Portfolio Optimization on the Dispersion Risk and the Asymmetric Tail Risk
aa r X i v : . [ q -f i n . P M ] S e p Portfolio Optimization on the Dispersion Risk and theAsymmetric Tail Risk
Young Shin Kim ∗ September 22, 2020
Abstract
In this paper, we propose a market model with returns assumed to follow a multivariatenormal tempered stable distribution defined by a mixture of the multivariate normal distribu-tion and the tempered stable subordinator. This distribution is able to capture two stylizedfacts: fat-tails and asymmetry, that have been empirically observed for asset return distribu-tions. On the new market model, we discuss a new portfolio optimization method, which isan extension of Markowitzs mean-variance optimization. The new optimization method con-siders not only reward and dispersion but also asymmetry. The efficient frontier is extended toa curved surface on three-dimensional space of reward, dispersion, and asymmetry. We alsopropose a new performance measure which is an extension of the Sharpe Ratio. Moreover, wederive closed-form solutions for two important measures used by portfolio managers in portfo-lio construction: the marginal Value-at-Risk (VaR) and the marginal Conditional VaR (CVaR).We illustrate the proposed model using stocks comprising the Dow Jones Industrial Average.First, perform the new portfolio optimization and then demonstrating how the marginal VaRand marginal CVaR can be used for portfolio optimization under the model. Based on the em-pirical evidence presented in this paper, our framework offers realistic portfolio optimizationand tractable methods for portfolio risk management.
Key words:
Portfolio Optimization, Asymmetry Risk Measure, Normal Tempered StableDistribution, Marginal Contribution, Portfolio Budgeting, Value at Risk, Conditional Value atRisk
It admits no doubt that the mean-variance model formulated by Harry Markowitz (1952) is amajor contribution to the portfolio theory in finance. Although some assumptions of the model ∗ College of Business, Stony Brook University, New York, USA ([email protected]). The author grate-fully acknowledges the support of GlimmAnalytics LLC and Juro Instruments Co., Ltd. The author is grateful toMinseob Kim, who reviewed this paper and corrected editorial errors. Also, all remaining errors are entirely my own. α -stable subordinated Gaussian distribution (Rachev and Mittnik (2000)), the inverse Gaussiansubordinated Gaussian distribution (Øig˚ard et al. (2005), Aas et al. (2006), Eberlein and Madan(2010)), the inverse Gamma subordinated Gaussian distribution (Stoyanov et al. (2013)) and thetempered stable subordinated Gaussian distribution (Barndorff-Nielsen and Shephard (2001) andBarndorff-Nielsen and Levendorskii (2001)).The second assumption is the use of the portfolio variance as a risk measure. Since assetreturn distribution does not follow Gaussian distribution but exhibits fat-tails and asymmetry, arisk measure would better be able to assess not only dispersion but also asymmetry. Portfoliooptimization with asymmetry risk measure has been discussed in King (1993) and Dahlquist et al. (2017). Since the asymmetric risk indirectly measured by the coherent risk measures such asthe conditional value at risk (CVaR) by Pflug (2000) and Rockafellar and Uryasev (2000, 2002),the portfolio optimization with coherent risk measure has been studied in Rachev et al. (2007),2ansini et al. (2007), Rachev et al. (2011), and Kim et al. (2012).In this paper, we propose a non-Gaussian market model that returns are assumed to follow thenormal tempered stable (NTS) distribution. The NTS distribution is the tempered stable subordi-nated Gaussian distribution. It is asymmetric and has exponential tails that are fatter than Gaussiantails and thinner than the power tails of α -stable distributions. For that reason, it can describe theasymmetric and fat-tail properties of the stock return distribution. Since it has exponential tails, ithas finite exponential moments and finite integer moments for all orders such as mean, variance,skewness, and kurtosis. By standardization, we obtain a NTS distribution having zero mean andunit variance. That can be used as the innovation distribution of a time series model including theARMA-GARCH model. Since the distribution is infinitely divisible, we can generate a continuous-time L´evy process on the NTS distribution. For this good properties, the distribution was popularlyused in finance; portfolio optimization(Eberlein and Madan (2010), Kim et al. (2012), Anand et al. (2016)), risk management (Kim et al. (2010), Anand et al. (2017), Kurosaki and Kim (2018)),option pricing (Boyarchenko and Levendorski˘i (2000), Rachev et al. (2011), Eberlein and Glau(2014), Kim et al. (2015)), term structure of interest rate model (Eberlein and ¨Ozkan (2005)), andcredit risk management(Eberlein et al. (2012), Kim and Kim (2018)).An important measure derived from portfolio optimization is the marginal risk