risk factors, copula dependence and risk sensitivity of a

Risk Factors, Copula Dependence and RiskSensitivity of a Large Portfolio

Catherine Bruneau∗, Alexis Flageollet†, Zhun Peng‡

June 2014

Abstract

In this paper we propose a flexible tool to estimate the risk exposure of a largedimensional portfolio composed of different classes of assets, especially in ex-treme risk circumstances. In such cases, the usual beta approach is no longerrelevant due to the complex- including taildependencies that usual correlationsare not able to account for. Thus we use the copulas’ theory but we have alsoto define a tractable and readable dependence structure. So we combine an exante interpretable factorial structure and a CVine copula model in the spirit of theCVine Market Sectors (CVMS) model introduced by Heinen and Valdesogo (2009)to build a CVine Risk Factors (CVRF) model. Our tool allows us to decomposethe risk of any asset and any portfolio into specific risk directions, like inflation,credit, emerging risks, and so on, depending on the context. Our approach is semi-parametric and we quantify the exact contribution of the different possible risksources to the risk premia of assets and to any usual risk measure applied to port-folios. In particular we refer to the CVaR measure, which is relevant in criticalcontexts. Illustrations are given which mainly refer to the credit risk but also to theemerging risk which have hit many investors in recent years. Obvious applicationsconcern regulation issues when the portfolio under study is the one of a financialinstitution.

Keywords: Complex dependence, Regular vine copula, Factors, Portfolio Management,Risk Management, Risk parity, Extreme Risks, Stress testing, Regulation.

JEL Classification: G11, G17, G32

∗University Paris I Panthéon-Sorbonne and Centre d’Economie de la Sorbone, 106-113 Bd de l’Hôpital75013 Paris, France.†[email protected], The views expressed are those of the author and do not reflect the

position of Natixis Asset Management.‡University of Evry and EPEE, Bâtiment IDF, Bd François Mitterrand 91025 Évry, France.

[email protected]

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1 IntroductionDiversification opportunities across asset classes can be limited, depending on the

market configuration (Clarke et al., 2005, Bender et al., 2010). As a consequence,the portfolios become more and more complex, with an increasing number of assetclasses. The portfolio manager is therefore facing multiple risk sources: without beingexhaustive, equity, interest rates, inflation, business cycle, emerging market, credit,or liquidity related risks. Globally speaking, the required risk premium for any assetdepends on its covariation with a stochastic discount factor (Cochrane, 2005, 2011)or more intuitively with “bad times” (Ilmanen, 2011). However we are not able toempirically identify one true factor representative of “bad times” 1. A multiple riskfactor model is thus used to deal with that issue (Ross, 1976). In what follows, weadopt this approach and refer to distinct factors to account for different dimensions ofbad times like Ilmanen, (2011), for example. . 2

Moreover, these different risk directions affect the assets in a complex way. Forinstance, inflation risk can prompt a positive correlation between stock and nomi-nal bond returns during high unexpected inflation periods (via positive risk premium)and the opposite during low unexpected inflation periods where nominal bonds areused to hedge equity risk (Campbell, Sunderam and Viceira, 2013). To deal with thiscomplexity, financial economists often consider time-varying risk discount rates (seeCochrane, 2011) which are for example driven by time-varying risk aversion (Cambpelland Cochrane, 1999) or by the business cycle (Fama and French, 1989). To our opin-ion, one particularly important issue is thus to account for asset dependencies whichincrease during financial turbulence periods and reduce diversification benefits at timeswhen these benefits are most needed.

Indeed, beyond diversification issues, being able to assess multiple interacting fi-nancial risk exposures is nowadays crucial because a poor risk management in a finan-cial institution can lead to an individual or potentially systemic default as witnessedduring the last crisis. Improving the risk measure of a portfolio is henceforth at thecore of regulation issues. In this regard, the question we mainly address in this paperis the following: what is the sensitivity of a large and complex multi-asset portfolio toextreme shocks related to multiple interacting risk sources? We propose a flexible toolto help answer such a question.

Usual risk measures such as correlations and variance are obviously not relevantfor that purpose. Indeed the Gaussian framework is not adapted to characterize therisk of a complex portfolio, in particular in extreme situations. We therefore refer tothe copula’s theory. However, the copulas at hand are thus high dimensional, whichis clearly intractable from a practical point of view. Fortunately, any joint densityfunction can be factorized as a product of bivariate copulas and marginal densities.The decomposition is not unique, but it is possible to characterize the dependencesfrom a regular vine, which corresponds to a particular factorization, as proposed by

1. Bad times refer for example to negative growth, high inflation, deflation, high volatility or correlationsbetween assets, illiquidity spiral, debt crises, etc.

2. Systematic factor risks must be rewarded because investors care about them. Indeed, investors whosupport factor risks expect a risk premium as they supply insurance to other ones against enduring losses inbad times.

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Bedford and Cooke (2001, 2002).In what follows, we will use a special case of a Regular vine, namely the Canon-

ical vine (C-Vine) studied by Brechman and Czado (2013), which is particularly welladapted for including factorial structures as the ones that are often used to decomposethe risk premia of financial assets.

Some authors address the statistical issue of finding the factorial structure that of-fers the best fit with the data. For example, Tumminello et al. (2007) apply a hierarchi-cal clustering procedure and estimate a hierarchically nested factor model. Brechmannand Czado (2013), who model market returns with a R-vine copula structure, use amaximum spanning tree and adopt a pure statistical approach to discover the link be-tween the assets. However the resulting relationships between financial series are oftennot easy to interpret.

Contrary to these authors, we do not aim at finding the best (statistical) factorialstructure, but we rather aim at proposing a tractable factorial dependence structurewhich combines a C-Vine factorization with asset’s return decompositions that are con-sistent from a financial point of view. Such models refer to a priori views but we checkthat the associated decompositions are well supported by the data inside the C-Vineframework. More precisely, the tool we propose allows for specifying and estimatingwhat we call a general C-Vine-Risk Factors (CVRF) dependence structure which is anextension of the Canonical Vine Market Sector (CVMS) specification introduced byHeinen and Valdesogo (2009). The C-Vine structure is thus organized and constrainedaccording to a factorial structure which we specify a priori.

Factor models state that the return of each asset results from a limited number ofrisk sources. Multi-factor models can include any type of factor since the theory giveslimited guidance (see Ilmanen, 2011). In the remaining of the paper, we focus on amulti-asset portfolio based on 35 different indexes (stocks, government or corporatebonds for various geographic area, currencies and commodities) to encompass a largevariety of risk sources. Of course, the ex ante factorial structure we retain in the follow-ing depends on the assets at hand but most of the risk factors we thus focus on cannotbe ignored in the current context. More precisely we aim at identifying 8 risk factors.Accordingly, we choose 8 indexes which can be viewed as common components formost of our 35 assets and are in the same time mainly driven by the different risk fac-tors we want to identify: according to the identification scheme we retain as shown infigure 2, we retain 3 global indexes that are mostly related to three risk factors, denotedin the following as real interest rates, inflation, and market risk factors and 5 additionalindexes which are specifically affected by (European) sovereign crisis, credit, USD,emerging and commodities related risks. The 5 latter indexes are used to emphasisthe possibility to deal with “custom risks” which are more investor’s portfolio specific.Finally, as risk factors are reputed to be regime dependent (Page and Taborsky, 2011),we focus on a recent period (2001-2013 i.e. the two last financial cycles) in order toavoid strong regime shifts.

For sake of illustration we can report to Figure 2 which summarizes the identi-fication scheme we adopt for the risk factors which are captured from representativeindexes. Note that we choose well diversified indexes because we assume that diver-sification magnifies embedded risk premiums by diminishing the noise (idiosyncraticrisk).

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The ordering described in the figure permits us to extract the risk factors we wantto capture and to organize the C-Vine copula in the same time. For example, we claimthat the two first indexes (World Inflation Linked Index and World Government BondsIndex (WGBI All maturities) which are mainly affected by interest rate and inflationrisks contribute to the risk premium of any asset of our data base. Moreover,we usethese two indexes to extract two risk factors, namely the real rate risk factor and theinflation risk factor.

More precisely, our CVRF model allows us to decompose the response of the re-turn of any asset to extreme shocks to any other asset. For each shock, the impact iscaptured through the changes in the distribution of the return after the shock, takinginto account not only its marginal distribution but also its co-movements with the otherassets through the factorial dependence structure. In this regard, all impact measuresare obtained from semi-parametric estimations.

More precisely, we successively focus on extreme shocks to the 8 indexes retainedas the main common components of the returns of our assets, and, each time, we de-compose the response of any asset and more specifically the change in its expectedreturn induced by the shock, into the contributions of the risk factors we want to cap-ture. Accordingly we are able to quantify the sensitivity of any asset to any extremeshock and to jointly decompose this sensitivity into the marginal contributions of therisk factors.

