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Risk Management for Hedge Funds Tackling Rare Events with an Incomplete History A. Jaun Risk Management for Hedge Funds Tackling Rare Events with an Incomplete History A. Jaun 1, 2, S. Umansky 1, H. El Showk 1 1 Signet Capital Management Limited 2 Assoc. Prof. Royal Institute Technology, Stockholm Contact [email protected] com Gdansk Conference, 11 -12 May 2007

Uncertain returns from markets Example: NYBOT coffee futures 1994 -2007 frost in Brazil Markowitz’N Uncertain returns from markets Example: NYBOT coffee futures 1994 -2007 frost in Brazil Markowitz’N 90: risk volatility = (ri -m)2 Engle’N 03: arbitrage free GARCH average … but frost only happens during the winter!

Maximum likelyhood historical fit with Normal-/Inverse Gaussian distributions Normal normal log NIG zoom stress Maximum likelyhood historical fit with Normal-/Inverse Gaussian distributions Normal normal log NIG zoom stress Adequate description of normal-, stress- and rare events? rare

The perception of risk evolves Volatility & kurtosis looking back 1 -12 years sugar The perception of risk evolves Volatility & kurtosis looking back 1 -12 years sugar coffee sugar Should coffee prices be getting more stable?

Modern tools for risk management prob weigthed returns • Value at risk Va. Ra Modern tools for risk management prob weigthed returns • Value at risk Va. Ra (not subadditive) • Expected shortfall 1 1 ES= 1 -a Va. Ru du probabitily 1 -a of losses > Va. Ra a (subadditive) • Simulation returns -Va. Ra -s m (historical 1 -10 years, Monte-Carlo) • Extreme Value Theory to model rare events (Generalized Pareto distribution is generic, Embrechts)

Example: trading coffee derivatives Daily risk budgeting Normal Stress Va. R 95 -1. 10% Example: trading coffee derivatives Daily risk budgeting Normal Stress Va. R 95 -1. 10% +0. 34% -1. 57% +0. 33% ES 95 -1. 38% +0. 35% -1. 84% +0. 36% ER -0. 0007% -0. 001 Worst case if DS=+15% 4% loss (no risk of frost in Apr)

And when there is not enough data Ex: avalanche risk • little/no history • And when there is not enough data Ex: avalanche risk • little/no history • incomplete data Take the right decision. . . before it is too late!

3 x 3 «orthogonal» qualitative factors • Global (from home) regional forecast. . . 3 x 3 «orthogonal» qualitative factors • Global (from home) regional forecast. . . . . 0 map, itinerary. . . 1 level of participants. . . . 0 • Local (from start) snow depth >15 cm. . . . 1 weather conditions. . . . 0 orientation (NE-NW). . . . 1 • Zonal (every step) slope >35 deg. . . 1 snow consistency. . . . 1 solidity test. . . . 1 Total. . . . . 6 > 4 too risky avoid

Optimize a fund of hedge funds Impossible to rely on the past perfomance Would Optimize a fund of hedge funds Impossible to rely on the past perfomance Would need > 140 years of monthly data (A. Lo) I. Check for structural risks People, organization, administrator, infrastructure II. Estimate aggregatable market risks Identify risk factors, limit and diversify exposures Estimate returns from worst case scenarios III. Maximize risk-adjusted expected returns Generalize Sharpe ratio: S = E[Return] / Risk Details of the process are propriatery, but…

Risk budgeting with uncertainty • Estimate optimization constraints – Exposures: gross, net, liquidity, geography, Risk budgeting with uncertainty • Estimate optimization constraints – Exposures: gross, net, liquidity, geography, strategy – Worst losses 9/11, stock crash, rate hikes, liquidity crisis • Account for uncertainties (work plan) Optimum with rigid constraints Range of optima with different confidence levels goal function constraint uncertainty

Risk-adjusted expected returns Returns from probability weighted scenarios E. g. 30% stagflation, 50% soft Risk-adjusted expected returns Returns from probability weighted scenarios E. g. 30% stagflation, 50% soft landing, 20% hard landing Risk from a fund = lack of confidence in Our own judgement (insufficient knowledge) Future expected returns (forward looking volatility) The preservation of capital (exposure to rare events) Estimates should be back-tested (work plan) How well does past performance match forecasts?

Example: fund of hedge funds Fixed income strategies fund • 50 hedge funds, 6 Example: fund of hedge funds Fixed income strategies fund • 50 hedge funds, 6 strategies, exposures, etc • Risk management process validated over 7 years • Low correlation to market and rare events Historical performance compared to indices

Conclusions Distribution of returns to describe market risks Max likelihood to fit Normal, NIG, Conclusions Distribution of returns to describe market risks Max likelihood to fit Normal, NIG, Pareto distributions Choice of the historical time span is the main issue When there is not enough data Identify aggregatable & orthogonal risk factors Bayesian estimate of returns for rare events Estimates can be back-tested and refined with time Rare events do happen and define our lives!

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