contribution.The marginal risk contribution help managers to make portfolio rebalancing decisions. This riskmeasure is the rate of change in risk, whether it is variance, Value-at-Risk(VaR), or CVaR, withrespect to a small percentage change in the size of a portfolio allocation weight. Mathematically itis defined by the first derivative of the risk measure with respect to the marginal weight. Becauseof the importance of this measure in portfolio decisions, a closed-form solution for this measureis needed. The general form of marginal risk contributions for the VaR and CVaR are provided inGourieroux et al. (2000). Moreover, the closed-form of the marginal risk contributions for VaR andCVaR are discussed under the skewed- t distributed market model in Stoyanov et al. (2013), underthe NTS market model in Kim et al. (2012), and under the Generalized Hyperbolic distribution3odel in Shi and Kim (2015).The contributions of this paper are as follows. First, we construct a market model using themultivariate NTS distributed portfolio returns. Different from the NTS market model in Kim et al. (2012), the market model in this paper is constructed by only the standard NTS distribution whichis a subclass of NTS distribution having zero mean and unit variance. Using the new NTS marketmodel, we discuss the portfolio optimization theory considering dispersion risk and asymmetryrisk. As the Markowitz model, the standard deviation is used for the dispersion risk measure. Inaddition to the dispersion risk measure, the asymmetric tail risk measure is proposed in order tocapture the asymmetry risk in portfolio optimization. It is a weighted mean of asymmetry param-eters of the standard NTS distribution. Using those two risk measures, we find an efficient frontiersurface which is an extension of Markowitz’s efficient frontier . Moreover, a new performancemeasure of a portfolio on an efficient frontier surface is presented. The new performance mea-sure is an extension of the Sharpe ratio (Sharpe (1966, 1994)). Finally, we provide closed-formsolutions of the marginal risk contribution for VaR and CVaR under the NTS market model. Themarginal VaR and marginal CVaR formula in this paper are simpler than the solutions presented inKim et al. (2012), and hence we can discuss the iterative risk budgeting which was not discussedin Kim et al. (2012).Empirical illustrations are provided for each topic in the paper with performance tests. Dataused in the empirical illustrations are historical daily returns of major 30 stocks in the U.S. mar-ket. We draw the efficient frontier surface based on the estimated parameters of the NTS marketmodel. The performance measure maximization strategy is also exhibited and it is compared tothe traditional Sharpe ratio maximization strategy by backtesting. We calculate Marginal VaR &CVaR and perform the risk budgeting using calculated marginal VaR & CVaR.The remainder of this paper is organized as follows. The NTS market model is presented inSection 2. The portfolio optimization with dispersion and asymmetry risk measures is discussed in The similar study has been presented in Shi and Kim (2015) under the Generalized Hyperbolic distribution model.
Let N be a finite positive integer and X = ( X , X , · · · , X N ) T be a multivariate randomvariable given by X = β ( T −
1) + diag ( γ ) ε √T , where • T is the tempered stable subordinator with parameters ( α, θ ) , and is independent of ε n forall n = 1 , , · · · , N . • β = ( β , β , · · · , β N ) T ∈ R N with | β n | < q θ − α for all n ∈ { , , · · · , N } . • γ = ( γ , γ , · · · , γ N ) T ∈ R N + with γ n = q − β n (cid:0) − α θ (cid:1) for all n ∈ { , , · · · , N } and R + = [0 , ∞ ) . • ε = ( ε , ε , · · · , ε N ) T is N -dimensional standard normal distribution with a covariance ma-trix Σ . That is, ε n ∼ Φ(0 , for n ∈ { , , · · · , N } and ( k, l ) -th element of Σ is given by ρ k,l = cov ( ε k , ε l ) for k, l ∈ { , , · · · , N } . Note that ρ k,k = 1 .In this case, X is referred to as the N -dimensional standard NTS random variable with parameters ( α , θ , β , Σ) and we denote it by X ∼ stdNTS N ( α , θ , β , Σ) (See more details in Appendix.).Consider a portfolio having N assets. The return of the assets in the portfolio is given by a The tempered subordinator is defined by the characteristic function (12) in the Appendix. R = ( R , R , · · · , R N ) T . We suppose that the return R follows R = µ + diag ( σ ) X (1)where µ = ( µ , µ , · · · , µ N ) T ∈ R N , σ = ( σ , σ , · · · , σ N ) T ∈ R N + and X ∼ stdNTS N ( α, θ, β, Σ) .Then we have E [ R n ] = µ n and var ( R n ) = σ n for all n ∈ { , , · · · , N } . This market model isreferred to as the NTS market model . Let w = ( w , w , · · · , w N ) T ∈ I N with I = [0 , be thecapital allocation weight vector . Then the portfolio return for w is equal to R P ( w ) = w T R . Thedistribution of R P ( w ) is presented in the following proposition whose proof is in Appendix. Proposition 2.1.
