Risk Factors, Copula Dependence and Risk Sensitivity of a Large Portfolio

Transcription

1 Risk Factors, Copula Dependence and Risk Sensitivity 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 large dimensional portfolio composed of different classes of assets, especially in extreme risk circumstances. In such cases, the usual beta approach is no longer relevant due to the complex- including tail- dependencies that usual correlations are not able to account for. Thus we use the copulas theory but we have also to define a tractable and readable dependence structure. So we combine an ex ante interpretable factorial structure and a CVine copula model in the spirit of the CVine Market Sectors (CVMS) model introduced by Heinen and Valdesogo (2009) to build a CVine Risk Factors (CVRF) model. Our tool allows us to decompose the 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 semiparametric and we quantify the exact contribution of the different possible risk sources to the risk premia of assets and to any usual risk measure applied to portfolios. In particular we refer to the CVaR measure, which is relevant in critical contexts. Illustrations are given which mainly refer to the credit risk but also to the emerging risk which have hit many investors in recent years. Obvious applications concern regulation issues when the portfolio under study is the one of a financial institution. 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, Bd de l Hôpital Paris, France. alexis.flageollet@am.natixis.com, 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 Évry, France. zhun.peng@univ-evry.fr 1

2 1 Introduction Diversification 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 asset classes. The portfolio manager is therefore facing multiple risk sources: without being exhaustive, equity, interest rates, inflation, business cycle, emerging market, credit, or liquidity related risks. Globally speaking, the required risk premium for any asset depends on its covariation with a stochastic discount factor (Cochrane, 2005, 2011) or more intuitively with bad times (Ilmanen, 2011). However we are not able to empirically identify one true factor representative of bad times 1. A multiple risk factor model is thus used to deal with that issue (Ross, 1976). In what follows, we adopt this approach and refer to distinct factors to account for different dimensions of bad times like Ilmanen, (2011), for example.. 2 Moreover, these different risk directions affect the assets in a complex way. For instance, inflation risk can prompt a positive correlation between stock and nominal bond returns during high unexpected inflation periods (via positive risk premium) and the opposite during low unexpected inflation periods where nominal bonds are used to hedge equity risk (Campbell, Sunderam and Viceira, 2013). To deal with this complexity, financial economists often consider time-varying risk discount rates (see Cochrane, 2011) which are for example driven by time-varying risk aversion (Cambpell and Cochrane, 1999) or by the business cycle (Fama and French, 1989). To our opinion, one particularly important issue is thus to account for asset dependencies which increase during financial turbulence periods and reduce diversification benefits at times when these benefits are most needed. Indeed, beyond diversification issues, being able to assess multiple interacting financial risk exposures is nowadays crucial because a poor risk management in a financial institution can lead to an individual or potentially systemic default as witnessed during the last crisis. Improving the risk measure of a portfolio is henceforth at the core of regulation issues. In this regard, the question we mainly address in this paper is the following: what is the sensitivity of a large and complex multi-asset portfolio to extreme shocks related to multiple interacting risk sources? We propose a flexible tool to help answer such a question. Usual risk measures such as correlations and variance are obviously not relevant for that purpose. Indeed the Gaussian framework is not adapted to characterize the risk of a complex portfolio, in particular in extreme situations. We therefore refer to the copula s theory. However, the copulas at hand are thus high dimensional, which is clearly intractable from a practical point of view. Fortunately, any joint density function can be factorized as a product of bivariate copulas and marginal densities. The decomposition is not unique, but it is possible to characterize the dependences from 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 correlations between assets, illiquidity spiral, debt crises, etc. 2. Systematic factor risks must be rewarded because investors care about them. Indeed, investors who support factor risks expect a risk premium as they supply insurance to other ones against enduring losses in bad times. 2

