1、TAIL RISK BUDGETING,R. Douglas Martin* Computational Finance Program Director Applied Mathematics and Statistics University of Washingtondougstat.washington.eduR-Finance Conference, Chicago, Ill., April 29-30, 2011* Parts of this presentation are due to joint work with Yindeng Jiang (UW Endowment Fu
2、nd), Minfeng Zhu (Aegon USA), and Nick Basch (Ph.D. student UW Statistics Dept.),Outline,Volatility Risk Budgeting2. Post-Modern Portfolio OptimizationTail Risk BudgetingFactor Model Monte CarloModern Portfolio Theory Inertia,2,1. Volatility Risk Budgeting,Portfolio construction that controls asset
3、volatility risk contributions to total riskBased on linear risk decompositions and reverse optimization Useful graphical displays for allocation guidance Well-suited to supporting investment committee decisions Alternative to black-box optimizersBut can be used as constraints in optimization. See Sc
4、herer and Martin (2005); Boudt, Carl and Peterson (2010),3,Litterman (1996), Grinold and Kahn, (2000), Sharpe (2002), Scherer(2002),4,The Additive Decomposition,Uses “MPT” Mean-Variance Foundation,Implied Returns (“Reverse MV Optimization”),5,EQUAL WEIGHTS: ORCL, MSFT, HON, LLTC , GENZ (20% EACH),6,
5、REBALANCED: ORCL 10%, MSFT 20%, HON 5%, LLTC 25% , GENZ 40%,.,2. POST-MODERN PORTFOLIO OPTIMIZATION,7,Martin et. al. (2003),Rockafellar and Uryasev (2000),Mean-vs-ETL Optimization (Current leading choice),8,Choice of Tail Probability,Martin and Zhang (2008),Guidance: Do not go too far into the tail,
6、 p not less than .05 to be safe!Note: The above large-sample results are quite accurate for finite sample sizes down to T = 40 for p = .05 and df 5 (not terrible at df = 3).,Hedge Fund Universe379 hedge funds selected from * Monthly returns 12/1991 to 11/2009Portfolios100 randomly selected with 20 h
7、edge funds eachPortfolio optimizationMinimum VoL Minimum ETL with 5% tail probability Monthly rebalancing on 5 years of returns,9,Fund-of-Hedge Funds Example,* Thanks to for providing the data,10,Mean of 100 Portfolio Values on a Monthly Basis,More detailed study: Martin and Zhu (2011) in preparati
8、on.,3. TAIL RISK BUDGETING,Q: What risk measures can give you an additive decomposition?,11,11,A: Euler: Any positive homogeneous risk measure,Works for: - Semi-standard deviation(SSD) - Value-at-Risk (VaR) - Expected-tail-loss (ETL),satisfies,ETL Risk Decomposition,(Tasche, 2000),Mean-ETL Implied R
9、eturns,(,MCETL vs. MCVOL Diagnostic Plot (Cognity*),13,* From FinAnalytica, Inc. with skewed t-distribution models,Reason for Differences (Cognity),The fat right tail influences volatility but not ETL,14,15,Example: Tail Risk Budget Rebalancing,5 years training, risk-budget guided rebalance once at
10、end of July 2008.,16,17,19,Need improved risk and performance estimates For risk analysis and portfolio constructionShort and unequal histories of returnsShort training periods for dynamic modelsBorrow strength from time series factor modelsUse factor model Monte Carlo (FMMC)Motivating work under no
11、rmal distributions: Stambaugh (1997) Pastor and Stambaugh (2002),20,4. FACTOR MODEL MONTE CARLO,21,Single Asset and p Risk Factors,Time series factor model:,Normal distribution MLEs (Anderson, 1957),Can get any normal distribution parameterized risk or performance measure, BUT GOOD ONLY FOR VOL, SHA
12、RPE, IR FOR FAT-TAILED RETURNS!,22,The FMMC Method,Factor model fit (LS and robust):,Estimate distribution of (large T, e.d.f. will suffice),Estimate distribution of,- Either fat-tailed skewed distribution fit, or e.d.f. (which?),Large Monte Carlo of,Estimate risk and performance measures from,large
13、,Jiang (2009) Jiang and Martin (2011),23,Simplest Version,Empirical Distributions OnlyFMMC = all unique combinations of and,10 years of risk factor data and 3 years of hedge fund returns:120 x 36 = 4320 samples (may often be good enough),Key Ingredient,Very good factor models!Need parsimonious from
14、large universe with high predictive powerLooking into methods such as Lasso, LARS, etc.,24,Date range: 2003/9 to 2006/8,R-squared = .86,Hedge Fund and Single Risk Factor,25,Risk Estimates and Bootstrap S.E.s,26,27,Risk Estimates and Bootstrap S.E.s,28,20 hedge funds 2000 through 2007 H 19 risk facto
15、rsMarket factors: SP500, DJIA, VIX, DAX, CAC 40, Nikkei 225 Hedge fund indexes: 12 DJ Credit Suisse Both robust and least squares fits 2004-7Initial universe reduction:Top 5 factors by LS R-squared then best subset R robust library model selection unreliable for larger p,29,MULTI-FACTOR MODELS FOR F
16、MMC,Basch and Martin (2011 current work),32,Model Comparisons Based on Bootstrapped Mean ETL,5. MPT INERTIA,At 50+ years old why is it still the dominant paradigm?Mathematically clean if no constraints (so what?) Entrenched in MBA Investments 500 (see Bodie, Kane & Marcus) Very costly for software v
17、endors to change (R&D, education) Markowitz knew better (SSD: but no nice math or easy compute)The post-modern foundations are in place:Artzner et. al. (1999) Coherent risk measures Rockafellar and Uryasev (2000) Mean-ETL optimization Considerable modern computing power Superior performance examples
18、 But more are needed Its time to move on!,33,MS Degree and Two Affiliated Certificateswww.amath.washington.edu/studies/compfin,34,The Computational Finance Certificate,SAMPLE R CODE,mctr.etl = function(returns, wts, gamma) returns.port = as.matrix(returns)%*%wtsmu.port = mean(returns.port)VaR.port =
19、 quantile(returns.port, gamma)index = which(returns.port = VaR.port)etl = -mean(returns.portindex)mctr = -apply(returnsindex, 2, mean)mu.imp = mu.port/etl*mctrreturn(list(mctr = mctr, mu.imp = mu.imp) ,MCETL and Implied Returns,Robust FM Fit: R package “robust”,library (robust) model.data = as.data.
20、frame(cbind(Returns,Factors) ) mod = lmRob(Returns., data = model.data) #Stepwise selectionmod.step = step.lmRob(mod, trace = FALSE) robust.coef= mod.step$coef robust.resid = resid(mod.step),Subset Model,library(glmnet) mod = regsubsets(x=Factors,y=Returns, nvmax = ncol(Factors)subset = summary(mod)
21、best.mod = which(subset$bic = min(subset$bic)subset.coef= as.vector(coef(mod,best.size),Simulate returns,fitted = robust.coef%*%t(Factors.full)#Factors.full is factor data for full time length r.sim = rep(0, times = 84*36)for(j in 1:84) for(k in 1:36)current = 36*(j-1) + kr.simcurrent = fittedj + residk ,
