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- 1. Post Modern Investment Management A Scientific Approach for the Complex World An Overview March 2010 Mikhail Munenzon, CFA, CAIA 1
- 2. Table of Contents • Returns 5 • Volatility 31 • Risk Measurement 39 • Risk and Return 52 • Dependencies 60 • Risk Management 73 • Markets 76 • Human Decision Making and Investor Edge 78 • Forecasting 81 • Optimization 88 • Hedging 101 • Asset Allocation 103 • Manager Evaluation and Selection 105 • Concluding remarks 108 • References 110 2
- 3. • Everything must be made as simple as possible but not simpler – Albert Einstein • I’d rather be vaguely right than precisely wrong – John Maynard Keynes • There are three kinds of lies: lies, damned lies and statistics – Benjamin Disraeli • Scientific approach – tools are empirically tested and measurable; applied consistently in a transparent manner to allow for improvement and outside verification 3
- 4. Format for each topic • Motivating question of practical significance for investor • Key classical tools and current practice • Key post modern tools • Particular attention paid to assumptions underlying various analytical tools and empirical support for them • Focus is on introducing principles of and intuition behind practically useful tools through verbal and visual methods and minimizing technical details that might obscure the value of such tools 4
- 5. behavior? features and understand its be extracted to of a security, what • Motivation – given a time series of returns useful information can SP500 Daily Return Cumulative Return 0 1 2 3 4 5 6 7 1/2/90 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 1/3/90 Daily Return SP500 1/2/91 Cumulative Return Returns 0 1 2 3 4 5 6 7 1/3/91 1/2/92 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 1/2/90 1/3/90 1/3/92 1/2/93 1/2/91 1/3/91 1/3/93 1/2/92 1/2/94 1/3/92 1/3/94 1/2/93 1/2/95 1/3/93 1/3/95 1/2/94 1/3/94 1/2/96 1/3/96 1/2/95 1/3/95 1/2/97 1/3/97 1/2/96 1/3/96 1/2/98 1/3/98 1/2/97 1/3/97 1/2/99 1/2/98 1/3/98 1/3/99 1/2/00 1/2/99 Date 1/3/99 1/3/00 Date 1/3/00 1/2/00 1/2/01 Date 1/3/01 Date 1/3/01 1/2/01 1/2/02 1/3/02 1/3/02 1/2/02 1/2/03 1/3/03 1/3/03 1/2/03 1/2/04 1/3/04 SP500 Daily Return - 1/2/1990 - 1/26/2010 1/3/04 1/2/04 SP500 Daily Return - 1/2/1990 - 1/26/2010 1/2/05 1/3/05 SP500 Cumulative Return - 1/2/1990 - 1/26/2010 1/3/05 1/2/05 SP500 Cumulative Return - 1/2/1990 - 1/26/2010 1/3/06 1/2/06 1/3/06 1/2/06 1/3/07 1/3/07 1/2/07 1/2/07 1/3/08 1/3/08 1/2/08 1/2/08 1/3/09 1/2/09 1/2/09 1/3/09 1/3/10 1/2/10 1/3/10 1/2/10 5
- 6. Returns (cont.) - Classical Perspective • Returns behavior follows a normal Histogram of 100,000 randomly generated value from the normal distribution (Gaussian) distribution 4500 – First introduced by DeMoirvre – Later further developed by Gauss to study and forecast planetary motions 4000 – Even if returns are not normal for a small sample, such time series will converge to 3500 normal distribution for large sample (via the Central Limit Theorem) • Implications 3000 – Only need the mean and volatility (will discuss later in more detail) of returns to assess 2500 probability of events – Symmetrical process (skewness is 0; returns greater than the mean are as likely as returns 2000 below the mean) – Extreme events are rare – any deviations 1500 from the mean within 2*volatility events will add up to about 95.5% of total probability and 3*volatility deviations add up to about 99.7% 1000 (kurtosis is 3 or excess kurtosis is 0 => measure of fat tails) 500 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
