Testing Downside-Risk Efficiency Under Distress

Transcription

1 Testing Downside-Risk Efficiency Under Distress Jesus Gonzalo Universidad Carlos III de Madrid Jose Olmo City University of London XXXIII Simposio Analisis Economico 1

2 Some key lines Risk vs Uncertainty. LP M vs Variance. LP M q = R p <τ P p (τ R p ) q. Portfolio optimization: min {w} s.t. LP M q (τ, w) = m w j R j = Rp; j=1 R p <τ P p (τ R p ) q m w j = 1, w j > 0. j=1 Utility function compatible with downside risk approach: U(R p ; q, τ) = R p k(τ R p ) q I(R p τ). XXXIII Simposio Analisis Economico 2

3 How do we compare portfolios? By Stochastic Dominance (SD). There is a close connection between LP M q measures and q stochastic dominance. Need for testing q-stochastic dominance. Thanks to a decomposition that we propose for LP M q we develop a new test for SD with a simpler asymptotic theory. When all of this is crucial? In distress periods: (R 1 u, R 2 u,..., R m u). We re-do the theory on stochastic dominance and hypothesis testing for distress periods. XXXIII Simposio Analisis Economico 3

4 Motivation Mean-Variance investment strategies are efficient only under restrictive assumptions. Mean-Variance investment strategies are not necessarily efficient under distress episodes of the market. Mean-Risk efficient sets are alternative investment strategies for investors concerned with downside risk, and where this is defined by a threshold. Downside-risk efficient strategies are therefore dependent on the level of investor s risk aversion and can vary heavily with the choice of the threshold. Stochastic dominant portfolios, instead, are optimal for every point in the domain of the relevant distributions. The concept of stochastic dominance can be extended to subsets of the domain of the distribution of the data. In our case to the left tail of the distribution (distress regimes). XXXIII Simposio Analisis Economico 4

5 Contributions Main aim of a diversified investor: To derive investment strategies that minimize risk for target return levels, and, ideally, under comovement episodes of the market. The benchmark being Mean-Variance strategies. In order to do this we need: To decompose the downside risk in a portfolio in downside risk due to the presence of comovements between the assets and due to the choice of optimal weights. To determine the relationship between mean-risk and stochastic dominance efficient sets under distress episodes of the market. To develop a novel test statistic for testing stochastic dominance that makes allowance for testing orders of dominance higher than zero and for general forms of dependence between portfolios. XXXIII Simposio Analisis Economico 5

6 Outline Relationship between mean-risk and stochastic dominance efficiency, and its extension to market distress. Develop a hypothesis test for stochastic dominance and conditional stochastic dominance. A Monte-Carlo experiment to study finite-sample performance of the test. An empirical comparison between mean-variance and stochastic dominance investment strategies. Conclusions. XXXIII Simposio Analisis Economico 6

7 Mean-Risk and Stochastic Dominance Efficient Sets Risk is based on the idea of dread events, see Hogan and Warren (1974), Bawa (1975), Arzac and Bawa (1977), or Bawa and Lindenberg (1977). The utility function corresponding to these investors is U(R p ; q, τ) = R p k(τ R p ) q I(R p τ), (1) with R p return on a portfolio P ; τ target return; k scaling parameter, I( ) an indicator function, and q investor s risk aversion level. The relevant risk measure is LP M q (τ) = τ (τ R p ) q df (x), (2) with F (x) = P (R p x). Result 1 (Fishburn, 1977): Portfolio A dominates Portfolio B in the mean-risk model if and only if µ(a) µ(b) and LP M q (A) LP M q (B) for q 0, with at least one strict inequality. XXXIII Simposio Analisis Economico 7

8 Stochastic Dominance Result 2 (Fishburn, 1977): A first stochastic dominates (FSD) B if and only if F A F B LP M0 B (τ) for all τ R, and LP M A 0 (τ) A second stochastic dominates (SSD) B if and only if F A F B and LP M A 1 (τ) LP M B 1 (τ) for all τ R, A third stochastic dominates (TSD) B if and only if F A F B, µ(a) µ(b), and LP M A 2 (τ) LP M B 2 (τ) for all τ R, with F A and F B the distribution functions of two portfolios A and B. Moreover, this author also shows that stochastic dominance of order one is sufficient for mean-risk efficiency for q 0 (risk-neutral and risk-averse investors). Also, stochastic dominance of order two is sufficient for mean-risk efficiency for q 1 (risk-averse investors). We extend these results to Distress periods. XXXIII Simposio Analisis Economico 8

