mathematical challenges in lecture 1: statistics of...

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MATHEMATICAL CHALLENGES IN FINANCEPathways Lecture Series

COE Keio University

June 12-14, 2007

UNDERSTAND • ANTICIPATE • ACT

Raphael Douady

http://www.riskdata.com

[email protected]

NYU Courant Institute

http://www.math.nyu.edu/seminars/math_finance_seminar.htmlPathways Lecture Series Keio Univ. June 12-14, 2007 2

Agenda

LECTURE 1: Statistics of Financial Markets - June 12Market DynamicsState Variables and Risk EvaluationTopological and Differential Structure of Interest Rates

LECTURE 2: Derivative Pricing - June 13Derivative SecuritiesControl and Calibration QuestionsNumerical Techniques for Parabolic PDE's

LECTURE 3: Optimal Investment Questions - June 14Portfolio OptimizationRisk BudgetingCross-asset Class OptimizationAlternative Investments

Pathways Lecture Series Keio Univ. June 12-14, 2007 3

Portfolio Optimization

Examples of Risk MeasuresStandard DeviationDownside Deviation (only negative returns)Value-at-Risk (VaR) = distribution percentileConditional VaR = return expectation conditional to < VaRGeneral class

Z = E[ (c – ∆P)p | ∆P < c’ ]1/p



Risk Measure UncertaintyUncertain VariancesUncertain correlationsNonlinear dependenciesRegime changes, shocks…

Ill-posed problemThe “optimal portfolio” is very sensitive to inputs as soon as the covariance matrix is badly conditionedNeed to account for uncertainty of statistics

Bayesian ApproachConsider the covariance matrix as random (and possibly other joint model parameters)Include this randomness in the computation of Z(∑qiSi)Find portfolio with marginal contributions proportional to return expectations

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Optimal portfolio depends on

Risk Measure characteristicsHorizon of timeTails weight vs. medium-size events

Model featuresExpected returnsLinear/Nonlinear relations between assetsFat tails modeling, shocks, etc.Liquidity modelingModel calibration process

Refrain from using raw history (e.g. “historical VaR”)!

Portfolio constraintsContractual or regulatory constraintsCost of tiltingLiquidity constraintsDelay between decision and action…


Agenda Lecture 3

Optimal Investment QuestionsPortfolio OptimizationRisk BudgetingCross-asset Class OptimizationAlternative Investments


Risk Budgeting

Risk Budgeting Process1. Identify Investment Universe2. Estimate expectations from each asset3. Choose a global level of Risk for the portfolio

and “allocate” portions of risk to each asset4. Compute target Marginal Risk Contributions5. Design portfolio to match targets


Risk Budgeting

Risk Budgeting Process in PracticeFocus on major mismatches, ignore small discrepancies (not worth the cost of tilting)The portfolio tail risk is, most of the time, almost equal to that of its riskiest componentsZero Marginal Contribution doesn’t exist, because of correlation uncertainty

Negative marginal contributions exist, when an asset acts as a hedge to others.

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Risk Budgeting


Risk Budgeting

Implied Expected ReturnsPortfolio is given (e.g. before tilting)Compute Marginal Risk ContributionsSet a target expected return of the portfolioAssume portfolio is optimal and deduce what should be the returns of the assetsIdentify unreasonable assumptions and correct portfolio accordingly


Agenda Lecture 3



Cross-asset Class Optimization

2 steps Traditional Investment ProcessStrategic allocation

Investment universe (bonds, stocks, alternatives…)Geographic allocation (developed, emerging…)

Tactical allocationSectorsChoice of specific assets (stock picking, etc.)

Implicit AssumptionImpact of Tactical allocation on global portfolio risk is of 2nd order with respect to Strategic

This is not always the case!

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The relation between asset classes is often

more complex than between assets in the

same classNonlinear (optional) relationsDifferent regimes with inversions of correlationsSpecific reactions to shocks, etc.



France Telecom CDS vs. Equity and Implied VolatilityFrance Telecom

0

10

20

30

40

50

60

70

0 100 200 300 400 500 600 700 800

CDS Spread

Stoc

k

0

20

40

60

80

100

120

140

StockImpl. Vol.

Impl

. Vol

.

12-sept-01 to 13-nov-02

Volatility 69.31% 59.12% 50.79%Correlation CDS Spread Stock Impl. Vol.CDS Spread 100% -54.60% 37.11%Stock -54.60% 100% -50.95%Impl. Vol. 37.11% -50.95% 100%



France Telecom

88

90

92

94

96

98

100

102

25 30 35 40 45 50 55

Stock price

Bon

d Pr

ice

(5Y

6% c

oup.

reca

lc. F

rom

CD

S)

FTE Bond

Regression

12-sept-01 to 13-nov-02

France Telecom Bond Price vs. Equity



Cross asset modelingAppropriate State VariablesIdentify systematic changes of correlation

Nonlinear modelingIdentify causes of regime change (e.g. a state variable passing a threshold)

WARNING!If asset prices have nonlinear relations, the Risk function Z is not necessarily convex with respect to positions qiThe optimal portfolio is not necessarily unique ⇒ Trading Discontinuity

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Agenda Lecture 3



Alternative investments

ExamplesHedge FundsPrivate equityReal estateEmerging marketsArt…

Common characteristicsNo liquidityLong lock-upsDecision under high uncertainty


Alternative Investments

Identify major risk driversRelated to MarketsOperationalCounterpartyHumanPoliticalLegal…

Quantify and Model when possibleMore often possible than expected!Often need s specific modeling and specific risk factors



Hedge Funds vs. MarketMedium/long term risk not so much related to current positionsNeed to mix all available info

Latest known positionsRisks internally identified by managersReturn series analysisBehavior of similar funds

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Hedge Fund Nonlinearities

Quadratic 35%

Linear 26%Cubic 39%

Primary reason is dynamic tradingBlack-Scholes: if positions have non-zero correlations with markets, the P/L is nonlinear


1000 Hedge FundsDistribution across Strategies similar to overall HF populationIncluding Dead Funds and their last returnMonthly Returns

Analysis PeriodJan 95 to Jun 05 (restricted to Fund existence)

MethodologySelect, among investable factors, the most explanatoryF-test of Quadratic and Cubic regression vs. Linear regressionIdentify Funds that reject Linear model with Confidence 95%



Conclusion

Did you lose your key here?

No, on the other side, but here, I

have light!

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