Management of Portfolio Volatility

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2 EDHEC-Risk Days 2012 London, March 29th, 2012, 10:45-12:00 Idiosyncratic Volatility Hedging for Better Management of Portfolio Volatility Lionel Martellini Professor of Finance, EDHEC Business School Scientific Director, EDHEC Risk Institute This research has been carried out within the FBF research chair on Structured Products and Derivative Instruments 2

3 Outline Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 3

4 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 4

5 Volatility as an Indicator vs. Volatility as an Asset Class Information about volatility is very useful in investment management, from both an asset pricing and a portfolio selection perspectives (see next presentation). More recently, investors have shown an increased interest in volatility as an asset class: one of the main motivations for trading in volatility is to diversify equity risk through long volatility exposure (Szado (2009)), with changes in volatility negatively correlated with stock index returns. In addition, negative correlation and high h volatility are particularly l pronounced in stock market downturns, offering protection against stock market losses when it is needed most, and when other forms of diversification are not very effective. 5

6 Explanations for Inverse Relationship Explanations behind the inverse relationship between volatility and equity returns: Leverage effect from an ex-post perspective (*): Black (1976), Christie (1982), Schwert (1989); Volatility feedback effect from an ex-ante perspective (**): French et al. (1987), Bekaert et tw Wu (2000), Wu (2001), Kim et al. (2004). The presence of profound economic reasons behind the inverse relationship between equity return and volatility is a comforting indication of robustness of expected diversification benefits, (*) The leverage effect : a decrease (respectively, an increase) in equity prices increases (respectively, decreases) the company s leverage, thereby increasing i (respectively, decreasing) the risk to equity holders and increasing i (respectively, decreasing) equity volatility. (**) The "volatility feedback effect assuming that volatility is incorporated in stock prices, a positive volatility shock increases the future required return on equity and stock prices are expected to fall simultaneously. 6

7 Other Motivations for Trading in Volatility Beyond diversification purposes, the demand for volatility exposure can be explained by hedging/speculative demands. Hedging demand: Implicit volatility exposure is a motivation for long volatility positions by fund managers who are often implicitly short volatility. In particular, benchmarked equity fund managers are short volatility because portfolio tracking error and rebalancing costs increase with an increase in the volatility of equity markets (Hill (2004)). Speculative/arbitrage demand: Directional bets on volatility changes; Non directional bets (e.g., implied versus historical, 3-months implied vs. 1 month-implied, etc.). 7

8 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 8

9 Volatility Indices Because information about changes in volatility is of a critical importance to market participants, a number of initiatives have been launched so as to make volatility measures available to investors in the form of volatility indices. Such indices are calculated based on option prices, and as such are based on implicit as opposed to physical volatility measures. The most popular volatility index is the VIX, which is built from prices of equity index options on the S&P 500, first introduced by the CBOE in 1993, with a methodology that t has subsequently been revised to make it a model free estimate for implied volatility. 9

10 Available (Implied) Volatility Indices Currently available volatility indices are implied volatility indices based on option prices Index Ticker Underlying Index Provider AMEX Volatility Index QQV QQQ AMEX CBOE Volatility Index VIX SPX CBOE CBOE DJIA Volatility Index VXD DJX CBOE CBOE NASDAQ-100 Volatility Index VXN NDX CBOE CBOE Russell 2000 Volatility Index RVX RUT CBOE CBOE S&P 100 Volatility Index VXO OEX CBOE CBOE S&P Month thvolatility Index VXV SPX CBOE CBOE VIX Premium Strategy Index VPD VIX CBOE CBOE Capped VIX Premium Strategy Index VPN VIX CBOE CBOE Crude Oil Volatility Index OVX USO CBOE CBOE Gold Volatility Index GVZ GLD CBOE CBOE EuroCurrency Volatility Index EVZ FXE CBOE AEX Volatility VAEX AEX Euronext BEL 20 Volatility VBEL BEL 20 Euronext CAC 40 Volatility VCAC CAC 40 Euronext FTSE 100 Volatility VFTSE FTSE 100 Euronext uo DAX Volatility VDAX-NEW DAX Deutsche Borse AG SMI Volatility VSMI SMI SIX Swiss Exchange AG EURO STOXX 50 Volatility VSTOXX EURO STOXX 50 STOXX Limited NIKKEI Volatility Index VNKY NIKKEI 225 Nikkei Inc. India NSE VIX INVIXN NIFTY 50 India NSE KOSPI 200 Volatility Index VKOSPI KOSPI 200 Korea Exchange Mexico Volatility Index VIMEX IPC MexDer 10

