A constant volatility framework for managing tail risk

Transcription

1 A constant volatility framework for managing tail risk QuantValley /QMI, Annual Research Conference, New York 2013 Nicolas Papageorgiou Associate Professor, HEC Montreal Senior Derivative Strategist, Montreal Exchange

2 Agenda The rational behind risk targeting strategies The challenges in maintaining constant risk exposure The Payoff Distribution Model (PDM) and the advantages of distributional targeting Extensions to bivariate setting Targeting dependence

3 How do we diversify equity risk? S&P/TSX Drawdown Dex Universe Return Canada 10-Year Yield Source: PIMCO Historically, Fixed Income has been a good hedge against Equity drawdowns However with long-term yields at historical lows, it would be fair to say that the potential hedge that fixed income can provide is greatly reduced

4 Risk management and asset allocation Diversification, both geographically and across assets, is neither a sufficient nor a reliable risk control mechanism. During severe market corrections, traditional relationships between assets break down: Volatility of risky assets tend to rise as there is greater uncertainty (and panic) Correlations across assets increase dramatically; In summary, the assumptions underpinning our asset allocation do not always hold, particularly when markets are stressed. Market risk should be actively managed to reduce the impact of market corrections on equity portfolio.

5 The pitfalls of employing long-term risk/return assumptions A portfolio designed based on the long-term risk/return characteristics 40% Risk/Return Reward High Vol State 30% 20% 10% Low Vol State Medium Vol State 0% -10% 47% 45% 8% -20% -30% -40% Return Volatility -50% Frequency Does not take full advantage of the available risk budget when volatility is low (and risk/return trade-off is most favorable) Is fully exposed to extreme market corrections when these events occur. 5

6 What makes a good hedge? For a tail risk hedge to be effective it should possess two important characteristics: the hedge must be negatively correlated to asset returns The hedge should exhibit convex behavior to the upside during periods of market stress.

7 The role of volatility in RM Cumulative returns with daily annualized volatility 400% 70% Daily cumulative performance 350% 300% 250% 200% 150% 100% 50% 0% 60% 50% 40% 30% 20% 10% Daily annualized volatility -50% 0% Volatility, or the measure of the variability of returns, is dynamic. Making an asset allocation decision using historical volatility measures ignores the time-varying nature of volatility.

8 Volatility products Product Investor Listed/OTC Charateristics Variance Swaps Hedge funds, Insurance companies, pension plans OTC -Exposure to Variance is constant and not path dependent. -Provides exposure to realized variance. -Counterparty and liquidity risk. VIX Hedge funds, Insurance companies, pension plans Exchange traded -Liquidity is good and improving. -Exchange traded with daily mark-tomarket. - Position requires rolling of futures position, cost-of-carry can be significant. -Provides exposure to a forward looking measure of volatility. Forward starting swaps Hedge funds, Insurance companies, pension plans OTC -Exposure is to implied volatility and does not accrue any realized volatility. -There is a cost of carry in maintaining exposure to forward implied volatility.

9 Hedging using volatility products If you are long risk premium in volatility, you must be selling insurance (receiving implied vol) and therefore are long tail risk. It is impossible to pocket the implied/realized premia and insure your portfolio. In order to hedge tail risk with VIX (implied) volatility, you need to be long volatility and there is a cost to maintaining this exposure. A good long volatility strategy will aim to minimize this cost by being dynamic (generating some alpha) Pick and choose: shape of term structure, price of convexity, etc. 9

10 Dynamic beta and volatility targeting If you have a static exposure, your risk is variable and changes with market conditions The most direct way to control risk is to render your exposure conditional to contemporaneous risk environment. Considerable litterature supporting volatility timing strategies Fleming, Kirby and Ostdiek (JF,2001, JFE,2003) Hocquard et al (JPM, 2013)

11 The volatility of volatility: A story of two tails Conditions in the markets change over time and, as a consequence, the risk (volatility) profile of a given asset also varies. As market volatility increases: the distribution of returns for the asset flattens; the tails fatten relative to their average historical distribution; and the historical probabilities are no longer representative of actual loss potential.

