- QUANT TOUCH - Performance and risk analysis of dynamic portfolio strategies Nicolas Gaussel Benjamin Bruder Université Paris Diderot 2 Mars 2012
— Dynamic portfolio analysis — Introduction (I): some issues • How is it possible for funds that have performed consistently to tumble in just a few months? • Are these brutal reversals only attributable to market factors or do certain investment behaviors generate Extreme Risks? • Which strategies benefit from market volatility? • What are the best and worse possible scenarios for a given strategy? • What risks are associated with strategies that play the mean-reversion theme?
— Dynamic portfolio analysis — Introduction (II): some econometrical approaches to “non linearity” in returns • Jondeau & Rockinger (2006): Portfolio allocation with higher moments • Harvey & Siddique (2000): explain fund returns with the square of asset returns • Agarwal and al. (2004): introduce factor mimicking call option returns • Fund and Hsieh (2001): benchmark CTA returns against lookback straddle options on MSCI World
— Dynamic portfolio analysis — Agenda • Introduction: Target profile and Trading impact • Examples of portfolio strategies analysis • Constant mix • Average down strategy • Mean-reversion strategy • Special case studies • Trend following strategies • Stop loss overlays
— Dynamic portfolio analysis — Option Profiles and Trading Impact
— Dynamic portfolio analysis — Framework • Analyze the behavior of systematic strategies: • Exposure depends only of the current wealth or risky asset value • Strong decomposition result: Option Intrinsic Time price Value Value Portfolio Option Trading strategy profile impact
— Dynamic portfolio analysis — Option profile • The number of risky asset in portfolio depends on its spot price: • Naïve interpretation as a curve integral: • The wealth is just a primitive of the exposure function: • We call this primitive the option profile
— Dynamic portfolio analysis — Trading impact appears with volatility • The number of risky asset in portfolio depends on its spot price • Trading impact: difference between wealth and payoff: Payoff • Option profile: Integrate the delta function • Apply Ito formula: Asset price
— Dynamic portfolio analysis — Analysis and extensions • Interpretations: • No need for probabilities… (except no jump assumption) • Path dependent trading impact, but European option profile. • May depend of realized variance AND spot trajectory. • Can be extended to: • Exposure depending on wealth (multiplicative gamma costs) • Model with interest rate • Other exposure policies (trend following strategies….)
— Dynamic portfolio analysis — Convex vs concave strategies
— Dynamic portfolio analysis — Popular strategies analysis • Constant mix • Average down strategy • Mean-reversion strategy
— Dynamic portfolio analysis — Example: Constant mix, CPPI • Constant exposure through time • Simple payoff formula: option profile trading impact • Only depends on: • Trading impact :accumulated variance • Option profile: Terminal risky asset value
— Dynamic portfolio analysis — Constant mix formula compared to real data… • Eurostoxx 50 PI, since 1987, with no interest rate, and 1Y maturity simulations. • Formula with average volatility of 20%. 50/50 constant mix 4 times leveraged Sensitivity to variance = 12.5% Sensitivity to variance = -600% Average trading impact= 50bps / Y Average trading impact = 24% / Y Final portfolio Formula Final portfolio Formula 140% 140% 120% 120% 100% 100% 80% 80% 60% 60% 40% 40% 20% 20% 0% 0% 0% 50% 100% 150% 200% 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%
— Dynamic portfolio analysis — Average down (i.e. doubling) strategy • Fixed wealth objective O=110%: • To be attained with a fixed performance of R=10%. • Necessary exposure is recalculated every day: 400% 300% 200% 100% 0% 70% 80% 90% 100% 110% 120%
— Dynamic portfolio analysis — Average down: option profile and trading impact • Terminal wealth is of the form: Strategy 6M Strategy 1M Risky asset • Example: 120% 100% • Objective=110% 80% • Expected asset perf=10% 60% 40% • Volatility=20% 20% 0% 60% 70% 80% 90% 100% 110% 120% 130% 140% Mostly attains objective, severe losses if not
— Dynamic portfolio analysis — 1Y rolling backtest on Eurostoxx 50 • Stable 10% returns… most of the time Positive 1Y return in 87% of cases