- 7. Returns (cont.) – Classical Perspective • Returns behavior follows a random walk – On any given day, price rise is as likely as price decline as rational actors (see more in human decision making section) fully and quickly incorporate all information in prices • Implications – Past prices cannot provide any valuable information with regard to future prices (serial correlation, relationship of current prices with its lagged values is non-existent or 0) – Any forecasting attempts are unlikely to add any value 7
- 8. Returns (cont.) – Empirical Evidence from Traditional Asset Classes plus VIX 1/2/1990-1/31/2010 SPX GSCI NAREIT JPMAGG VIX daily data Arithmetic avg return 0.0357% 0.0253% 0.0554% 0.0264% 0.1763% Compounded avg return 0.0291% 0.0158% 0.0426% 0.0261% 0.0068% max 11.6% 7.8% 18.4% 1.3% 64.2% min -9.0% -16.9% -19.5% -1.5% -25.9% vol 1.2% 1.4% 1.6% 0.3% 5.9% Skewness 0.00 -0.40 0.47 -0.14 1.23 Kurtosis 12.71 10.74 33.85 4.77 10.91 Number of days 5,238 5,238 5,238 5,238 5,238 Normality at 95% confidence level? No No No No No p-values 0.1% 0.1% 0.1% 0.1% 0.1% No serial correlation at 95% confidence level? No No No No No p-values 0.0% 0.0% 0.0% 0.0% 0.0% Cumulative Return 358.7% 129.2% 833.0% 292.9% 42.8% % of days with positive returns 51.7% 49.3% 51.9% 53.0% 45.8% 8
- 9. Returns (cont.) – Empirical Evidence from Traditional Asset Classes plus VIX Notes: Jarque-Bera test was used to evaluate normality of a time series; null hypothesis is stated in the question. Ljung-Box test with 20 lags was used to evaluate serial correlation of a time series; null hypothesis is stated in the question. SPX - SP500 Total Return GSCI - SP GSCI NAREIT - FTSE EPRA/NAREIT US Total Return JPMAGG - JPM Morgan Aggregate Bond Total Return VIX - VIX Index 9
- 10. Returns (cont.) – Empirical Evidence from Alternative Investment Strategies CISDM Alternative Strategies Indices 12/31/1992 - 12/31/2009 monthly data Convert Merger Long/ TOTAL EW Arb Distressed Arb CTA Macro Short FOF EMN EM FI Arb Arithmetic avg return 1.03% 0.79% 0.91% 0.80% 0.70% 0.78% 0.96% 0.62% 0.67% 0.92% 0.55% Compounded avg return 1.00% 0.78% 0.90% 0.79% 0.67% 0.77% 0.93% 0.61% 0.67% 0.85% 0.54% max 8.37% 4.71% 5.26% 4.74% 7.86% 8.61% 9.40% 4.50% 2.76% 12.13% 5.21% min -8.82% -11.49% -10.59% -5.61% -5.43% -5.36% -9.42% -6.40% -2.10% -26.25% -9.44% vol 2.16% 1.44% 1.83% 1.12% 2.49% 1.62% 2.20% 1.42% 0.58% 3.80% 1.40% Skewness -0.63 -3.92 -1.94 -0.80 0.41 1.19 -0.25 -1.31 -0.43 -2.12 -2.92 Kurtosis 6.49 33.46 13.38 8.64 3.06 7.67 5.75 8.29 6.23 15.94 24.28 cumulative return 762.97% 433.42% 589.06% 449.07% 319.22% 420.27% 641.14% 275.27% 319.76% 518.67% 118.47% Notes: 10 fixed income arbitrage is from 12/31/1997.