9 Definition of a Distress Measure In our framework we will identify the presence of distress comovements with the event: {R 1 u,..., R m u} with probability P {R 1 u,..., R m u} := λ(u). The relevant new risk measure is LP M P q,u(τ) = τ (τ x) q df u (x), (3) where F u (x) := P {R p x R 1 u,..., R m u}. proposition 1: Let LP M P q ( ) and LP M P q,u( ) for q 0 be the downside risk measures defined in (2) and (3) respectively. Then, under some regularity assumptions, LP M P q (τ) = E[(τ R p ) q R p τ]lp M P 0 (τ), (4) and LP M P q,u(τ) = E[(τ R p ) q R p τ, R 1 u,..., R m u]lp M P 0,u(τ). (5) XXXIII Simposio Analisis Economico 9

10 Corollary 1: The unconditional downside risk measure of interest can be decomposed in terms of the downside risk measures conditional on distress comovements and noncomovements regimes. LP M q (τ) = λ(u)γ q,u (τ)lp M q,u (τ) + (1 λ(u))γ q,ū (τ)lp M q,ū (τ), (6) with γ q,u (τ) = E[(τ R p ) q R p τ] E[(τ R p ) q R p τ,r 1 u,r 2 u,...,r m u], γ q,ū (τ) = E[(τ R p ) q R p τ] E[(τ R p ) q R p τ,r 1 >u or R 2 >u or...or R m >u], and LP M q,ū the risk measure conditional on {R 1 u, R 2 u,..., R m u} C. Remark: This corollary shows that A more efficient than B unconditionally does not imply that A more efficient than B under distress periods. In fact, there can other allocations more efficient under distress episodes of the market. XXXIII Simposio Analisis Economico 10

11 Conditional Stochastic Dominance Under Distress With this decomposition in place and the conditional downside-risk measure we can define conditional stochastic dominance. Definition 1: A first conditional stochastic dominates (FCSD) B if and only if F A u F B u and LP M A 0,u(τ) LP M B 0,u(τ) for all τ u, A second conditional stochastic dominates (SCSD) B if and only if F A u F B u and LP M A 1,u(τ) LP M B 1,u(τ) for all τ u, A third conditional stochastic dominates (TCSD) B if and only if Fu A µ u (B), and LP M2,u(τ) A LP M2,u(τ) B for all τ u, F B u, µ u (A) XXXIII Simposio Analisis Economico 11

12 theorem 1: If A FCSD B then A dominates B in the mean-risk model defined by LP M q,u measures for all q 0. If A SCSD B then A dominates B in the mean-risk model defined by LP M q,u measures for all q 1, except when µ u (A) = µ u (B) and LP M A q,u(τ) = LP M B q,u(τ) for all τ u. If A TCSD B then A dominates B in the mean-risk model defined by LP M q,u measures for all q 2, except when µ u (A) = µ u (B) and LP M A q,u(τ) = LP M B q,u(τ) for all τ u. In order to make the conditions stated before statistically testable we develop in what follows hypothesis tests for stochastic and conditional stochastic dominance of different orders. XXXIII Simposio Analisis Economico 12

13 Hypothesis Tests for Stochastic Dominance Our test statistic is of Kolmogorov-Smirnov type and shares the spirit of the test statistics proposed in in McFadden (1989), Anderson (1996), Davidson and Duclos (2000), Barret and Donald (2003) or Linton, Maasoumi and Whang (2005) among others. The results in Fishburn (1977) allow us to focus on the hypothesis test { H0,γ : LP Mγ A (τ) LP Mγ B (τ), for all τ R, H 1,γ : LP Mγ A (τ) > LP Mγ B (τ), for some τ R, (7) rather than on the strict inequality, for testing first (γ = 0), second (γ = 1) and third (γ = 2) stochastic dominance between two portfolios A and B. Alternatively, one can write the test as { H0,γ : D γ (τ) 0, for all τ R, H 1,γ : D γ (τ) > 0, for some τ R, (8) with D γ (τ) := LP M A γ (τ) LP M B γ (τ). XXXIII Simposio Analisis Economico 13

