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This paper introduces an investor-specific risk measure derived from the linear-exponential (linex) utility function. It combines the notions of risk perception and risk aversion. To make this measure interpretable and comparable with others like variance or value-at-risk, it is translated into an Equivalent Risky Allocation (ERA), where the risk value is matched with the one of a selected benchmark. We demonstrate that portfolio allocations are sensitive to risk perception. The linex risk measure provides more stable allocations and is closer to the target risk profile than the variance, while it provides better consistency of risk exposures over time than the value-at-risk.

The Modern Portfolio Theory introduced by Markowitz [

We define this notion of perception as the subjective judgment of an investor over the characteristics and severity of a potential loss. Undeniably, investors display heterogeneous attitudes towards the notion of “risk”. We can identify two extreme behaviors. Some investors put a very strong emphasis on the stability of returns around their mean and put a significantly less weight on extreme but rare losses. These are close to traditional “mean- variance investors”, in the scope of the Modern Portfolio Theory framework. At the other end of the spectrum, some investors primarily care about tail risk. They are sensitive to the threat of a shortfall with respect to a threshold level of wealth. These investors fall closer to the “mean-VaR (i.e., Value-at-Risk) investors” such as in the framework of Favre and Galeano [

The objectives of our paper are twofold. We first aim to propose a risk measure derived from the Expected Utility Theory that explicitly takes into account the risk perception of the investor. This measure is applicable to various kinds of investor types, from one extreme to another. It is derived from Bell’s linear plus exponential (linex) utility function [^{1}. We demonstrate that for a same value of ERA, there exist several optimal portfolio allocations depending on the risk perception of the investor. Our second objective is to examine, with this new risk measure, the relevance of explicitly accounting for investors’ risk perceptions in their portfolio allocation decisions, both static and dynamic. Considering several investors with different risk profiles (risk perception and risk aversion) but confronted with the same set of asset classes and allocation constraints, we assess the consistency of using one-size-fits-it-all measures of risk, such as the variance or the value-at-risk (VaR), and compare the results with utility-based allocation schemes with the use of Bell’s measure.

We observe that, under a passive buy-and-hold strategy, the ERA obtained with Bell’s Risk Measure allows a better control of actual risk exposures than with the variance, which only addresses the stability of returns. Meanwhile, the VaR measure produces much more conservative allocations, at the expense of a lower variety of asset classes in the optimal allocations.

Under an active strategy with portfolio rebalancing every four weeks, applying the ERA with the proper risk perception enables the portfolio manager to tightly monitor and control her risk exposure. It is particularly relevant in the European MiFID directive context where taking account the proper risk perception of investors is one of the most valuable criteria. Portfolio risk, measured four weeks after the allocation, never moves further than 1% from its target. We also illustrate the influence of a different perception of risk on the allocations of the optimal portfolios. Finally, we show the impact of a wrong profiling through both an in-sample and out-of-sample check of the portfolio ERA measures with another risk profile than the one used for portfolio allocations. This acid test emphasizes the strong consistency of Bell [

This paper is organized as follows. Section 2 introduces the two-dimensional framework with risk aversion and risk perception. In Section 3, we discuss the data and methodology for the empirical investigation of optimal allocations over time. All results are reported and analyzed in the fourth section. Section 5 discusses managerial implications, followed by the concluding section.

Bell [^{2} and will approach risk neutrality for small gambles when extremely rich, is the linex utility function:

where W is the wealth level, b the risk aversion coefficient and c the risk perception coefficient. Both coefficients are positive and investor-specific.

Financial decision making under uncertainty is an issue of trading off risk against return. For that purpose, Bell [_{0}. Writing the evaluation of an alternative as follows:

The only definition of risk compatible with Bell’s assumptions is the following:

Because of parameter c which varies from one person to another, this measure is not unique but specific to each investor. The main advantage of this risk measure is that it includes, as special cases, many other measures of risk previously proposed in the literature.

Hlawitschka [_{0} and the global amount invested in the risky asset I. The risk premium on this amount is equal to x = θ ? r, where θ is the return of the risky asset, and r is the risk-free return. The expected utility becomes:

The Taylor series expansion of this expression around the mean is then:

where

From this equation, we get the risk measure

An example is helpful in order to better understand the role of the risk perception parameter C. Suppose three different investors, respectively characterized by C equal to 5, 17 and 34^{3}. Each one is asked to choose between two assets characterized by the same expected return and the same variance (equal to 0.0029), but with different skewness and kurtosis.

Perception parameter, C | Coefficients Multiplying: | ||
---|---|---|---|

Variance | Skewness | Kurtosis | |

5 | 0.50 (21%) | −0.83 (35%) | 1.04 (44%) |

17 | 0.50 (3%) | −2.83 (18%) | 12.04 (78%) |

34 | 0.50 (1%) | −5.67 (10%) | 48.17 (88%) |

Comparing two assets with the same expected return and the same variance (and so the same risk in the mean-variance framework), but with different skewness and kurtosis, we can see the evolution of the risk measures (see

In order to make the Bell Risk Measure easily interpretable and comparable, we propose to express this measure in terms of an Equivalent Risky Allocation (ERA). The ERA of a portfolio is the percentage invested in a specified benchmark (the rest being invested in the risk-free asset) that delivers the same risk as this portfolio. The benchmark can be any portfolio used as a reference, and is not constrained to be ex-ante or ex-post efficient. This makes our measure very simple and practical. In the mean-variance context, the ERA can be compared to a generalization of the weight put in a benchmark portfolio along the Capital Allocation Line^{4}.

