Portfolio Construction and Risk Management under Non-Normality Fiduciary Investors Symposium, Beijing - China October 23 rd 26 th, 2011 Ho Ho Quantitative Portfolio Manager, CalPERS The views expressed in this presentation are strictly presenter s view.
What is The Probability? 2
Topics of Discussion Dream World vs. Real World Real World Return Distribution and Characteristics Portfolio Construction, Analytic and Risk Management System Example Conclusion Q & A 3
Dream World vs. Real World 4
Dream World Asset/strategy returns are normally distributed. Mean and standard deviation are sufficient to describe return distribution. Standard deviation is sufficient to describe risk. Returns from period to period are independent and identically distributed (no serial correlation). Relationship between assets/strategies are assumed linear (measured by simple correlation). Joint distribution of asset returns is multivariate normal. Extreme negative events are highly improbable. 5
Real World Asset/strategy returns are NOT normally distributed. Return distributions exhibit considerable leptokurtosis (fat-tailed) and skew (lack of symmetry, negatively skewed most of the time). Evidence of serial correlation exists in illiquid asset/strategy (some hedge fund strategies, private equities, and real estate). Standard deviation will be under-estimated Sharpe Ratio will be over-estimated The relationships between assets/strategies are NOT always linear. The joint distribution of asset returns is NOT multivariate normal. 6
Real World (cont.) Using linear correlation matrices will underestimate the probability of joint negative returns under extreme conditions. Periods of intense and high volatility tend to cluster together (volatility clustering). Volatility has slowly decaying autocorrelations (volatility persistence). Extreme negative events happen more often than normal distribution suggested. 7
Real World Return Distributions and Characteristics 8
Random Walk Down Wall Street? Belief system in equilibrium changes due to random shocks Results in prediction that huge events are extremely rare Contradicts empirical data: Source: Steve Keen PhD University of Western Sydney. Magnitude +/- Events Gaussian Ratio Actual/Random 1% 17813 18648 0.96 2% 3818 3447 1.11 3% 780 719 1.09 4% 257 67 3.83 5% 106 2.79 38 6% 57 0.0511 1,114 7% 22 0.000411 53,464 8% 11 0.00000144 7,613,560 9% 3 0.0000000022 1,363,030,944 10% 8 0.00000000000 11% 1 0.00000000000 12% 2 0.00000000000 13% 2 0.00000000000 Gaussian prediction is zero to 20+ decimal places Daily movements in DJIA (1914 to 1999) Any size crash feasible Likelihood far higher than predicted by random/equilibrium model Crashes not aberrations but normal behaviour 9
Z-Score Z-Score Actual vs Normal Distributions Russell 3000 Daily Return (1/2/85 to 12/31/10) 13.00 8.00 3.00-2.00-7.00-12.00-17.00 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Russ 3K Daily Return Russell 3000 Daily Return (1/2/85 to 12/31/10) -- Normal Distribution Simulation 13.00 8.00 3.00-2.00-7.00-12.00-17.00 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Russell 3000 Daily Return (Normal Distribution Simulation) 10
Return Distributions Statistics Russell 3000 MSCI EAFE Barclay US Agg Bond S&P Commodity FTSE NAREIT Hedged Systematic Trend Following Equity Market Neural Mean 0.0063 0.0031 0.0055 0.0059 0.0092 0.0097 0.0059 Std. Dev. 0.0441 0.0506 0.0133 0.0624 0.0473 0.0215 0.0092 Skew -0.7188-0.4561-0.2235-0.1327-0.3745 0.1123-0.0997 Kurtosis 4.3371 4.1254 4.0083 5.0982 9.5772 2.6845 4.2323 Source: Hedge Fund Research Inc., FTSE NAREIT Index, and Datastream (Jan. 1990 to Jul. 2011) 11
Return Distributions Statistics Equity Long/Short Merger Arbitrage Distressed RV Convert Arbitrage RV Fixed Income Arbitrage RV Multi- Strategy HFoF Diversified Mean 0.0111 0.0073 0.0101 0.0074 0.0068 0.0072 0.0061 Std. Dev. 0.0263 0.0119 0.0189 0.0193 0.0193 0.0128 0.0175 Skew -0.2143-2.1839-1.0181-3.1912-1.3607-2.1811-0.4809 Kurtosis 4.9771 12.0766 8.1547 33.0812 10.0430 16.9892 7.0391 Source: Hedge Fund Research Inc., FTSE NAREIT Index, and Datastream (Jan. 1990 to Jul. 2011) 12
Return Distributions Statistics In general, merger arbitrage, distressed, relative value (RV) convertible arbitrage, RV fixed income arbitrage, RV multi-strategy, and fund of hedge fund have higher mean returns and lower standard deviations than equity indices returns but their skews and kurtosis are much worse. Mean-variance approach to portfolio construction will lead to sub-optimal portfolios with significantly underestimated downside risk. Trade off between moments. 13