This decomposition requires simulations of the returns of all assets after drawingextreme values of non conditional and conditional distribution functions in the CVRFmodel framework. For the latter case we develop an original algorithm.

This is our CVRF‘s core application from which we propose risk sensitivity anal-ysis for different benchmark portfolios. First, by referring to risk budgeting strategies,we examine three cases: an equal risk contributions (ERC) portfolio, a high risk and alow risk budgeting portfolio. Finally, we consider the most standard capital budgetingportfolio used in the USA (60% risky assets/40% bonds).

The paper is organized as follows. In section 2, we present the principles of C-vinecopulas and our C-vine risk factor model. Section 3 is devoted to the practical imple-mentation with a presentation of the data, of the factorial structure and the principlesof the different simulations that are used later. Section 4 is devoted to the risk sensi-tivity analysis developed for the returns of the assets of our database and for differentbenchmark portfolios. Section 5 concludes.

2 Dependence structure: Copulas, Canonical Vine andFactors

In this section we recall the definition of copulas, the principles of a CVine depen-dence structure and, finally, we show how to include a factorial structure to get ourCVRF model.

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2.1 Canonical vineA n-dimensional Copula C(u1, ...un) is a cumulative distribution function (cdf)

with uniformly distributed marginals U(0, 1) on [0,1].A copula is useful to characterize the dependence structure of several random vari-

ables whatever their marginal distribution. Indeed, according to the Sklar’s theorem(Sklar, 1959) a multivariate cdf F of n random variables X = (X1, ..., Xn) withmarginals F1(x1),...,Fn(xn) can be written as:

F (x1, ..., xn) = C(F1(x1), ..., Fn(xn)), (1)

where C(F1(x1), ..., Fn(xn)) = F (F−11 (u1), ..., F−1n (un)) is some appropriate n-dimensional copula and F−1i s denoting the quantile functions of the marginals. Ac-cordingly, modeling of margins and dependence can be separated by the way of cop-ulas. Moreover, for an absolutely continuous F with strictly increasing, continuousmarginal cdf Fi, we get the joint density function f by differentiating (1),

f(x1, ..., xn) = c1:n(F1(x1), ..., Fn(xn)) · f1(x1) · · · fn(xn), (2)

which is the product of the n-dimensional copula density c1:n(·) and the marginal den-sities fi(·).

Then, the n− dimensional density c1:n can be decomposed as a product of bivariatecopulas. The decomposition is not unique (See a possible decomposition in the trivari-ate case in Appendix). To help organize the possible factorizations of the joint density,Bedford and Cooke (2001,2002) have introduced a graphical model denoted the reg-ular vine. Regular vines (R-vines) are a convenient graphical model to hierarchicallystructure pair copula constructions. A special case of regular vines is the canonicalvine where certain variables play a leading role.

More precisely, according to Kurowicka and Cooke (2006) a regular vine (R-vine)on n variables consists first of a sequence of linked trees T1, ..., Tn−1 with nodes Niand edges Ei for i = 1,..., n, where T1 has nodes N1 = 1, ..., n and edges E1, and for i= 2,..., n-1, Ti has nodes Ni = Ei−1 . Moreover, two edges in tree Ti are joined in treeTi+1 only if they share a common node in tree Ti (See Brechman and Czado (2013)for a detailed presentation). A special case of R-vines which is often considered arecanonical vines (C-vines). A C-vine is a R-vine if each tree Ti has a unique node withdegree d− i, the root node.

The general n-dimensional canonical vine (CVine) copula density can be written asfollowing:

c1:n(F1(x1), ..., Fn(xn)) =

n−1∏j=1

n−j∏i=1

cj,j+i|1,...,j−1 (3)

(F (xj |x1, ..., xj−1), F (xj+i|x1, ..., xj−1))

where cj,j+i|1,...,j−1 denotes the bivariate copula between the distributions of xj andxj+i taken conditionally on x1, ..., xj−1.

Figure 1 shows a canonical vine with five variables. From the figure, we observethat the variable 1 at the root node is a key variable that plays a leading role in governinginteractions in the data set.

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Figure 1: A five dimensional canonical vine tree

In the first tree, all nodes are associated with theX1, ..., X5 variables. For example,the edge 12 corresponds to the copula c(F1(x1), F2(x2). In the second tree, the edge23|1 denotes the copula c(F2|1(x2|x1), F3|1(x3|x1)). The following trees are builtaccording to the same rules.

In order to organize the dependence structure, it is useful to recall how to charac-terize the independence of two variables in terms of copula.

2.2 Conditional independence in canonical vineFor a complete n-dimensional canonical vine, there are n(n− 1)/2 bivariate copu-

las. This means that the numbers of parameters to estimate is very high for a large sizeportfolio. In order to simplify the structure, some conditional independence assump-tions may be useful.

If one refers to the three dimensional case (See Appendix), assuming that X1 playsa leading role leads to the following factorization:

c23|1(F2|1(x2|x1), F3|1(x3|x1)) = 1

which means that x2 and x3 are independent, conditionally on x1. Hence, the structuresimplifies to:

c(F1(x1), F2(x2), F3(x3)) = c12(F1(x1), F2(x2)) · c13(F1(x1), F3(x3)).

Generally speaking, for a set of conditioning variables, υ and two variables X , Y ,assuming that X and Y are conditionally independent given υ, gives:

cxy|υ(Fx|υ(x|υ), Fy|υ(y|υ)) = 1. (4)

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Table 1: Factorial dependence matrix Ms

CC1 CC2 CC3 CC4 a1 a2 a3 a4f1 1f2 1 1f3 1 0 1f4 1 0 1 1a1 1 1 0 0 1a2 1 0 1 0 0 1a3 1 0 1 1 0 0 1a4 1 0 1 1 0 1 0 1

Heinen and Valdesogo (2009) use this property to develop a simplified version ofcanonical vine, namely the Canonical Vine Market Sector (CVMS) model. This two-factor model assume that each asset depends on the market and on its own sector. Toinclude this model into a canonical vine structure with the market and the sectors asroot nodes, some conditional independence assumptions need to be introduced: condi-tionally on the market, sectoral returns are assumed to be independent and asset returnsare independent once they belong to different sectors. The remaining dependence ofasset returns taken conditionally on the market and the respective sectors is modeledwith a multivariate Gaussian copula. The example given in Appendix illustrate thissimplified structure.

Our CVRF model is an extension of the CVMS model.

2.3 CVine-Risk-Factors model(CVRF)Referring to Heinen and Valdesogo (2009), we introduce a C-vine copula based

factor model. Thus we assume that asset returns depend on several risk factors whichmainly explain their dependence structure. Further, we loosen the usual conditionalindependence assumptions and assume that the risk factors can depend on each otherwhile asset returns can depend on one or several risk factors at the same time. Thespecification of the factorial dependence structure is therefore more flexible than in theCVMS setting and can be used in accordance with any particular view of a portfoliomanager.

As shown for example in table 1, the unconditional and conditional dependencestructure can be specified in a symmetric matrix with dummy variables. Among then = 8 assets in the table, we distinguish between CC-type assets which denotes in-dexes that are common components - or "factors" in the usual sense - for all assets anda-type ones which refer to the other asset of the database. The random variables arethe corresponding returns, ri; i = 1, ..., 8 . If the dummy variable in the ith row andjth column dij is equal to 1, the return of asset aj (or of common component CCj)is related to the return of asset ai (or common component CCi), conditionally on thereturns of any asset (or common component) preceding aj (e.g., rj−1,rj−2,...,r1). Ifdij = 0, the pair is conditionally independent, and the density of the associated copulais equal to one.

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Constraining the previous matrixMs allows us to impose any dependence structurespecified "a priori". All diagonal entries are equal to 1 since each asset is obviouslylinked with itself, but imposing that all elements of the first column are equal to 1,di,1 = 1 means that the returns of all assets (including the ones of the common com-ponents CC2, CC3andCC4) depend on the first common component CC1. We canimpose conditional independence or dependence between the common components;here, d3,2 = 0 and d4,3 = 1, respectively mean that CC2 and CC3 are independent,conditionally on CC1, and CC4 and CC3 are dependent, conditionally on CC1.

Moreover, each asset can share just one common component or several ones: forexample, a1 is only related to CC2 conditionally on CC1 and a3 to CC3 and CC4,conditionally on CC1. In the same way, assets can be dependent or independent oneach other conditionally on the common components: for example, d8,6 = 1 meansthat a2 is related to a4 given the 4 common components while d8,7 = 0 means thata3 and a4 are independent, conditionally on these 4 components. Moreover, for eachpair of related assets (di,j = 1), the dependence is further characterized by one copulachosen in a set of various bivariate copulas.