- 11. Returns (cont.) – Empirical Evidence from Alternative Investment Strategies Histogram of monthly returns for CISDM Convertible Arb Index Histogram of monthly returns for CISDM Distressed Index 12/31/1991 - 12/31/2009 12/31/1991 - 12/31/2009 140 60 120 50 100 40 80 30 60 20 40 10 20 0 0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 Histogram of monthly returns for CISDMCEW CTA Index Historgram of monthly return for CISDM Macro Index 12/31/1991 - 12/31/2009 12/31/1991 - 12/31/2009 40 90 35 80 70 30 60 25 50 20 40 15 30 10 20 5 10 0 0 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 11
- 12. Returns (cont.) – Empirical Evidence from Alternative Investment Strategies • Note on the prior page how most of the colored graph for convertible arb and distressed is below the mean (negative skewness) and the left part of the graph is thick and long (fat tails or large extreme losses • Results for CTA and macro are exactly the opposite • For more detail on traditional asset classes, see Munenzon (2010) 12
- 13. Returns (cont.) – Empirical Evidence • Significant non-normality – skewness and kurtosis; mean and volatility are not enough to describe return distribution • Negative skewness and leptokurtosis (fat tails), especially for such alternative strategies like – the opposite of what investors generally prefer: consistent, positive returns • Convergence to normal distribution does not occur for large samples • Short term persistence – Significant serial correlation as measured by autocorrelation coefficients • For liquid asset classes at daily frequency, the magnitude of the first order autocorrelation is generally close to 0 (e.g, for SP500, it is -.0589) • For other time horizons and securities, it can be much larger (e.g., 0.58 for Convertible arb, .43 for Distressed vs 0.15 for Macro, 0.01 for CTA) – Positive momentum effects when measured 9-24 month period => securities with strong performance over the past 6-12 months are likely to continue to outperform in the next 3-12 months (research starting with Jegadeesh et al (1993) for US; later extended to international securities) • Over the very short term (week and month), returns may show negative momentum – ‘Smoothness’ of data (strong positive autocorrelation) implies that volatility on raw data will understate the true range of outcomes (more later in risk section) • Long term persistence – Losers of the past 3-5 years are likely to be winners of the next 3-5 years – Can be measured by Hurst exponent (.5 => no persistence; <.5 =>anti persistence; >.5 => positive persistence); using the full sample at daily frequency = > • SP500 => .18 • GSCI => .2 • NAREIT => .22 • JPMAGG => .18 • VIX => .19 13
- 14. Returns (cont.) – Empirical Evidence from Traditional Asset Classes plus VIX • Volatility of returns is 3.50% Rolling 250 day daily volatility of SP500 time varying and 3.00% 2.50% exhibits a tendency to 2.00% Volatility 1.50% cluster (also can be 1.00% 0.50% seen in the daily 0.00% 12/18/90 12/18/91 12/18/92 12/18/93 12/18/94 12/18/95 12/18/96 12/18/97 12/18/98 12/18/99 12/18/00 12/18/01 12/18/02 12/18/03 12/18/04 12/18/05 12/18/06 12/18/07 12/18/08 12/18/09 Date return chart on p.6; VIX - 1/2/1990 - 1/26/2010 90 more in the section on 80 70 60 volatility) 50 VIX 40 30 20 10 0 1/2/90 1/2/91 1/2/92 1/2/93 1/2/94 1/2/95 1/2/96 1/2/97 1/2/98 1/2/99 1/2/00 1/2/01 1/2/02 1/2/03 1/2/04 1/2/05 1/2/06 1/2/07 1/2/08 1/2/09 1/2/10 Date 14
- 15. Returns (cont.) – Empirical Evidence from Traditional Asset Classes plus VIX • Are different levels of volatility associated with meaningfully levels of returns? • Use VIX to assess impact across asset classes; can be refined further through more tailored volatility measures, prior states and other factors 15
