Beta is not Sharpe Enough …. September 2010 Steven P. Greiner, Ph.D. sgreiner@factset.com 0101.312.566.5109 Sharpe-r Risk Measures Agenda • • • • • • • • • Tracking Error Measures FactSet’s Balanced Risk Module in PA Tracking Error Forecasts Introducing the “g-Factor”, a robust Volatility Measure Value-at-Risk Stress Testing: Time & Event Stress Testing: Black Swan Event VAR Extreme Event Stress Testing Fat-Tail VAR Raising the IQ of the Intelligent Investor Sharpe-r Risk Measures… Ben Graham said: In a Barron’s article, he said that what bothered him is that authorities equate beta with the concept of risk. Price variability yes, risk no Excerpt from Barron’s, Sept 23, 1974, Dow Jones and Company Real risk he wrote, is measured not by price fluctuations but by a loss of quality and earnings power through economic or management changes As for variance or standard deviation of return being a useful risk measure, in the same Barron’s article he says that the idea of measuring investment risks by price fluctuations is repugnant to him, because it confuses what the stock market says with what actually happens to the owner’s stake in the business Tracking Error: What it isn’t! • Usually, TE is reported as meaning that the portfolio’s return is “bounded” by being within +/- TE of the Benchmark 67% of the time. Is this True? 3.5E-02 3.0E-02 2.5E-02 2.0E-02 1.5E-02 σ = sqrt[Σ{(x-μ)/(n-1)}^2] 1.0E-02 5.0E-03 0.0E+00 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Avg(x) = μ = 2.0 4.0 5.0 6.0 7.0 Tracking Error: What it isn’t! • Using the math from the previous slide, substitute “P-BM” for “x” and re-plot the graph… 0.04 • This then implies TE is the stdev about the average value of the XS return, 0.04 not the BM return.. 0.03 0.03 0.02 σ’ = sqrt[Σ{((P-BM)-μ’)/(n-1)}^2] = TE 0.02 0.01 0.01 -3.00 -2.00 -1.00 0.00 0.00 1.00 2.00 3.00 4.00 Avg(P-BM) = XS Ret = μ’ = 1.4 5.00 6.00 7.00 Tracking Error… What it is!! + Consider the impact this has on interpretation + There can be considerable asymmetry around bench returns using Port = XS + Bench…. Annualized long term numbers.. Port Bench XS TE 4.0 3.0 1.0 4.0 4.0 5.0 -1.0 4.0 so 68% of time XS is: XS-TE XS+TE -3.0 5.0 -5.0 3.0 Port's Abs Ret Bounds Lwr Bnd Upr Bnd 0.0 8.0 0.0 8.0 Portfolio about Bench Lwr Bnd Upr Bnd -3.0 5.0 -5.0 3.0 6.0 6.0 5.0 7.0 1.0 -1.0 6.0 6.0 -5.0 -7.0 7.0 5.0 0.0 0.0 12.0 12.0 -5.0 -7.0 7.0 5.0 8.0 8.0 7.0 9.0 1.0 -1.0 9.0 9.0 -8.0 -10.0 10.0 8.0 -1.0 -1.0 17.0 17.0 -8.0 -10.0 10.0 8.0 Empirical Data for: S&P, 2 SPDR’s, Exxon, ISRG, LCV, Magellan, Nikkei & 2 Hypothetical's • Weekly returns downloaded from FactSet from December 31st, 2006 to August 31st, 2010 Empirical FactSet Data for: S&P, 2 SPDR’s, Exxon, ISRG, LCV, Magellan, Nikkei & 2 Hypothetical's Data is Weekly Returns From 12/31/2006 to 8/31/2010… S&P500 TE=> XS=> Mean Ret Stdev Ret True Defn => Usual Defn => -0.07 2.85 XLK XLF XO M ISRG LCV 1.42 0.09 0.04 2.89 3.92 -0.21 -0.34 5.59 2.61 0.02 -0.00 3.15 6.65 0.93 1.11 7.53 1.03 -0.11 -0.19 3.20 Magellan Nikkei 1.29 -0.05 -0.11 3.40 Normal t-Dist-12 2.67 -0.17 -0.24 3.43 5.46 0.06 -0.15 4.44 5.95 0.06 -0.18 4.61 68.8% 81.8% % of time XS return is found within it's average value and +/- TE 76.0% 82.3% 77.1% 81.8% 87.5% 81.8% 68.8% % of time Portfolio Return is found between S &P500 Ret and +/-TE 39.1% 64.1% 60.9% 74.5% 29.2% 32.3% 62.0% 78.6% 88.5% Empirical Data for: S&P, 2 SPDR’s, Exxon, ISRG, LCV, Magellan, Nikkei & 2 Hypothetical's + For smaller TE, the effect is more pronounced! Tracking Error Measures…. + If the TE is large and the abs(XS) return is small, you can stick to the old paradigm + If in 2008 one lowered TE hoping to lower relative risk while