In what follows, we retain the simplified structure which is summarized by graph 1in Appendix; we describe it in details in the following section.

3 Practical implementationIn what follows, we work with a database composed of 35 indexes (stocks, bonds,

currencies and commodities); the observation frequency is weekly over the period01/05/2001 to 09/27/2013. We suppose that we have a particular (factorial) structurecapturing the following risk directions: real (interest) rates, inflation, global equity,credit, emerging equity, commodities, USD. Within the C-vine structure, each condi-tioning is associated with an underlying factor and block independence implies thatassets earn only up to 4 risk premiums according to the "ladder" structure presented ingraph 2.

3.1 Marginal DistributionFirst, we have to specify the marginal distributions and the copulas to characterize

the joint distribution of the returns of all indexes. Concerning the marginal distribu-tions, there are different approaches. We have retained a usual ARMA-GARCH spec-ification with a GED Distribution of the standardized residuals. As mentioned before,any other characterization of the marginal distributions could be retained. The detailsare given in Appendix. See in particular Table 6. For almost all indexes (31 of 35),we find that the ARMA(0,0)-GARCH(1,1) and the GED give the best specification.We also observe that the parameters (ν) of the GED distributions are smaller than 2 inmost cases. This means that most of the distributions have thicker tails than the normaldistribution.

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3.2 The choice of copulas and the tail dependenciesHaving estimated the marginal parameters, we transform the standardized residu-

als into uniform residuals by using the approach proposed by Meucci(2006). In thesecond step, we fit a bivariate copula model to the standardized residuals by integrat-ing the structure in section 2.3 into a C-vine copula described in the section 2.1. Thebivariate copulas can be chosen from a set of families: Gaussian, Student t, Clayton,Frank. Gaussian and Frank copulas do not allow for any tail dependencies contrary tothe Student and Clayton ones which allow for symmetric and lower tail dependenciesrespectively.

Based on the dependence structure, it’s naturally to assume that most tail depen-dencies are captured by the first two global indexes. Therefore, we consider only thebivariate copulas between each of the three global indexes and others indexes. About50% of the bivariate copulas chose copula families with tail dependance as the bestfitted model in which about 10% chose Clayton copula with lower tail dependance and40% chose student t copula with two sides tail dependencies. We find indeed evidencesof tail dependence between indexes.

Frequency in %Gaussian 34 34.3%Student t 40 40.4%Clayton 8 8.1%Frank 17 17.2%

99 100%

Table 2: Results of the bivariate copulas between the two global indexes and othersindexes

Finally, we have to choose the "ex ante" factorial structure.

3.3 Factorial structureAsset returns can be decomposed as the sum of a risk free rate and the expositions

to several risk s. We use a long term Treasury bond instead of cash to get the risk“free” rate as it better matches the investment horizon of a strategic allocation. Specificrealized bond premia (Credit, Emerging, etc.) and realized equity premium are oftencalculated as the spread between the corresponding asset returns and the nominal bondreturn of a benchmark (US, AAA-rated, etc.). Regarding commodities and currencies,the link with risk free rate is not clear or nil and must be confirmed empirically (seebelow). Furthermore nominal bond risk embodies real rate and inflation risks; that iswhy we consider the real rate risk as the first pivotal factor and then the inflation riskas a second one.

Once we remove the nominal bond rate factors (split into real rate and inflationfactors), we got a set of risk factors (Equity, Credit, etc.). We choose the equity risk asthe next pivotal factor since stock markets best reveal the risk aversion of the investors( whatever the risk). That global market risk factor thus captures the last link with all

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remaining risk factors and permits us to spread a negative shock from any other riskfactors to all asset returns included in our database.

Finally, we define specific or residual factors as those obtained once we have con-trolled assets returns from the first three risks (labeled global factors). Assets in ourportfolio permit us to study credit risk, Euro debt concerns factor, USD, commoditiesand emerging equity risk.

Thus we retain eight factors. Among them, we have three global common factorswhich are expected to influence most of indexes in the portfolio. The five other factorsare more specific because they are shared by a limited number of indexes.

The structure can be associated with a tree which helps showing different groupsof indexes. At a given level, one index is associated with a particular node and itis related to the factors whose nodes share a branch with its node. However someparticular indexes can belong to different groups in the view of the portfolio manager.For example the FX Emerging index is not only linked with the emerging index butalso with the currency index. The complete dependence structure is described in Table3 and 4 in Appendix.

The table 5 displays Kendall’s taus which is an average rank correlation betweenassets and the considered factor (labeled by column). We should keep in mind when wemove to the next column we measure the average rank correlation between the factorand assets both of them conditioned on former factors (if not independent).

The first global factor is the real rate factor effect which is assessed by the returnof the world inflation protected government bonds 3. TIPS (US case) are the risk-less assets for long-term investors (US) who care about real return (Ilmanen, 2011).We think inflation protected bonds are a good proxy for an imperfectly estimated realinterest rate risk, especially because they can be temporarily prone to liquidity concernsas in 2008. In addition, bond returns are negatively related to interest rate thus ourfactor is “minus real rate” because the positive Kendall’s tau. That factor separatesbonds from equities (developed more than emerging) but the rank correlation is quiteobviously much higher for bonds than equities (in absolute value). Basic theory statesthat equity is negatively related to real rate (as bonds) because higher real rate increasethe discount rate reducing the price. In our sample government bonds (best rated) areconsidered as safe-haven assets because of financial crises and low and stable inflation,are flight-to-quality periods the sole explanation to a positive relation between equityreturns and real interest rate (i.e. negative rank correlation)? Surprisingly, high-yieldbond returns are not related to real rate because rank correlations are not significantlydifferent from zero. Credit premium is the main driver (see below). Commodities areglobally unrelated to real rate. Currencies (against USD) are also not linked to realrate except Europe ones. This result may come from different business cycle phases(and monetary policy reactions). US economic cycle tends to lead European one in oursample.

The second global factor is the (minus) inflation factor effect. As expected, US andUK inflation government bond indexes (most represented in the world index) have noinflation sensibility while Euro index shows some residual inflation sensibility. How-ever, this negative relation to inflation (positive rank correlation) is much lower than

3. The three first most important exposures are US, UK and France.

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traditional government bonds (nominal rate). This phenomenon may be due to higherliquidity concerns in the Euro inflation index than those affecting the world inflation in-dex. Government bond indexes are almost equally sensitive to inflation and to real rate.Nevertheless, empirical results reveal higher Euro and German inflation sensibilitieswhich are intuitive outcomes (Central bank inflation fear is well known). Regardingrisky assets, equities are similarly positively related to inflation. The link betweenequities and inflation depends mainly on the source of inflation: a demand driven in-flation causes a positive relation with stock returns; a supply driven inflation causes anegative correlation (Lee, 2009). In our sample inflation is mainly driven by demand.Interestingly, oil has the highest positive rank correlation with inflation. This result isconsistent with the fact that oil has the best inflation hedging ability. Precious metalindex (Gold) has a weak link with inflation confirming the fact that gold is not reallyan inflation hedge. Gold is regarded as a safe haven against financial turmoil and USdollar weakness (Ilmanen, 2011).

The goal of the third global factor is to capture risk aversion through stock market(equity risk factor). All risky assets are positively related to that factor. We notice thatgovernment bonds have a weak but negative rank correlation with risk. We expectedsuch a relation because the studied period encompassed flight-to-quality episodes. Onthe contrary, euro government bonds are positively related to risk aversion reflectingthe euro-area debt concerns which occurred at the end of the sample. As expected,corporate and emerging bonds (premia) have positive rank correlations with equityrisk. The high-yield sensitivity is naturally higher than investment grade sensitivity.

We now turn to residual specific factors 4. Euro high-yield bonds are surprisinglynot sensitive to the residual euro debt factor. We linked emerging bonds with creditfactor because emerging bond spread is generally viewed as a measure of an emergingeconomy’s creditworthiness. Besides, the rank correlation is slightly higher than theone with the emerging factor. All currencies or basket of currencies have a negativecorrelation with the USD factor because all studied currencies are short USD whereasour factor is long USD. It is worth noting that gold could be seen as a currency andseems to be negatively related to the USD factor confirming its dollar hedge ability incase of USD weakness. Asian stocks show the most important rank correlation withemerging equity risk reflecting the weight of Asian countries within the emerging in-dex. As previously, oil has the highest rank correlation with commodity factor becauseof its weight in the global index.

3.4 Return simulationsWe implement two types of simulations with "not conditional" and "conditional"

shocks.

3.4.1 General Simulation

To run the simulations, we proceed as follows. First, by using the estimated pa-rameters for the different copulas and the algorithm 2 described in Aas et al. (2009),

4. Except for emerging equity risk, the word « residual » is suitable because we have taken our riskaversion factor into account in each specific risk.

