Measures of Contribution for Portfolio Risk

X Workshop on Quantitative Finance Milan, January 29-30, 2009

Agenda Coherent Measures of Risk Spectral Measures of Risk Capital Allocation Euler Principle Application Risk Measurement Risk Attribution Risk Contributions

Coherent Measures of Risk A risk measure ρ : V R is called coherent if it is monotonic translation invariant ρ(x ) ρ(y ) ρ(x + l) = ρ(x) + l X Y l R positive homogeneous subadditive ρ(λx ) = λρ(x ) λ 0 ρ(x + Y ) ρ(x ) + ρ(y ) where we identify each portfolio X, Y V with its loss function.

Coherent Measures of Risk COHERENT MEASURES NON-COHERENT MEASURES Semi-Standard Deviation Expected Shortfall Value at Risk (not subadditive) Standard Deviation (not monotonic) Can we create more coherent measures?

Spectral Measures A risk measure M ϕ (X ) is defined spectral if M ϕ (X ) = 1 0 ϕ(u)f X (u)du (1) where ϕ : [0, 1] R is the risk spectrum (or risk aversion function). Eq. 1 is coherent iff ϕ(u) satisfies: Positivity ϕ(u) 0 for all u [0, 1] Normalization 1 0 ϕ(u)du = 1 Monotonicity (weakly increasing) ϕ(u 1 ) ϕ(u 2 ) for all 0 u 1 u 2 1

Spectral Measures - Risk Spectrum How much do we consider the loss?

Spectral Measures - Risk Spectrum And in which way? (456-782954:;-<=;3>4:;-!-?-"-?-#-?-@!A- (123*45/62178*9:80;178*!*<*"*<*#*<*=">* % #,-#!!. % % )*#!!+ #$"!$+ #$" #$" #!$*!$) # #!$"!$%!$"!$"!! "!!! #!!!! &'(!! "!!! #!!!! /0!! "!!! #!!!! &'(!! "!!! #!!!!,- #,-#!!. %,-#!!. % )*#!!+? )*#!!+!$+ #$" #$" +!$*!$) # # %!$%!$"!$" #!! "!!! #!!!! 0123/,1!! "!!! #!!!! 0123B=>!! "!!! #!!!! -./0,).!! "!!! #!!!! -./0@:; After having computed the portfolio risk in a consistent and axiomatic way, can we do the same for its decomposition in order to know what is the attribution to risk of each single asset?

Coherent Capital Allocation A coherent capital allocation is a function Λ ρ from V V to R s.t. X V, Λ(X, X ) = ρ(x ) and is linear diversifying Λ(X, X ) = d i=1 λ iλ(x i, X ). Λ(X, Y ) Λ(X, X ). translation invariant Λ(X + a, Y + b) = Λ(X, Y ) + a a, b R monotonic Λ(X, Z) Λ(Y, Z) X Y, Z V

Existence of a Capital Allocation If there exists a linear, diversifying capital allocation Λ with the relative risk measure ρ, then ρ is positively homogeneous and subadditive. If ρ is positively homogeneous and subadditive then Λ ρ is a linear, diversifying capital allocation with the related risk measure ρ. If, moreover, Λ ρ is also continuous at a portfolio Y V lim ɛ 0 Λ ρ (X, Y + ɛx ) = Λ ρ (X, Y ) Λ ρ is given by the Euler principle (or allocation by the gradient)

Euler Principle If ρ is positive homogeneous and differentiable at λ R we have where ρ(λ) = d i=1 λ i ρ λ i (λ) (2) Marginal risk: ρ λ i (λ) = Λ(X i, X ) Risk Contribution: κ i = λ i ρ λ i (λ) The risk capital k := ρ(x ) of the portfolio X to its subportfolios is completely allocated (i.e. κ 1,..., κ d of X 1,..., X d with κ = κ 1 + + κ d.

Euler Principle for Elliptical loss distributions Theorem Consider the special case of an elliptical loss distribution X E d (0, Σ, ψ) The risk contribution for each asset is always the same, independent if we use Euler principle with the standard deviation, VaR, ES, Spectral measures or other positive homogeneous risk measures. The relative risk contribution is given by Λ ρ i Λ ρ j d k=1 = Σ ik d k=1 Σ, 1 i, j d. (3) jk

Derivatives of the Risk Measures Covariance principle κ i = λ i Λ ρ SD i = λ i cov(x i, X ) var(x ) c + E(X i ) Value at Risk Spectral Measures κ i = λ i Λ ρ VaR i = λ i E(X i X = VaR α (X )) κ i = λ i ( c E(X i X = ES 0 ) (1 c) Expected Shortfall 1 κ i = λ i Λ ρα ES i = λ i E(X i X VaR α (X )) 0 ) E(X i X = FX (u))ϕ(du)

Application - Historical We apply the Euler Principle to a portfolio of equities: 5 Italian stocks, Enel, Fiat, Generali, Luxottica and Telecom Italia; Observation period from 01/01/2001 to 13/10/2008, daily frequencies (1974 obs); Value of the portfolio 1000 euro.

