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## UBI Pramerica SGR

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**UBI Pramerica SGR**Implementation of portfolio optimization with spectral measures of risk Roberto Strepparava January 2009**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**Introduction and motivation**• Efficient frontier in the Markowitz plane using different risk measures: using VaR the problem is impossible to solve: problem non convex, plagued by local fake minima, due to non-subadditivity of VaR. • Portfolio optimization issue for portfolio managers (stable weights, i.e. well-posedness of the problem). • Need to solve efficient frontier with more general risk measures ρ: all advantages of VaR and none of the shortcomings: • ρuniversal measure (= applies to any kind of risk) • ρglobal measure (= “sums” different risks into a single number) • ρprobabilistic (= provides probabilistic info on the risk) • ρ expressed in units of “lost money”**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**Optimization of Spectral Measures of Risk**• Coherent Risk Measures (CRM) if for all X,Y portfolio’s P&L r.v.’s • Subadditivity related to the “risk diversification principle”. Hedging benefit: • Given a prob. measure P, a CMR is said to be law-invariant(LI)if (X) is in fact a functional of the distribution function FX(x) only. • A measure of risk is said to be Comonotonic Additive (CA) if adding together two comonotonic risks X and Y provides NO HEDGING AT ALL**Optimization of Spectral Measures of Risk**Kusuoka (2001) showed the class of LI CA CRMs is given by all convex combinations of possible ES’s at different conf. levels: Acerbi (2001) introduced the same class of CMRs calling it “spectral measures of risk” (SM) which is a CMR if the “risk spectrum” :[0,1] satisfies • 0 • = 1 • weakly decreasing**Optimization of Spectral Measures of Risk**The spectral measure with spectrum is the (p)-weighted average loss in all cases (p-quantiles p[0,1]) of the portfolio: subadditivity (through condition 3.) imposes to give larger weights to worse cases. To this class belongs the Expected Shortfall ES (flat spectrum with domain [0,]) and even VaR which however is not a CMR because it fails to satisfy 3. (Dirac- spectrum peaked on ):**Optimization of Spectral Measures of Risk**Optimization of ES: Uryasev et al. (2000, 2001) develop an efficient procedure for the minimization of ES, avoiding to deal with ordered statistics (read: sorting operations). Theorem: let a portfolio with weights . Define Then • a • b • c**Optimization of Spectral Measures of Risk**In an N-scenarios pdf this problem is a nonlinear convex optimization of the form: Notice the absence of sorting operations (read: ordered statistics). The objective function is piecewise linear in and w**Optimization of Spectral Measures of Risk**But the problem can be mapped again into a linear programming (LP) optimization problem of the form : Where linearity has been bought at the price of introducing N new variables z**Optimization of Spectral Measures of Risk**Optimization of general Spectral Measures: Acerbi and Simonetti (2002) extend the method above to a general SM. The objective function in this case takes the form: and therefore, in the general case the additional parameter is a whole function (t).**Optimization of Spectral Measures of Risk**The N-scenarios optimization problem can be cast again into a nonlinear convex program: where we have discretized the risk spectrum φ to a piecwise-constant function with J jumps**Optimization of Spectral Measures of Risk**And again the problem can be mapped into a linear program in a (generally) huge number of variables:**Optimization of Spectral Measures of Risk**Theorem (risk-reward optimization for Spectral Measures): The optimal portfolios of the ( ,E(X)) risk-reward constrained optimization problem are the solutions of the unconstrained minimization problem of the SMs: Defined for all , where Most useful for implementation: • we get all and only optimal portfolios and no dominated frontier • the range of the parameter to vary is exactly known [0,1]**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**α-ES efficient frontier via parametric method**Constrained α-ES minimization drawbacks: • chosen constraint value μ incompatible with efficient frontier • even if all compatible constraints, portion of dominated frontier As Parametric α-ES minimization advantages: • all and only optimal portfolios retrieved • range of the Lagrange parameter λ exactly known**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**Efficient frontier with general Spectral Measures**Two-percentile SM: piecewise constant SM with parametric method (tipically α=1%, β=5%) interesting features • Measures interpolate extrema α-ES and β-ES, belonging to class of weighted V@R measures -Cherny (2006)- • Well-possess of the optimization problem: how smoothly portfolio weights depend on risk spectrum φ • Solve practical issue: risk manager that finds 1%-ES too loose and 5%-ES too strict can calibrate the measure . Especially useful in the present context of financial crisis**Efficient frontier with general Spectral Measures**Well-posedness of optimization: smooth dependence of portfolio weights on shape of the spectrum, within a certain range M For minimum risk portfolio new weights come into play.**Efficient frontier with general Spectral Measures**Open problems (suitable theorems needed?): • Optimization fails (problem unbounded) as soon as measure becomes slightly incoherent. • Well-posedness for ptf with derivatives –Alexander et al. (2006) • Success of LP optimization with simplex method, failure with interior point method of the same problem**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**Backtest on real portfolios**Case study: portfolio of an Asset Management company • 63 assets in portfolio (Italian stocks) • Medium-depth HS (764 daily observations) • 3 months backtest • Simplest SM: 5% Expected Shortfall • Inclusion of transaction costs (1 BP per transaction) + management fees Results: even including costs + fees, high risk portfolio beats NAV of the fund, while minimum risk portfolio stays very close to NAV**Summary**• Introduction and motivation • Optimization of Spectral Measures of risk • α-ES efficient frontier via parametric method • Efficient frontier with general Spectral Measures • Backtest on real portfolios • Conclusions and future work**Conclusions and future work**• Feasibility of more general Spectral Measures than simple ES: now that ES is being widely used (thereby slowly replacing VaR), SMs await their turn. • SMs both interesting theoretically and useful to risk managers • Parametric method of efficient frontier construction very efficient and fast (all simulations on laptop, Pentium processor 1.7GHz with Matlab™ R2007b on) • Numerical simulations hint at theorems that need to be properly stated Agenda: • Analysis of SMs with more general risk aversion functions φ (discrete exponential spectrum). • Analysis on real portfolio requiring major changes in constraints: leveraged portfolios, portfolios short of derivatives. • Heavy MC simulations to see computational burden on the optimizer. • Rolling analysis on assets’ HS to see well-posedness of the problem w.r.t. changes in empirical distribution of the portfolio P&L random variable