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we simulate N samples from an I dimensional canonical vine for the next period,i.e. u1:I,T+1. Then, the inverse error distribution functions (G−1) produce a sampleof standardized residuals, i.e. z1:I,T+1 = G−1(u1:I,T+1). Finally, according to theGARCH equation ( 6) in Appendix, the observed initial values and the estimatedARMA-GARCH parameters are used to compute the return forecasts for i = 1 : I ,

ri,T+1 = µi + φiri,T + θiσi,T zi,T + σi,T+1zi,T+1

with the variance forecast,

σ2i,T+1 = ωi + αiσ

2i,T z

2i,T + βiσ

2i,T

3.4.2 Simulation with extreme non conditional shocks

In the following, we focus on the simulations of uniforms from the vine structurewhile the process transformation from uniforms to returns remains the same as before.First, we introduce the simulations with non conditional shocks.

With the tool we can implement simulations in accordance to an extreme behaviourof one index. Indeed, instead of drawing all the u1:I between 0 and 1, we draw sam-ples from a extreme zone (for example from 0 to 0.05) for the stressed variable ui,i ∈ {1, ..., I}. Since the dependence structure is supposed to be unaffected by theshock, a stress situation for one factor impacts not only the variables which are directlyrelated to this factor but also others variables in an indirect way, by affecting the keyfactor at the root node of the C-vine that are related to all variables. This means that asharp decrease of one factor can cause the distress of the whole portfolio if other assetsdepend positively on this factor. The algorithm is given in Brechmann et al. (2013).

3.4.3 Simulation with extreme conditional shocks

Moreover we can apply shocks from conditional distributions which are interpretedas shocks to specific risk sources. First of all, some definitions of risk sources need tobe clarified. The non conditional distribution of an index-factor f summarizes a set ofdifferent risk sources, whereas the conditional distribution of factor fi given anotherfactor fj can be interpreted as a combination of the remaining risk sources when therisk associated with fj has been removed. By considering the gap between the returnsassociated with the non-conditional and conditional distributions, we can isolate theeffect of a specific risk.

More generally speaking, if we want to apply a shock to the i − th specific risk,it has to involve conditional cumulative distribution functions F (xi|x1, x2, ..., xc). Weadapt here the simulations for C-vine copulas involving conditional distributions. Forj ≤ c or j > i, the sampling procedure from F (xj |x1, x2, ..., xj−1) is the same asthe one described before. However, sampling from F (xj), c < j ≤ i given theF (xi|x1, x2, ..., xc) has to be modified. We develop a new algorithm to specificallydeal with simulations involving conditional shocks. See Appendix.

In the next section, we propose two applications of the tool to portfolio manage-ment.

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4 Applications to portfolio management in critical con-texts

In this section we use the CVRF model for portfolio management purposes. First,we show how to measure the sensitivity of any asset to extreme shocks to any otherasset and we decompose the corresponding response into the marginal contributionsof the different risk factors. In what follows we just focus on extreme shocks to the 8first indexes of our data base, because they mainly drive the comovments of the assetsand may accordingly dramatically increase the risk of a portfolio in case of extremeevents. Indeed, limiting the stress tests to this type of extreme shocks is natural whenwe extend the same type of analysis to portfolios as presented in a second stage.

4.1 Sensitivity analysis for the assetsOur analysis is based on the factorial structure summarized by the ladder structure

displayed in Figure 2 in Appendix. We proceed as follows.We successively consider extreme shocks to each of the 8 first indexes, i = 1, ..., 8.For each of these shocks, we decompose the responses of any index j of our data

base into the marginal contributions of the different risk factors. Note that this decom-position depends on the origin of the shock.

Suppose that we want to measure the total sensitivity of index j to a shock to oneof the first 8 indexes index i; we proceed as follows:

First we compute the "total sensitivity" of index j to an extreme shock to index i:

Total Sensitivityj→i = E(Rj |F (Ri) < 5%)− E(Rj)

where E(Rj |F (Ri) < 5%)) denotes the expected return of asset j when the (nonconditional) distribution F (Ri) is stressed in its extreme negative part and E(Rj) isthe return obtained from a general simulation without shock.

In case of an extreme shock to i, the shock simply corresponds to a draw in theextreme (negative) part of F (Ri) and all other returns are simulated conditionally onthis extreme draw.

The expected returnE(Rj) gives us a benchmark value corresponding to a situationwithout shock; the sensitivity of index j to any extreme shock to index i is simplycalculated as the deviation from this benchmark value due to the shock.

Then we can decompose the previous sensitivity into the marginal contributions ofthe different risk factors, which have an effective impact on the asset. 5

For 4 ≤ i ≤ 8, the decomposition of the total sensitivity can be obtained as follows:First, we can calculate the marginal sensitivity related to the real rate risk factor:

Real rate risk contributionj = E(Rj |F (Ri) < 5%)− E(Rj |F (Ri|R1) < 5%)

5. For example, if the shock comes from the first index, we just identify the contribution of the real raterisk factor. In that case, we can not measure the contributions of the other risk factors because they do notimpact the first index according to Figure 2.

13

where E(Rj |F (Ri|R1) < 5%) is the return of index j when the conditional distribu-tion F (Ri|R1) is stressed. In the case where i = 5, there are only three risk factorsunderlying the conditional distribution F (Ri|R1): Inflation, Market and Credit. Byconditioning on the first index which is a proxy for the Real Rate risk, the correspond-ing risk is indeed removed from index i. Consequently, the sensitivity of index j to theconditional shock to index i given R1 comes only from the exposition to the three re-maining risk factors. The difference between the responses obtained for the two casesgives us the marginal contribution of the first Real rate risk factor.

We can proceed in the same way to capture the marginal contribution of the inflationrisk factor, that is:

Inflation risk contributionj = E(Rj |F (Ri|R1) < 5%)− E(Rj |F (Ri|R1, R2) < 5%)

with E(Rj |F (Ri|R1, R2) < 5%) denoting the expected return of index j when theconditional distribution F (Ri|R1, R2) is stressed.

Still referring to index i = 5, when conditioning on R1, R2, there are only tworisk factors underlying the distribution F (Ri|R1, R2) -Market and Credit-, as the twofirst risk factors (Real rate and Inflation) are removed by conditioning on the first twoindexes. In this case, the sensitivity of index j comes from the exposition to the tworemaining risk factors affecting index i. The difference between the two responses thusgive us the marginal contribution of the Inflation risk factor to the total sensitivity ofindex j to an extreme shock to index i.

Lastly, we can measure the marginal contribution of the Market risk factor accord-ing to the same principle:

Market risk contributionj = E(Rj |F (Ri|R1, R2) < 5%)− E(Rj |F (Ri|R1, R2, R3) < 5%)

withE(Rj |F (Ri|R1, R2, R3) < 5%) denoting the expected return of index j whenthe conditional distribution F (Ri|R1, R2, R3) is stressed.

Finally we obtained the marginal contribution of the ith-specific risk (1 ≤ i ≤ 4)as follows:

ith-specific risk contributionj = E(Rj |F (Ri|R1, R2, R3) < 5%)− E(Rj)

The total sensitivity is obviously obtained as the sum of the previous marginal con-tributions.

Table 7 and 8 summarize the results for the decompositions of sensitivity we obtain.Here we give some comments for shocks on each of the 8 indexes:

1. shocks on World inflation linked bond index affect directly all other indexes inline with Kendall’s tau (cf. section 3.2);

2. shocks on World government bonds index spread clearly through the inflationfactor for risky assets;

3. the impact of the shocks on Eurozone sovereign bond index are smaller becauseof the positive correlation between the specific factor Euro and risky assets ;

14

4. specific shock credit lowers the stocks, the effect of the inflation factor is lessimportant than before (negative shock on credit are not conducive to increases ofinflation);

5. when the World equity market index is stressed, the real rate effects are negligiblefor the risky assets. Fixed-income assets increase through the first two factors tothe same extent;

6. when we stress the USD index (the performance of the USD against a basket ofcurrencies, long USD), the principal effects here are specific effects, we noticethat emerging markets have the highest sensitivity to a falling of dollar;

7. the effects of the shocks on Emerging stock market index are similar to the effectsin 5);

8. while we stress commodity index, we have positive impacts for the bonds throughthe inflation factor and negative impacts for risky assets through the risk aversionfactor;

For sake of illustration let us focus on the effects of a particular shock. Let usconsider a negative shock to developed equity markets. We find from our historicalsample (2001-2013) that equity is expected to lose around 18% ( -17,7%) a month andUS government bond (7-10 years) is expected to rise about 3% (%2,9) a month. USgovernment bonds are considered as safe haven assets in our sample. We found that+3% is decomposed into +1.5% for real rate marginal effect, +1% for inflation marginaleffect and 50bp for equity risk marginal effect. See Table 8.