Return Distributions Statistics 14
Systematic Return HFRI Systematic vs Russell 3000 0.080 0.060 0.040 0.020 0.000-0.020-0.040-0.060-0.200-0.150-0.100-0.050 0.000 0.050 0.100 0.150 Russell 3000 Return Systematic Fitted_Jul11 Linear (Systematic) 15
Long/Short Return HFRI Long/Short vs Russell 3000 0.150 0.100 0.050 0.000-0.050-0.100-0.150-0.200-0.200-0.150-0.100-0.050 0.000 0.050 0.100 0.150 Russell 3000 Return Long/Short Fitted_Jul11 16
Merger Arb Return HFRI Merger Arbitrage vs Russell 3000 0.040 0.020 0.000-0.020-0.040-0.060-0.080-0.200-0.150-0.100-0.050 0.000 0.050 0.100 0.150 Russell 3000 Return Merger Arb Fitted_Jul11 Linear (Merger Arb) 17
Convertible Arb HFRI Convertible vs Russell 3000 0.150 0.100 0.050 0.000-0.050-0.100-0.150-0.200-0.200-0.150-0.100-0.050 0.000 0.050 0.100 0.150 Russell 3000 Return Convertible Fitted_Jul11 Linear (Convertible) 18
FI Arb Return HFRI Fixed Income Arbitrage vs Russell 3000 0.150 0.100 0.050 0.000-0.050-0.100-0.150-0.200-0.150-0.100-0.050 0.000 0.050 0.100 0.150 Russell 3000 Return RV FI - Corp Fitted_Jul11 Linear (RV FI - Corp) 19
Relative Value Return Correlation of Non-linear Strategies HFRI Relative Value vs HFRI Long/Short 0.055 0.035 0.015-0.005-0.025-0.045-0.065-0.085-0.120-0.070-0.020 0.030 0.080 0.130 Scatter Plot Fitted Linear Eq L/S (long bias) Return Dwn Mkt: 0.75 Up Mkt: 0.43 Linear: 0.65 20
Portfolio Construction, Analytic and Risk Management System 21
Portfolio Construction, Analytic, and Risk Management System Analysis, Modeling, and Evaluation Forecasting Higher-Moment Optimization Not Satisfied Rebalance Risk Management Satisfied Data Storage Monitor 22
Portfolio Construction and Risk Management The total process is composed of six distinct steps Portfolio return distribution profile assessment Asset/strategy/manager returns modeling and evaluation Forecasting Higher-moments portfolio optimization Risk Management Monthly performance and risk monitoring 23
Portfolio Return Distribution Profile Assessment Determine the acceptable range of the portfolio return distribution moments Expected: return, volatility, skew, and kurtosis Determine the acceptable level of market exposure in the portfolio (portfolio market beta). 24
Returns Modeling and Evaluation Determine which strategies have the desired statistical characteristics established in the first step. Quantitative analysis: Multifactor analysis with factors designed for specific strategies. Statistical, time series, econometric, and nonlinear modeling. Identify the return drivers and leverage of the strategies at different states of the market. Must understand the intuition of the return distribution and linear/nonlinear relationship with systematic factors. 25
Returns Modeling and Evaluation (cont.) Qualitative & operational analysis Investment process Portfolio construction (& hedging) process Trading procedures Articulation of alpha generation Amount of leverage Risk management Stability of the organization Experience and quality of investment professionals Compliance and accounting (reconciliation and pricing) 26
Forecasting System of models and methods for the following: Trend identification Structural break identification Identification of signal frequency content Bubble identification Prediction of expected return distribution 27
Higher-Moments Portfolio Optimization 1,2 Higher-moments optimization Expected return Variance-covariance matrix Skew-coskew matrix Kurtosis-cokurtosis matrix Directly takes skew and kurtosis into consideration Objectives of the optimization are to maximize expected return and skew, and to minimize volatility and kurtosis. 1. Mark Anson, Ho Ho, Kurt Silberstein, Hedge Fund Optimization using Higher Order Moments of the Distribution, Manual Hedge Funds: Opportunities, Risks, and use in Asset Allocation, Uhlenbruch Verlag, May 2005. 2. Mark Anson, Ho Ho, Kurt Silberstein, Building a Hedge Fund Portfolio with Kurtosis and Skewness, The Journal of Alternative Investments, summer 2007. 28
