Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization Jean- Damien Villiers ESSEC Business School Master of Sciences in Management Grande Ecole September 2013 1
Non Linear Dependence Structures: a Copula Opinion Approach in Portfolio Optimization Abstract The recent financial crisis brought out the Risk- on/risk- off notion, characterized by periods where the market is not driven any more by the fundamental analysis. Consequently, a question about the effectiveness of the classic portfolio optimization models has risen: the strong rise of cross asset correlations suggests integrating new market anomalies into models, such as the non- normality of returns or the non- linear dependence structures. The aim of this paper is to provide a methodology to implement a copula approach model, derived from the Black- Littermann methodology. The model outperforms the Black- Littermann model, and reduces sensitively the risk of our portfolio. INTRODUCTION Markowitz proved in 1954 that an appropriate pool of assets could reduce the systematic risk of a portfolio without impacting its return expectancy. However, several criticisms have been made on this model, discussing the simplicity of the optimization based on only mean and variance, the normality of returns assumed, and its operational non- stability making the model hardly usable in practice. Since then, several models have emerged. The Black- Littermann model is one of them, widely used by asset managers: based on both Bayesian markets forecasts and mean- variance optimization, it solves the issue of stability on expected returns, and proposes a way to rely both on fundamental and quantitative analysis. Despite the democratization of this methodology, several weaknesses still remain: the normality of asset returns and their linear codependence modeled by a variance- covariance matrix are not well adapted for the stressed scenarios we ve observed over the previous years. Since the bankruptcy of Lehman Brothers a new constraint must be taken into account: we ve observed a sharp rise of correlations between assets historically uncorrelated, and it seems necessary to widen the possibilities of the Black- Littermann model to take these anomalies into account. In this research paper, we propose a methodology based on codependence structures and marginal distribution functions to model the market, keeping the strengths of the Black- Littermann method. More particularly, we define marginal distribution functions for each type of asset, and quantify the correlations between them through copulas. Finally, we make our asset pooling according to the optimization of Omega ratio, a risk measure depending on the whole distribution of returns. 2
METHODOLOGY The first objective of the model is to simulate the expected future evolution of the market ( Posterior Market Distribution ), mixing the results of a quantitative model based on historical data ( Prior Market Distribution ) and the forecasts of an investor ( Views Distribution ). The cumulative distribution functions are then averaged according to weights depending on the confidence levels the investor has on his views (See Appendix 1). The Prior Market Distribution is the result of Monte- Carlo simulations based on independent marginal distributions for each asset, and copulas to model the interdependency. As we intend to quantify non- linear dependencies between assets in cases of highly stressed markets, it is necessary to use marginal distributions adapted to leptokurtic returns. We assume log normal distributions for FX and Credit products, and use the Heston model for Commodities and Equities products, a stochastic volatility model widely used for its calibration easy to implement, defined by the following stochastic differential equations: = " + = ( ) " + where is the stock price, the instantaneous variance, "# are Wiener processes with correlation ρ, μ is the rate of return of the asset, θ is the long term expected variance, κ the rate at which V t reverts to θ, and ξ is the volatility of volatility. The dependence structure between assets is then modeled by the Student copula (t- copula), extracted from the bivariate Student distribution, according to the Sklar s theorem (see Appendix 2):, ;, =, (, ) with ρ the correlation coefficient and, the bivariate Student distribution with a correlation matrix depending on ρ and k, the degrees of freedom. The choice of this copula is justified by its cumulative distribution function easy to implement and the presence of lower and upper tails dependence, enabling to model non linear dependence of rare events (see Appendix 3). Both copula and marginal distributions are then calibrated on historical data with a maximum likelihood algorithm, and the Prior Market Distribution can finally be built with Monte- Carlo simulations, and stored in a matrix M "# where J is the number of simulations and N the number of assets. 3
