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UBI Pramerica SGR Implementation of portfolio optimization with spectral measures of risk Roberto Strepparava January 2009

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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

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2 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: 1.ρ universal measure (= applies to any kind of risk) 2.ρ global measure (= “sums” different risks into a single number) 3.ρ probabilistic (= provides probabilistic info on the risk) 4.ρ expressed in units of “lost money”

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3 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

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4 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 F X (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

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5 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 1. 0 2. = 1 3. weakly decreasing

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6 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 ):

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7 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 1.a 2.b 3.c

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8 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

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9 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

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10 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).

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11 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

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12 Optimization of Spectral Measures of Risk And again the problem can be mapped into a linear program in a (generally) huge number of variables:

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13 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]

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14 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

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15 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

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16 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

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17 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

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18 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.

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19 Efficient frontier with general Spectral Measures Open problems (suitable theorems needed?): 1.Optimization fails (problem unbounded) as soon as measure becomes slightly incoherent. 2.Well-posedness for ptf with derivatives –Alexander et al. (2006) 3.Success of LP optimization with simplex method, failure with interior point method of the same problem

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20 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

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21 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

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22 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

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23 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

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