Equivalent Risky Allocation S. Plunus – R. Gillet – G. Hübner

Motivation of the paper MIFID Directive Financial institutions must guarantee: A better protection to investors More transparency The Mifid Implementation directive requires the financial institutions to collect: –informations on the investment horizon, –The investors preferences towards risk, –His risk profile; and, –His investment objectives.

Risk profile The risk profile of an investor determine the risk measure(s) the investor uses to take his decisions : Rendu de Lindt (2002) Investors use simultaneously several risk measures and their risky investment choices are directly influenced by their perception of risk : Veld and Veld-Merkoulova (2008)

Risk Perception Risk perception is the subjectif judgment of an investor regarding the caracteristics and the severity of a potential loss. Perception of risk is different from risk aversion defining the return the investor requires in exchange of the risk taken.

Portfolio Selection Relevance of higher moment of distribution in portfolio selection: Samuelson (1970), Kraus and Litzenberger (1976), … Risk measures developped: Coombs and Lehner (1981), Favre and Galeano (2002) None of them take into account the risk perception

Bells Risk Measure Derived from the utility function theory Allows for risk aversion and risk perception distinction Allows the integration of higher and partial moments of distribution

Bells linex utility function u(w) = w–be -cw Respect the axioms of the expected utility theory of von Neumann and Morgenstern Respect the decreasing risk aversion condition of Arrow Satisfy the one-switch rule of Fishburn (1984) These features are consistent with a risk-return interpretation

Bells risk measure Bell writes the evaluation of an alternative as follows: The only definition of risk compatible with Bells assumptions is the following: or a monotonic transformation thereof.

Fourth-order Taylor series expansion of the Bell function around the expected value of the wealth : With I the amount invested in the risky asset, the expected utility is then: From this, we get the risk measure for one unit of wealth: where

Mesure de risque de Bell - Illustration Paramètre de perception C Coefficients multiplicateur VarianceSkewnessKurtosis (21%)-0.83 (35%) 1.04 (44%) (3%)-2.83 (18%)12.04 (78%) (1%)-5.67 (10%)48.17 (88%) VarianceSkewnessKurtosis Mesure de Risque de Bell C = 5C =17C =34 Asset A Asset B

Allocation équivalente en risque Afin de rendre la mesure de Bell facilement interprétable et comparable, nous lexprimons en terme dallocation équivalente en risque (ERA): LERA est le pourcentage à investir dans un portefeuille de référence (le reste étant investi au taux sans risque) pour obtenir le même risque (quelque soit sa mesure) que lactif étudié.

Equivalent Risky Allocation Where R x is the risk value of the asset and, R B is the risk value of the selected benchmark.

Data and methodology 9 equity indices: S&P500, S&P500 Growth, S&P500 Value, S&P400, S&P400 Growth, S&P400 Value, S&P600, S&P600 Growth, S&P600 Value 1 bond index: US Aggregate bond index Weekly returns, with Expected Returns adjusted through Black-Litterman (1992) 12 investors: 4 defensive, 4 medium and 4 agressive 3 Markowitz, 3 protective, 3 stable and 3 MVaR Rolling window of 18 months

Etude pratique Optimisation de portefeuilles pour des investisseurs ayant la même aversion au risque (ERA ==), mais une perception différente du risque: –Le risque est la volatilité des rendements (investisseur « Markowitz ») –Le risque est une combinaison de la volatilité, de lasymétrie des rendements positifs et négatifs et de la probabilité des pertes extrêmes mais: Est surtout défini par la volatilité (investisseur progressif) Est surtout défini par la probabilité des pertes extrêmes (investisseur protecteur) Est défini à travers la Modified Value-at-risk

Tests Optimisation en début de période pour chaque investisseur et chaque niveau de risque, puis observation de lévolution Optimisation « refaite » toutes les 4 semaines pendant les 8 ans –Observation de lévolution des allocations –Observation de la variation de la mesure de risque en 4 semaines –Observation de la mesure de risque « après coup »

Optimal portfolios on June 29th, 2001 ERA Risk Measure InvestorsWeights ProtectiveMedianProgressiveMarkowitzS&P400 S&P400 Growth S&P400 Value S&P400 S&P400 Growth S&P400 Value S&P600 S&P600 Growth S&P600 Value Corp Bond Defensive (ERA = 50%) Var 73%45%50% 0,030,040,110,050,070,140,040,110,140,278,0% Bell C=5 72%45%50% 0,00 0,110,0000,010,200,030,150,200,313,2% Bell C=17 76%50%55% 0,00 0,100,0040,010,200,050,150,200,293,7% Bell C=34 50%19%23% 0,03 0,090,040,060,110,010,040,100,497,3% Medium (ERA = 75%) Var 89%73%76%75%0,040,050,120,08 0,170,070,130,150,128,2% Bell C=5 88%71%75%74%0,00 0,130,000,020,200,090,190,200,173,3% Bell C=17 89%75%78%77%0,00 0,110,02 0,200,100,190,200,163,8% Bell C=34 75%48%53% 0,040,050,110,060,070,140,020,110,140,258,2% Agressive (ERA = 100%) Var 101%102%101%100%0,020,040,100,080,120,180,120,140,170,027,2% Bell C=5 100% 99%0,00 0,120,050,060,200,120,20 0,053,7% Bell C=17 100% 99%97%0,000,0030,110,060,030,200,140,20 0,063,9% Bell C=34 100%99%97%96%0,030,050,130,100,090,180,100,140,150,037,6%