Risk Management Dynamic simulation that captures Negatively skewed, and fat-tailed distribution Clustering of volatility Inter-dependence of return, volatility, and random shock Volatility spread across markets Time-varying correlations and volatility structures Linear/nonlinear relationship between asset/strategy returns, systematic factors, and random shocks. Forward looking risk reports, return distribution statistics including confidence interval. Hedge/overlay design if necessary 29
Z-Score Z-Score Risk Management (cont.) Russell 3000 Daily Return (1/2/85 to 12/31/10) 13.00 8.00 3.00-2.00-7.00-12.00-17.00 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 Russ 3K Daily Return MSCI EAFE Daily Return (1/2/85 to 12/31/10) 10.00 5.00 0.00-5.00-10.00-15.00 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 MSCI EAFE Daily Return 30
Z-Score Z-Score Risk Management (cont.) S&P Commodity Daily Return (1/2/85 to 12/31/10) 10.00 5.00 0.00-5.00-10.00-15.00 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Jan-07 Jan-09 S&P Commodity Daily Return Barclay US Treasury Daily Return (3/1/94 to 12/31/10) 8.00 6.00 4.00 2.00 0.00-2.00-4.00-6.00-8.00 Mar-94 Mar-96 Mar-98 Mar-00 Mar-02 Mar-04 Mar-06 Mar-08 Mar-10 Barclay US Treasury 31
Risk Management (cont.) Inputs of the dynamic simulator Outputs from the higher-moments optimization Weight of each asset/strategy in the portfolio Systematic factors Proper asset/strategy return models structure (linear/nonlinear) Outputs of the dynamic simulator Forward looking risk reports 32
Risk Management (cont.) Forward looking risk reporting All reports available per simulated period Portfolio level can drill down to strategy/manager level Return distribution statistics including confidence interval VaR, marginal VaR, component VaR, CVaR, marginal CVaR, and component CVaR Max drawdown distribution 33
Monthly Performance & Risk Monitoring Monitoring items Performance relative to peers Return distribution statistics Style drift Changes in the factor exposures Qualitative and operational changes Possible rebalance Fund new managers/strategies Terminate existing managers/strategies Shift managers/strategies allocations 34
Asset Allocation Example for a Hypothetical Endowment Fund 35
Asset Allocation Example Endow Wts. HMOM Wts. Domestic Equity (Russell 3000) 17.00% 17.04% Foreign Equity (MSCI World EX US) 16.00% 15.33% Fixed Income (CGBI USBIG Inv Grade) 6.00% 6.39% Real Estate (FTSE NAREIT Hedged) 30.00% 0.00% Alternative 31.00% 61.24% Convertible Arbitrage 3.44% 3.41% Fixed Income Arbitrage 3.44% 0.00% Event Driven 3.44% 4.47% Market Neutral 3.44% 11.58% Multi-Strategy Fund 1 3.44% 3.19% Multi-Strategy Fund 2 3.44% 3.83% CTA Fund 1 3.44% 11.58% CTA Fund 2 3.44% 11.58% CTA Fund 3 3.44% 11.58% Total 100.00% 100.00% 36
Asset Allocation Example (cont.) In-Sample (Jan. 90 to Dec. 07) Out-of-Sample (Jan. 08 to Dec. 10) In-Sample Endowment In-Sample HMOM Opt. Out-of-Sample Endowment Out-of-Sample HMOM Opt. Mean 0.0075 0.0094 0.0028 0.0036 Std. Dev. 0.0237 0.0204 0.0442 0.0237 Skew -0.8340-0.1723-0.9454-0.1095 Kurtosis 4.3929 2.4697 4.9617 2.1157 Sharpe Ratio 0.3171 0.4610 0.0645 0.1514 37
Return Asset Allocation Example (cont.) Cumulative Return 0.150 0.100 0.050 0.000-0.050-0.100-0.150-0.200-0.250-0.300-0.350 Nov-07 Feb-08 Jun-08 Sep-08 Dec-08 Mar-09 Jul-09 Oct-09 Jan-10 May-10 Aug-10 Nov-10 Feb-11 Endowment Example HMOM Opt Cumulative return as of December 2010 Endowment: 6.97% HMOM Opt.: 12.64% 38
Asset Allocation Example (cont.) Risk analysis Dynamic simulation 60 months and 5,000 paths Endowment HMOM Opt Average Diversified VaR @ 95% (mo.) -0.0430-0.0278 Average Diversified CVaR @ 95% (mo.) -0.0594-0.0375 Average Minimum Return (mo.) -0.1345-0.0812 Average Maximum Return (mo.) 0.1560 0.1608 Mean (mo.) 0.0080 0.0095 Standard Deviation (mo.) 0.0297 0.0228 Skew -0.2688 0.1445 Kurtosis 3.7920 4.2932 Sharpe Ratio 0.2705 0.4178 Average Max Drawdown -0.1280-0.0731 Average Max Drawdown Duration (mo.) 9.54 6.47 39
Asset Allocation Example (cont.) Endowment s worst max drawdown: -0.39 HMOM Opt. s worst max drawdown: -0.22 40
Asset Allocation Example (cont.) Endowment s worst max drawdown duration: 60 months HMOM Opt. s worst max drawdown duration: 43 months 41
Conclusion 42
Conclusion Asset/strategy returns are NOT normally distributed. Portfolio construction must take skew and kurtosis into consideration. Extreme negative events happen much more often than normal distribution predicted. Risk management should be part of the portfolio construction process, NOT an after thought. Risk management is more than risk monitoring. 43