In the last section we measure and compare the risk sensitivities of different port-folios, when they are exposed to different types of extreme shocks.

4.2 Sensitivity analysis for portfoliosAs we retain three portfolios composed according to risk budgeting rules we first

briefly recall the general principles of such portfolio allocation strategies.

4.2.1 Principles of risk budgeting

Risk budgeting is an approach to investment portfolio management which focuseson allocation of risk rather than allocation of capital. The risk of a position is usuallyassessed in terms of measures as the standard deviation, the Value-at-Risk (VaR) ofthe expected shortfall or Conditional VaR, (CVaR). The risk R(w) of a portfolio is afunction of the composition w. All previous measures are homogeneous. Accordingly,the following identity holds:

R(w) =

n∑i=1

wi∂R(w)

∂wi(5)

In other words, total risk can be expressed as the sum of the contributions fromthe different assets, when the generic i-th contribution is the product of the "per unit"marginal contribution ∂R(w)

∂wiand the "amount" of the i-th asset, as represented by wi.

15

In what follows, we retain the CVaR as risk measure. Indeed, its allows us to betterexploit our characterization of the dependencies, including the tail ones. Moreover,choosing this risk measure avoids to excessively focus on the specific risk when look-ing for an optimal portfolio with the standard mean-variance criteria (Roncalli,2013).Thus, according to Mausser (2003), Epperlein and Smillie (2006), Meucci et al. (2007),the marginal contributions are given by:

∂R(w)

∂wi= −q

′

cSw

where qc is a step function that jumps from 0 to 1/cJ at a rescaled confidence levelcJ of the expected shortfall(CVaR) and Sw is a JxN panel, with the generic j-columndefined as the j-th column of the JxN panel F , obtained by Monte Carlo simulations,sorted as the order statistic of the J-dimensional vector −Fw.

When the number N of assets is large, practitioners prefer to analyze risk at anaggregated level. We follow this practice and define 4 buckets each composed of sim-ilar assets that are mainly exposed to the same type of risk; Other bonds, Governmentbonds, Equity and Commodity.

It is thus natural (see Meucci, 2007) to define the contribution to the risk of the k-thbucket as the sum of the individual contributions from the assets in this bucket:

Ck =∑i∈Nk

∂R(w)

∂wiwi

In what follows,we propose to compare the risk sensitivity of different types ofportfolios to extreme shocks.

4.2.2 Risk sensitivity of different portfolios to extreme shocks

The portfolios we retain are the following:– Equal Risk Contribution (ERC) Portfolio;– High Risk Budgeting portfolio with risk budget as (10%, 10%, 70%, 10%) re-

spectively for the 4 groups;– Low Risk Budgeting portfolio with risk budget as (30%, 50%, 15%, 5%) respec-

tively for the 4 groups;– Weight budgeting portfolio with high risk allocation as (20%, 20%, 55%, 5%)

respectively for the 4 groupsAs before, we examine the responses of these portfolios to extreme shocks to the 8

first indexes (1 ≤ i ≤ 8).We apply the same type of sensitivity analysis as the ones previously developed

for single assets and we decompose, as before, the total sensitivity of each portfolioto each extreme shocks into the marginal contributions of our 8 risk factors. Thisdecomposition can be summarized as follows:

16

Real rate riskp = E(Rp|F (Ri) < 5%)− E(Rp|F (Ri|R1) < 5%)

Inflation risk contributionp = E(Rp|F (Ri|R1) < 5%)− E(Rp|F (Ri|R1, R2) < 5%)

Market riskp = E(Rp|F (Ri|R1, R2) < 5%)− E(Rp|F (Ri|R1, R2, R3) < 5%)

ith-Specific riskp = E(Rp|F (Ri|R1, R2, R3) < 5%)− E(Rp)

where Rp denotes the return of portfolio p.Our 4 allocations cover 2 types of portfolios: defensive (risk budgeting based ones)

and aggressive. Allocation in risky assets (equities and commodities) ranges from 15%to 30% for defensive portfolios and is equal to 60% for the aggressive one (table 9).

We notice from every stressed index results in table 10 that ERC portfolio is nearlyinsensitive to inflation risk whereas aggressive allocation is rather insensitive to realrate risk compared to the other ones.

It is worth noting that defensive portfolios are not sensitive to inflation risk in a sim-ilar way. High risk budgeting portfolio is positively sensitive to inflation risk whereaslow risk budgeting portfolio is negatively related.

Interestingly, the expected return of the aggressive portfolio (labeled high risk cap)decrease roughly in the same magnitude than defensive portfolios when we stress ourcorporate bond index. But the spreading is completely different. The main channelsfor defensive portfolios are real rate risk and market equity risks with marginal effectswhich contribute slightly differently depending on risky asset weights in the portfo-lio. However, within the aggressive portfolio, the negative corporate bond shock onlyspreads through equity market risk via an increase of risk aversion or downward growthconcerns associated to a negative credit risk shock. This result is consistent with eco-nomic intuition.

All residual marginal effects are obviously proportional to the weight of the mostrepresentative assets of the stressed index in the portfolio. Moreover, residual marginaleffects are globally quite low compared to other risks. However, it is worth noting thatwe can have a portfolio with a relative smaller weight in a specific asset (thus smallerresidual effect) but facing larger spillover effect through equity market risk (see Eurogovernment bonds and commodity stress scenarios).

Finally, in our example and with traditional asset classes, we show that inflation riskcould be diversified away with an appropriate balanced portfolio while (real) rate riskand equity risk remain the most important risk concerns which can only be diversifiedaway within a more trivial and more highly risk concentrate portfolio.

5 ConclusionThe aim of this paper was to show the practical usefulness of vine copula based

models for portfolio management in the case of a large number of assets. We haveproposed a CVFR model combining a Canonical Vine and a factorial-type dependencestructure specified a priori. Accordingly a portfolio manager can easily use this modelto impose any dependence structure reflecting his own risk perception and to decom-pose the returns into risk factors which are crucial to his opinion (bond, equity, infla-

17

tion, credit for example), while taking into account complex relationships between thedifferent assets that can not be summarized by simple correlations.

As an application, we have examined the case of a set of 35 indexes of differenttypes - stock, bonds, commodities, currencies. The marginal distributions of the weeklyreturns mostly obey to an AR(0,0)-GARCH(1,1) with a generalized distribution (GED)of the residuals which is preferred to the Gaussian and the Student distributions. Theshape parameter of the GED show that most of the marginal distributions have thickertails than the normal distribution. As to copula results, evidences of tail dependenceare found between a significant number of indexes.

The factorial-type structure we have specified a priori includes 8 indexes as com-mon components from which we have identified 8 different risk factors correspondingto real rate, inflation, market, credit, (European) sovereign debt, UDS, Emerging andCommodity risks.

The core applications of our model are sensitivity analysis for each asset of ourdata base to extreme shocks to any other asset and particularly to the 8 indexes whichmainly account for the co-movements of the assets. Moreover our model allows usto decompose the total sensitivity of each asset into the marginal contributions of therisk factors we are able to identify. All these sensitivity analysis take into accountthe complex dependence structure among the 35 indexes we retain, including the taildependencies which are particularly crucial in case of extreme shocks. All results weobtained from simulations. In this regard our approach is semi-parametric.

We have applied the same type of sensitivity analysis to portfolios and comparedthe sensitivity of several portfolios to different types of extreme shocks.

All these stress test exercises show that our model is well adapted to provide aportfolio manager with a general measure of the exposition of a wide range of assetsand portfolios to various risk sources especially in critical (extreme risk) circumstances.Moreover, the decompositions of the sensitivities we propose into the contributions ofthe risk factors tant be identified should help him more effectively choose a mix ofasset classes that best diversifies his risks while also reflecting his views on the globaleconomy and financial markets, as summarized by the factorial-type structure he retainsa priori.

References[1] Aas, K., C. Czado, A. Frigessi and H. Bakken, 2009. “Pair-copula constructions

of multiple dependence.” Insurance: Mathematics and Economics, Elsevier, vol.44(2), 182-198, April.

[2] Adler, T. and M. Kritzman, 2007. “Mean-Variance versus Full-Scale Optimisa-tion: In and Out of Sample.” Journal of Asset Management, Vol. 7, No. 5, 302-311, 2007.

[3] Ang, Andrew, 2010. “The Four Benchmarks of Sovereign Wealth Funds.”Columbia Business School and NBER, September 2010.

18

[4] Batiz-Zuk, E., G.Christodoulakis and S.H. Poon, 2013, "Structural Credit LossDistributions under Non-Normality", the JOURNAL Journal of Fixed Income,Summer, Vol. 23, No. 1: pp. 56-75.