underperforming, one actually increased the likelihood of continued underperformance, hence risk actually increased. + This is because as TE goes down for a given XS return, one draws a narrower range around (P-BM) where the portfolio spends the majority of time in. If (P-BM) is negative, you lose the opportunity to out-perform as TE decreases. + If you have negative XS return, increase your TE to lower risk. E.g. XS Ret = -200 bps & TE = 4%; -6% < Port Ret < 2% (67% of the time) XS Ret = -200 bps & TE = 6%; -8% < Port Ret < 4% (67% of the time) + If you have positive XS return, decrease your TE to lower risk. XS Ret = 200 bps & TE = 6%; -4% < Port Ret < 8% (67% of the time) XS Ret = 200 bps & TE = 4%; -2% < Port Ret < 6% (67% of the time). FactSet’s Balanced Risk Module….. + Components Uniting Equities, Fixed Income and Currencies.. Monte Carlo Value-At-Risk Stress Testing 1: Time & Event Weighting (Equity Only) Stress Testing 2: Extreme Event (Equity Only) MC Extreme Event Risk Global Equities, Corp, Hi-Yld, Agencies, Tips, U.S.Treas, Sovereign, U.S. MBS, Exc-Trad Options + Four Equity Vendor Risk Models, Plus Factset’s Own.. SUNGUARD – APT Axioma MSCI-Barra Northfield Inf. Services MAC-ST + (Country, Regional, & Global, MT & ST, Equity only) (Global, EMG, Euro, U.S., Canada & Japan, Equity only) (Country, Regional and Global, Equity only) (Country, Regional, Global, MT and ST, Equity only) (Included in Balanced Risk Product & Required for FI) Fast Re-Pricing Algorithm for FI.. Yield Curve (Int. Rate) Risk Specific to Underlying Currency of Security 17 KR Dur specified by 4 PCA of 6 Libor & Govt Curves (U.S., Can, Aus, EUR, Jap, UK) Each Major Asset Class Has its Own Spread Model 3 Base Currency (USD, EURO, GBP) Reporting Options w/ Exp. to 13 Currencies Available + Fully Integrated with Portfolio Attribution.. Example of Global Equity Portfolio… + Construct Global Portfolio and Compare VAR and TE computed through FactSet Balanced Risk Module Percent of Total Holdings GLOB_EQUITY vs. MSCI EAFE MAC Global Multi-Asset Class Model (USD) U.S. Dollar 12/31/2008 Asset Class Port. Weight Bench. Weight Difference MC % Value at Risk 22 Day, 95% Total 100.00 100.00 -- 12.26 Equity 96.26 United States 26.14 Japan 15.57 France 11.82 Germany 8.66 Netherlands 6.02 Sw itzerland 4.59 Australia 4.12 Hong Kong 3.95 China Mobile Ltd. 3.73 Hutchison Whampoa Ltd. 0.20 Lenovo Group Ltd. 0.02 United Kingdom 2.86 Canada 2.45 Sw eden 2.07 Spain 1.72 Finland 1.46 Denmark 1.26 Brazil 1.01 Portugal 0.77 Israel 0.59 Peru 0.55 Italy 0.51 Singapore 0.14 Singapore Telecommunications Ltd.0.14 [Cash] 3.74 Euro 1.32 British Pounds 1.17 U.S. Dollar 1.14 Japanese Yen 0.10 100.00 -25.25 10.50 8.74 2.52 8.41 5.94 2.01 -0.14 -19.88 -1.97 4.53 1.39 0.84 -0.33 --3.62 1.06 0.18 ------ -3.74 26.14 -9.68 1.33 -0.09 3.50 -3.82 -1.82 1.93 3.73 0.06 0.02 -17.01 2.45 0.10 -2.82 0.07 0.42 1.01 0.44 0.59 0.55 -3.11 -0.92 -0.04 3.74 1.32 1.17 1.14 0.10 12.18 2.85 1.09 1.95 1.06 0.89 0.51 0.61 0.54 0.51 0.02 0.00 0.38 0.67 0.37 0.26 0.23 0.21 0.11 0.10 0.10 0.13 0.10 0.02 0.02 0.10 0.05 0.04 -0.00 -0.00 MC % Marginal Value at Risk 22 Day, 95% 0.13 0.11 0.07 0.17 0.12 0.15 0.11 0.15 0.14 0.14 0.10 0.05 0.13 0.27 0.18 0.15 0.15 0.17 0.10 0.14 0.17 0.24 0.19 0.11 0.11 0.03 0.04 0.04 -0.00 -0.02 MC % MC % Standalone Relative Value at Risk Tracking Error (StDev) 22 Day, 95% 22 Day 12.26 3.55 12.65 16.71 17.75 18.90 19.84 18.04 15.77 21.84 22.03 22.31 21.55 35.61 18.13 36.72 23.70 19.61 20.66 23.59 40.18 21.61 35.90 46.63 30.37 20.29 20.29 3.77 6.01 6.23 -0.00 5.95 ------------------------------- Tracking Error Forecasts…. + Computed TE using VAR and Historical (black) for Global Portfolio