Evolution de lERA pour un portefeuille passif (ERA = 75%) ProtecteurMarkowitz

Time consistency with rebalancing Evolution of the portfolio allocation of defensive portfolios Protective investor Stable investor

ERA for defensive investors MVaRProtectiveStableMarkowitz Panel A - Optimization with Markowitz ERAt 73,2%45,8%50,5%50,0% ERAt+4 73,4%46,1%50,7%50,2% bias 0,6%2,9%0,8%0,6% RMSE 5,7%9,0%3,7%3,3% Panel B - Optimization of Bell Utility Function with C = 5 ERAt 72,3%44,6%50,0%49,7% ERAt+4 72,7%45,1%50,4%50,0% bias 0,8%3,4%0,8%0,6% RMSE 6,0%11,3%3,8%3,5% Panel C - Optimization of Bell Utility Function with C = 17 ERAt 75,8%50,0%55,3%54,9% ERAt+4 76,2%50,6%55,7%55,3% bias -0,1%1,1%-1,1%-1,2% RMSE 5,1%7,2%8,3%8,9% Panel D - Optimization of Bell Utility Function with C = 34 ERAt 50,0%18,9%23,2%23,3% ERAt+4 50,2%19,0%23,3%23,4% bias 0,6%3,6%0,1%-0,1% RMSE 4,7%22,5%14,2%13,9%

ERA for median-risk investors MVaRProtectiveStableMarkowitz Panel A - Optimization with Markowitz ERAt 89,0%74,2%76,1%75,0% ERAt+4 89,1%74,4%76,3%75,2% bias 0,4%1,6%0,5%0,4% RMSE 4,2%7,4%2,5%2,2% Panel B - Optimization of Bell Utility Function with C = 5 ERAt 87,8%71,6%75,0%74,2% ERAt+4 88,1%72,2%75,4%74,6% bias 0,6%2,3%0,6%0,5% RMSE 4,5%8,8%2,8%2,6% Panel C - Optimization of Bell Utility Function with C = 17 ERAt 89,4%75,0%77,9%77,1% ERAt+4 89,7%75,6%78,4%77,5% bias 0,1%0,7%-0,5%-0,6% RMSE 2,8%4,6%5,9%6,6% Panel D - Optimization of Bell Utility Function with C = 34 ERAt 75,0%48,4%53,7%53,3% ERAt+4 75,3%48,8%53,9%53,5% bias 0,4%2,3%0,0%-0,2% RMSE 2,8%12,3%10,0%10,3%

ERA for agressive investors MVaRProtectiveStableMarkowitz Panel A - Optimization with Markowitz ERAt 101,0%102,7%101,5%100,0% ERAt+4 101,1%102,7%101,6%100,1% bias 0,2%0,4%0,2% RMSE 2,0%4,3%1,6%1,5% Panel B - Optimization of Bell Utility Function with C = 5 ERAt 100,2%100,5%100,0%98,6% ERAt+4 100,4%100,9%100,3%98,9% bias 0,3%0,8%0,3% RMSE 2,5%5,5%1,8%1,9% Panel C - Optimization of Bell Utility Function with C = 17 ERAt 100,2%100,0%98,8%97,3% ERAt+4 100,3%100,5%99,2%97,7% bias 0,1%0,4%-0,1%-0,2% RMSE 1,6%3,3%5,4%6,3% Panel D - Optimization of Bell Utility Function with C = 34 ERAt 100,0%99,3%97,8%96,2% ERAt+4 100,1%99,5%98,0%96,5% bias 0,1%0,2%-0,2% RMSE 1,1%3,4%6,4%7,2%

Portfolio Consistency Risk MeasureERA Investors Max spread ProtectiveMedianProgressiveMarkowitz Variance Defensive 0,840,440,51 40% Median 0,890,670,74 22% Agressive 0,950,991,001,015% Bell_5 Defensive 0,830,460,540,5538% Median 0,910,740,790,8017% Agressive 0,961,001,021,037% Bell_17 Defensive 0,910,570,64 34% Median 0,960,810,850,8614% Agressive 1,001,04 1,054% Bell_34 Defensive 0,650,190,25 45% Median 0,760,500,56 26% Agressive 0,860,970,99 13%

Conclusions de létude Pour une même quantité de risque (même aversion), lallocation des portefeuilles optimaux pour différentes perceptions sont significativement différentes La mesure de risque de Bell est fiable (objectifs respectés) Utiliser une « mauvaise » mesure de risque pour un investisseur peut lui faire prendre trop de risque ou le faire passer à côté dun rendement potentiellement plus élevé Si lon se « trompe » dans le profilage dun investisseur, il sera moins affecté si on optimise son portefeuille avec la mesure de risque de Bell plutôt quavec une simple variance, et il aura plus de chances de rester sous le seuil critique imposé par son aversion au risque.