Let µ ∈ R N , σ ∈ R N + , and X ∼ stdNTS N ( α, θ, β, Σ) . Suppose a N -dimensionalrandom variable R is given by (1) and w ∈ I N with I = [0 , . Then R P ( w ) d = ¯ µ ( w ) + ¯ σ ( w )Ξ for Ξ ∼ stdNTS ( α, θ, ¯ β ( w ) , , (2) where ¯ µ ( w ) = w T µ, ¯ σ ( w ) = p w T Σ R w, ¯ β ( w ) = w T diag ( σ ) β ¯ σ ( w ) and Σ R is the covariance matrix of R . According to Proposition 2.1, we need only Σ R , which is covariance matrix of R , and we donot need to know Σ , which is the covariance matrix for ǫ , when we study the portfolio return of theNTS market model. Empirical Illustration
We fit the NTS market model to 30 major stocks in the U.S. stock market. The 30 stocks areselected based on the components for Dow Johns Industrial Average (DJIA) index, since April 6,2020. However, Dow Inc.(DOW) in the components is replaced by DuPont de Nemours Inc.(DD), In this paper, we consider the long only portfolio. and the empirical CDF obtained by the kernel density estimation. Inorder to find α and θ , we use the DJIA index data and find other parameters as the followingtwo-step method: Step 1
Find ( α DJ , θ DJ , β DJ ) using the curve fit method between the empirical CDF andstdNTS CDF for the standardized return data of DJIA index. Step 2
Fix α = α DJ and θ = θ DJ . Find β n using the curve fit method again for each n -thstock returns n ∈ { , , · · · , } with fixed α and θ .The estimated parameters of µ n , σ n and β n are presented in Table 2. The fixed α and θ are in CDF of stdNTS distribution is obtained by the fast Fourier transform method by Gil-Pelaez (1951). µ n (%) σ n (%) β n (%) KS p -value (%) AAPL .
13 1 . − .
55 0 .
022 83 . AXP .
073 1 . − .
90 0 .
036 29 . BA .
11 1 .
68 0 .
50 0 .
038 22 . CAT .
071 1 . − .
93 0 .
035 29 . CSCO .
071 1 . − .
56 0 .
015 99 . CVX .
018 1 . − .
65 0 .
032 41 . DD − .
020 1 .
70 0 .
32 0 .
034 32 . DIS .
046 1 .
24 2 .
00 0 .
029 55 . GS − . . − .
01 0 .
043 11 . HD .
074 1 . − .
01 0 .
044 10 . IBM − .
013 1 . − .
31 0 .
012 99 . INTC .
075 1 . − .
19 0 .
024 74 . JNJ .
041 1 . − .
53 0 .
020 91 . JPM .
072 1 .
21 3 .
85 0 .
042 13 . KO .
050 0 . − .
67 0 .
032 41 . MCD .
076 1 . − .
17 0 .
024 77 . MMM . . − .
55 0 . . MRK .
067 1 . − .
68 0 .
037 24 . MSFT .
13 1 . − .
47 0 .
027 65 . NKE .
093 1 .
52 0 .
44 0 .
030 50 . PFE .
037 1 . − .
19 0 .
032 41 . PG .
064 1 . − .
26 0 .
025 71 . RTX .
050 1 . − .
41 0 .
024 77 . TRV .
025 1 . − .
52 0 .
036 26 . UNH .
086 1 . − .
41 0 .
029 52 . V .
12 1 . − .
07 0 .
023 82 . VZ .
033 1 . − .
12 0 .
040 16 . WBA − .
036 1 . − .
62 0 .
021 88 . WMT .
083 1 . − .
64 0 .
019 93 . XOM − .
019 1 . − .
16 0 .
