EDHEC-Risk Days Europe 2014 London, March 26, 2014, 14:15-15:30. New Frontiers in Risk Parity

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1 EDHEC-Risk Days Europe 2014 London, March 26, 2014, 14:15-15:30 New Froniers in Risk Pariy Lionel Marellini Professor of Finance, EDHEC Business School Scienific Direcor, EDHEC-Risk Insiue Senior Scienific Advisor, ERI Scienific Bea This research has been carried ou as par of he LYXOR "Asse Allocaion Soluions" Research Chair 1

2 Ouline Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 2

3 Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 3

4 Scienific versus Naive Diversificaion The benefis of diversificaion are inuiively clear: a reducion of unrewarded risk leading o an increase in reward per uni of risk. Maximizing he Sharpe raio, however, is no a fully operaional objecive because of he presence of parameer uncerainy, especially for expeced reurn esimaes (Meron (1980)). Naive diversificaion is ofen a useful subsiue, or a leas complemen, o scienific diversificaion (weigh consrains lead o using a naively diversified porfolio as an anchor poin see Jagannahan and Ma (2003), DeMiguel e al. (2009)). Recen finding: here are wo forms of naive diversificaion. Equal dollar conribuion (equally-weighed porfolio); Equal risk conribuion (risk pariy porfolio). 4

5 Equally-Weighed (EW) versus Risk Pariy (RP) Porfolios A porfolio ha seems o be well-balanced in erms of dollar conribuions can be exremely concenraed in erms of risk conribuions because of differences in volailiy and pairwise correlaion levels amongs he consiuens. The risk pariy (RP) scheme, which uses he covariance marix as inpu, aims a equaing conribuions of consiuens o porfolio risk (Qian (2006), Maillard e al. (2010)). In his conex, risk pariy appears o be a more meaningful approach o naive diversificaion compared o he equallyweighed (EW) approach; a RP porfolio is max Sharpe Raio if he porfolio consiuens have idenical Sharpe raios and pairwise correlaions. 5

6 Formal Definiion of Risk Pariy Consider he following decomposiion of porfolio variance: N N N N 2 p wi wj ij wi wj ij i1 j1 i1 j1 Then define he conribuion o risk (variance or volailiy) as: N N var var 2 i i jij wih i p var j1 i1 i i N N 2 v ol wi vol p i jij wih i p p j1 i1 c w w w c w c w w c w Risk pariy porfolio is defined as: v ol c w c w v ol var ci w ci w 2 p p p 1 for all N i Key propery: RP porfolio ends o overweigh low volailiy asses w.r.. high volailiy ones. 6

7 Limis of Risk Pariy Porfolios The radiional approach o RP, which relies on hisorical vol and correlaion esimaes, suffers from 3 ypes of risks (a.k.a. S-risks) : Sample risk: esimaors depend on a paricular hisorical scenario; Saionariy risk: pas values of rue parameers curren values; Specificaion risk: no disincion beween upside & downside risk. In an asse allocaion conex, his approach, which has performed well over a sample ha conains he longes bull bond marke in hisory due o a subsanial over-weighing of bonds (less volaile han equiies), raises a general quesion and a opical concern: The quesion is ha leverage is ofen needed o generae a arge expeced reurn consisen wih invesors expecaions; The concern (relaed o S-risks) is he massive overweighing of bonds when bond yields are hisorically low, wih an increase in riskiness ha may no be accouned for by hisorical vol esimaes. 7

8 Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 8

9 From Uncondiional o Condiional Risk Pariy (CRP) The purpose of his research projec is o analyze how risk pariy porfolios can be made more efficien in responding o changes in marke condiions, as measured hrough key sae variables such as bond yields. Rolling-window (RW) volailiy esimaes, and even GARCHbased esimaes, heavily depend on hisorical bond reurns, even hough hey induce more ime-variaion compared o he use of long-erm esimaes. The quesion of how o generae an insananeous and observable measure of bond volailiy, so as o make i even more responsive o changes in marke condiions, can be addressed by relaing bond volailiy o bond duraion, and relaing bond duraion o bond yields. 9

10 Bond Volailiy and Bond Duraion Consider he model-free firs-order approximaion o he reurn on a coupon-paying bond P: P D q P where D = duraion and q = yield-o-mauriy, defined as: m jq mq P h e Fe, wih h coupon paymens, F face value j j j1 j m 1 jq mq D h je j Fe m P j1 j In a one facor model for he yield curve, his suggess an esimae for bond volailiy aken o be a funcion of duraion and yield volailiy: D (1) P q 10