- 16. State = > VIX <= 20 SPX GSCI NAREIT JPMAGG VIX daily Arithmetic avg return 0.0983% 0.0411% 0.1004% 0.0254% -0.1932% Compounded avg return 0.0962% 0.0351% 0.0976% 0.0252% -0.3318% max 2.9% 6.8% 3.6% 1.0% 64.2% min -3.5% -4.6% -4.8% -1.1% -25.9% vol 0.7% 1.1% 0.7% 0.2% 5.3% Skewness 0.02 0.18 -0.33 -0.04 1.25 Kurtosis 4.16 5.06 7.03 4.77 13.94 Number of days 2,945 2,945 2,945 2,945 2,945 Normality at 95% confidence level? No No No No No p-values 0.1% 0.1% 0.1% 0.1% 0.1% No serial correlation at 95% confidence level? Yes Yes No No No p-values 49.5% 13.0% 1.1% 0.9% 0.0% Cumulative Return 1595.7% 181.1% 1669.5% 109.9% -100.0% % of days with positive returns 55.3% 49.3% 55.2% 52.7% 43.5% 16
- 17. State => 25 < VIX <= 30 SPX GSCI NAREIT JPMAGG VIX daily Arithmetic avg return -0.0776% -0.0227% -0.0296% 0.0230% 0.4789% Compounded avg return -0.0872% -0.0373% -0.0392% 0.0226% 0.2802% max 5.0% 5.9% 8.8% 0.9% 40.7% min -3.8% -16.9% -7.0% -0.9% -20.0% vol 1.4% 1.7% 1.4% 0.3% 6.4% Skewness 0.27 -1.76 0.20 -0.36 0.72 Kurtosis 3.23 19.72 9.96 3.68 6.01 Number of days 590 590 590 590 590 Normality at 95% confidence level? No No No No No p-values 1.9% 0.1% 0.1% 0.1% 0.1% No serial correlation at 95% confidence level? Yes Yes No Yes Yes p-values 11.0% 12.8% 0.0% 84.1% 56.6% Cumulative Return -40.2% -19.8% -20.7% 14.3% 421.0% % of days with positive returns 44.6% 48.8% 45.6% 54.6% 49.3% 17
- 18. State => VIX > 40 SPX GSCI NAREIT JPMAGG VIX daily Arithmetic avg return -0.5372% -0.4839% -0.7983% 0.0562% 1.5559% Compounded avg return -0.5940% -0.5300% -1.0065% 0.0555% 1.1540% max 11.6% 7.5% 18.4% 1.3% 34.5% min -9.0% -8.1% -19.5% -1.0% -24.7% vol 3.4% 3.0% 6.5% 0.4% 9.2% Skewness 0.41 0.12 0.49 0.12 0.66 Kurtosis 4.22 3.25 3.64 4.36 4.77 Number of days 158 158 158 158 158 Normality at 95% confidence level? No Yes No No No p-values 0.7% 50.0% 1.9% 0.9% 0.1% No serial correlation at 95% confidence level? Yes Yes Yes No No p-values 62.0% 33.2% 30.2% 0.6% 1.9% Cumulative Return -61.0% -56.8% -79.8% 9.2% 512.8% % of days with positive returns 39.2% 37.3% 34.2% 53.8% 51.3% 18
- 19. Cumulative Return 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 2 4 6 8 10 12 14 16 18 20 '8/7/1990' '1/3/1990' '10/10/1990' '4/15/1991' '10/31/1990' '9/9/1991' '1/16/1991' '2/12/1992' '11/17/1997' '7/7/1992' '12/2/1992' SP 500 Total Return '8/24/1998' '4/26/1993' '10/19/1998' SP 500 Total Return '9/16/1993' '1/13/1999' '2/8/1994' '2/18/1999' '7/6/1994' '3/22/2001' FTSE EPRA/NAREIT US Total Return '11/28/1994' '9/11/2001' '4/20/1995' '10/4/2001' FTSE EPRA/NAREIT US Total Return '9/12/1995' '10/23/2001' '2/2/1996' '7/10/2002' '6/28/1996' Date Date '8/29/2002' '12/2/1996' '10/21/2002' '2/18/1998' '9/4/2000' '11/5/2002' '8/11/2003' SP GSCI '1/24/2003' Cumulative Return - State of VIX (<=20) '1/13/2004' '2/6/2003' '6/11/2004' '2/19/2003' Cumulative Return - State of VIX (>30 and <=35) '11/3/2004' '3/5/2003' '3/28/2005' SP GSCI '3/18/2003' '8/18/2005' '3/17/2008' '1/10/2006' '5/4/2009' '6/2/2006' '5/15/2009' '10/30/2006' '6/3/2009' '3/22/2007' JP Morgan US Aggregate Bond Total Return '7/8/2009' '10/10/2007' JP Morgan US Aggregate Bon Cumulative Return Cumulative Return 0 0.2 0.4 0.6 0.8 1 1.2 0.5 0.7 0.9 1.1 1.3 '8/31/1998' '1/5/1990' '9/10/1998' '2/13/1990' '10/5/1998' '3/23/1990' '10/12/1998' '7/23/1990' '9/21/2001' '11/23/1990' '8/6/2002' '12/21/1990' SP 500 Total Return '10/9/2002' '2/11/1991' '10/7/2008' '8/19/1991' SP 500 Total Return '10/14/2008' '7/16/1996' '10/21/2008' '2/5/1997' '10/28/2008' '11/4/2008' '4/1/1997' FTSE EPRA/NAREIT US Total Return '11/11/2008' '6/3/1997' FTSE EPRA/NAREIT US Total Return '11/18/2008' '7/24/1997' '11/25/2008' '8/22/1997' '12/2/2008' '9/25/1997' '12/9/2008' Date '12/5/1997' Date '12/16/2008' '1/29/1998' '12/23/2008' '4/2/1998' '12/30/2008' '5/6/1998' SP GSCI '1/13/2009' '6/11/1998' SP GSCI '1/20/2009' '11/24/1998' Cumulative Return - State of VIX (>25 and <=30) Cumulative Return - State of VIX (>40) '1/27/2009' Returns in Different States '3/5/1999' '2/4/2009' '4/12/1999' '2/11/2009' '6/18/1999' Returns (cont.) – Asset Class '2/18/2009' '7/30/1999' '2/25/2009' '3/4/2009' '9/9/1999' '3/11/2009' '10/27/1999' '3/18/2009' '11/29/1999' '3/25/2009' '12/28/1999' JP Morgan US Aggregate Bond Total Return '4/1/2009' '2/2/2000' JP Morgan US Aggregate Bond Total Return 19