3 Bedford and Cooke (2001, 2002). In what follows, we will use a special case of a Regular vine, namely the Canonical vine (C-Vine) studied by Brechman and Czado (2013), which is particularly well adapted for including factorial structures as the ones that are often used to decompose the risk premia of financial assets. Some authors address the statistical issue of finding the factorial structure that offers the best fit with the data. For example, Tumminello et al. (2007) apply a hierarchical clustering procedure and estimate a hierarchically nested factor model. Brechmann and Czado (2013), who model market returns with a R-vine copula structure, use a maximum spanning tree and adopt a pure statistical approach to discover the link between the assets. However the resulting relationships between financial series are often not easy to interpret. Contrary to these authors, we do not aim at finding the best (statistical) factorial structure, but we rather aim at proposing a tractable factorial dependence structure which combines a C-Vine factorization with asset s return decompositions that are consistent from a financial point of view. Such models refer to a priori views but we check that the associated decompositions are well supported by the data inside the C-Vine framework. More precisely, the tool we propose allows for specifying and estimating what we call a general C-Vine-Risk Factors (CVRF) dependence structure which is an extension of the Canonical Vine Market Sector (CVMS) specification introduced by Heinen and Valdesogo (2009). The C-Vine structure is thus organized and constrained according to a factorial structure which we specify a priori. Factor models state that the return of each asset results from a limited number of risk sources. Multi-factor models can include any type of factor since the theory gives limited guidance (see Ilmanen, 2011). In the remaining of the paper, we focus on a multi-asset portfolio based on 35 different indexes (stocks, government or corporate bonds for various geographic area, currencies and commodities) to encompass a large variety of risk sources. Of course, the ex ante factorial structure we retain in the following depends on the assets at hand but most of the risk factors we thus focus on cannot be 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 for most of our 35 assets and are in the same time mainly driven by the different risk factors we want to identify: according to the identification scheme we retain as shown in figure 2, we retain 3 global indexes that are mostly related to three risk factors, denoted in the following as real interest rates, inflation, and market risk factors and 5 additional indexes which are specifically affected by (European) sovereign crisis, credit, USD, emerging and commodities related risks. The 5 latter indexes are used to emphasis the 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 ( i.e. the two last financial cycles) in order to avoid strong regime shifts. For sake of illustration we can report to Figure 2 which summarizes the identification scheme we adopt for the risk factors which are captured from representative indexes. Note that we choose well diversified indexes because we assume that diversification magnifies embedded risk premiums by diminishing the noise (idiosyncratic risk). 3

4 The ordering described in the figure permits us to extract the risk factors we want to capture and to organize the C-Vine copula in the same time. For example, we claim that the two first indexes (World Linked Index and World Government Bonds Index (WGBI All maturities) which are mainly affected by interest rate and inflation risks contribute to the risk premium of any asset of our data base. Moreover,we use these two indexes to extract two risk factors, namely the real rate risk factor and the inflation risk factor. More precisely, our CVRF model allows us to decompose the response of the return of any asset to extreme shocks to any other asset. For each shock, the impact is captured through the changes in the distribution of the return after the shock, taking into account not only its marginal distribution but also its co-movements with the other assets through the factorial dependence structure. In this regard, all impact measures are obtained from semi-parametric estimations. More precisely, we successively focus on extreme shocks to the 8 indexes retained as the main common components of the returns of our assets, and, each time, we decompose the response of any asset and more specifically the change in its expected return induced by the shock, into the contributions of the risk factors we want to capture. Accordingly we are able to quantify the sensitivity of any asset to any extreme shock and to jointly decompose this sensitivity into the marginal contributions of the risk factors. This decomposition requires simulations of the returns of all assets after drawing extreme values of non conditional and conditional distribution functions in the CVRF model framework. For the latter case we develop an original algorithm. This is our CVRF s core application from which we propose risk sensitivity analysis 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 a low risk budgeting portfolio. Finally, we consider the most standard capital budgeting portfolio used in the USA (60% risky assets/40% bonds). The paper is organized as follows. In section 2, we present the principles of C-vine copulas and our C-vine risk factor model. Section 3 is devoted to the practical implementation with a presentation of the data, of the factorial structure and the principles of the different simulations that are used later. Section 4 is devoted to the risk sensitivity analysis developed for the returns of the assets of our database and for different benchmark portfolios. Section 5 concludes. 2 Dependence structure: Copulas, Canonical Vine and Factors In this section we recall the definition of copulas, the principles of a CVine dependence structure and, finally, we show how to include a factorial structure to get our CVRF model. 4