Measured with various risk models………Which is right? Tracking Error Forecasts with CI’s…. + Which is right? Most are, whence you compute the 95% Confidence Interval on the Historical….Note Asymmetry… Tracking Error… Bias A cross-section of the TE at a point in time has the following form.. bootstrap : test : var 0.2 0.4 0.6 0.8 1.0 Observ ed Mean 0.0 Density + 1 2 v ar1.1 3 Using Betas for measures of Volatility… + What is the impact of the correlation on one’s interpretation of how volatile a stock or portfolio is? + Beta’s ~ ISRG: 1.2 XLK: 0.9, XOM: 0.7 Using Betas for measures of Volatility… + So a portfolio that has next to no correlation with it’s bench then, has essentially no volatility? + Beta’s ~ Norm: 0.08 & t-Dist-12: 0.01 The Way to a Better Volatility Measure…g-Factor A question we might ask is, what’s the amount of time the bench & portfolio spend in a constant vicinity of their mean return? Stdev of Bench = SD + Form the distribution of returns for a time period + Measure the area under curve between Mean +/- SD for both Bench and Portfolio….. Use the Bench’s SD for each… + Ratio of Bench area to Portfolio area is g-Factor The “g-Factor”… + The g-Factor is independent of the correlation and just compares the amount of time the benchmark and portfolio “spend” within an identical distance of their mean values g-Factor: (% of time Bench within +/- SD) / (% of time Port within +/-SD) SP50 XLK XLF XOM ISRG LCV 11.752 11.752 11.455 10.535 42.698 19.429 12.160 8.515 62.615 14.982 14.132 12.342 16.049 12.999 14.653 9.582 20.003 0.974 24.018 0.164 76.6% g-Factor 1.000 Beta 1.000 76.0% 1.007 0.896 0.83 58.9% 1.301 1.653 0.76 69.8% 1.097 0.725 0.51 50.0% 1.531 1.275 0.31 74.5% 1.028 1.050 0.93 71.4% 1.073 1.106 0.91 72.4% 1.058 0.815 0.54 56.3% 1.361 0.083 0.00 80.7% 0.948 0.014 0.00 Variance Covariance Magellan Nikkei Normal t-Dist-12 % of Time Ret is Spent within +/- SD of its Mean R^2 on Beta Issues for Value-at-Risk.. + Trading or portfolio positions change over time, thus the longer horizon VAR calculated, the less realistic it’s going to be, which is why we use daily VAR + VAR techniques are subject to model risk. In particular, the parametric model used for the drawing in Monte Carlo influences the value of the VAR calculated, hence there’s no “correct” VAR, it’s just an estimate + VAR isn’t effective when macro-risks, extreme events (Black Swans or ELE) are occurring. The returns distribution obtained from either a covariance based method or a copula, predicated on modeling the past years dependencies, isn’t representative of how the returns will behave in extreme events. Even in a copula fitting of the factor returns with an attempt to garner non-linear dependencies in the tail, VAR will not show how the dependency really behaves during a Black Swan event Existing VAR models reflect risks that are not useful during transition periods or when “broken” correlation structures occurs across assets + For a given covariance matrix, there are many, many datasets whose variance or covariance will satisfy it. There is no unique set of factor returns for a given covariance matrix (or copula) Value-at-Risk Example_1 Stress Testing One: Time & Event Weighting.. + Pick a “shock”, any risk model factor or exogenous factor that has a timeseries (obviously, cause & effect economic variables, not weather forecasts) + Determine covariance/correlation of this “shock” to all risk model factors + Compute “Beta” between shocked factor “K” and all risk model factors from the covariance measurements + New Factor Return = Beta * Shock Earnings/Price Book/Price Trading Activity Log of Market Cap Earnings Variability EPS Growth Rate Revenue/Price