12 Targeting a constant volatility There is a systematic and predictable component to volatility that managers/investors should exploit. Market exposure should be managed actively so as to keep the portfolio volatility at the target (desired) level. Market exposure is removed if volatilty rises above the target level. Market exposure is increased if volatility falls below the target level. This eliminates fat-tails by ensuring that the drawdowns are consistent with the volatility targets. 12

13 Implementation: Target volatility index fund or overlay Asset allocation Asset/Liability Modelling Risk Tolerance MPT/Efficient Frontier Tactical 25% 25% S&P % S&P/TSX MSCI EAFE Short Long Implementation Overlay Equity Index Futures contracts are managed on top of an existing equity portfolio to maintain the portfolio volatility at the targeted level The result is that manager Alpha is preserved while market volatility remains constant

14 Historical performance of a constant volatility strategy Monthly cumulative performance 500% 400% 300% 200% 100% 0% -100% Equity Proxy Equity + Volatility Overlay Performance comparison 100% Futures exposure Daily Exposure 60% 20% -20% -60% -100% * Exposure is shown as a % of the equity fund value on a daily basis.

15 Targeting a constant volatility Rather than target an asset mix, the plan/manager should target a level of volatility that is constant and consistent with the assumptions that were used to define the reference (benchmark) portfolio. This approach requires a little more effort: Risk targets need to be established A dynamic measure of risk needs to be estimated A framework for translating volatility to exposure needs to be derived A re-balancing rule needs to be established

16 Measuring volatility/risk There are a number of volatility measures: Historical standard deviation (Moving average, Exponential Moving Average, etc ) Econometric models (GARCH, NGARCH etc.) Implied Volatility measures (VIX, VIXC, VSTOXX etc ) VaR, CVaR, EVT Volatility is a mathematical construct and there are an unlimited number of volatility models/measures. We need to be aware of the implications regarding our choice of risk (volatility) parameter.

17 Translating volatility into exposure Having established the volatility measure and volatility target, we still need to calculate the required exposure. There are a number of possibilities that present themselves to us: Ratio of actual to target volatility Ratio of actual to target variance Distributional targeting

18 Payoff distribution model (PDM) Dybvig (JB,1988) first introduced the PDM concept. Given an underlying asset S uuuuu with monthly returns R uuuuu, and a monthly target distribution F tttttt, there exists a function g R uuuuu such that the distribution of g. is the same as the distribution F tttttt. This payoff function g. is calculated using the distribution F uuuuu of the underlying asset and the marginal distribution function of the target distribution F tttttt. The expression for g. is given by: g x = F tttttt 1 F uuuuu x ; x R

19 Optimal dynamic hedging strategy Having solved for the payoff function g(.), we need to derive an optimal dynamic trading strategy that will replicate the payoff function. In order to achieve this, we develop extensions of the results of Schweizer (1995). We minimize the expected square hedging error Note that there is no risk-neutral measure in our approach and all calculations are carried out under the objective probability measure.

20 Extending the PDM to incorporate dependence Payoff function g(x,y) can be used to define the target payoff as a function of two assets returns (x,y). 1 g( x, y) = FTrgt asset 2( Funder asset 2( y x) x) The payoff is a function of two joint-distributions: - The joint distribution F tttt aaaaaa needs to be specified (not estimated) and therefore we have consideraby freedom in selecting the statistical properties that are most desirable. - The joint distribution F uuuuu aaaaaa needs to be estimated and temporal aggregation properties need to be taken into account.

21 Applications Constant volatility funds (or volatility management via overlays) Risk parity strategies Gliding path funds

22 Conclusion Tail risk is part of the risk of investing in any risky asset. Over the long run, we expect be compensated for experiencing the occasional drawdown. The problem is that asset returns exhibit fat tails. The frequency and severity of the drawdowns are not consistent with our assumptions about risk (volatility). Dynamic beta (volatility targeting) is an effective way of eliminating the fat without incurring the cost associated with traditional tail hedging (portfolio insurance) techniques.

23 APPENDIX

24 Comparison of volatility targeting with a collar strategy 6,00% 5,00% 4,00% 24 3,00% 2,00% 1,00% Source: Bloomberg and Brockhouse Cooper. 0,00% -35% -29% -23% -18% -12% -6% 0% 6% 12% 18% Monthly Return