5 2.1 Canonical vine A n-dimensional Copula C(u 1,...u n ) 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 variables whatever their marginal distribution. Indeed, according to the Sklar s theorem (Sklar, 1959) a multivariate cdf F of n random variables X = (X 1,..., X n ) with marginals F 1 (x 1 ),...,F n (x n ) can be written as: F (x 1,..., x n ) = C(F 1 (x 1 ),..., F n (x n )), (1) where C(F 1 (x 1 ),..., F n (x n )) = F (F1 1 (u 1 ),..., Fn 1 (u n )) is some appropriate n- dimensional copula and F 1 i s denoting the quantile functions of the marginals. Accordingly, modeling of margins and dependence can be separated by the way of copulas. Moreover, for an absolutely continuous F with strictly increasing, continuous marginal cdf F i, we get the joint density function f by differentiating (1), f(x 1,..., x n ) = c 1:n (F 1 (x 1 ),..., F n (x n )) f 1 (x 1 ) f n (x n ), (2) which is the product of the n-dimensional copula density c 1:n ( ) and the marginal densities f i ( ). Then, the n dimensional density c 1:n can be decomposed as a product of bivariate copulas. The decomposition is not unique (See a possible decomposition in the trivariate 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 regular vine. Regular vines (R-vines) are a convenient graphical model to hierarchically structure pair copula constructions. A special case of regular vines is the canonical vine 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 T 1,..., T n 1 with nodes N i and edges E i for i = 1,..., n, where T 1 has nodes N 1 = 1,..., n and edges E 1, and for i = 2,..., n-1, T i has nodes N i = E i 1. Moreover, two edges in tree T i are joined in tree T i+1 only if they share a common node in tree T i (See Brechman and Czado (2013) for a detailed presentation). A special case of R-vines which is often considered are canonical vines (C-vines). A C-vine is a R-vine if each tree T i has a unique node with degree d i, the root node. The general n-dimensional canonical vine (CVine) copula density can be written as following: c 1:n (F 1 (x 1 ),..., F n (x n )) = n 1 n j j=1 i=1 (F (x j x 1,..., x j 1 ), F (x j+i x 1,..., x j 1 )) c j,j+i 1,...,j 1 (3) where c j,j+i 1,...,j 1 denotes the bivariate copula between the distributions of x j and x j+i taken conditionally on x 1,..., x j 1. Figure 1 shows a canonical vine with five variables. From the figure, we observe that the variable 1 at the root node is a key variable that plays a leading role in governing interactions in the data set. 5

6 Figure 1: A five dimensional canonical vine tree In the first tree, all nodes are associated with the X 1,..., X 5 variables. For example, the edge 12 corresponds to the copula c(f 1 (x 1 ), F 2 (x 2 ). In the second tree, the edge 23 1 denotes the copula c(f 2 1 (x 2 x 1 ), F 3 1 (x 3 x 1 )). The following trees are built according to the same rules. In order to organize the dependence structure, it is useful to recall how to characterize the independence of two variables in terms of copula. 2.2 Conditional independence in canonical vine For a complete n-dimensional canonical vine, there are n(n 1)/2 bivariate copulas. This means that the numbers of parameters to estimate is very high for a large size portfolio. In order to simplify the structure, some conditional independence assumptions may be useful. If one refers to the three dimensional case (See Appendix), assuming that X 1 plays a leading role leads to the following factorization: c 23 1 (F 2 1 (x 2 x 1 ), F 3 1 (x 3 x 1 )) = 1 which means that x 2 and x 3 are independent, conditionally on x 1. Hence, the structure simplifies to: c(f 1 (x 1 ), F 2 (x 2 ), F 3 (x 3 )) = c 12 (F 1 (x 1 ), F 2 (x 2 )) c 13 (F 1 (x 1 ), F 3 (x 3 )). 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: c xy υ (F x υ (x υ), F y υ (y υ)) = 1. (4) 6

7 Table 1: Factorial dependence matrix M s CC 1 CC 2 CC 3 CC 4 a 1 a 2 a 3 a 4 f 1 1 f f f a a a a Heinen and Valdesogo (2009) use this property to develop a simplified version of canonical vine, namely the Canonical Vine Market Sector (CVMS) model. This twofactor model assume that each asset depends on the market and on its own sector. To include this model into a canonical vine structure with the market and the sectors as root nodes, some conditional independence assumptions need to be introduced: conditionally on the market, sectoral returns are assumed to be independent and asset returns are independent once they belong to different sectors. The remaining dependence of asset returns taken conditionally on the market and the respective sectors is modeled with a multivariate Gaussian copula. The example given in Appendix illustrate this simplified 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 which mainly explain their dependence structure. Further, we loosen the usual conditional independence assumptions and assume that the risk factors can depend on each other while asset returns can depend on one or several risk factors at the same time. The specification of the factorial dependence structure is therefore more flexible than in the CVMS setting and can be used in accordance with any particular view of a portfolio manager. As shown for example in table 1, the unconditional and conditional dependence structure can be specified in a symmetric matrix with dummy variables. Among the n = 8 assets in the table, we distinguish between CC-type assets which denotes indexes that are common components - or "factors" in the usual sense - for all assets and a-type ones which refer to the other asset of the database. The random variables are the corresponding returns, r i ; i = 1,..., 8. If the dummy variable in the ith row and jth column d ij is equal to 1, the return of asset a j (or of common component CC j ) is related to the return of asset a i (or common component CC i ), conditionally on the returns of any asset (or common component) preceding a j (e.g., r j 1,r j 2,...,r 1 ). If d ij = 0, the pair is conditionally independent, and the density of the associated copula is equal to one. 7