Application - Historical ENEL FIAT GENERALI LUXOTTICA TELECOM IT. Mean 6.60E-05-5.30E-04-0.0002 7.18E-05-0.0007 Median 0.00014 0-0.0003 0.0004 0 Maximum 0.17 0.13 0.08 0.11 0.13 Minimum -0.09-0.12-0.09-0.15-0.17 Std. Dev. (annual) 0.22 0.36 0.25 0.31 0.33 Skewness 0.30 0.09-0.15-0.12-0.58 Kurtosis 18.90 5.63 7.35 7.30 11.09 Jarque-Bera 20769 574 1564 1525 5494 Probability 0.00 0.00 0.00 0.00 0.00 Observations 1974 1974 1974 1974 1974 Table 1: Statistics of the returns of the 5 assets. ENEL FIAT GENERALI LUXOTTICA TELECOM ENEL 1.00 0.36 0.50 0.35 0.45 FIAT 0.36 1.00 0.47 0.38 0.40 GENERALI 0.50 0.47 1.00 0.38 0.44 LUXOTTICA 0.35 0.38 0.38 1.00 0.34 TELECOM 0.45 0.40 0.44 0.34 1.00 Table 2: Correlation matrix.

Application - Historical Figure 1: Single risk measures and portfolio ones with equal weights. Enel is the least risky and Fiat and Telecom are the riskiest at 95% confidence level. On aggregate terms, SpecPut says that at the 95% probability we will lose in a day max 37euro.

Application - Historical Figure 2: Risk contributions for an equally weighted portfolio, at 95% and 99% confidence level. Similar pattern as the risk measures (innocent dependence structure). VaR is really sensitive; non-sense results increasing the confidence level.

Application - Historical Figure 3: Spectral measure contributions given by expected shortfalls with increasing confidence levels.

Application - Historical Figure 4: Reducing the amount invested in Fiat and Telecom.

Application - Historical Before Less risky variation % ES 32.4 31.3-3.5 VaR 22.9 21.8-4.8 SpecExp 28.2 27.3-3.2 SpecPut 37.4 36.4-2.7 SD 13.3 12.8-3.8 SDminus 8.5 8.2-3.5 Table 3: % variations of the risk measures (expressed in euro after a change in the amount invested in each component in order to decrease the portfolio risk). % Variation ES VaR SpecExp SpecPut SD SDminus Enel 25-15 28 27 31 30 Fiat -18 44-19 -20-21 -20 Generali 10 44 11 10 11 11 Luxottica 10-71 11 9 12 11 Telecom Italia -13-69 -15-14 -14-15 Table 4: Variation of risk contributions for all the measures after the reduction in the amount of Fiat and Telecom.

Application - MonteCarlo Figure 5: Multivariate gaussian, 10x10.000 simulation runs. Flat correlation matrix (ρ = 0.8), µ = 0, decreasing λ i, α = 95%, β = γ = 90%. Gaussian case, non-symmetric problem. Equal contributions.

Application - MonteCarlo Figure 6: 10x10.000 simulation runs for a Multivariate t (7 dof), equal λ i, µ = 0, increasing vols, flat corr matrix (ρ = 0.5) except first asset (ρ = 0.5). For risk measures with γ = 90%, α = 95%, β = 99% and for Spectral measures. Student t case, non-symmetric problem. VaR instable to input. Should have a huge number of simulation for convergence. Equal contributions.

Application - MonteCarlo Figure 7: Increasing µ, remember volatilities are also increasing.

Application - MonteCarlo What happens in a non-elliptical case?

Application - MonteCarlo Figure 8: Meta-Gumbel distribution (Gumbel copula and normal margins, µ = 0 and equal volatilties 0), first asset negative dependent, 10.000 simulations, equal λ i.

Application - MonteCarlo Figure 9: Meta-Gumbel distribution with Student t margins, µ = 0, equal volatilities 0 and 15 dof. The hedging asset is really hedging!

Conclusion Value at Risk provides misleading results. Spectral measures of risk show a great stability and give a better understanding of what is going on in the tail; Covariance allocation is not monotonic; VaR instead is not diversifying; The archimedean copula seems more appropriate since it has right tail dependence. Multivariate t admits also left tail dependence that is not really a stylized fact. Future developments: what if we include liquidity risk?

References Acerbi, C. (2002). Spectral measures of risk: a coherent representation of subjective risk aversion, Journal of Banking & Finance 25(7), 1505-1518. Artzner, P., Delbaen, F., Eber, J.M., Heath, D. (1999). Coherent measures of risk, Mathematical Finance 9, 203 228. Denault, M. (2001). Coherent allocation of risk capital, Journal of Risk 4(1), 1 34. Kalkbrener, M. (2005). An axiomatic approach to capital allocations, Mathematical Finance 15(3), 425 437. Kalkbrener, M. (2007). An axiomatic characterization of capital allocations of coherent risk measures. Mina, J. (2002). Risk attribution for asset managers, RiskMetrics Journal 3(2), 33 56 Tasche, D. (2000). Risk contributions and performance measurement, Technical report, Department of Mathematics, TU-Munchen.