14 Asymptotic theory theorem 2: Under some regularity assumptions, n sup ( D γ (τ) D γ (τ)) τ R d sup G γ (τ), (9) τ R with G γ (τ) a Gaussian process with zero mean and covariance function given by E[G γ (τ i )G γ (τ j )] = ( k2γ(τ A i τ j )F A (τ i τ j ) kγ A (τ i )F A (τ i )kγ A (τ j )F A (τ j ) ) ( + k B 2γ (τ i τ j )F B (τ i τ j ) kγ B (τ i )F B (τ i )kγ B (τ j )F B (τ j ) ) ( k A,B γ (τ i, τ j )F A,B (τ i, τ j ) kγ A (τ i )F A (τ i )kγ B (τ j )F B (τ j ) ) ( k A,B γ (τ j, τ i )F A,B (τ j, τ i ) k A γ (τ j )F A (τ j )k B γ (τ i )F B (τ i ) ), (10) for all τ i, τ j R. Notation: F A,B (τ, τ) := P (R A p τ, RB p τ), ki γ (τ) = E[(τ Ri p )γ R i p and k A,B γ (τ, τ) = E[(τ R A p )γ (τ R B p )γ R A p τ, RB p τ] τ] with i = A, B XXXIII Simposio Analisis Economico 14

15 A new test statistic for H 0,γ Our family of test statistics is defined by T n,γ := n sup τ R D γ (τ). The null hypothesis is the equality of functions LP M A γ (τ) = LP M B γ (τ) for every τ R. Under some regularity assumptions, and H 0,γ, T n,γ d sup G γ (τ). (11) τ R Further, the asymptotic critical values of these tests indexed by γ are given by c γ (1 α) := inf x R {x P with α denoting the significance level. ( ) sup G γ (τ) x τ R 1 α}, (12) XXXIII Simposio Analisis Economico 15

16 Consistency of the tests proposition 3: Under some regularity assumptions and the test statistic T n,γ, then: (i) Under H 0,γ, lim P (reject H 0,γ) = lim P (T n,γ > c γ (1 α)) α, (13) n n with equality when F A (τ) = F B (τ) for every τ R. (ii) If H 0,γ is false, lim P (reject H 0,γ) = lim P (T n,γ > c γ (1 α)) = 1. (14) n n XXXIII Simposio Analisis Economico 16

17 P-value approximations for stochastic dominance tests of orders higher than zero 2,..., x(j) n ), j = 1,..., be a collection of random samples of dimension n 2 drawn from a bivariate distribution F A,B (τ, τ). Let T n,γ (j) be the test statistic associated to each sample, and c (j) γ (1 α) the corresponding critical proposition 4: Let x (j) n := (x (j) 1, x(j) values obtained from the corresponding estimated functional of G γ. Then (i) Under H 0,γ, lim P (reject H (j) 0,γ) = lim P (T n,γ > c (1) γ (1 α)) α, (15) n n almost surely for every random sample x (j) n, and with equality when F A (τ) = F B (τ). (ii) If H 0,γ is false, lim P (reject H (j) 0,γ) = lim P (T n,γ > c (1) γ (1 α)) = 1, n n almost surely for every random sample x (j) n. XXXIII Simposio Analisis Economico 17

18 Hypothesis Test for Stochastic Dominance Under Distress { H0,γ,u : D γ,u (τ) 0, for all τ R, (16) H 1,γ,u : D γ,u (τ) > 0, for some τ R, where D γ,u (τ) = LP M A γ,u(τ) LP M B γ,u(τ). theorem 3: Under some regularity assumptions, nu sup ( D γ,u (τ) D γ,u (τ)) τ (,u] d sup G γ,u (τ), (17) τ (,u] with G γ,u (τ) a Gaussian process with zero mean and covariance function given by E[G γ,u (τ i )G γ,u (τ j )] = ( k2γ,u(τ A i τ j )Fu A (τ i τ j ) kγ,u(τ A i )Fu A (τ i )kγ,u(τ A j )Fu A (τ j ) ) ( + k B 2γ,u (τ i τ j )Fu B (τ i τ j ) kγ,u(τ B i )Fu B (τ i )kγ,u(τ B j )Fu B (τ j ) ( k A,B γ,u (τ i, τ j )Fu A,B (τ i, τ j ) kγ,u(τ A i )Fu A (τ i )kγ,u(τ B j )Fu B (τ j ) ) ( k A,B γ,u (τ j, τ i )Fu A,B (τ j, τ i ) kγ,u(τ A j )Fu A (τ j )kγ,u(τ B i )Fu B (τ i ) ), for all τ i, τ j u. XXXIII Simposio Analisis Economico 18