8 Constraining the previous matrix M s allows us to impose any dependence structure specified "a priori". All diagonal entries are equal to 1 since each asset is obviously linked with itself, but imposing that all elements of the first column are equal to 1, d i,1 = 1 means that the returns of all assets (including the ones of the common components CC 2, CC 3 andcc 4 ) depend on the first common component CC 1. We can impose conditional independence or dependence between the common components; here, d 3,2 = 0 and d 4,3 = 1, respectively mean that CC 2 and CC 3 are independent, conditionally on CC 1, and CC 4 and CC 3 are dependent, conditionally on CC 1. Moreover, each asset can share just one common component or several ones: for example, a 1 is only related to CC 2 conditionally on CC 1 and a 3 to CC 3 and CC 4, conditionally on CC 1. In the same way, assets can be dependent or independent on each other conditionally on the common components: for example, d 8,6 = 1 means that a 2 is related to a 4 given the 4 common components while d 8,7 = 0 means that a 3 and a 4 are independent, conditionally on these 4 components. Moreover, for each pair of related assets (d i,j = 1), the dependence is further characterized by one copula chosen in a set of various bivariate copulas. In what follows, we retain the simplified structure which is summarized by graph 1 in Appendix; we describe it in details in the following section. 3 Practical implementation In what follows, we work with a database composed of 35 indexes (stocks, bonds, currencies and commodities); the observation frequency is weekly over the period 01/05/2001 to 09/27/2013. We suppose that we have a particular (factorial) structure capturing the following risk directions: real (interest) rates, inflation, global equity, credit, emerging equity, commodities, USD. Within the C-vine structure, each conditioning is associated with an underlying factor and block independence implies that assets earn only up to 4 risk premiums according to the "ladder" structure presented in graph Marginal Distribution First, we have to specify the marginal distributions and the copulas to characterize the joint distribution of the returns of all indexes. Concerning the marginal distributions, there are different approaches. We have retained a usual ARMA-GARCH specification with a GED Distribution of the standardized residuals. As mentioned before, any other characterization of the marginal distributions could be retained. The details are 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 in most cases. This means that most of the distributions have thicker tails than the normal distribution. 8

9 3.2 The choice of copulas and the tail dependencies Having estimated the marginal parameters, we transform the standardized residuals into uniform residuals by using the approach proposed by Meucci(2006). In the second step, we fit a bivariate copula model to the standardized residuals by integrating the structure in section 2.3 into a C-vine copula described in the section 2.1. The bivariate 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 to the Student and Clayton ones which allow for symmetric and lower tail dependencies respectively. Based on the dependence structure, it s naturally to assume that most tail dependencies are captured by the first two global indexes. Therefore, we consider only the bivariate copulas between each of the three global indexes and others indexes. About 50% of the bivariate copulas chose copula families with tail dependance as the best fitted model in which about 10% chose Clayton copula with lower tail dependance and 40% chose student t copula with two sides tail dependencies. We find indeed evidences of tail dependence between indexes. Frequency in % Gaussian % Student t % Clayton 8 8.1% Frank % % Table 2: Results of the bivariate copulas between the two global indexes and others indexes Finally, we have to choose the "ex ante" factorial structure. 3.3 Factorial structure Asset 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. Specific realized bond premia (Credit, Emerging, etc.) and realized equity premium are often calculated as the spread between the corresponding asset returns and the nominal bond return 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 (see below). Furthermore nominal bond risk embodies real rate and inflation risks; that is why we consider the real rate risk as the first pivotal factor and then the inflation risk as a second one. Once we remove the nominal bond rate factors (split into real rate and inflation factors), we got a set of risk factors (Equity, Credit, etc.). We choose the equity risk as the 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 9