Debt/Equity Industry Current Factor Exposures Current Factor Returns Return Forecast 2.62 Factor Contribution to Return Forecast Oil Shock Magnitude -30% Beta between Oil Shock and Factor Return New Factor Return w/Oil Shock Shock Return Forecast -0.08 Factor Contribution to Shock Return Forecast Beta Table 1.1 EXAMPLE OF STRESS TEST 0.103 0.802 0.082 3.2% 0.658 0.848 0.558 21.3% 0.085 0.851 0.072 2.8% 0.587 1.153 0.677 25.9% 0.720 0.557 0.401 15.3% 0.711 1.033 0.734 28.1% 0.022 1.066 0.024 0.9% 0.132 0.822 0.108 4.1% -0.158 0.687 -0.109 -4.2% 0.049 1.390 0.069 2.6% 0.185 -0.056 -0.006 7.5% 0.139 -0.042 -0.027 36.3% -0.038 0.011 0.001 -1.3% -0.015 0.005 0.003 -3.5% 0.208 -0.062 -0.045 59.3% 0.032 -0.010 -0.007 8.9% 0.014 -0.004 0.000 0.1% 0.000 0.000 0.000 0.0% 0.106 -0.032 0.005 -6.6% -0.037 0.011 0.001 -0.7% Stress Testing One: Time vs. Event Weighting.. Test Name: Report Date: Report Currency: Risk Model: Time Decay: Event Decay: Factor: S&P 500 30% Decline 8/23/2010 U.S. Dollar NIS US Fundamental Model 0.98 0.94 Shock % -30.00% Date Factor Chg (%) Tim e Weight (%) # Date Factor Chg (%) Event Weight (%) 7/30/2010 6/30/2010 5/28/2010 4/30/2010 3/31/2010 2/26/2010 1/29/2010 12/31/2009 11/30/2009 10/30/2009 "" "" "" "" 7/31/2006 6/30/2006 5/31/2006 4/28/2006 3/31/2006 2/28/2006 1/31/2006 12/30/2005 11/30/2005 10/31/2005 9/30/2005 8/31/2005 7.01 -5.24 -7.98 1.58 6.03 3.10 -3.60 1.93 6.00 -1.86 "" "" "" "" 0.62 0.14 -2.88 1.34 1.25 0.27 2.65 0.04 3.78 -1.67 0.81 -0.91 2.01 1.97 1.93 1.89 1.85 1.82 1.78 1.75 1.71 1.68 "" "" "" "" 0.76 0.75 0.73 0.72 0.70 0.69 0.68 0.66 0.65 0.64 0.62 0.61 1 2 3 4 5 6 7 8 9 10 "" "" "" "" 49 50 51 52 53 54 55 56 57 58 59 60 10/31/2008 8/31/1998 9/30/2002 2/27/2009 2/28/2001 8/31/1990 9/30/2008 6/30/2008 1/30/2009 9/28/2001 "" "" "" "" 1/31/2005 6/29/2001 4/30/1993 5/28/1999 5/31/2000 8/31/1992 12/31/1996 2/28/2007 3/31/1992 2/28/2002 4/29/2005 2/29/2000 -16.79 -14.46 -10.87 -10.65 -9.12 -9.11 -8.91 -8.43 -8.43 -8.08 "" "" "" "" -2.44 -2.43 -2.42 -2.36 -2.05 -2.05 -1.98 -1.96 -1.95 -1.93 -1.90 -1.89 6.00 5.64 5.30 4.98 4.68 4.40 4.14 3.89 3.66 3.44 "" "" "" "" 0.31 0.29 0.27 0.26 0.24 0.23 0.21 0.20 0.19 0.18 0.17 0.16 Stress Testing One: Example Percent of Total Holdings 50 notsonifty and 50 sp100 eq.wgt vs. Russell 1000 9/07/2010 R-Squared Daily Global Equity Model (USD) U.S. Dollar FX Rate - US$ per E uro (!XRE UR) D aily from 31-Aug-2007 to 23-Aug-2010 U.S. D olla r (Spli t / Spinoff - Adjusted) USD/EUR FX Rate 30% Decline High: 1.60 Low: 1. 19 BenchmarkLast: 1.27 Benchmark Percent Standalone Return (Time Wght) Percent Return (Event Wght) Percent Standalone Return (Event Wght) Percent Return (Time Wght) Percent 1.60 Return (Event Wght) Economic Sector Port. Weight Bench. Weight Difference Percent Return (Time Wght) Total 100.00 100.00 -- -21.19 -21.19 -18.62 -18.62 -19.17 -16.34 6.19 5.99 13.03 10.27 14.37 4.75 8.93 1.86 19.54 15.07 4.07 10.42 10.96 10.55 18.01 3.93 11.66 3.09 10.95 16.36 2.12 -4.43 2.08 -0.28 -3.64 0.82 -2.74 -1.22 8.59 -1.29 -1.29 -0.61 -2.97 -2.60 -3.22 -0.76 -1.25 -0.03 -4.32 -4.15 -20.80 -10.22 -22.80 -25.33 -22.38 -15.99 -13.96 -1.50 -22.12 -27.53 -1.00 -0.48 -2.64 -2.00 -2.97 -0.61 -0.89 -0.11 -4.00 -3.91 -16.20 -8.06 -20.22 -19.48 -20.69 -12.88 -9.98 -6.14 -20.48 -25.95 -0.85 -1.12 -2.66 -2.40 -3.38 -0.53 -1.68 -0.39 -2.30 -3.85 -0.67 1.45 -1.00 -2.42 1.40 -1.93 -2.84 1.35 -0.46 -1.24 1.30 -0.36 -2.08 1.25 -3.32 Materials Consumer Staples Industrials Energy Information Technology Utilities Health Care Telecommunication Services Consumer Discretionary Financials 1.55 1.50 1.20 Holdings Data As Of 50 notsonifty and 50 sp100 eq.wgt 12/31/2008 Russell 1000 9/07/2010 Risk Model As