- 20. Returns (cont.) – Asset Class Returns in Different States • Key observations from the prior pages – Gains and losses are quite concentrated (e.g., for SPX in VIX <=20, the probability of positive days is only 55% but the magnitude of positive days is very large compared to negative days) – Normality is very rare even in individual states and serial correlation generally remains (past information is likely to be useful in predicting the future) – Very consistent behavior among asset classes, particularly at the negative extreme => diversification benefits are possible but may be highly concentrated (e.g., aggregate bond market or more specifically government bonds) 20
- 21. Returns (cont.) – Approaches to data evaluation • Frequentist school of statistics – Derive the probability distribution of a sample from as large a sample of data – Appropriate for large data sample produced by stable process – Data points are independent and identically distributed – Example: probability of positive return days for SP500 is 51.7% • Baysian school of statistics – There’s uncertainty about the true distribution of a process – Pick your initial distributional shape and update as new relevant information arrives – Can better incorporate limited data and unstable processes but more vulnerable to incorrect, starting distributional choices and more computationally intensive – Example: probability of positive return days for SP500 given that VIX >40 is 39.2% 21
- 22. Returns (cont.) – Approaches to data evaluation • In the example of SP 500 for Frequentist Bayesian the full sample of 1/2/1990 – perspective perspective 1/29/2010 Probability that daily return is positive 51.7% 51.7% – Frequentist predictions are different from empirical results Probability that daily return is positive if (especially within each state), Last trading day was positive 51.7% 50.6% whose uncertainty can be Last 2 trading days were positive 51.7% 49.8% handled with Bayesian tools Last 3 trading days were positive 51.7% 46.8% • As returns are assumed to be Last 4 trading days were positive 51.7% 44.8% independent and identically distributed, the probability of positive return on any given Probability of 2 consecutive positive days 26.7% 26.2% day is the same Probability of 3 consecutive positive days 13.8% 13.0% • Probability of n consecutive Probability of 4 consecutive positive days 7.2% 6.1% positive days is .517^n Probability of 5 consecutive positive days 3.7% 2.7% 22
- 23. Returns (cont.) – More on Approaches to Data Evaluation • In the SP500 price series sample from 1928- Decade return for SP500 price index - first decade calculation starts 2009 (excludes dividends and inflation) at the end of 1937 going to 1928, etc 1928 - 2009 – While the probability that each decade return is positive is very high (almost 85%) 400.00% 350.00% – There are distinct period of high and low 300.00% Prior Decade Return returns and 250.00% – When decade returns start declining, such 200.00% 150.00% periods take years to complete, though there 100.00% are only 2 such periods in this sample 50.00% • Economic and sentiment factors affecting 0.00% performance may take a long time to reverse -50.00%1937 1942 1947 1952 1957 1962 1967 1972 1977 1982 1987 1992 1997 2002 2007 • Positive returns are possible; however, they are -100.00% replacing years that were better and may not be Year enough to offset negative years • Given the current state in this historical sample, implications for performance the next few years are not positive SP500 price index annual return - 1928 - 2009 – The concept of sample error (also see 60.00% manager evaluation section) • Error cause by observing the limited sample 40.00% rather than the true population • How likely that the sample one has for analysis 20.00% Annual Return is unrepresentative of the true population data? 0.00% – 1 / square root of the number of observations 1928 1933 1938 1943 1948 1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003 2008 -20.00% -40.00% -60.00% Year 23