10 remaining risk factors and permits us to spread a negative shock from any other risk factors to all asset returns included in our database. Finally, we define specific or residual factors as those obtained once we have controlled assets returns from the first three risks (labeled global factors). Assets in our portfolio permit us to study credit risk, Euro debt concerns factor, USD, commodities and emerging equity risk. Thus we retain eight factors. Among them, we have three global common factors which are expected to influence most of indexes in the portfolio. The five other factors are more specific because they are shared by a limited number of indexes. The structure can be associated with a tree which helps showing different groups of indexes. At a given level, one index is associated with a particular node and it is related to the factors whose nodes share a branch with its node. However some particular 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 but also with the currency index. The complete dependence structure is described in Table 3 and 4 in Appendix. The table 5 displays Kendall s taus which is an average rank correlation between assets and the considered factor (labeled by column). We should keep in mind when we move to the next column we measure the average rank correlation between the factor and 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 return of the world inflation protected government bonds 3. TIPS (US case) are the riskless 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 real interest rate risk, especially because they can be temporarily prone to liquidity concerns as in In addition, bond returns are negatively related to interest rate thus our factor is minus real rate because the positive Kendall s tau. That factor separates bonds from equities (developed more than emerging) but the rank correlation is quite obviously much higher for bonds than equities (in absolute value). Basic theory states that equity is negatively related to real rate (as bonds) because higher real rate increase the discount rate reducing the price. In our sample government bonds (best rated) are considered 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 equity returns and real interest rate (i.e. negative rank correlation)? Surprisingly, high-yield bond returns are not related to real rate because rank correlations are not significantly different from zero. Credit premium is the main driver (see below). Commodities are globally unrelated to real rate. Currencies (against USD) are also not linked to real rate 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 our sample. The second global factor is the inflation factor effect. As expected, US and UK inflation government bond indexes (most represented in the world index) have no inflation sensibility while Euro index shows some residual inflation sensibility. However, this negative relation to inflation (positive rank correlation) is much lower than 3. The three first most important exposures are US, UK and France. 10

11 traditional government bonds (nominal rate). This phenomenon may be due to higher liquidity concerns in the Euro inflation index than those affecting the world inflation index. Government bond indexes are almost equally sensitive to inflation and to real rate. Nevertheless, empirical results reveal higher Euro and German inflation sensibilities which are intuitive outcomes (Central bank inflation fear is well known). Regarding risky assets, equities are similarly positively related to inflation. The link between equities and inflation depends mainly on the source of inflation: a demand driven inflation causes a positive relation with stock returns; a supply driven inflation causes a negative correlation (Lee, 2009). In our sample inflation is mainly driven by demand. Interestingly, oil has the highest positive rank correlation with inflation. This result is consistent with the fact that oil has the best inflation hedging ability. Precious metal index (Gold) has a weak link with inflation confirming the fact that gold is not really an inflation hedge. Gold is regarded as a safe haven against financial turmoil and US dollar 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 that government bonds have a weak but negative rank correlation with risk. We expected such a relation because the studied period encompassed flight-to-quality episodes. On the contrary, euro government bonds are positively related to risk aversion reflecting the euro-area debt concerns which occurred at the end of the sample. As expected, corporate and emerging bonds (premia) have positive rank correlations with equity risk. The high-yield sensitivity is naturally higher than investment grade sensitivity. We now turn to residual specific factors 4. Euro high-yield bonds are surprisingly not sensitive to the residual euro debt factor. We linked emerging bonds with credit factor because emerging bond spread is generally viewed as a measure of an emerging economy s creditworthiness. Besides, the rank correlation is slightly higher than the one with the emerging factor. All currencies or basket of currencies have a negative correlation with the USD factor because all studied currencies are short USD whereas our factor is long USD. It is worth noting that gold could be seen as a currency and seems to be negatively related to the USD factor confirming its dollar hedge ability in case of USD weakness. Asian stocks show the most important rank correlation with emerging equity risk reflecting the weight of Asian countries within the emerging index. As previously, oil has the highest rank correlation with commodity factor because of its weight in the global index. 3.4 Return simulations We implement two types of simulations with "not conditional" and "conditional" shocks General Simulation To run the simulations, we proceed as follows. First, by using the estimated parameters 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 risk aversion factor into account in each specific risk. 11