Of R-Squared Daily Global Equity Model (USD) 9/06/2010 Market Portfolio: Russell 1000 Volume in Thousands (max/avg) 0 08 Data Source: IDC / Exshare 09 10 Stress Testing One: Example …same data, but perspective has changed… Stress Testing Two: Extreme Event Stress.. + Extreme Event Stress let’s us go back in time and measure the current portfolio’s response to factor returns garnered from the past + It’s like using the cross-security relationships, the dependence structure from the past, because the factor returns used from a chosen historical stressed market environment, were those used to construct the covariance matrix at that time + In this module, we use past factor returns, multiplied by current exposures to allow us to examine how a portfolio today might behave should history “almost” repeat itself Stress Testing Two: Example Percent of Total Holdings + + GLOB_EQUITY vs. MSCI EAFE 9/23/2010 NIS Global Model U.S. Dollar What’s the Internet Bubble’s impact on Global Equity Portfolio, “Today”? Borrow factor returns from April 2000 Ru ssell 1000 (R.1000) D a il y fro m 3 1 -D e c- 1 9 96 to 3 1 -D e c- 2 0 09 To ta l R e tu rn Internet Bubble (4/2000) Bench. Weight Difference Total 100.00 100.00 -- -6.71 -6.71 -7.03 0.32 Equity United States Japan France Netherlands Germany Switzerland Australia United Kingdom Sweden Canada Hong Kong Denmark Spain Finland Israel Italy Portugal Peru Brazil Singapore [Cash] 97.06 25.36 12.51 11.80 6.29 6.25 5.40 4.54 4.51 3.59 3.46 3.24 1.84 1.80 1.78 0.94 0.91 0.91 0.89 0.89 0.14 2.94 40.99 000 100.00 -21.62 9.52 2.75 7.75 7.85 8.59 21.72 3.02 -2.66 1.00 3.72 1.08 0.84 2.70 0.27 --1.64 -- -2.94 25.36 -9.11 2.28 3.54 -1.50 -2.45 -4.05 -17.22 0.57 3.46 0.58 0.84 -1.91 0.70 0.10 -1.79 0.63 0.89 0.89 -1.49 2.94 -6.64 -1.38 -1.23 -0.74 -0.42 -0.35 -0.32 -0.11 -0.24 -0.31 -0.21 -0.36 -0.27 -0.19 -0.14 -0.07 -0.08 -0.05 -0.11 -0.05 -0.01 -0.07 -6.84 -5.42 -9.87 -6.31 -6.60 -5.64 -5.86 -2.44 -5.42 -8.62 -5.97 -11.11 -14.70 -10.38 -7.95 -7.73 -8.45 -5.78 -12.55 -5.26 -5.81 -2.37 0.39 -1.38 0.81 -0.09 -0.21 0.29 0.03 0.27 0.64 -0.08 -0.21 -0.09 -0.19 0.21 -0.04 -0.03 0.15 -0.03 -0.11 -0.05 0.14 -0.07 ----- 0.99 0.98 0.88 0.08 -0.02 -0.05 0.00 -0.00 -1.91 -4.74 0.01 -5.09 -7.03 --2.04 -0.66 -0.21 -0.65 -0.35 -0.39 -0.89 -0.23 --0.27 -0.08 -0.39 -0.10 -0.04 -0.22 -0.02 ---0.15 ------ H ig h : 3 8 5 0 .8 6 L o w : 1 4 8 7.2 9 L a st: 2 9 0 9 .1 8 0.98 0.88 3500 0.08 2500 2000 1500 99 00 01 02 03 Port vs Bench Percent Difference (Event) Port. Weight 3000 98 Benchmark Percent Return (Event) Asset Class British Pounds Euro U.S. Dollar Japanese Yen 97 Data Source: Russell Percent Standalone Return (Event) Percent Return (Event) 04 05 06 07 08 09 -0.02 -0.05 0.00 -0.00 Stress Testing Two: Example + Internet Bubble’s impact on Global Equity Portfolio is… + Wouldn’t be the worst since 1997!! EAFE -18.224 -14.397 -13.050 -11.580 -11.116 -9.936 -9.670 -9.208 -9.182 -8.498 -8.424 -8.381 -7.570 -7.340 -7.329 -55.591 -50.777 -38.937 -32.961 -31.239 -27.722 -27.132 -26.812 -24.273 -22.718 -20.163 -17.746 -16.541 -14.758 -14.715 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 -6.708 -14.467 -13.900 -13.750 -13.077 -11.255 -11.179 -10.883 -10.370 -10.235 -9.749 -9.663 -9.494 -9.382 -9.199 -8.093 -7.492 32 33 34 35 -2.957 -2.909 -2.636 -2.602 -6.470 -6.364 -5.572 -4.903 -4.153 -4.128 -3.902 -3.798 -3.630 -3.497 -3.207 -3.189 -3.162 -3.150 -3.107 -7.030 -5.315 -5.281 -4.397 45.0 40.0 Monthly Returns: 3/1997 to 8/2010 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0 -5.0 -10.0 -15.0 -20.0 -25.0 -30.0 -35.0 -40.0 -45.0 -50.0 -55.0 -60.0 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 Global Equity 