R-Squared 28

30 Comments on TE Decompostons The frst, usng Absolute Covarances, treats cash as rskless, and so attrbutes all the rsk to the stock bet The second uses Absolute Holdngs, and attrbutes most of the rsk to the neutral poston n bonds!! Ths second verson wll also say that a zero holdng n the portfolo has no contrbuton to Trackng Error, even f the asset s held n the benchmark The thrd verson s (usually) the most ntutve, and n ths case gves a very sensble answer R-Squared 29

31 Margnal Contrbutons to TE - 1 Changng an Absolute Holdng by a small amount s the same as changng a Relatve Holdng by a small amount, snce the Benchmark Holdng s fxed Thus, we only get two dfferent sets of results for the Margnal contrbutons to Trackng Error The frst verson s gven by :- MCVRA TV TV = = = 2Σ x d d C = 2 ( C C ) B R-Squared 30

32 Margnal Contrbutons to TE 2 & 3 The second and thrd versons are gven by :- MCVAR TV = = 2Σ x x Ĉ = 2Ĉ MCVRR = TV d = 2 Σ d Cˆ = 2Cˆ whch are the same. R-Squared 31

33 Margnal Contrbutons to TE - 1 Margnal Contrbutons to Trackng Error usng Absolute Covarance matrx Stocks Bonds Cash MCR() = (0.150) MCR() = (0.032) MCR() = R-Squared 32

34 Margnal Contrbutons to TE 2 & 3 Margnal Contrbutons to Trackng Error usng Relatve Covarance matrx Stocks Bonds Cash MCR() = (0.047) MCR() = MCR() = R-Squared 33

35 Comment on Margnal Contrbutons At frst sght these results seem contradctory - The frst verson says f we ncrease the bond holdng, the TE wll decrease - The second and thrd versons say f we ncrease the bond holdng, the TE wll ncrease However, note that we stll have the budget constrant, so we stll get the same net result In the frst case : (-0.150) = In the other cases : (-0.047) = R-Squared 34

36 Conclusons Most nsttutonal nvestors are concerned wth Trackng Error rather than Absolute Rsk Rsk Decomposton s at least as mportant as the overall level of rsk n an actvely-managed portfolo However, there are dfferent answers! The ndustry default s the frst decomposton, but ths s almost certanly not the best, or most ntutve Caveat Manager!! R-Squared 35

37 Contact Informaton R-Squared Rsk Management Lmted The exus Buldng, Broadway, Letchworth Garden Cty, Hertfordshre, SG6 3TA, Unted Kngdom Lakeland Street, Grosse onte, MI 48230, U. S. A Emal: R-Squared Rsk Management

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