12 we simulate N samples from an I dimensional canonical vine for the next period, i.e. û 1:I,T +1. Then, the inverse error distribution functions (G 1 ) produce a sample of standardized residuals, i.e. ẑ 1:I,T +1 = G 1 (û 1:I,T +1 ). Finally, according to the GARCH equation ( 6) in Appendix, the observed initial values and the estimated ARMA-GARCH parameters are used to compute the return forecasts for i = 1 : I, with the variance forecast, ˆr i,t +1 = ˆµ i + ˆφ i r i,t + ˆθ iˆσ i,t ẑ i,t + ˆσ i,t +1 ẑ i,t +1 ˆσ 2 i,t +1 = ˆω i + ˆα iˆσ 2 i,t ẑ 2 i,t + ˆβ iˆσ 2 i,t Simulation with extreme non conditional shocks In the following, we focus on the simulations of uniforms from the vine structure while 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 behaviour of one index. Indeed, instead of drawing all the û 1:I between 0 and 1, we draw samples from a extreme zone (for example from 0 to 0.05) for the stressed variable û i, i {1,..., I}. Since the dependence structure is supposed to be unaffected by the shock, a stress situation for one factor impacts not only the variables which are directly related to this factor but also others variables in an indirect way, by affecting the key factor at the root node of the C-vine that are related to all variables. This means that a sharp decrease of one factor can cause the distress of the whole portfolio if other assets depend positively on this factor. The algorithm is given in Brechmann et al. (2013) Simulation with extreme conditional shocks Moreover we can apply shocks from conditional distributions which are interpreted as shocks to specific risk sources. First of all, some definitions of risk sources need to be clarified. The non conditional distribution of an index-factor f summarizes a set of different risk sources, whereas the conditional distribution of factor f i given another factor f j can be interpreted as a combination of the remaining risk sources when the risk associated with f j has been removed. By considering the gap between the returns associated with the non-conditional and conditional distributions, we can isolate the effect 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 (x i x 1, x 2,..., x c ). We adapt here the simulations for C-vine copulas involving conditional distributions. For j c or j > i, the sampling procedure from F (x j x 1, x 2,..., x j 1 ) is the same as the one described before. However, sampling from F (x j ), c < j i given the F (x i x 1, x 2,..., x c ) has to be modified. We develop a new algorithm to specifically deal with simulations involving conditional shocks. See Appendix. In the next section, we propose two applications of the tool to portfolio management. 12

13 4 Applications to portfolio management in critical contexts 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 other asset and we decompose the corresponding response into the marginal contributions of the different risk factors. In what follows we just focus on extreme shocks to the 8 first indexes of our data base, because they mainly drive the comovments of the assets and may accordingly dramatically increase the risk of a portfolio in case of extreme events. Indeed, limiting the stress tests to this type of extreme shocks is natural when we extend the same type of analysis to portfolios as presented in a second stage. 4.1 Sensitivity analysis for the assets Our 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 decomposition depends on the origin of the shock. Suppose that we want to measure the total sensitivity of index j to a shock to one of 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 Sensitivity j i = E(R j F (R i ) < 5%) E(R j ) where E(R j F (R i ) < 5%)) denotes the expected return of asset j when the (non conditional) distribution F (R i ) is stressed in its extreme negative part and E(R j ) is the return obtained from a general simulation without shock. In case of an extreme shock to i, the shock simply corresponds to a draw in the extreme (negative) part of F (R i ) and all other returns are simulated conditionally on this extreme draw. The expected return E(R j ) gives us a benchmark value corresponding to a situation without shock; the sensitivity of index j to any extreme shock to index i is simply calculated as the deviation from this benchmark value due to the shock. Then we can decompose the previous sensitivity into the marginal contributions of the 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 contribution j = E(R j F (R i ) < 5%) E(R j F (R i R 1 ) < 5%) 5. For example, if the shock comes from the first index, we just identify the contribution of the real rate risk factor. In that case, we can not measure the contributions of the other risk factors because they do not impact the first index according to Figure 2. 13