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Global Equity EAFE Monte-Carlo Extreme Event Risk.. + Monte-Carlo Extreme Event Risk is enabling and is a unique combination of FactSet’s stress testing platform combined with Value-at-Risk methodologies + Go back in time and literally take the covariance matrix from the past, decompose it via “Cholesky”, while separately and simultaneously, Monte Carlo-generated scenarios are made, and multiplied by this historically fashioned Cholesky matrix to compute “factor returns” + The Monte Carlo VaR is computed by multiplying each set of Monte-Carlo generated factor returns by the current exposure matrix In this way, we use the dependence structure from a “Black Swan” event and past co-variances to see what a current portfolio’s VaR would look like under that past stressed situation Monte-Carlo Extreme Event Risk Example.. + LTCM occurred August of 1998 + Retns ~ -10% to -20% Monte-Carlo Extreme Event Example.. Percent of Total Holdings 50 notsonifty and 50 sp100 eq.w gt vs. Russell 1000 9/07/2010 NIS US Fundam ental Model U.S. Dollar Current Sim Economic Sector Port. Weight Bench. Weight Difference MC % Value at Risk 1 Day, 95% Total Consum er Discretionary Consum er Staples Energy Financials Health Care Industrials Inform ation Technology Materials Telecom m unication Services Utilities 100.00 19.51 5.99 10.26 15.02 8.94 13.05 14.40 6.19 1.88 4.76 100.00 10.94 10.43 10.53 16.32 11.69 10.97 18.03 4.07 3.09 3.93 -8.57 -4.43 -0.27 -1.31 -2.75 2.08 -3.63 2.12 -1.21 0.83 2.65 0.58 0.08 0.25 0.51 0.11 0.29 0.49 0.18 0.07 0.09 Holdings Data As Of 50 notsonifty and 50 sp100 eq.w gt 12/31/2008 Russell 1000 9/07/2010 Risk Model As Of NIS US Fundamental Model 8/31/2010 Market Portfolio: Russell 1000 + When LTCM happened, the covariance matrix defined more lepokurtic return distributions + Whereas now, it shows a much broader distribution of returns + So today’s VAR is greater than that of this past extreme event LTCM (8/1998) - Sim MC % Expected Tail Loss 1 Day, 95% MC % Standalone Value at Risk 1 Day, 95% ST % Value at Risk 1 Day, 95% ST % Expected Tail Loss 1 Day, 95% ST % Standalone Value at Risk 1 Day, 95% 3.27 ----------- 2.65 3.08 1.59 2.57 3.47 1.62 2.31 3.81 3.27 4.95 2.30 1.95 0.44 0.06 0.16 0.36 0.08 0.22 0.39 0.13 0.04 0.07 2.45 ----------- 1.95 2.34 1.41 2.21 2.58 1.42 1.80 3.14 2.52 4.30 1.81 Monte-Carlo Extreme Event Risk Example Two + Credit Crisis of November 2008 Monte-Carlo Extreme Event Risk Example Two.. + Using Global Portfolio of Equities, FI, Options and Currencies (Balanced..) + Examine impact of Credit Crisis (11/30/2008) on VaR Percent of Total Holdings GLOB_BAL_MAND vs. MSCI EAFE 9/23/2010 U.S. Dollar Report Asset Class Port. Weight Bench. Weight Difference Total Equity United States Japan France 100.00 73.60 25.15 13.10 9.34 100.00 100.00 -21.62 9.52 --26.40 25.15 -8.52 -0.18 Germany Australia Canada United Kingdom Sweden Hong Kong Hutchison Whampoa Ltd. Sun Hung Kai Properties Ltd. Hang Seng Bank Ltd. Swire Pacific Ltd. China Mobile Ltd. Lenovo Group Ltd. Netherlands Finland Switzerland Italy Singapore United Overseas Bank Ltd. Singapore Telecommunications Ltd. Ireland Israel Fixed Income Corporate Canada South Korea Australia France United States United Kingdom Italy Spain Hungary Japan Honda Bank Gmbh 0.0% 12-oct-2010 Toyota Motor Credit Corp. 0.0% 04-jan-2011 Toyota Finance Australia Ltd. 4.12% 31-jul-2017 Toyota Capital Malaysia Sdn. Bhd. 4.2% 02-jul-2014 Government Related United States Sovereign United States Derivatives Metlife Inc Call DEC10 36 Factset Research S Call DEC10 80 State Street Corp Put JAN11 32 Bank Of Ny Mellon Put DEC10 22.5 Costco Whsl Corp N Put JAN11 65 