11 Available Derivatives Contracts Trading in options is not a clean bet on volatility alone (vega but also delta/gamma exposure); volatility can now be directly traded through OTC swaps, and more recently through exchange-traded futures and options on volatility indices. Institution Index Ticker (Owner) Futures Ticker VX (VIX Futures), VM CBOE Volatility Index VIX CBOE (Mini-VIX Futures) CBOE DJIA Volatility Index VXD CBOE DV CBOE NASDAQ-100 Volatility Index VXN CBOE VN CBOE Russell 2000 Volatility Index RVX CBOE VR VDAX- Deutsche Borse DAX Volatility NEW AG FVDX (delisted) SIX Swiss SMI Volatility VSMI Exchange AG FVSM (delisted) EURO STOXX 50 VSTOX STOXX FVSX (delisted), FVS Volatility X Limited (Mini-Futures) Options on Futures Available - Available Available

12 Issues with Volatility Indices Volatility indices are not available for an extensive set of markets, because they require the presence of a liquid option market; for example, relatively few volatility indices exist for Asian markets, or other emerging markets. In developed markets, they only exist at the aggregate country large cap stock level: no volatility index available for small cap stocks, growth/value stocks, sectors, etc. Beside, where and when they exist, implied volatility estimates may be plagued by option-market problems that have little to do with underlying equity markets. 12

13 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 13

14 Introducing a New Volatility Index Series The focus of this research project is to introduce a new form of volatility indicator, which is meant to be : Contemporaneous; Model-free; Based on equity market data; Available for all markets/sectors at all frequencies. References: Idiosyncratic Risk and the Cross-Section of Stock Returns, by René Garcia, Daniel Mantilla and Lionel Martellini. Introducing a New Form of Volatility Index: the Cross-Sectional Volatility Index (CVIX), by Felix Goltz, Renata Guobuzaite and Lionel Martellini. 14

15 On the Cross-Sectional Distribution of Stock Returns Key insight: we propose to exploit informational content in the cross-sectional distribution of stock returns, i.e., the dispersion of stock returns on a given date, to design a novel form of volatility indicator. When the S&P 500 index goes down by 5% on a given date, one may be looking at very different situations: All stocks in the S&P500 universe went down by 5%. Half of the stocks went down by 20% and half went up by 10%. The performance of an index is nothing but the average value of a distribution, and as such a very imperfect proxy for the whole distribution. 15

16 Cross-Section of Stock Returns Low Dispersion 50% Cross-Sectional Distribution of Returns (Minimum Value) 45% 40% 35% 30% Percentage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Returns Cross-sectional distribution ib ti of daily stock return data for the S&P500 universe on December 28, 2008 (corresponding to the lowest dispersion for the sample period ranging from January 1990 to November 2010). 16

17 Cross-Section of Stock Returns Average Dispersion 50% Cross-Sectional Distribution of Returns (Median Value) 45% 40% 35% 30% Percentage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Returns Cross-sectional distribution ib ti of daily stock return data for the S&P500 universe on September 24, 1997 (corresponding to the median dispersion for the sample period ranging from January 1990 to November 2010). 17

18 Cross-Section of Stock Returns High Dispersion 50% Cross Sectional Distribution of Returns (Maximum Value) 45% 40% 35% 30% Percentage 25% 20% 15% 10% 5% 0-35% -30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% Returns Cross-sectional distribution ib ti of daily stock return data for the S&P500 universe on Nov 24, 2008 (corresponding to the highest dispersion for the sample period ranging from January 1990 to November 2010). 18

19 Information Content of Cross-Sectional Variance 19

20 Introducing CSV Index Broadly speaking, we propose to use the cross-sectional dispersion of returns at any given date and frequency as an observable, model-free, proxy for volatility. Formally, define for a given universe of N t stocks the return on a portfolio weighting scheme w t as (e.g., CW or preferably EW): ( wt ) r t N = w The CSV index is defined as the square-root root of the crosssectional variance, a.k.a. variety (*): Nt ( w ) 2 t CSV w ( r r ) t it it t i= 1 t i= 1 it r it (*) In what follows, we will loosely use the acronym CSV for both cross-sectional variance or volatility. 20