- 24. Returns (cont.) – How to evaluate drivers of returns? • Classical perspective – CAPM: 1 factor model using a market index • Advantages: simplicity • Disadvantages: – A extensive list of anomolies and deviations from empirical data - e.g., Fama French factors, calendar effects – Cannot be tested as the true market index is hard to specify (Roll critique) – Time variance of beta and risk premium – APT: multi factor models • Advantages: more flexibility in describing return drivers • Disadvantages: – Which factors should be included? – How many factors should be included? 24
- 25. Returns (cont.) – Returns and economic growth • Current practice – Invest in countries that show high rates of economic growth • Evidence – Zero to slightly negative relationship between returns and economic growth over multi year period (3+ or the length of a business cycle) • Investors herd and overbid • In order to take advantage of economic growth, companies will require capital, driving down returns for existing shareholders • Benefits of economic growth don’t accrue just to companies already listed on exchanges – Positive or negative relationship possible at certain stages of the economic cycle for short to mid term; however, must have flexibility and tools to appropriately time entry and exit to take advantage 25
- 26. Returns (cont.) – Practical Issues • Jarque Bera test – measures if data follows normal distribution; visual tools include histrogram, normal probability and quantile/quantile plots • Ljung Box Q test – measures if data exhibits serial correlation • Durbin Watson test – measures if error terms of a a regression are serially correlated • Classical regression – Y = intercept + beta * X +error – Linear – Beta is estimated via the ordinary least squares method • Beta obtained is the BLUE estimator as long as at least the following holds (there are assumptions not covered here): – Error terms are uncorrelated with each other – Error terms have the same volatility (homeskedastic) – BLUE – best (with the lowest variance than other estimators); linear; unbiased (expectation of the estimator is the true value of the coefficient); estimator • Practical issues with regression approach – Most financial data has time varying volatility (see prior pages; heteroskedastic; if not adjusted for heteroskedasticity (White method), the significance of estimators is likely to be overestimated – Most financial data is serially correlated (see prior pages); if not adjusted for serial correlation (Newey West method, which also corrects for heteroskedasticity), the statistical significance of estimators will be overestimated – The greater the deviations (particularly for many alternative investment strategies), the less the output from classical regression can be relied on to assess the statistical significance of estimators – More advanced method like generalized least squares embeds these issues within the estimation process 26
- 27. Returns (cont.) – Practical Issues • Practical issues with regression approach (cont.) – Which factors and how many • Current practice – The user selects factors for a regression » Capture what occurred vs capture what the user thinks should have occurred » No methodology to decide which and how many factor • Should reflect non-linearity and non-normality of observed returns – e.g., option based strategy factors • Can be selected via stepwise or related algorithm from a universe of factors based on their statistical significance and contribution to the overall explanatory power of a regression • From random matrix theory and tests on financial data, most of return variance of any security can be explained by 4 - 8 factors; the rest is noise • Large number of factors lead to overfitting of historical data and are likely to produce multicollinearity (high correlation among factors), reducing the forecasting power of the model and statistical significance of factors • Ridge or orthogonal regression can be used if multicollinearity problems are severe – May use nonlinear