19 The family of test statistics for H 0,γ,u are T nu,γ := n u sup τ (,u] D γ,u (τ). Under some regularity assumptions, and H 0,γ,u, with u R, satisfy T nu,γ d sup G γ,u (τ). τ (,u] The asymptotic critical values of these tests are given by c γ,u (1 α) := inf x R {x P ( sup G γ,u (τ) x τ (,u] ) 1 α}, (18) with α denoting the significance level. Simulation procedures as a p-value transformation or bootstrap can be proposed to approximate the critical value of the test. Alternatively, we propose to estimate the asymptotic covariance function from the data and approximate the relevant critical value by Monte-Carlo simulation of the restricted supremum of the estimated gaussian process. XXXIII Simposio Analisis Economico 19

20 Finite Sample Performance: Size In Table 1 empirical size of the different hypothesis tests H 0,γ, γ = 0, 1: First and Second Order Stochastic Dominance. In Table 2 empirical size of the different hypothesis tests H 0,γ,u, γ = 0, 1, u = 0: First and Second Order Stochastic Dominance Under Distress. Two portfolios A and B are generated from a Student-t distribution with ν degrees of freedom. Different scenarios of dependence between portfolios are analyzed by increasing the correlation ρ between marginal Student-t distributions. The accuracy of the approximation given by the estimated critical values is assessed by using different sample sizes: n = 50, 100, 500. For the conditional test under distress we use n = 200, 400, 2000 in order to have roughly the same sample sizes as for the unconditional experiment: n u = 50, 100, 500. XXXIII Simposio Analisis Economico 20

21 Table 1. Empirical size for H 0,γ, γ = 0, 1 for a standardized bivariate Student-t with ν = 5 d.f and correlation ρ. Gp : asymptotic p-value, p : Multiplier method p-value. n sample size. B = 1000 Monte-Carlo simulations to approximate asymptotic critical value. mc = 500 Monte-Carlo iterations to approximate the nominal size. m = 100 partitions of the real line to generate observations from the corresponding asymptotic Gaussian process with a given covariance matrix. ν = 5 Method γ = 0 γ = 1 ρ = 0 10% 5% 1% 10% 5% 1% n = 50 Gp-value p-value n = 100 Gp-value p-value n = 500 Gp-value p-value XXXIII Simposio Analisis Economico 21

22 ν = 5 Method γ = 0 γ = 1 ρ = % 5% 1% 10% 5% 1% n = 50 Gp-value p-value n = 100 Gp-value p-value n = 500 Gp-value p-value ν = 5 Method γ = 0 γ = 1 ρ = % 5% 1% 10% 5% 1% n = 50 Gp-value p-value n = 100 Gp-value p-value n = 500 Gp-value p-value XXXIII Simposio Analisis Economico 22

23 Size Conditional Test Under Distress Table 2. Same as before. Threshold value: u = 0. ν = 5 Method γ = 0 γ = 1 ρ = 0 10% 5% 1% 10% 5% 1% n = 200 Gp-value (n u 50) p-value n = 400 Gp-value (n u 100) p-value n = 2000 Gp-value (n u 500) p-value ν = 5 Method γ = 0 γ = 1 ρ = % 5% 1% 10% 5% 1% n = 200 Gp-value (n u 50) p-value n = 400 Gp-value (n u 100) p-value n = 2000 Gp-value (n u 500) p-value XXXIII Simposio Analisis Economico 23

24 ν = 5 Method γ = 0 γ = 1 ρ = % 5% 1% 10% 5% 1% n = 200 Gp-value (n u 50) p-value n = 400 Gp-value (n u 100) p-value n = 2000 Gp-value (n u 500) p-value XXXIII Simposio Analisis Economico 24

25 Finite-Sample Performance: Power In Table 3 empirical power of the different hypothesis tests H 0,γ, γ = 0, 1: First and Second Order Stochastic Dominance. In Table 4 empirical power of the different hypothesis tests H 0,γ,u, γ = 0, 1, u = 0: First and Second Order Stochastic Dominance Under Distress. We generate two portfolios A and B, where F B freedom, and F A (τ) = F B (τ) + cf B (τ) n alternatives. is a Student-t with ν degrees of with c = 0.5, 1, 5, characterizing a family of local Different scenarios of dependence between portfolios are analyzed by increasing the correlation ρ between marginal Student-t distributions. The accuracy of the approximation given by the estimated critical values is assessed by using different sample sizes: n = 50, 100, 500. XXXIII Simposio Analisis Economico 25