— Dynamic portfolio analysis — Mean reverting strategy: volatility statistical arbitrage • Clear view on the average value of forward variance: • Strategy: Buy when cheaper, sell when more expensive: 150% 100% 50% Strategy vega 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% -50% Example with m=1 : -100% -150% Volatility
— Dynamic portfolio analysis — Resulting terminal wealth: • Option profile: • Additive trading impact: • Example: • 15% average volatility • 100% volatility of variance -> 50% volatility of volatility • Average volatility move: 7,5% p.a. • Trading impact: 1,25% p.a. for m=1
— Dynamic portfolio analysis — Risk analysis: • Wins when near the average value • Severe drawdowns when going far from that value Target payoff 1M 3M 6M 101% 100% 99% 98% 97% 96% 0% 5% 10% 15% 20% 25% 30% 35%
— Dynamic portfolio analysis — Other Case Studies • Trend following strategies • Stop loss overlays
— Dynamic portfolio analysis — Preliminary remarks • CTA represents 15% of the total hedge Fund industry and an average $290 Bn AUM (Barclayhedge) • Longstanding history of so-called « Dow theory » (B. Graham, 1949) • Abundant trader memories (e.g. Turtle.org) but limited academic literature
— Dynamic portfolio analysis — Trend following strategy : discrete case From: Trend followers lose more often than they gain (J.P. Bouchaud and M. Potters, Capital Fund Management) • Simple model: • Discrete time, discrete space : +1% or -1% each day 3 2 1 1 0 0 -1 -1 -2 -3 • Simple strategy: • Buy until first negative performance
— Dynamic portfolio analysis — Trend following strategy : discrete case • Terminal wealth analysis: 5 4 3 3 2 2 1 1 0 0 -1 1 1 1 1 1 2 4 8 16 32 • Asymmetric behavior: • High probability (50%) of small losses : -1% • Low probability (25%) of high gains : (2% on average) • 25% probability to have P&L = 0
— Dynamic portfolio analysis — Continuous time framework • Standard trend indicator: Exponentially weighted average return. • Standard Merton allocation procedure: • Expression of the terminal wealth:
— Dynamic portfolio analysis — Asymmetric return profile • Option profile: short term variations but stationary on the long term • Cumulative trading impact with asymmetric effect: • Can be very high due to term • Losses per year can not be above • Long term gains if the ex post squared Sharpe ratio is above • Example: Gains if the absolute Sharpe ratio is above 100% for a 6 month moving average.
— Dynamic portfolio analysis — Trend following on the vol-targeted Eurostoxx 50 • Eurostoxx 50 exposure is adjusted to keep a 15% volatility • 6M moving average, strategy calibrated for a 5% vol Cumulated trading impact vs. Daily trading impact Actual NAV
— Dynamic portfolio analysis — Stationary distribution of trading impact • The distribution of instantaneous trading impact can be computed once the true historical model is specified. • Example for the former strategy, depending on the true value of in the model:
— Dynamic portfolio analysis — Case study : The stop loss strategy From : The Stop-Loss Start-Gain Strategy and Option Valuation (P. Carr, R. Jarrow) • Two ways to protect a portfolio : • Buying a put option • Sure protection, but expensive • Stop loss strategy • A free Put Option ? A sure protection? • Is the target payoff component a put option? • What about trading impacts?
— Dynamic portfolio analysis — Two kind of stop loss strategies Delta (right scale) Asset price Wealth Definitive stop 120% 100% • Misses rebound 110% 100% • Effectively limits losses 90% 80% Re-exposure 70% 0% • Takes advantage of rebound 120% • Further losses may occur 100% 110% 100% 90% 80% 70% 0%
— Dynamic portfolio analysis — Stop loss : continuous time analysis Delta • Exposure function: • Option profile: asset + put option: 1 • Result of the strategy: K S • Negative trading impact taken when crossing the strike • Average trading impact proportional to the option gamma
— Dynamic portfolio analysis — Stop gain: continuous time analysis • Exposure function: opposite of stop loss: • Option profile: asset – call option: Delta 1 • Result of the strategy: K S • Trading impact: gains taken when crossing the strike
— Dynamic portfolio analysis — The slippage problem • Last move before exposure is cut: Asset move Wealth Hedged profile Losses Asset Price Stop loss level
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