11 Bond Duraion and Bond Yield Duraion of a coupon bond is a decreasing funcion of he yield: D 0 for las coupon dae (2) q For example, Campbell, Lo and MacKinlay (1997) show ha for a bond wih annual paymens, selling close o par and mauring in m years: D 1 e 1 e mq q (3) Taken ogeher, (1) and (2) sugges ha (a) an insananeous volailiy esimae can be consruced from he empirical duraion (*); and (b) his esimae will be decreasing in bond yields (**). (*) I can also be consruced from duraion aken o be a funcion of yield levels using (3). (**) Tha is unless ime-variaion in yield volailiy conflics wih his effec. 11

12 Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 12

13 Exending Risk Pariy o Downside Risk Measures Whaever he esimaion mehodology, volailiy does no disenangle downside risk from upside risk; in paricular, i does no reflec an increased downside risk for bonds when ineres raes are low and expeced o mean-rever back o higher levels. The naural approach o address his issue is o replace volailiy by a dissymmeric risk measure (semi-volailiy, VaR, ec.). This approach requires an expeced reurn esimae, and as such is relaed o he analysis in Roncalli (2013) of he following class of measures: R w w' g w' w N i1 w w wi i g ci R c i 13

14 Gaussian Semi-Volailiy and VaR Under he assumpion ha asse reurns are Gaussian, we obain explici expressions for he semi-volailiy and he VaR: GSV GVaR w 2 P 2 P P N P 1 w P N P P P Pn P wih p and p being he porfolio expeced reurn and volailiy. Boh measures are decreasing in expeced reurn: 14

15 Non-Gaussian VaR The Gaussian assumpion is hardly he bes approach o VaR esimaion; in wha follows, we consider a Cornish-Fisher approximaion o he VaR (Cornish and Fisher (1937)): NGVaR w q Sk Ku Sk wih: q N q 1 q 3q 2q 5q 2 P P Sk Ku E E w ' X, 1 3 P w ' X, 1 4 P P P Esimaing conribuions o risk requires esimaes for higher-order comomens (see Boud e al. (2008)); one may focus on he inverse CF VaR heurisic sraegy as a firs sep for simpliciy.(*) (*) One may also ake consan sk and kur parameers for simpliciy in a firs sep. 15

16 Dissymmeric Risk Measure and Reurn Predicabiliy While dissymmeric risk measures are meaningful for invesors, hey do depend on expeced reurns, which are hard o esimae. Key difference beween MSR porfolios and improved condiional risk pariy porfolios: hey boh rely on expeced reurn esimaes, bu CRP sraegies rea hese esimaes as (direcional) risk measures, implying a lower sensiiviy o esimaion risk (see laer). Expeced reurns can be relaed o observable variables: Equiy risk premium is ime-varying and depends on dividend-price raios (Fama and French (1988), Cochrane (2010)); Bond risk premium is ime-varying, and depends on bond yields, as well as forward raes (Fama and Bliss (1987), Campbell e al. (1997), Cochrane and Piazzesi (2005)) - Since bond expeced reurns and volailiy can be regarded as funcions of bond yields, one can consruc a CRP sraegy based on a dissymmeric risk measure aken as a (decreasing) funcion of bond yields. 16

17 Bond & Sock Reurns are Predicable Equiy risk premium is ime-varying and depends on dividend-price raios (Fama and French (1988), Cochrane (2010)). 17

18 Bond & Sock Reurns are Predicable Following Fama and French (1988), we use he dividend-price raio as a predicor ( ) for sock reurns, and obain a 9% adjused R-squared for 1Y horizon. We regress fuure bond index oal reurns on curren yields, and obain a posiive relaionship and an adjused R-squared ha increases wih ime-horizon, and already very high for 1Y (31.7%). 18

19 Dissymmeric Risk Measures Semi-volailiy Gaussian VaR 99% Non-Gaussian VaR 99% For each of he hree risk measures, he sock index appears o be more risky han he bond index, bu he relaive disance is smalles for he non-gaussian VaR a 99%. This is likely o imply a lower bond allocaion wih he non- Gaussian VaR. 19

20 Risk Pariy Sraegies (Jan Dec. 2012) The condiional risk pariy (CRP) sraegy based on Non- Gaussian VaR esimaes differs subsanially from he sandard uncondiional risk pariy (URP) sraegy based on rolling window volailiy esimaes; in paricular, i has a subsanially lower bond allocaion a he las rebalancing (end of December 2012): 53.3% versus 79.8%. 20