21 Asymptotic Properties of CSV Assume without any loss of generality the following single factor model for (excess) stock returns: r it = β F + ε Further make two assumptions (to be discussed later): it t Homogeneous beta assumption: β it = β t for all i; Homogeneous residual variance assumption: E(ε 2 it)=σ 2 ε(t) for all i. it Under these assumptions, we have (for all weighting schemes) that: Nt ( wt CSV ) w r r ( wt ) 2 2 = t t i= 1 it ( ) () it t N t σ ε 21

22 Finite Distance Properties of CSV Under the same assumptions, we have that: N [ ( w )] = ( ) t t 2 E CSV t σ ε t 1 i= 1 w ( ) 2 Nt Nt Nt w t Var[ CSV ] = ( ) + t 2σ ε t wit wit 2 wit i= 1 i= 1 i= 1 The CSV based on an equal-weighting g scheme is theoretically (and empirically) shown to be an asymptotically unbiased, efficient (minimum bias for EW) and convergent estimator for specific volatility. [ ] EW E CSVt = σ ε () t 1 σ () t N ε N t t Var 2 it 1 Nt 2 Nt [ ] EW 4 CSV = 2σ 0 t ε t N t 22

23 Relaxing the Homogeneous Residual Variance Assumption If we relax the homogeneous residual variance assumption, we have that: N t N t ( wt ) E CSV = w σ t w t [ ] ( ) ( ) t i= 1 it ε i itσ ε i i= 1 Hence, the CSV appears as a (biased) estimator for the average idiosyncratic variance. If we focus on the EW scheme, we obtain: Nt 1 = N t i [ ] EW 2 t σ ( t) N ε i E CSV = t 23

24 Relaxing the Homogeneous Beta Assumption If we relax the homogeneous beta assumption, we have that: where: N N = it ε it σ i ε i i= 1 i= 1 2 Nt Nt β t = wit β it w it β it i= 1 i= 1 [ ( ) ] t t wt β ( ) ( ) [ ] E CSV w σ t w t + E F CSV t CSV When the homogenous beta assumption does not hold, the CSV is not an unbiased estimator for average specific variance, even for an increasingly large number of stocks. t t Empirically, however, this bias is found to be small - less than 1% on average (cross-sectional dispersion of betas is quite large, but becomes small after multiplication by squared factor return). 24

25 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 25

26 VIX versus CVIX While representing a priori two different underlying risk measures, systematic and average specific volatility indicators are expected to be highly correlated. Indeed, they both reflect the aggregate uncertainty faced by investors at a given point in time regarding economic fundamentals. In what follows, we confirm this intuition and find high correlation level between the VIX index, an imperfect measure of systematic ti risk based on option prices, and the CSV index, a model-free efficient and unbiased proxy for average specific risk. 26

27 CSV as a Measure of Economic Uncertainty Time evolution of the CSV index and recession periods Daily time series for the CSV index. The shaded areas are the NBER recessions. The sample period is January 1990 to November

28 Comparison with VIX Time evolution of the CSV index and the VIX index Quantile-based CSV index performance, compared with the VIX index, based on daily stock return data for the S&P500 universe on the sample period ranging from January 1990 to November

29 Increasing Correlation Time evolution of the CSV index and the VIX index Quantile-based CSV index correlation with the VIX index, based on 10-year daily stock return rolling window data on the sample period ranging from January 1990 to November

30 Volatility Indices and Corresponding CSV Indices Index Ticker Correlation Sample period CBOE Volatility Index VIX AEX Volatility VAEX BEL 20 Volatility VBEL CAC 40 Volatility VCAC FTSE 100 Volatility VFTSE DAX Volatility VDAX-NEW SMI Volatility VSMI EURO STOXX 50 Volatility VSTOXX NIKKEI Volatility Index VNKY India NSE VIX INVIXN KOSPI 200 Volatility Index VKOSPI Mexico Volatility Index VIMEX