26 Table 3. Empirical power for H 0,γ, γ = 0, 1. Alternative hypotheses given by F A (τ) = F B (τ)+ cfb (τ) n with F B and f B a Student-t distribution and density function with ν = 5 and c = 0.5, 1, 5. The correlation parameter is rho. n sample size. B = 1000 Monte-Carlo simulations to approximate asymptotic critical value. mc = 500 Monte-Carlo iterations to approximate the nominal size. m = 100 partitions of the real line to generate observations from the corresponding asymptotic Gaussian process with a given covariance matrix. ν = 5, α = 0.05 γ = 0 γ = 1 ρ = 0 / c = n = n = n = XXXIII Simposio Analisis Economico 26

27 ν = 5, α = 0.05 γ = 0 γ = 1 ρ = 0.4 / c = n = n = n = ρ = 0.8 / c = n = n = n = XXXIII Simposio Analisis Economico 27

28 Power Conditional Test Under Distress Table 4. Empirical power for H 0,γ,u, γ = 0, 1, u = 0. Alternative hypotheses given by F A (τ) = F B (τ)+ cfb (τ) n with F B and f B a Student-t distribution and density function with ν = 5 and c = 0.5, 1, 5. The correlation parameter is rho. n sample size. B = 1000 Monte-Carlo simulations to approximate asymptotic critical value. mc = 500 Monte-Carlo iterations to approximate the nominal size. m = 100 partitions of the real line to generate observations from the corresponding asymptotic Gaussian process with a given covariance matrix. ν = 5, α = 0.05 γ = 0 γ = 1 ρ = 0 / c = n = n = n = XXXIII Simposio Analisis Economico 28

29 ν = 5, α = 0.05 γ = 0 γ = 1 ρ = 0.4 / c = n = n = n = ρ = 0.8 / c = n = n = n = XXXIII Simposio Analisis Economico 29

30 An Empirical Study of Mean-risk Efficiency Consider a portfolio of risky and heavily traded stocks in the US economy given by Microsoft (MSFT), General Electric (GE), Bank of America Corporation (BAC) and Verizon Communications (VZ) spanning the period 02/01/ /12/ u= LPM 0 (τ) 0.5 LPM 0,u (τ) τ τ Left chart: No possible ranking of portfolios in terms of first order stochastic dominance. Right chart (Under distress): This plot tells a different story... XXXIII Simposio Analisis Economico 30

31 This is formalized with the results below of the tests corresponding to the three possible combinations between portfolios. Unconditional test for First stochastic dominance: H 0,0 : A B A : LP M0 P LP M0 P - σp 2 1 A : σp Unconditional test for Second Stochastic Dominance: H 0,1 : A B A : LP M0 P LP M0 P - σp 2 1 A : σp Conditional test for First Stochastic Dominance Under Distress: H 0,0,0 : A B A : LP M0,0 P LP M0,0 P - σp 2 0 A : σp XXXIII Simposio Analisis Economico 31

32 Conclusions In this paper we have extended the concept of stochastic dominance to define it under distress, and have exploited the relationship between mean-risk efficiency and stochastic dominance to derive efficient portfolios in distress episodes of the market. The efficiency of these portfolios depends on several factors; namely, the order of conditional stochastic dominance, the degree of risk aversion of the investor and the definition of the threshold defining distress comovements. The asymptotic distribution of our tests for stochastic dominance conditional on distress takes a simple and estimable form for any order of dominance, and more importantly, unlike most of the seminal papers in the literature, our method makes allowance for general forms of dependence between portfolios. XXXIII Simposio Analisis Economico 32

33 Our method to approximate the asymptotic critical value confirm our asymptotic theory and show the limitations of the p-value transformation under dependence between portfolios to derive the correct asymptotic critical value. This phenomenon is magnified for stochastic dominance under Distress periods. Finally, the empirical study makes us conclude that investment strategies alternative to mean-variance methods need to be carefully considered when financial markets go through distress episodes. Further, the adequacy of different investment strategies should be assessed by means of statistical tests for stochastic dominance. Future research: Tests of contagion between markets based on tail probabilities. Dynamic tests based on realized lower partial moments. Endogenous determination of threshold u defining distress commovements. XXXIII Simposio Analisis Economico 33