regression to more explicitly incorporate non-linearity – Robust statistics (see optimization section) • Reduce the impact of outliers on metrics, making them more stable to small changes in environment • Separates analysis of regular and extreme events • Examples – Use median, trimmed mean or other metrics that are less sensitive to the presence of outliers – Robust regression to estimate coefficient of parameters – May use other approaches to minimize sampling error (Bayesian tools or shrinkage estimators; see optimization section) – May find spurious relationships among random time series if they are integrated (see forecasting section) 27
- 28. Returns (cont.) – Practical Issues • Other econometric approaches (more in forecasting section) – Key modeling approach => return today is related to the prior return • Easy to apply • Good results for trending series • No need to pick factors • May add other info – e.g., deviations from the mean • May be refined further by modeling deviations separately via GARCH models (see next section) and extended to include multi- security relationships via VAR models (see forecasting section) • May incorporate both momentum and reversal • Some assumptions may be too restrictive • Depending the model chosen, may be too computationally intensive if there are many securities (VAR) or only applicable to a few time series (cointegration) • Or incorporate statistical learning, Bayesian type methods (more in forecasting section) 28
- 29. Returns (cont.) – More Empirical Evidence • Significant majority of portfolio return and its variance is explained (starting with Brinson et al (1986)) by – Decisions on which asset classes to invest in – Decisions on which sectors within asset classes to invest in – Decisions on individual securities/managers contribute the least amount (e.g., 10%) and sometimes contribute negatively – Current practice • Investors spend most of their time evaluating individual security/manager factors • Sector vs country – For developed markets, industry factors are the primary driver of returns – For developing counties, country decisions are more important in explaining returns • Value effect (tendency for value stocks to outperform growth stocks) – Concentrated among the smallest companies (~$100mil market cap or less) with little analyst coverage (~3 analysts or fewer) and low institutional ownership (up to 50% owned) 29
- 30. Returns (cont.) – Summary • Key empirical facts – Returns exhibit positive momentum over the short to mid term but mean reversion over mid to long term => return processes are time varying with different distribution characteristics, particularly at extremes (see risk section on extreme value theory) – Returns are significantly non-normal with negative skews and large tails as measured by kurtosis – Losses and gains are concentrated – Return volatility is time varying and has a tendency to cluster (next section) – Returns across countries are not related to economic growth => zero to slightly negative relationship – Return of a portfolio and its variance are primarily explained by decisions on asset classes and their sectors; security selection is the least important part • Implications – Incorporate non-normal distributions to model returns to capture skews and fat tails => e.g., t distribution better captures empirical features of tails than normal distribution – There’s uncertainty as to the true return distribution due to time variance of returns and different statistical properties of regimes – Because of a wide range of outcomes and regimes, the reliance on the concept of average in analysis will not appropriately incorporate complex reality – Situational awareness => attempt to identify the current environment – Potential value to incorporating time varying volatility in analysis – Regression output must be adjusted for serial correlation and changes in volatility to properly assess the statistical significance of estimators – Methods other than regression may incorporate empirical features better because of their greater flexibility 30