051 4 . α = 0 . and θ = 0 . Table 2: NTS parameter fit using 1-day-returns from 1/1/2017 to 12/31/2019.8he bottom of the table. Those parameters are estimated using 1-day log-returns of the 30 stocksfrom 1/1/2017 to 12/31/2019. We can see that only 5 stocks (BA, DD, DIS, JPM, NKE) out ofthose 30 stocks have positive betas in this table. That means, 25 stocks follow left-skewed returndistributions, while only 5 stocks follow positively skewed distributions. The parameter α closesto 1, and θ is less than 1. That means the 30-dimensional distribution for those 30 stock returns hasfat-tails . We perform the Kolmogorov-Smirnov (KS) goodness of fit test. KS statistic (KS) valuesand those p -values are presented in the table too. According to the KS p -values in the table, themarginal NTS distributions are not rejected at 5% significant level except XOM. At 4% significantlevel, the marginal NTS distribution is not rejected for XOM either. We do not need to estimate thecovariance matrix Σ for ε , since the covariance matrix Σ R of R is enough to analysis the portfolioreturn as discussed in Proposition 2.1. In traditional mean-variance portfolio optimization, the investor finds the optimal portfoliowhich maximizes the reward and minimizes the dispersion risk. The reward is the expected returnof the portfolio and the dispersion risk is the standard deviation (or the variance) of the portfolio.However, the dispersion risk is only one feature of the portfolio risk, while there are many otherrisks for instance the asymmetric tail risk. If there are two portfolios having same mean andvariance, and one of them follows a skewed right distribution and the other follows a skewedleft distribution, then the skewed left distributed portfolio is more risky than the skewed rightdistributed portfolio. We take the NTS market model and measure both the dispersion risk andasymmetric tail risk of a given portfolio.The classical dispersion risk measure introduced by Markowitz (1952) is the standard deviation More precisely, it has the semi-fat-tails having the exponential decaying tails
9f the portfolio defined as follows: Disp. ( w ) = p w T Σ R w where w is the capital allocation weight of the portfolio. The asymmetric tail risk is related toasymmetric tails of the NTS distribution defined as follows:Asym. ( w ) = w T β. Since | β n | < q θ − α for all n ∈ { , , · · · , N } by the definition of the NTS market model, | Asym. ( w ) | cannot be larger than q θ − α , in ≤ w n ≤ for all n ∈ { , , · · · , N } . A portfoliohaving positive Asym. ( w ) follows the right-skewed distribution, and a portfolio having negativeAsym. ( w ) follows the left-skewed distribution. If there are two portfolios, A and B, whose cap-ital allocation weight vectors are w A and w B respectively, and Asym. ( w A ) < Asym. ( w B ) , thenportfolio A is more risky than portfolio B. That will be discussed again in Example 4.1.Based on those two risk measures, we set a nonlinear programming problem for the portfoliooptimization as min w ( Disp. ( w )) subject to N X n =1 w n = 1 w n ≥ for all n ∈ { , , · · · , N } Asym. ( w ) = w T β ≥ b ∗ Reward ( w ) = w T µ ≥ m ∗ where the benchmark values for Portfolio reward and asymmetric tail risk are m ∗ ∈ [min( µ ) , max( µ )] and b ∗ ∈ [min( β ) , max( β )] , respectively. 10igure 1: Surface of efficient frontiersUsing the parameters in Table 2, we perform the portfolio optimization for 51 points of b ∗ in { b = min( β ) + k · (max( β ) − min( β )) / | k = 0 , , , · · · , } and for 51 points of m ∗ in { m = min( µ ) + k · (max( µ ) − min( µ )) / | k = 0 , , , · · · , } . We finally obtain theefficient frontier surface on the three-dimensional space in Figure 1. The dot-points of the surfaceis ( Disp. ( w ) , Asym. ( w ) , Reward ( w )) for the optimal capital allocation weight w . The mash surfaceof the figure is an interpolation surface of the dot-points.In order to look into the surface, Figure 2 provides three examples of efficient frontier curvesbetween the dispersion risk and the reward. The three curves are extracted from Figure 1. Forinstance, we fix b ∗ = − . , and perform the optimization for m ∗ ∈ [min( µ ) , max( µ )] =[ − . , . , present it to the dash curve of the figure. Using the same method, we draw thedash-dot curve and the solid curve for b ∗ = 0 . and b ∗ = 2 . , respectively. Figure 3 presentsanother examples of efficient frontier curves between the asymmetric tail risk and the reward.Those three curves are extracted from Figure 1, too. The solid curve is for ( Asym. ( w ) , Reward ( w )) having Disp. ( w ) = 0 . . Also, the dash-dot and the dashed curves are for Disp. ( w ) = 0 . .006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 Disp. R e w a r d -4 Asym. = -1.64e-02Asym. = 1.87e-03Asym. = 2.02e-02
Figure 2: Efficient frontiers between the dispersion risk and the reward. We can observe that themean-variance efficient frontier is not unique but changes with respect to the asymmetric tail risk.and for Disp. ( w ) = 1 . , respectively. According to those two figures, we observe that theclassical (mean-variance) efficient frontier is not unique but there are many various form of efficientfrontier curves with respect to the asymmetric tail risk. Moreover, we can observe that the rewardis decreasing when the asymmetric tail risk is increasing under the fixed dispersion risk. In this section, we discuss performance measure of portfolios. We present a new performancemeasure on the NTS Market model. The new performance measure is an extension of the Sharperatio (Sharpe (1966, 1994)), and it measures performance of portfolios considering not only thedispersion risk but also asymmetric tail risk. 12
Asym. R e w a r d -4 Disp. = 8.40e-03Disp. = 9.61e-03Disp. = 1.09e-02