The Views Distribution is another input of the model, expressed as a combination of two matrix: - Two vectors contain the lower and upper bounds of the forecasts on returns. For example, if we consider the vectors lb=(0%, 1%) et ub=(2%, 4%), the investor expresses 2 forecasts, in the ranges [0%: 2%] and [1%: 4%]. The assets are assumed to follow uniform distributions between these bounds. In this research paper, the forecasts are voluntarily subjective views on the market and are not the result of any model. - A Pick matrix ℳ"# where K is the number of views and N the number of assets, giving the weights on the assets affected by the kth view (lower and upper bounds vectors above) The Posterior Market Distribution: we finally define a vector ℳ" with weights between 0% and 100%, whether we trust or not our forecasts. 0% would mean a model fully relying on the Views Distribution. We estimate this vector using confidence intervals. More precisely, we deduce the confidence level of each forecast from its variance, which leads to build a volatility model. We choose the heteroscedastic model E- Garch(1,1), described by the following equations: = + ù = log = + log + [ ]+ where and are the return and the variance of the considered asset. Once we have calibrated each view, we estimate the variance of the residuals which follow a Gaussian distribution, and the confidence level from the confidence interval: 1, = "#$( + ) where and are respectively the return and the forecast for each asset. We finally apply this for a Gaussian distribution: =, +, where, is the Gaussian cumulative distribution function, with parameters (0, ), and T a defined threshold. We finally build the final Posterior Market Distribution, which is a weighted average of the Prior Market Distribution and Views Distribution, according to the following formula:,, + (1 ) (, ) where is Prior cumulative distribution function, the Views cumulative distribution function, the confidence in the kth view, and W the sorted returns of the J simulations. Quant Awards 2013 September 2013 4
OPTIMIZATION CRITERIA: THE OMEGA RATIO Once we ve computed the Posterior Market Distribution, we have J simulations of the possible evolution of the market, according to both quantitative ( Prior ) and fundamental ( Views ) analysis. We choose the parametric ratio Omega as criteria to optimize our portfolio and select allocation weights: Ω H = 1 F x dx F x dx Where F is the CDF of the portfolio returns, and H the targeted return. Example of Cumulative Distribution Function: The Omega ratio, introduced by Keating and Shadwick, is interpreted as the ratio of performances over the threshold H divided by the performances under the threshold. If the average return of the portfolio is higher than the defined threshold, the ratio will tend to increase. Consequently, we optimize our portfolio increasing this ratio as much as possible. The strength of this ratio is to take into account more than two moments of the distribution, since it is derived from the CDF. This risk measure is consequently more precise than the Sharpe Ratio. From an algorithm point of view, we are facing a multiple parameter optimization problem, which is solved by a simplex method: the Nelder- Mead optimization technique. 5
INPUTS OF THE MODEL To build the Prior Market Distribution we use the daily closing prices of cross asset single stocks and index, between 14/11/2007 and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he Views Distribution is chosen voluntarily subjective but could be also the result of econometric forecasts, or any external model. The lower and upper bounds are the same for each forecast but we consider 6 different economic scenarios, implying different weights on the 11 assets: "#$% &#'()*+#,-. /00()*+#,-. 1234 " #" /5 " $" 567 " #$" 681 " %" 9:1 " &" ;<*=(>"*?42).@ " &" ;<*=(>"*?A#$2A@ " &" =<*;B,C"C(3 " %" ;<*;B,C"C(3 " %" 1#DD#.C"C(3 " &" 8#A. " &" Pick Matrix: "#$%&'() "#$%&'("%)*+,)(-&. /0 /0 /0 /0 /0 /0 /0 1/0 2/0 1/0 /0,($3'$(456 /0 /0 /0 /0 /0 /0 /0 2/0 2/0 7/0 /0 8)9"3 /0 /0 /0 /0 7/0 /0 /0 2/0 2/0 /0 /0 :&%;"%&'(" /0 /0 /0 1/0 1/0 /0 /0 /0 2/0 /0 /0 :&%;#$%&'(" <=0 /0 7=0 /0 /0 /0 1/0 /0 /0 /0 1/0 >94966'(" /0 =/0 7/0 /0 /0 /0 /0 /0 /0 /0 1/0 6