- 31. Volatility • From the prior section – Volatility is time varying – Volatility has a tendency to cluster – Different volatility regimes are associated with different return processes • Motivation – Can volatility be forecasted? – If yes, which methods are most appropriate? 31
- 32. Volatility (cont.) – Classical perspective • Use historical volatility figures (full historical sample or, more rarely, rolling window as on p.6) on an equal weighted basis – the dominant practitioner approach – Assumes that returns are independent and identically distributed (iid) – Behavior of future volatility is the same as its past behavior – May lead to sharp changes as crashes enter and exit data series – Silent on the length of window • Use weighted average volatility, giving more weight to recent figures (e.g., exponentially weighted moving average (EWMA) approach via a decay constant,lambda, used by RiskMetrics) – gaining share – Volatility is related to the prior return and volatility that are weighted • vol^2 = (1-lambda)*r(t-1)^2+lambda*vol(t-1)^2; – Assumes that returns are iid – Behavior of future volatility is the same as its past behavior – For mathematical reasons, the same decay factor is used for all securities/asset classes – Persistence in volatility of a security (lambda) is closely related to its sensitivity to market shock (1-lambda) – No methodology for choosing the decay constant – RiskMetrics uses .94 for daily data and .97 for monthly data – Can’t incorporate mean reversion as long term volatility is undefined 32
- 33. Volatility (cont.) – Empirical evidence – Volatility can be forecasted over a limited time horizon – Has a tendency to cluster – One approach to model clustering is through Markov chains and transition probability matrices (see next page) – Mean reversion (e.g., see p.13,32) – Model should incorporate volatility changes due to market shocks and persistence in its level as separate variables 33
- 34. States: 1-VIX<=20, 2-20<VIX<=25, 3-25<VIX<=30,4- 30<VIX<=35,5-35<VIX<=40,6-VIX>40; based on daily data Transition probability matrix for VIX Next day State Average Maximum % of all days Current State 1 2 3 4 5 6 VIX State Duration Duration in State 1 95.9% 4.1% 0.0% 0.0% 0.0% 0.0% 1 24.3 578 56.2% 2 9.9% 82.0% 7.9% 0.2% 0.0% 0.0% 2 5.6 37 23.1% 3 0.0% 16.3% 75.1% 8.5% 0.2% 0.0% 3 4.0 16 11.3% 4 0.0% 0.4% 22.1% 68.5% 8.1% 0.9% 4 3.2 25 4.5% 5 0.0% 0.0% 0.0% 22.0% 64.0% 14.0% 5 2.8 10 1.9% 6 0.0% 0.0% 0.0% 0.0% 10.1% 89.9% 6 9.9 64 3.0% 34
- 35. Volatility (cont.) – Volatility as a random process • GARCH (generallized autoregressive conditional heteroskedasticity) models – Original version (Bollerlev (1986)) – • Volatility is related to the prior return and volatility but the model has important differences compared to the weighting approach – Vol^2 = c+alpha*error(t-1)^2+beta*vol(t-1)^2 • Long term or unconditional volatility is defined as c/(1-(alpha+beta) • Allows to model sensitivity to market shocks and persistence in volatility separately • Can incorporate serial correlation in returns and return volatility clustering • Can be estimated separately for each security/asset • Symmetrical response to positive and negative volatility • Introduces some skewness and kurtosis – More advanced versions (A-GARCH, GJR-GRACH, Student t-GARCH, etc) allow to • Capture assymmetric responses to negative vs positive shocks • More fully include skewness and kurtosis • Model different transition probabilities (original model implies that the probability of switching from one regime to another constant though the conditional probability of being in a given regime varies over time) – Greater computational and mathematical complexity as the cost of more appropriate description of complex reality – Trade-off between capturing time variance nature of volatility and incorporating tails • Other models to model volatility as a random process – e..g, Heston model via stochastic calculus 35

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