[5] Bedford, T. and R.M. Cooke, 2001. “Probability density decomposition for condi-tionally dependent random variables modeled by vines.” Ann. Math. Artif. Intell.32 (1-4), 245-268.

[6] Bedford, T. and R. M. Cooke, 2002. “Vines-a new graphical model for dependentrandom variables.” Ann. Statist., 30 (4), 1031-1068.

[7] Bender, J., R. Briand and F. Nielsen, 2010. “Portfolio of Risk Premia: A NewApproach to Diversification.” Journal of Portfolio Management, Vol. 36, No. 2,Winter 2010.

[8] Brechmann, E.C. and C. Czado, 2013. “Risk Management with High-Dimensional Vine Copulas: An Analysis of the Euro Stoxx 50.” Statistics & RiskModeling, to appear.

[9] Brechmann, E.C., K.Hendrich and C. Czado, 2013. “Conditional copula simula-tion for systemic risk stress testing.” Insurance: Mathematics and Economics, 53,722–732.

[10] Brechmann, E.C. and U.Schepsmeier, 2013. “Modeling Dependence with C- andD-Vine Copulas: The R Package CDVine.” Journal of Statistical Software, 52(3),1-27. URL http://www.jstatsoft.org/v52/i03/.

[11] Campbell, J., Y. and J., H., Cochrane,1999. “By force of habit: A consumption-based explanation of aggregate stock market behavior”, Journal of Political Econ-omy 107, 205-251.

[12] Campbell, J.Y., L.M. Viceira, A. Sunderam, 2013. “Inflation Bets or DeflationHedges? The Changing Risks of Nominal Bonds.” 2013.

[13] Chow, G., 1995. “Portfolio Selection Based on Return, Risk, and Relative Perfor-mance.” Financial Analysts Journal, Vol. 51, No. 2, March/April 1995.

[14] Chua, D., M. Kritzman, and S. Page, 2009. “The Myth of Diversification.” TheJournal of Portfolio Management, Vol. 36, No. 1, Fall 2009.

[15] Clarke, R. G., H. de Silva and R. Murdock, 2005. “A Factor Approach to AssetAllocation.” The Journal of Portfolio Management, Vol. 32, No. 1., Fall 2005.

[16] Cochrane, J., H., 2011. “Discount Rates”, The Journal of Finance, 66(4), 1047-1109.

[17] Epperlein, E.and A. Smillie, 2006. “Cracking VaR with Kernels”, Risk Magazine,19, pp. 70-74.

[18] Fama, E. and K. French, 1989. “Business conditions and expected returns onstocks and bonds,” Journal of Financial Economics, Vol. 25, 23-49.

[19] Fama, E. F., French, K. R., 1993 “Common Risk Factors in the Returns on Stocksand Bonds.”, Journal of Financial Economics, Vol. 33, No. 1.

[20] Gordon, J. A., A. M. Baptista, 2002. “Economic Implications of Using a Mean-VaR Model for Portfolio Selection: A Comparison With Mean-Variance Analy-sis.” Journal of Economic Dynamics & Control, Vol. 26, No. 7-8, pp. 1159-1193.

19

[21] Heinen, A. and A. Valdesogo, 2009. “Asymmetric CAPM dependence for largedimensions: the Canonical Vine Autoregressive Model.” CORE Discussion Pa-pers 2009069, Université catholique de Louvain, Center for Operations Researchand Econometrics (CORE).

[22] Idzorek,T.M. and M. Kowara, 2013, "Factor-Based Asset Allocation vs. Asset-Class-Based Asset Allocation", Financial Analysts Journal, Vol 69, N˚ 3.

[23] Ilmanen, A., 2003. “Stock-Bond Correlations”, Journal of Fixed Income, Vol.13(2), 55-66.

[24] Ilmanen, A., 2011. “Expected Returns”, Wiley.[25] Lee, Bong-Soo, 2009. “Stock returns and inflation revisited”, working paper,

available at SSRN: http://ssrn.com/abstract=1326501[26] Maillard, S., T. Roncalli, and J. Teiletche, 2008. “On the Properties

of Equally-Weighted Risk Contributions Portfolios”. Available at SSRN:http://ssrn.com/abstract=1271972 or http://dx.doi.org/10.2139/ssrn.1271972.

[27] Mausser, H., 2003. “Calculating Quantile-Based Risk Analytics with L-estimators”, Journal of Risk Finance, pp. 61-74.

[28] Meucci, A., 2006. “Beyond Black-Litterman in Practice: A Five-StepRecipe to Input Views on Non-Normal Markets.” Available at SSRN:http://ssrn.com/abstract=872577 or http://dx.doi.org/10.2139/ssrn.872577.

[29] Meucci, A., 2007. “Risk contributions from Generic User-defined Factors” sym-mys.com

[30] Meucci, A., 2010. “The Black-Litterman Approach: Original Model and Exten-sions”, The Encyclopedia of Quantitative Finance, Wiley.

[31] Meucci, A., Y. Gan, A. Lazanas and B. Phelps, 2007. “A Portfolio managers guideto Lehman Brothers Tail Risk Model.” Lehman Brothers Publications.

[32] Page, S. and M.Taborsky, 2011. “The myth of diversification: Risk Factors vs.Asset Classes.c The Journal of Portfolio Management, Vol. 37, No. 4, pp. 1–2,Summer 2011.

[33] Roncally, T., 2013. “Introduction to Risk Parity and Budgeting”, Chapman &Hall/CRC Financial Mathematics Series .

[34] Ross, S., 1976. “The arbitrage theory of capital pricing”, Journal of EconomicTheory, 13, 341-360.

[35] Sklar, A., 1959. “Fonctions de répartition à n dimensions et leurs marges”, Pub-lications de l’Institut de Statistique de l’Université de Paris 8, 229–231.

[36] Tumminello, F. L. and R. N. Mantegna, 2007. "Hierarchically nested factor modelfrom multivariate data.” EPL (Europhysics Letters), 78 30006. Structural CreditLoss Distributions under Non-Normality

A Factorization of a trivariate densityHere we present in details a possible factorization of a joint three-dimensional den-

sity function as a product of bivariate copulas and marginal densities.

20

f(x1, x2) = c12(F1(x1), F2(x2)) · f1(x1) · f2(x2),

and the first conditional density is:

f(x2|x1) = c12(F1(x1), F2(x2)) · f2(x2).

In the same way, one possible decomposition of the second conditional density f(x3|x1, x2)is:

f(x3|x1, x2) = c23|1(F2|1(x2|x1), F3|1(x3|x1)) · f(x3|x1),

where c23|1 is the bivariate copula, applied to the transformed variables F2|1(x2|x1)and F3|1(x3|x1)). Decomposing f(x3|x1) further, leads to:

f(x3|x1, x2) = c23|1(F2|1(x2|x1), F3|1(x3|x1)) · c13(F1(x1), F3(x3)) · f3(x3).

Combining the equations above, one obtains the joint density of the three variables asa product of marginal densities and bivariate conditional copulas:

f(x1, x2, x3) = c23|1(F2|1(x2|x1), F3|1(x3|x1)) · c12(F1(x1), F2(x2))

·c13(F1(x1), F3(x3)) · f1(x1) · f2(x2) · f3(x3).

Note that Joe (1996) showed that conditional cdf’s of the form F (x|υ) where υ is avector, can be derived recursively from marginal cdf’s by

h(x, υ,Θ) = F (x|υ) =∂Cx,υj |υ−j

(F (x|υ−j), F (υj |υ−j))∂F (υj |υ−j)

,

where υ−j denotes the set υ without the jth index. h(·) is the conditional distributionfunction and Θ denotes the set of parameters for the copula of the joint distributionfunction of x and υ. Let h−1(u, υ,Θ) be the inverse of the conditional distributionfunction that will be used in the simulation.

B Five-dimensional CVMS modelLet rM denote the market return, rA and rB be the returns of two sectors. Further,

let rA1 , rA2 , rB1 and rB2 be those of assets belonging to sectors A and B, respectively.According to the independence assumptions, we have: (i) rA is independent of rBgiven rM , and (ii) rA1 and rA2 are independent of rB conditioning on rM , while rB1 andrB2 are independent of rA conditioning on rM .

