Performance and risk analysis of dynamic portfolio strategies - - PowerPoint PPT Presentation

• QUANT TOUCH -

Performance and risk analysis of dynamic portfolio strategies

Nicolas Gaussel Benjamin Bruder Université Paris Diderot 2 Mars 2012

—Dynamic portfolio analysis—

• 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?

Introduction (I): some issues

—Dynamic portfolio analysis—

• 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
• ptions on MSCI World

Introduction (II): some econometrical approaches to “non linearity” in returns

—Dynamic portfolio analysis—

• 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

Agenda

—Dynamic portfolio analysis—

Option Profiles and Trading Impact

—Dynamic portfolio analysis—

• Analyze the behavior of systematic strategies:
• Exposure depends only of the current wealth or risky asset value
• Strong decomposition result:

Framework

Portfolio strategy Option profile Trading impact

Option price Intrinsic Value Time Value

—Dynamic portfolio analysis—

• 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

Option profile

—Dynamic portfolio analysis—

Asset price Payoff

Trading impact appears with volatility

• The number of risky asset in portfolio depends on its spot price
• Trading impact: difference between wealth and payoff:
• Option profile: Integrate the delta function
• Apply Ito formula:

—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

• ption profile

trading impact

• Constant exposure through time
• Simple payoff formula:
• 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

Sensitivity to variance = 12.5% Average trading impact= 50bps / Y

4 times leveraged

Sensitivity to variance = -600% Average trading impact = 24% / Y

0% 20% 40% 60% 80% 100% 120% 140% 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% Final portfolio Formula 0% 20% 40% 60% 80% 100% 120% 140% 0% 50% 100% 150% 200% Final portfolio Formula

—Dynamic portfolio analysis—

• Fixed wealth objective O=110%:
• To be attained with a fixed performance of R=10%.
• Necessary exposure is recalculated every day:

Average down (i.e. doubling) strategy

0% 100% 200% 300% 400% 70% 80% 90% 100% 110% 120%

—Dynamic portfolio analysis—

• Terminal wealth is of the form:
• Example:
• Objective=110%
• Expected asset perf=10%
• Volatility=20%

Average down: option profile and trading impact

Mostly attains objective, severe losses if not

0% 20% 40% 60% 80% 100% 120% 60% 70% 80% 90% 100% 110% 120% 130% 140% Strategy 6M Strategy 1M Risky asset

—Dynamic portfolio analysis—

• Stable 10% returns… most of the time

1Y rolling backtest on Eurostoxx 50

Positive 1Y return in 87% of cases

—Dynamic portfolio analysis—

• Clear view on the average value of forward variance:
• Strategy: Buy when cheaper, sell when more expensive:

Mean reverting strategy: volatility statistical arbitrage

Example with m=1 :

• 150%
• 100%
• 50%

0% 50% 100% 150% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% Strategy vega Volatility

—Dynamic portfolio analysis—

• 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

Resulting terminal wealth:

—Dynamic portfolio analysis—

• Wins when near the average value
• Severe drawdowns when going far from that value

Risk analysis:

96% 97% 98% 99% 100% 101% 0% 5% 10% 15% 20% 25% 30% 35% Target payoff 1M 3M 6M

—Dynamic portfolio analysis—

Other Case Studies

• Trend following strategies
• Stop loss overlays

—Dynamic portfolio analysis—

• 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

Preliminary remarks

—Dynamic portfolio analysis—

• Simple model:
• Discrete time, discrete space : +1% or -1% each day
• Simple strategy:
• Buy until first negative performance

Trend following strategy : discrete case

From: Trend followers lose more often than they gain (J.P. Bouchaud and M. Potters, Capital Fund Management) 1

• 1
• 2

2 1

• 1

3

• 3

—Dynamic portfolio analysis—

• Terminal wealth analysis:
• 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

Trend following strategy : discrete case

1

• 1

2 1 3 2 4 3 5

2 1 4 1 8 1 16 1 32 1

—Dynamic portfolio analysis—

• Standard trend indicator: Exponentially weighted average

return.

• Standard Merton allocation procedure:
• Expression of the terminal wealth:

Continuous time framework

—Dynamic portfolio analysis—

• 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.

Asymmetric return profile

—Dynamic portfolio analysis—

Trend following on the vol-targeted Eurostoxx 50

Cumulated trading impact vs. Actual NAV Daily trading impact

• Eurostoxx 50 exposure is adjusted to keep a 15% volatility
• 6M moving average, strategy calibrated for a 5% vol

—Dynamic portfolio analysis—

• 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:

Stationary distribution of trading impact

—Dynamic portfolio analysis—

• 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?

Case study : The stop loss strategy

From : The Stop-Loss Start-Gain Strategy and Option Valuation (P. Carr, R. Jarrow)

—Dynamic portfolio analysis—

Two kind of stop loss strategies

Definitive stop

• Misses rebound
• Effectively limits losses

Re-exposure

• Takes advantage of rebound
• Further losses may occur

0% 100% 70% 80% 90% 100% 110% 120% 0% 100% 70% 80% 90% 100% 110% 120% Delta (right scale) Asset price Wealth

—Dynamic portfolio analysis—

• Exposure function:
• Option profile: asset + put option:
• Result of the strategy:
• Negative trading impact taken when crossing the strike
• Average trading impact proportional to the option gamma

Stop loss : continuous time analysis

K S Delta 1

—Dynamic portfolio analysis—

• Exposure function: opposite of stop loss:
• Option profile: asset – call option:
• Result of the strategy:
• Trading impact: gains taken when crossing the strike

Stop gain: continuous time analysis

K S Delta 1

—Dynamic portfolio analysis—

• Last move before exposure is cut:

The slippage problem

Asset move Losses Hedged profile Stop loss level Wealth Asset Price

—Dynamic portfolio analysis—

• Binomial tree example:
• Each day, asset price moves of h or –h
• For a yearly volatility we have:
• Stop loss at
• Each time barrier is crossed:
• Looses w.r.t. asset + put profile
• Average number of times barrier is crossed during 1 year:
• Behaves like
• Average trading impact , price of the put option!

Average slippage cost

1 1+h 1-h 1 1-2h 1+2h 1+h 1-h 1+3h 1-3h

—Dynamic portfolio analysis—

Conclusion

Portfolio strategy Option profile Trading impact