Finally, the confidence vector is the output of the E- Garch(1,1) model, using confidence intervals at 99%: "#$#%&"'(")$*+&# "#$%&'#(' "#$%&'("%)*+,)(-&. /0123,($4'$(567 80/23 9):"4 /0;23 <&%="%&'(" 10>23 <&%=#$%&'("?10?23 @:5:77'(" 10A23 RESULTS To measure the improvement of the copula approach, we define as benchmark the output of the Black- Littermann model, assuming similar views and economic scenarios. The model is run for 100,000 simulations and the ratio Omega is optimized with a threshold of 1%. We finally get the following basket allocation: ""#$%&'#( )"%$*+,'&&-./%((+)-($0/%.* 1#23"%+42'('#(+22.#%$0 52.-%6 1%70 " #" " 859 $" #" %#" 9:;5 &" ##" %'" :<+1=>?:9 $" (" %(" @A+1=>?:9 '$" &" " >B+,41, )" '" *" >B+@=? (" #" &"?B+>C8:9:>5 $" (" %(" >B+>C8:9:>5 #" #(" %#)" 14BB4 " +" %*" <4,? $" #," %#," 9#&%" #$$" #$$" The backtesting of the model is made out of the calibration window, between December 2011 and September 2012. The portfolio weights are kept unchanged during this period. We get a higher return from 5,43% to 8,39% whilst the volatility and the Omega ratio of our portfolio strongly decrease. ""#$%&'#()*#+," -,&./()#()&0,) 1,/'#+ ((.%"'2,+) 3#"%&'"'&4 5*,6%)-%&'#) 7#8,9&)/,&./( "#$%&'())*+,#--&*-$.,#+% /0123 40513 60227 860253 60243 9:;<"#&=;(-(:-&>;;+:#$. 702?3 20463 6066/ 8504@3 50713 :'60,/)/,&./( 7
CONCLUSION This research paper examines whether the modeling of non- linear term structure dependences between assets can reduce the systematic risk of a portfolio, without affecting its return. The model presented outperforms the Black- Littermann benchmark, both on return and risk measures. Compared to the Black- Littermann model, we mainly highlight three improvements: - The use of copulas enables to model the Risk- on Risk- off phenomenon: in case of high stressed scenarios, the fundamental stock analysis is less effective and a copula opinion pooling would be more reliable. - The simulation of assets based on marginal distributions combined with the Monte- Carlo method widens the scope, letting the possibility to choose a different diffusion process for each asset. - The confidence matrix is one of the hardest parameter to estimate in the Black Littermann model. We propose a way to compute it, based on a volatility model, letting to the investor the forecasts on returns as only subjective inputs. The model presented enables to capture different Market phenomenon as the leptokurticity of returns, the non- linearity of correlations, the heteroscedasticity and asymmetry of the volatility, whilst we let the possibility to take into account any fundamental analysis on stocks. For practical reasons, we ve chosen to apply the model in a very specific case (Student Copula, Heston and EGARCH diffusion process ), but the model can be seen as a generic methodology with further possible improvements (implementation of Jump process, calibration on Market prices, discussion about the copula ). In any case, we suggest measuring the effectiveness of a model with a risk ratio depending on the whole distribution of returns, such as the Omega ratio we ve presented. 8
APPENDIX Appendix 1: The cumulative distribution function of the Posterior Market Distribution is the weighted average between the Prior Market Distribution and the View Distribution. Appendix 2: A copula is a multivariate cumulative distribution function whose marginal distributions are uniform random variables on [0,1]. If C is a copula on R, we can find a random vector (U,, U ) such as: P U u,, U u = C(u,, u ) P U u = u for all i [0,1] A copula can be determined by the previous definition, or using pre- existing multivariate distribution. In this latter case, we use the Skal s theorem, explaining the link between the copula C and the multivariate distribution F, depending on its marginal univariate distributions and. Sklar s Theorem: 9
Let F be a bivariate distribution with marginal distribution and. The copula C associated is defined by: C u, u = C F x, F x = F F u, F u = F(x, x ) C is unique when the margins and are continuous. Appendix 3: Source : «Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications Markus Junker, Alex Szimayer, Niklas Wagner Lower Tail dependence: (,) A copula C has a lower tail dependence if : = lim exists and ]0,1] Upper Tail dependence: (,) A copula C has a upper tail dependence if : = lim exists and ]0,1] 10
REFERENCES Markus Junker, Alex Szimayer, Niklas Wagner, 2004, Nonlinear Term Structure Dependence: Copula Functions, Empirics, and Risk Implications Attilio Meucci, 2005, Beyond Black- Litterman: Views on Non- Normal Markets Attilio Meucci, 2006, Beyond Black- Litterman in Practice: a Five- Step Recipe to Input Views on non- Normal Markets Michael Stein, 2008, Copula Opinion Pooling in Asset Allocation Attilio Meucci, 2011, A New Breed of Copulas for Risk and Portfolio Management Arthur Charpentier, 2010, Copules et risques corrélés, Journées d Études Statistique Fischer Black and Rober Litterman, 1992, Global Portfolio Optimization, Financial Analysts Journal Con Keating and William F. Shadwick, 2002, An Introduction to Omega S.J. Kane and M.C. Bartholomew- Biggs, M. Cross and M. Dewar, OPTIMIZING OMEGA Joanna Gatz, 2007, Properties and Applications of the Student T Copula 11