In a canonical vine structure, if the order is set like: rM , rA, rB , rA1 , rA2 , rB1 , rB2 ,where the market is the first root node and the sectors are the second and third rootnodes, the first tree T1 gives the dependencies between the market and the sectors, aswell as those between the market and the stocks.

cM,A(F (rM ), F (rA)) · cM,B(F (rM ), F (rB)) · cM,1A(F (rM ), F (rA1 ))

·cM,2A(F (rM ), F (rA2 )) · cM,1B(F (rM ), F (rB1 )) · cM,2B(F (rM ), F (rB2 ))

21

Specific Risk Factors Related Asset Classes

Real

RatesWorld Inflation Linked Index

Real

RatesInflation WGBI All Maturities

Real

RatesInflation

Equity Market

Risk AversionMSCI World Index

Real

RatesInflation

Equity Market

Risk AversionEuro Debt Eurozone Sovereign

Real

RatesInflation

Equity Market

Risk AversionCredit Average Corporate Bonds

Real

RatesInflation

Equity Market

Risk AversionLong USD Dollar Index

Real

RatesInflation

Equity Market

Risk AversionEmerging MSCI Emerging Market

Real

RatesInflation

Equity Market

Risk AversionCommidity DJUBS Commodity

Global Risk Factors

Figure 2: Risk exposure and asset classes

23

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Tabl

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Mat

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ofst

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26

Real rate

(m

inu

s)

In

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(m

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s)

Eq

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Ris

k A

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Eu

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Deb

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Cred

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US

D

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Citigroup WGBI All Maturities 0.60 NA NA NA NA NA NA NA

MSCI World Index Daily Net TR (0.21) (0.20) NA NA NA NA NA NA

IBOXX € EZSOV OA TR 0.51 0.54 0.17 NA NA NA NA NA

Average Corporate Bonds 0.38 0.17 0.29 - NA NA NA NA

DOLLAR INDEX SPOT (0.14) 0.05 (0.11) - - NA NA NA

MSCI Daily TR Net Emerging Mar (0.12) (0.19) 0.50 - - - NA NA

DJUBS Commodity TR ns (0.20) 0.15 - - - - NA

Euro MTS Inflation Linked Inde 0.45 0.14 0.12 0.40 - - - -

Citigroup US Inflation Linked 0.67 ns ns - - - - -

Barclays UK Inflation Linked B 0.62 ns (0.07) - - - - -

IBOXX € LQD CRP TR 0.46 0.34 0.16 0.33 0.40 - - -

iBoxx $ Liquid Investment Grad 0.47 0.30 0.15 - 0.40 - - -

CS Western Euro High Yield Ind ns (0.06) 0.22 ns 0.35 - - -

IBOXX USD Liquid High Yield Index ns ns 0.30 - 0.41 - - -

Citigroup US GBI 7 to 10 Year 0.55 0.47 (0.09) - - - - -

Citigroup UK GBI 7 to 10 Year 0.56 0.42 (0.08) - - - - -

Citigroup Germany GBI All Matu 0.54 0.54 (0.06) - - - - -

EUR-USD Carry Return 0.12 ns 0.09 - - (0.81) - -

CHF-USD Carry Return 0.18 ns ns - - (0.66) - -

GBP-USD Carry Return 0.08 ns 0.08 - - (0.52) - -

MSCI Daily TR Net North Americ (0.19) (0.17) 0.82 - - - - -

MSCI Daily TR Net EMU Local (0.22) (0.17) 0.71 - - - - -

MSCI Daily TR Net Europe Ex EM (0.21) (0.19) 0.70 - - - - -

MSCI Daily TR Net Pacific Ex J (0.10) (0.17) 0.48 - - - - -

JPM EMBI Global Diversified 0.15 ns 0.32 - 0.24 - 0.18 -

MSCI EM LATIN AMERICA (0.10) (0.17) 0.50 - - - 0.38 -

MSCI EM ASIA (0.08) (0.17) 0.40 - - - 0.73 -

MSCI EM EUROPE (0.05) (0.18) 0.38 - - - 0.38 -

FX COMMO ns (0.12) 0.25 - - (0.51) - 0.13

FX EMERGENT ns (0.11) 0.33 - - (0.36) 0.27 -

DJUBS PrcMtl 0.11 (0.09) 0.06 - - (0.31) - 0.24

DJUBS IndMtl ns (0.17) 0.24 - - - - 0.37

DJUBS Agri ns (0.11) 0.12 - - - - 0.43

DJUBS Pet 0.06 (0.21) 0.07 - - - - 0.54

ns: non significant

Table 5: Kendall’s tau

D Characterization of the marginal distributionsHere we present the different results concerning the best marginal distributions for

each of the 34 indexes.

D.1 ARMA-GARCH specificationFor each index, we specified an ARMA(1,1)- GARCH(1,1) with generalized error

distribution (GED) residuals model can be described as follow: let rti denote the return

27

of the ith asset at time t, then

ri,t = µi + φiri,t−1 + θiεi,t−1 + εi,t

εi,t = σi,tzi,t, zi,t ∼ GEDi(ν) (6)σ2i,t = ωi + αiε

2i,t−1 + βiσ

2i,t−1

For each marginal model, we have a list of parameters for different equations: (µ, φ, θ)in the mean equation, (ω, α, β) in the variance equation and (ν) for the innovationdistribution. Standardized residuals from the model are given by

zi,t =1

σi,t(ri,t − µi − φiri,t−1 − θiσi,t−1zi,t−1)

By using the Bayesian information criterion (BIC), we select the best model from alist of possible model. The possible mean specification are ARMA(0,0), ARMA(1,0),ARMA(0,1), ARMA(1,1) while for the volatility specification we choose the mostcommonly used GARCH(1,1). Besides, the innovation distribution can be selectedamong Gaussian, Student t and generalized error distribution.

28

1 2 3 4 5 6 7µ 0.0013 0.0007 0.0031 0.0010 0.0009 -0.0005 0.0042ϕ NA NA NA NA 0.3663 NA NAθ NA NA NA NA NA NA NAω 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001α 0.0658 0.0530 0.2165 0.0910 0.1288 0.0634 0.1399β 0.8984 0.9084 0.7300 0.8500 0.8490 0.9005 0.7802ν 1.4240 1.9765 1.5156 1.5658 1.4853 1.9604 1.4112

Distribution GED GED GED GED GED GED GED

8 9 10 11 12 13 14µ 0.0013 0.0011 0.0017 0.0013 0.0010 0.0013 0.0007ϕ NA NA NA NA NA NA 0.6482θ 0.0487 NA NA NA NA NA -0.3948ω 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000α 0.0600 0.1350 0.1116 0.0755 0.0781 0.0756 0.1260β 0.9177 0.8139 0.8459 0.9137 0.8894 0.9032 0.8782ν 1.4971 1.5076 1.4061 1.7364 1.7187 1.3298 1.1360


15 16 17 18 19 20 21µ 0.0013 0.0011 0.0010 0.0008 0.0009 0.0007 0.0006ϕ 0.2983 NA NA NA NA NA NAθ NA NA NA NA NA NA NAω 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000α 0.1387 0.0446 0.0493 0.0641 0.0580 0.0845 0.1041β 0.8518 0.9423 0.9415 0.9212 0.9149 0.8595 0.8460ν 1.0000 1.6950 1.8226 1.9914 1.9323 1.8675 2.0141


22 23 24 25 26 27 28µ 0.0027 0.0032 0.0033 0.0037 0.0025 0.0050 0.0042ϕ NA NA NA NA NA NA NAθ NA NA NA NA NA NA NAω 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001α 0.1836 0.1788 0.2859 0.1449 0.3071 0.1120 0.1692β 0.7654 0.8008 0.6819 0.8219 0.6567 0.8249 0.7692ν 1.4215 1.5452 1.4311 1.4534 1.1061 1.4655 1.5285


29 30 31 32 33 34 35µ 0.0053 0.0021 0.0021 0.0036 0.0010 0.0000 0.0032ϕ NA NA NA NA NA NA NAθ NA NA NA NA NA NA NAω 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001α 0.1110 0.0916 0.1500 0.0552 0.0997 0.1144 0.0836β 0.8283 0.8657 0.7908 0.9161 0.8783 0.8480 0.8807ν 1.3216 1.4167 1.3428 1.3648 1.7478 1.6127 1.6461


Table 6: Marginal distribution parameters. Most of the 35 indexes choose the samespecification: AR(0,0)-GARCH(1,1)-GED. From distribution parameters, we observethat nearly all the distributions have a thicker tails than the normal distribution.

29

D.2 Generalized Error DistributionThe density of a GED random variable normalized to have a mean of zero and a

variance of one is given by:

f(x) =νexp[− 1

2 |x/λ|ν]

λ2(1+1/ν)Γ(1/ν)

where Γ(•) is the gamma function, and

λ ≡ [2(−2/ν)Γ(1/ν)

Γ(3/ν)]1/2.

The parameter ν measure the tail-thickness of the distribution. The standard normaldistribution has a parameter ν equal to 2. For ν < 2, the distribution has thicker tailsthan the normal distribution and for ν > 2, the distribution has thinner tails than thenormal distribution.

E Algorithm for conditional simulationsLet us consider the i − th specific risk characterized by the conditional cumu-

lative distribution functionF (xi|x1, x2, ..., xc) with 1 < c < i. First, sample ωj ,j = 1, ..., i− 1, i+ 1, ..., n independent uniform on [0,1], and ωi uniform on a definedinterval [Min,Max] ∈ [0, 1]. Then, we set

ω1 = F (x1)

ω2 = F (x2|x1)

.. = ...