14 where E(R j F (R i R 1 ) < 5%) is the return of index j when the conditional distribution F (R i R 1 ) is stressed. In the case where i = 5, there are only three risk factors underlying the conditional distribution F (R i R 1 ):, Market and Credit. By conditioning on the first index which is a proxy for the Real Rate risk, the corresponding risk is indeed removed from index i. Consequently, the sensitivity of index j to the conditional shock to index i given R 1 comes only from the exposition to the three remaining risk factors. The difference between the responses obtained for the two cases gives 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 inflation risk factor, that is: risk contribution j = E(R j F (R i R 1 ) < 5%) E(R j F (R i R 1, R 2 ) < 5%) with E(R j F (R i R 1, R 2 ) < 5%) denoting the expected return of index j when the conditional distribution F (R i R 1, R 2 ) is stressed. Still referring to index i = 5, when conditioning on R 1, R 2, there are only two risk factors underlying the distribution F (R i R 1, R 2 ) -Market and Credit-, as the two first risk factors (Real rate and ) are removed by conditioning on the first two indexes. In this case, the sensitivity of index j comes from the exposition to the two remaining risk factors affecting index i. The difference between the two responses thus give us the marginal contribution of the risk factor to the total sensitivity of index j to an extreme shock to index i. Lastly, we can measure the marginal contribution of the Market risk factor according to the same principle: Market risk contribution j = E(R j F (R i R 1, R 2 ) < 5%) E(R j F (R i R 1, R 2, R 3 ) < 5%) with E(R j F (R i R 1, R 2, R 3 ) < 5%) denoting the expected return of index j when the conditional distribution F (R i R 1, R 2, R 3 ) is stressed. Finally we obtained the marginal contribution of the ith-specific risk (1 i 4) as follows: ith-specific risk contribution j = E(R j F (R i R 1, R 2, R 3 ) < 5%) E(R j ) The total sensitivity is obviously obtained as the sum of the previous marginal contributions. 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 in line with Kendall s tau (cf. section 3.2); 2. shocks on World government bonds index spread clearly through the inflation factor for risky assets; 3. the impact of the shocks on Eurozone sovereign bond index are smaller because of the positive correlation between the specific factor Euro and risky assets ; 14

15 4. specific shock credit lowers the stocks, the effect of the inflation factor is less important than before (negative shock on credit are not conducive to increases of inflation); 5. when the World equity market index is stressed, the real rate effects are negligible for the risky assets. Fixed-income assets increase through the first two factors to the same extent; 6. when we stress the USD index (the performance of the USD against a basket of currencies, long USD), the principal effects here are specific effects, we notice that 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 effects in 5); 8. while we stress commodity index, we have positive impacts for the bonds through the inflation factor and negative impacts for risky assets through the risk aversion factor; For sake of illustration let us focus on the effects of a particular shock. Let us consider a negative shock to developed equity markets. We find from our historical sample ( ) that equity is expected to lose around 18% ( -17,7%) a month and US government bond (7-10 years) is expected to rise about 3% (%2,9) a month. US government 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 marginal effect and 50bp for equity risk marginal effect. See Table 8. In the last section we measure and compare the risk sensitivities of different portfolios, when they are exposed to different types of extreme shocks. 4.2 Sensitivity analysis for portfolios As we retain three portfolios composed according to risk budgeting rules we first briefly recall the general principles of such portfolio allocation strategies Principles of risk budgeting Risk budgeting is an approach to investment portfolio management which focuses on allocation of risk rather than allocation of capital. The risk of a position is usually assessed in terms of measures as the standard deviation, the Value-at-Risk (VaR) of the expected shortfall or Conditional VaR, (CVaR). The risk R(w) of a portfolio is a function of the composition w. All previous measures are homogeneous. Accordingly, the following identity holds: R(w) = n i=1 w i R(w) w i (5) In other words, total risk can be expressed as the sum of the contributions from the different assets, when the generic i-th contribution is the product of the "per unit" marginal contribution R(w) w i and the "amount" of the i-th asset, as represented by w i. 15

16 In what follows, we retain the CVaR as risk measure. Indeed, its allows us to better exploit our characterization of the dependencies, including the tail ones. Moreover, choosing this risk measure avoids to excessively focus on the specific risk when looking 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) w i = q cs w where q c is a step function that jumps from 0 to 1/cJ at a rescaled confidence level cj of the expected shortfall(cvar) and S w is a JxN panel, with the generic j-column defined 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 F w. When the number N of assets is large, practitioners prefer to analyze risk at an aggregated level. We follow this practice and define 4 buckets each composed of similar assets that are mainly exposed to the same type of risk; Other bonds, Government bonds, Equity and Commodity. It is thus natural (see Meucci, 2007) to define the contribution to the risk of the k-th bucket as the sum of the individual contributions from the assets in this bucket: C k = R(w) w i i N k In what follows,we propose to compare the risk sensitivity of different types of portfolios to extreme shocks 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%) respectively for the 4 groups; Low Risk Budgeting portfolio with risk budget as (30%, 50%, 15%, 5%) respectively for the 4 groups; Weight budgeting portfolio with high risk allocation as (20%, 20%, 55%, 5%) respectively for the 4 groups As 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 portfolio to each extreme shocks into the marginal contributions of our 8 risk factors. This decomposition can be summarized as follows: w i 16