Kraft Foods Inc Put DEC10 28 Bristol-Myers Squi Put DEC10 23 Astrazeneca Plc Put OCT10 28 Standard Chartered Plc Put OCT10 14 [Cash] U.S. Dollar British Pounds Euro Japanese Yen 7.34 4.38 3.48 2.81 1.60 1.59 0.48 0.31 0.28 0.26 0.20 0.06 1.32 0.95 0.84 0.73 0.38 0.26 0.11 0.32 0.28 20.69 14.05 4.60 2.18 2.12 2.01 1.95 1.04 0.12 0.02 0.02 0.02 0.01 0.01 0.01 0.01 5.08 1.07 0.48 0.48 5.39 1.71 1.57 1.02 0.40 0.25 0.25 0.15 0.02 0.01 0.32 0.10 0.09 0.08 0.06 7.75 8.59 -21.72 3.02 2.66 0.18 0.22 0.11 0.10 --2.75 1.08 7.85 2.70 1.64 0.16 0.18 0.23 0.84 ------------------------------------ -0.41 -4.21 3.48 -18.91 -1.42 -1.07 0.31 0.09 0.17 0.16 0.20 0.06 -1.43 -0.14 -7.01 -1.97 -1.26 0.10 -0.07 0.09 -0.56 20.69 14.05 4.60 2.18 2.12 2.01 1.95 1.04 0.12 0.02 0.02 0.02 0.01 0.01 0.01 0.01 5.08 1.07 0.48 0.48 5.39 1.71 1.57 1.02 0.40 0.25 0.25 0.15 0.02 0.01 0.32 0.10 0.09 0.08 0.06 Holdings Data As Of GLOB_BAL_MAND 12/31/2009 MSCI EAFE 9/23/2010 Hidden: Benchm ark Only Securities and Groups Monte-Carlo Extreme Event Risk Example Two.. + It’s clear that if the crisis of 2008 were to occur again, the addition of derivatives in the portfolio would offer a strong hedge against losses Percent of Total Holdings GLOB_BAL_MAND vs. MSCI EAFE 9/23/2010 Factset/R-Squared Daily Global Multi-Asset Class Model (USD) U.S. Dollar Credit Crisis ST % Value at Risk Difference 22 Day, 95% ST % Standalone Value at Risk 22 Day, 95% MC % Value at Risk 22 Day, 95% MC % Marginal Value at Risk 22 Day, 95% MC % Standalone Value at Risk 22 Day, 95% Asset Class Port. Weight Bench. Weight Total 100.00 100.00 -- 8.78 8.78 6.88 Equity Fixed Income 73.60 20.69 100.00 -- -26.40 20.69 8.78 0.63 14.05 7.29 5.89 0.23 0.07 0.02 8.76 3.35 Derivatives [Cash] 5.39 0.32 0.10 0.09 0.08 0.06 ------- 5.39 0.32 0.10 0.09 0.08 0.06 -0.62 42.31 2.59 -0.01 6.04 5.38 5.91 0.73 0.00 -0.00 0.00 0.00 -0.00 0.12 0.01 -0.00 0.02 0.03 -0.00 48.48 2.11 -0.01 3.94 4.25 4.32 U.S. Dollar British Pounds Euro Japanese Yen 0.01 -0.00 0.00 0.00 -0.00 6.88 Exchange Traded Options + Barone-Adesi & Whaley (JOF Vol42, No.2, June 1987) 1. 2. 3. Analytical approximation of American option pricing starting with European formula Many times faster than most other methods Loses accuracy for long dated options unfortunately (e.g. LEAPS) but acceptable accuracy for short to mid-maturity options “They” used a normal approximation for the implied volatility, but that was written in 1987 before the 19 Oct 1987 “Black Monday” event inaugurated the volatility “smile” Therefore FactSet uses an implied vol that’s fit to “f(strike/price, time to maturity)” from stock’s option chain, incorporating the observation that implied vols vary as the stock’s price varies from the option strike (volatility smile effects). This is a very smart methodology 1. 2. 3. The option pricing first involves solving iteratively for a critical stock price (Eq. 19 in their paper), below which the option’s call value is given by the Black-Scholes equation and above which the option’s call value is given by its exercisable proceeds (Price-Strike) The critical price solution is placed into an analytical expression involving the addition of a early exercise premium to the Black-Scholes equation (Eq. 20 of their paper) The next step, given option strike, vols, risk-free rate, time to maturity and stock price from the MC generating process, is simply to “plug-and-chug” to compute the option’s price Ramifications for Fixed Income.. + Due to liquidity issues, seldom have real FI security returns to regress against factor exposures to compute Betas + Hence we used previously calculated “sensitivities” (dur, convex..) + Monte-Carlo generated Interest Rate (yield curve) moves, spread and currency changes + Fast Re-Pricing (Taylor Series expansion) schema utilizes these changes to price “FI” instruments along with time decay 1. All securities of same currency have same yield curve exposure to the same set of 17 key rate risk factors (6 Libor & Govt Curves: U.S., Can, Aus, EUR, Japan, UK) 2. Different types of instruments have differing spread models, currently configured for: Corporates High Yield Agency U.S. Treasuries Sovereigns (that we have yield curves for) U.S. MBS Treasury Inflation-Protected Securities Exchange Traded Derivatives Rotund Posteriors, Hefty Backsides & Pudgy Extremities.. + Fat-Tails should be considered when skewness &/or kurtosis are prevalent Rotund Posteriors, Hefty Backsides & Pudgy Extremities.. S FY UB S I IN D B -1 -1 0 1 2 -2 Quant iles of St andard Normal 1 0. 2 -1 1 2 -2 1 2 1 2 -2 -1 0 1 2 Quant iles of St andard Normal GA M Ordered R et urns 0. 0 -0. 4 -0. 10 Ordered R et urns 0. 2 0. 4 0. 3 0. 1 0. 0 Ordered R et urns 0 0 P E TM -0. 2 -1 -1 Quant iles of St andard Normal 0. 2 0. 8 0. 6 0. 4 0. 2 Ordered R et urns 0 Quant iles of St andard Normal H MN Quant iles of St andard Normal 2 -0. 2 -2 D Y II -2 1 0. 1 Ordered R et urns 0. 1 0. 0 Ordered R et urns -0. 1 0. 1 2 0 LD G -0. 2 0 -1 Quant iles of St andard Normal HR 0. 0 Ordered R et urns -1 Quant iles of St andard Normal 0. 1 0. 2 -2 -0. 2 -2 0. 0 Ordered R et urns 0. 2 2 0. 2 1. 0 0. 5 0. 0 Ordered R et urns 1 R OL -0. 2 Most stocks are non-normal, the evidence is overwhelming.. 0 Quant iles of St andard Normal RS A S + 0. 1 -0. 2 -2 0. 0 2 -0. 1 1 -2 -1 0 1 Quant iles of St andard Normal 2 0. 05 0 Quant iles of St andard Normal 0. 0 -1 -0. 2 -0. 4 -2 0. 0 Ordered R et urns 0. 2 0. 0 Ordered R et urns -0. 2 0. 1 0. 0 Q-Q Plots of 12 randomly selected small cap stocks -0. 1 + Ordered R et urns 0. 4 0. 3 0. 2 0. 3 GB C I -2 -1 0 1 Quant iles of St andard Normal 2 -2 -1 0 1 Quant iles of St andard Normal 2 VAR techniques are subject to model risk so the parametric model used for drawing in Monte Carlo influences the value of the VAR calculated.. + + Fat-Tails underestimate 95% VAR, but are closer to it than normal method + Normal approximation leads to overly optimistic forecasts at 99% VAR + 95% VAR - 95% VAR Empirical Normal is acceptable at 95% VAR Fat-Tails generally result in conservative and accurate 99% VAR YEN/GBP GBP/USD S&P 500 DAX 30 CAC 40 NIKKEI 225 DJI ..mean.. Fat-Tails-> Normal 1 -0.017 -0.070 0.039 -0.057 0.113 -0.075 0.115 -0.059 0.124 -0.063 0.073 -0.125 0.208 -0.035 0.094 -0.069 2 -0.135 -0.094 -0.076 -0.057 -0.085 -0.190 -0.072 -0.101 3 -0.108 -0.052 -0.065 -0.056 -0.085 -0.016 -0.055 -0.062 99% VAR - 99% VAR Empirical YEN/GBP GBP/USD S&P 500 DAX 30 CAC 40 NIKKEI 225 DJI ..mean.. Fat-Tails-> Normal 1 -0.451 0.268 -0.248 0.447 -0.162 0.266 -0.258 -0.100 -0.308 0.127 -0.691 1.408 -0.249 0.393 -0.338 0.401 2 0.133 0.426 -0.093 -0.189 -0.049 0.414 0.232 0.125 3 0.515 0.894 0.691 0.182 0.076 2.585 0.550 0.785 Fat-Tailed & Skewed Asset Return Distributions; Frank Fabozzi Series; Wiley Finance 2005, pg 237 VAR techniques are subject to model risk so the parametric model used for drawing in Monte Carlo influences the value of the VAR calculated.. + Our own work suggests that normal method under-estimates the VAR compared to Fat-Tailed methods, even at 95% confidence level Fat-Tail Value-at-Risk + No “magic bullet” as it doesn’t capture correlation structural changes which occur in real “Black Swan” events (not modeling the volatility of volatility) Currently @ FactSet + Internal discussions on methodology + Robustness tests, ease of use, computation time + On-going development continuing..