31 Volatility Indices and Corresponding CSV Indices

32 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 32

33 Data Set Our sample includes all publicly-traded stocks in 24 developed markets, constituent for the MSCI Developed Markets Index, with returns denominated in US Dollars. The exchange with the largest number of traded stocks is selected (except for China (Shanghai and Shenzen exchanges), Japan (Osaka and Tokyo exchanges), and the Unites States (NYSE, AMEX and NASDAQ). The sample period ranges from Jan-1980 to Sep-2011 (except for Finland, Greece, Israel, New Zealand, Portugal, and Spain which begins in mid-1980s). 33

34 Data Set Con t Country Stock Exchange Start Date End Date # of Stocks Listed Australia Australian 01/ /2011 3,738 Austria Vienna 01/ / Belgium Euronext Brussels 01/ / Canada Toronto 01/ /2011 4,361 Denmark Copenhagen 01/ / Finland Helsinki 01/ / France Euronext Paris 01/ /2011 2,342 Germany Frankfurt 01/ /2011 1,790 Greece Athens 01/ / Hong Kong Hong Kong 01/ /2011 1,469 Ireland Dublin 01/ / Israel Tel Aviv 01/ / Italy Borsa Italiana 01/ / Japan Osaka /Tokyo 01/ /2011 3,461 Netherlands Euronext Amsterdam 01/ / New Zealand New Zealand 01/ / Norway Oslo 01/ / Portugal Euronext Lisbon 01/ / Singapore Singapore 01/ / Spain Madrid 03/ / Sweden Stockholm 01/ /2011 1,521 Switzerland SIX Swiss 01/ / United Kingdom London 01/ /2011 6,711 United States NYSE/AMEX/NASDAQ 01/ / ,113 TOTAL: 48,882 34

35 Correlation of CSV Indices Australia Austria Belgium Canada Denmark Finland France Germany Greece Hong Kong Ireland Israel Italy Japan Netherla nds New Zealand Norway Portugal Singapore Spain Sweden Switzerland UK US

36 Principal Component Analysis of Country CSV 90 90% Variance Expla ained % 70% 60% 50% 40% 30% 20% 10% Principal Component 0% 36

37 Correlation of Broad Country Indices with Vol Indices Correlation Country CSV Index VIX Index Australia Austria Belgium Canada Denmark Finland France Germany Greece Hong Kong Ireland Israel Italy Japan Netherlands New Zealand Norway Portugal Singapore Spain Sweden Switzerland United Kingdom United States

38 Turning to Sector CSV Series: Correlation Matrix (US) Oil&Gas Basic Materials Industrials Consumer Goods Healthcare Consumer Services Telecommun ications Utilities Financials Technology

39 Principal Component Analysis of US Sector Portfolios Varianc ce Explained (%) % 90% 80% 70% 60% 50% 40% 30% 20% 10% Principal Component 0% 39

40 Correlation of US Sector Portfolios with Vol Indices Correlation CSV Index VIX Index Oil & Gas Basic Materials Industrials Consumer Goods Healthcare Consumer Services Telecommunications Utilities Financials Technology

41 Correlation of CSV Index across World Sectors Oil&Gas Basic Materials Industrials Consumer Goods Healthcare Consumer Services Telecomm unications Utilities Financials Technology

42 Principal Component Analysis of World Sector Indices Varianc ce Explained (%) % 90% 80% 70% 60% 50% 40% 30% 20% 10% Principal Component 0% 42

43 Correlation of World Sector Indices with Vol Indices Correlation CSV Index VIX Index Oil&Gas Basic Materials Industrials Consumer Goods Healthcare Consumer Services Telecommunications Utilities Financials Technology

44 Volatility: Risk Indicator versus Asset Class Benefits and Limits of Existing Volatility Indices A New Form of Volatility Index: the CSV Approach Properties of the CSV Index Factor Analysis of CSV International Series Conclusion 44

45 Wrap-Up We introduce a new form of volatility index, the crosssectional volatility index or CSV index. Through formal central limit arguments, we show that the cross-sectional dispersion of stock returns can be regarded as an efficient estimator for the average idiosyncratic volatility of stocks within the universe under consideration. Our results suggest that the CSV index is closely related to other volatility measures where and when such measures are available, and that t it can be used as a reliable proxy for volatility when such measures are not available. 45