17 Real rate risk p = E(R p F (R i ) < 5%) E(R p F (R i R 1 ) < 5%) risk contribution p = E(R p F (R i R 1 ) < 5%) E(R p F (R i R 1, R 2 ) < 5%) Market risk p = E(R p F (R i R 1, R 2 ) < 5%) E(R p F (R i R 1, R 2, R 3 ) < 5%) ith-specific risk p = E(R p F (R i R 1, R 2, R 3 ) < 5%) E(R p ) where R p 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 nearly insensitive to inflation risk whereas aggressive allocation is rather insensitive to real rate risk compared to the other ones. It is worth noting that defensive portfolios are not sensitive to inflation risk in a similar way. High risk budgeting portfolio is positively sensitive to inflation risk whereas low 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 our corporate bond index. But the spreading is completely different. The main channels for defensive portfolios are real rate risk and market equity risks with marginal effects which contribute slightly differently depending on risky asset weights in the portfolio. However, within the aggressive portfolio, the negative corporate bond shock only spreads through equity market risk via an increase of risk aversion or downward growth concerns associated to a negative credit risk shock. This result is consistent with economic intuition. All residual marginal effects are obviously proportional to the weight of the most representative assets of the stressed index in the portfolio. Moreover, residual marginal effects are globally quite low compared to other risks. However, it is worth noting that we can have a portfolio with a relative smaller weight in a specific asset (thus smaller residual effect) but facing larger spillover effect through equity market risk (see Euro government bonds and commodity stress scenarios). Finally, in our example and with traditional asset classes, we show that inflation risk could be diversified away with an appropriate balanced portfolio while (real) rate risk and equity risk remain the most important risk concerns which can only be diversified away within a more trivial and more highly risk concentrate portfolio. 5 Conclusion The 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 have proposed a CVFR model combining a Canonical Vine and a factorial-type dependence structure specified a priori. Accordingly a portfolio manager can easily use this model to impose any dependence structure reflecting his own risk perception and to decompose the returns into risk factors which are crucial to his opinion (bond, equity, infla- 17

18 tion, credit for example), while taking into account complex relationships between the different assets that can not be summarized by simple correlations. As an application, we have examined the case of a set of 35 indexes of different types - stock, bonds, commodities, currencies. The marginal distributions of the weekly returns 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. The shape parameter of the GED show that most of the marginal distributions have thicker tails than the normal distribution. As to copula results, evidences of tail dependence are found between a significant number of indexes. The factorial-type structure we have specified a priori includes 8 indexes as common components from which we have identified 8 different risk factors corresponding to real rate, inflation, market, credit, (European) sovereign debt, UDS, Emerging and Commodity risks. The core applications of our model are sensitivity analysis for each asset of our data base to extreme shocks to any other asset and particularly to the 8 indexes which mainly account for the co-movements of the assets. Moreover our model allows us to decompose the total sensitivity of each asset into the marginal contributions of the risk factors we are able to identify. All these sensitivity analysis take into account the complex dependence structure among the 35 indexes we retain, including the tail dependencies which are particularly crucial in case of extreme shocks. All results we obtained from simulations. In this regard our approach is semi-parametric. We have applied the same type of sensitivity analysis to portfolios and compared the sensitivity of several portfolios to different types of extreme shocks. All these stress test exercises show that our model is well adapted to provide a portfolio manager with a general measure of the exposition of a wide range of assets and portfolios to various risk sources especially in critical (extreme risk) circumstances. Moreover, the decompositions of the sensitivities we propose into the contributions of the risk factors tant be identified should help him more effectively choose a mix of asset classes that best diversifies his risks while also reflecting his views on the global economy and financial markets, as summarized by the factorial-type structure he retains a priori. References [1] Aas, K., C. Czado, A. Frigessi and H. Bakken, Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, Elsevier, vol. 44(2), , April. [2] Adler, T. and M. Kritzman, Mean-Variance versus Full-Scale Optimisation: In and Out of Sample. Journal of Asset Management, Vol. 7, No. 5, , [3] Ang, Andrew, The Four Benchmarks of Sovereign Wealth Funds. Columbia Business School and NBER, September