ωc = F (xc|x1, x2, ..., xc−1)

ωc+1 = F (xc+1|x1, x2, ..., xc, xi).. = ...

ωi−1 = F (xi−1|x1, x2, ..., xi−2, xi)ωi = F (xi|x1, x2, ..., xc)

ωi+1 = F (xi+1|x1, x2, ..., xi).. = ...

ωn = F (xn|x1, x2, ..., xn−1)

V = (νj,k), j ∈ 1, ..., n; k ∈ 1, ..., j is an lower triangular matrix to store theconditional distribution function and Θ = (θj,k), j ∈ 1, ..., n; k ∈ 1, ..., j is the ma-trix of parameters. We can easily get xi = wi. Like in the general algorithm, thefirst for loop runs over the variables from 2 to c. In this for loop, we have two othersub-for loops. The first one samples the variable xj , j ∈ 2, ..., c with the h−1 func-tion and the second one gives the conditional distribution function needed for samplingthe xj+1 by using the h-function. The variable xi is sampled in the following proce-dure xi = F−1(wi|x1, x2, ..., xc). The second for loop runs over the variables from

30

c + 1 to n. For sampling the variable xj , c < j < i, we need the correspondingconditional distribution which is computed in the If loop, F (xj |x1, x2, ..., xj−1) =F−1(wj , F (xi|x1, x2, ..., xj)) where the argument F (xi|x1, x2, ..., xj) is computed atthe end of the last for loop. Then, a for loop samples the variable xj , c < j < nwith the h−1 function. The remaining part of the algorithm provides the conditionaldistribution functions as arguments required for sampling the next variable.

With this algorithm, we can sample from a C-vine model given that the conditionaldistribution F (xi|x1, x2, ..., xc) belongs to a given interval. This means that we cancapture the outcome for all variables (returns) in the extreme case where the value ofthe conditional distribution is drawn between 0% and 5% for instance.

31

Algorithm 1 Simulate sample from a C-vine model given the conditional distributionGenerates one sample x1,x2...xn.

Sample ω1,ω2,...,ωi−1,ωi+1,...,ωn independent uniform on [0,1].Sample ωi uniform on a defined interval [Min,Max] ∈ [0, 1].x1 = ν1,1 = wifor j ← 2, ..., cνj,1 = wjfor k ← j − 1, ..., 1νj,1 = h−1(νj,1, νk,k, θj,k)

end forxj = νj,1for l← 1, ..., j − 1νj,l+1 = h(νj,l, νl,l, θj,l)

end forif j=c thenνi,1 = wifor k ← c, ..., 1νi,1 = h−1(νi,1, νk,k, θi,k)

end forxi = νi,1for p← 1, ..., cνi,p+1 = h(νi,p, νp,p, θi,p)

end forend if

end forfor j ← c+ 1, ..., i− 1, i+ 1, ..., nνj,1 = wjif j < i thenνj,1 = h−1(νj,1, νi,j , θi,j)

end iffor k ← j − 1, ..., 1νj,1 = h−1(νj,1, νk,k, θj,k)

end forxj = νj,1if j < n then

for l← 1, ..., j − 1νj,l+1 = h(νj,l, νl,l, θj,l)

end forend ifif j < i thenνi,j+1 = h(νi,j , νj,j , θi,j)

end ifend for

32

F Risk decomposition

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xes

35

G Portfolio study

32.4%

45.5%

11.0%

11.0%

ASSET ALLOCATION

ERC BUDGETING

Other bonds (Inflation,

credit, HY)

Governement Bonds

(nominal)

Equity

Commodity

21.9%

48.5%

22.9%

6.7%

ASSET ALLOCATION

HIGH RISK BUDGETING


credit, HY)

Governement Bonds

(nominal)

Equity

Commodity

30.3%

54.9%

10.0%

4.8%

ASSET ALLOCATION

LOW RISK BUDGETING


credit, HY)

Governement Bonds

(nominal)

Equity

Commodity

20%

20%55%

5%

ASSET ALLOCATION

HIGH RISK CAP


credit, HY)

Governement Bonds

(nominal)

Equity

Commodity

Table 9: Portfolio allocations

36

TO

TA

LR

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IFIC

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risk

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ion

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IFIC

LOW

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K

BU

DG

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-3.2

9%

-3.2

9%

NA

NA

NA

-1.0

8%

0.7

4%

0.2

9%

-2.1

2%

NA

ER

C

BU

DG

ET

ING

-3.0

2%

-3.0

2%

NA

NA

NA

-1.8

3%

0.7

1%

0.1

2%

-2.6

6%

NA

HIG

H R

ISK

BU

DG

ET

ING

-2.3

3%

-2.3

3%

NA

NA

NA

-3.5

8%

0.5

9%

0.0

4%

-4.2

1%

NA

HIG

H R

ISK

CA

P-0

.09

%-0

.09

%N

AN

AN

A-9

.60

%0

.21

%-0

.56

%-9

.25

%N

A

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

LOW

RIS

K

BU

DG

ET

ING

-3.0

2%

-2.3

5%

-0.6

7%

NA

NA

-1.2

5%

0.4

2%

0.2

8%

-1.3

2%

-0.6

3%

ER

C

BU

DG

ET

ING

-2.4

3%

-2.4

1%

-0.0

2%

NA

NA

-1.8

8%

0.3

9%

0.1

0%

-1.6

8%

-0.6

9%

HIG

H R

ISK

BU

DG

ET

ING

-1.6

3%

-2.0

3%

0.4

0%

NA

NA

-3.6

7%

0.3

1%

0.0

3%

-2.5

6%

-1.4

4%

HIG

H R

ISK

CA

P1

.65

%-1

.29

%2

.94

%N

AN

A-9

.54

%0

.04

%-0

.56

%-5

.55

%-3

.46

%

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

LOW

RIS

K

BU

DG

ET

ING

-3.0

7%

-1.9

9%

-0.2

3%

-0.5

1%

-0.3

4%

-1.0

6%

-0.2

3%

0.2

4%

-0.4

6%

-0.6

2%

ER

C

BU

DG

ET

ING

-2.5

8%

-1.9

3%

0.3

1%

-0.6

4%

-0.3

2%

-2.1

3%

-0.2

2%

0.0

8%

-0.5

6%

-1.4

4%

HIG

H R

ISK

BU

DG

ET

ING

-2.0

1%

-1.4

3%

0.7

2%

-1.0

2%

-0.2

8%

-1.9

8%

-0.1

5%

-0.0

5%

-0.9

1%

-0.8

7%

HIG

H R

ISK

CA

P0

.44

%-0

.10

%2

.97

%-2

.27

%-0

.16

%-3

.41

%0

.04

%-0

.77

%-2

.03

%-0

.65

%

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

TO

TA

LR

ea

l ra

te

(min

us)

Infl

ati

on

(min

us)

Ma

rke

t e

qu

ity

risk

av

ers

ion

SP

EC

IFIC

LOW

RIS

K

BU

DG

ET

ING

-2.4

6%

-1.1

6%

-0.1

1%

-0.6

2%

-0.5

8%

1.1

0%

0.6

4%

-0.0

5%

0.3

7%

0.1

5%

ER

C

BU

DG

ET

ING

-2.4

1%

-1.0

4%

0.0

2%

-0.7

8%

-0.6

2%

1.3

6%

0.5

7%

0.0

0%

0.4

5%

0.3

5%

HIG

H R

ISK

BU

DG

ET

ING

-2.3

2%

-0.7

1%

0.1

0%

-1.2

5%

-0.4

6%

1.3

7%

0.4

0%

0.0

3%

0.7

3%

0.2

1%

HIG

H R

ISK

CA

P-2

.30

%0

.40

%0

.58

%-2

.77

%-0

.51

%1

.85

%-0

.16

%0

.22

%1

.63

%0

.16

%

US

D B

ask

et

Fa

cto

r D

eco

mp

osi

ton

of

Va

ria

tio

n o

f E

xp

ect

ed

Re

turn

Fo

llo

win

g S

tre

ss S

cen

ari

os

Wo

rld

In

fla

tio

n B

on

ds

Wo

rld

De

ve

lop

ed

Eq

uit

ies

Wo

rld

Go

ve

rne

me

nt

Bo

nd

sE

me

rgin

g E

qu

itie

s

Eu

ro G

ov

ern

em

en

t B

on

ds

Co

mm

od

ity

Av

era

ge

Are

a C

orp

ora

te B

on

ds

Tabl

e10

:Por

tfol

iofo

llow

ing

stre

sssc

enar

ios

38

risk factors, copula dependence and risk sensitivity of a

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