21 Benefis of CRP Sraegies The CRP approach based on CF VaR recognizes ha bonds are more risky when yields are low, which implies ha hey avoid massive over-weighing of bonds in such circumsances Hisorical risk esimaes, on he oher hand, are backward-looking and reac much more slowly o changes in marke condiions. In wha follows we simulae 2,000 possible scenarios for bond yields, saring wih he curren value, assuming mean-reversion owards he long-erm mean value for he yield, and also assuming ha volailiy is an inverse funcion of yield. 21

22 Characerisics of Simulaed Pahs We consider wo economic scenarios wih more or less fas mean reversion in he yield and differen impacs on equiy marke. In he firs (resp. second) scenario i akes 5 (resp. 2) years o rever back o a level of 6%, wih no impac on equiy markes (resp. wih a sharp following drop in equiy expeced reurns). (*) Ploed values prior Dec (verical line) are eiher observed or esimaed from hisorical daa. Ploed values afer Dec are averages across 2,000 simulaed pahs. 22

23 Simulaed Performances of Sraegies in Scenario 1 Sraegies are rebalanced every quarer as of Dec. 31, The lef able shows he average yield and he average excess reurn of each sraegy (annualized), and he righ able shows he annualized volailiies of he yield change and he reurns on he sraegies. Each analyics is compued along each simulaed pahs, and he numbers repored in he able are averages across he 2,000 pahs. Bond reurns are srongly negaive in he firs year, and end o improve as he yield sabilizes around he long-erm value. The URP sraegy underperforms he CRP sraegies on average over he 2,000 scenarios. CRP-NGVAR99 ouperforms CRP-VOL every year. 23

24 Simulaed Performances of Sraegies in Scenario 2 Sraegies are rebalanced every quarer as of Dec. 31, The lef able shows he average yield and he average excess reurn of each sraegy (annualized), and he righ able shows he annualized volailiies of he yield change and he reurns on he sraegies. Each analyics is compued along each simulaed pahs, and he numbers repored in he able are averages across he 2,000 pahs. The bond index experiences severe losses in 2013 due o he rapid increase in ineres raes, bu i recovers posiive reurns more quickly han in he previous scenario. As in Scenario 1, CRP sraegies sill ouperform URP in , bu hey now underperform i in 2015 because hey are penalized by heir higher sock allocaion in his year. 24

25 Inroducing Commodiies wih Condiional Risk Pariy Sraegies As in he wo-asse case, CRP-NGVAR99 implies he lowes bond allocaion a he las rebalancing dae in he sample (end of December 2012). 25

26 Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 26

27 MSR Porfolio as CRP Porfolio The (long-only) MSR porfolio is defined as: w arg max w ' μ MSR w '11 w 'Σ w 0 w I is a risk pariy porfolio, in he sense of volailiy, if all correlaions (always verified if he asses are wo) and Sharpe raios are equal (Maillard e al. (2010)), in which case: 1 MSR wi α σi In he absence of prior informaion on he Sharpe raios of he sock and he bond indices, a reasonable agnosic assumpion is ha hey have he same long-erm values; on he oher hand, he wo Sharpe raios may be differen a any given poin in ime depending on curren marke condiions proxied by bond yields and dividend yields. 27

28 Weighs of MSR Sraegies Jan Dec On Dec. 31, 2012, he MSR porfolio has a higher bond allocaion (68.1%) compared o CRP porfolios wih non-gaussian measures. The MSR sraegy has a large urnover, which suggess ha he use of expeced reurn esimaes involves a sronger concern over lack of robusness (corner soluions) compared o he CRP approach. 28

29 Benefis and Limis of Risk Pariy Porfolio Sraegies From Uncondiional o Condiional Risk Pariy Sraegies Exending Risk Pariy o Downside Risk Measures Risk Budgeing Sraegies wih Time-Varying Risk Premia Conclusions and Exensions 29

30 Conclusions & Exensions When hisorical volailiy is used as a risk measure, RP porfolios are largely dominaed by bonds, regardless of marke condiions. The use of a risk pariy sraegy based on condiional dissymmeric risk measures: 1. Allows o alleviae he pure reliance on he sample; 2. Makes he RP porfolio more sensiive o ineres rae changes; 3. Leads o lower bond allocaion in a low-rae environmen. This mehodology uses condiional expeced reurn esimaes in a more robus manner compared o mean-variance opimizaion; i can be used as a reasonable, neural, saring poin, and acive views on expeced reurn esimaes can be inroduced o generae furher ouperformance. 30