As the risk perception differs from one investor to another, there will be as many ERAs associated to a portfolio as the number of different investors. This measure provides a single index value for each portfolio, while properly accounting for the heterogeneity of the investors. The ERA can be applied to any measure of risk, as long as one uses the same measure to measure the risk of the studied asset and the risk of the benchmark. If the risk measure is homogeneous of degree 1 with the weight in the risky asset, such as the variance and the Bell Risk Measure, the formula of the ERA is:

where R_{B} is the risk value of the selected benchmark

To illustrate this measure, consider Asset X in the previous example. Take the S&P500 index as benchmark, and the variance as the risk measure. The monthly variance of the S&P500 is equal to 0.0023. The ERA is then:

meaning that asset X is 26% riskier than the benchmark in terms of volatility, or that one needs to invest 126% of her wealth in the S&P500 (and borrow the 26% at the risk-free rate) in order to obtain the same risk level (measured by variance in this case) as her asset X.

This approach enables us to measure the risk of any portfolio with a single metric, irrespective of how the investor perceives the notion of risk. This is helpful in order to characterize the evolution of portfolios allocated with different risk-return optimization rules, as we do in the empirical section.

To compute optimal portfolios, we consider the weekly returns of nine equity indices and one bond index, for the period of January 7th, 2000 to November 5th, 2010. The equity indices are S&P500, S&P500 Growth, S&P500 Value, S&P400, S&P400 Growth, S&P400 Value, S&P600, S&P600 Growth and S&P600 Value total return indices. The bond index is JPM US Aggregate bond index total return.

Variance (×100) | Skewness (×1000) | Kurtosis (×10,000) | Bell Risk Measure (×100) | |||
---|---|---|---|---|---|---|

C = 5 | C = 17 | C = 34 | ||||

Asset X | 0.29 | −0.20 | 2.00 | 0.18 | 0.44 | 1.22 |

Asset Y | 0.29 | 0.10 | 0.50 | 0.14 | 0.18 | 0.33 |

^{5}, and the views we have of the market returns.

To specify our view, we go along with the Fama and French’s study [

Once the expected returns are smoothened with the Black-Litterman model, we construct defensive, median-risk and aggressive optimal portfolios, that is, with a maximum ERA of respectively 50%, 75% and 100%. The defensive (resp. median-risk, aggressive) portfolio is therefore constructed such as its risk is half (resp. 75%, 100% of) the risk of an equally weighted portfolio of the 9 equity indices (our benchmark)^{6}.

We build these portfolios for four types of investor’s profiles:

- A “MVaR” investor, for whom we construct an optimal portfolio maximizing her expected return for a given Modified Value-at-Risk^{7};

- A protective investor, more affected by extreme losses than variability. Her portfolio is constructed using the Bell risk measure with a high C, equal to 34;

- A stable investor, more affected by variability than extreme events, characterized by a low C, equal to 5;

- A Markowitz investor, for whom we construct an optimal portfolio maximizing his expected return for a given variance.

In order to compare our twelve optimal portfolios computed with 4 different risk measures for three levels of risk, the risk values are systematically expressed in ERA. In the first sub-section we test the time consistency of the Bell Measure for a fixed-weights portfolio, by observing the evolution of the ERA over time. Then, we test the time consistency of the measure for a rebalanced portfolio. Finally, we test the portfolio consistency by recomputing the risk measures for the rebuilt historical of the rebalanced portfolios. Note that all portfolios are constructed under the constraint that the maximum weight of each equity index is set to 20%.

Mean | Min | Max | Standard Deviation | Stand. Skewness | Stand. Kurtosis | |
---|---|---|---|---|---|---|

S&P500 | 0.0000 | −0.2002 | 0.1141 | 0.0276 | −0.8464 | 6.7649 |

S&P500 Growth | 0.0003 | −0.2066 | 0.1369 | 0.0361 | −0.6857 | 4.2316 |

S&P500 Value | 0.0014 | −0.2391 | 0.1912 | 0.0373 | −0.6070 | 7.0428 |

S&P400 | 0.0014 | −0.1853 | 0.1537 | 0.0318 | −0.6541 | 5.0299 |

S&P400 Growth | 0.0017 | −0.1759 | 0.1396 | 0.0360 | −0.4798 | 3.0560 |

S&P400 Value | 0.0020 | −0.1860 | 0.1972 | 0.0359 | −0.5534 | 6.6720 |

S&P600 | 0.0014 | −0.1605 | 0.1427 | 0.0333 | −0.5618 | 3.2121 |

S&P600 Growth | 0.0018 | −0.1700 | 0.1375 | 0.0353 | −0.5738 | 3.2364 |

S&P600 Value | 0.0019 | −0.2064 | 0.1977 | 0.0411 | −0.4647 | 4.5362 |

US Corp. Bond | 0.0013 | −0.0215 | 0.0186 | 0.0055 | −0.5204 | 1.1015 |

View 1 | View 2 | View 3 | View 4 | View 5 | |
---|---|---|---|---|---|

S&P500 | −1/3 | 0 | 0 | −1/3 | 0 |

S&P500 Growth | −1/3 | 0 | −1/3 | 0 | 0 |

S&P500 Value | −1/3 | 0 | 1/3 | 1/3 | 0 |

S&P400 | 0 | −1/3 | 0 | −1/3 | 0 |

S&P400 Growth | 0 | −1/3 | −1/3 | 0 | 0 |

S&P400 Value | 0 | −1/3 | 1/3 | 1/3 | 0 |

S&P600 | 1/3 | 1/3 | 0 | −1/3 | 0 |

S&P600 Growth | 1/3 | 1/3 | −1/3 | 0 | 0 |

S&P600 Value | 1/3 | 1/3 | 1/3 | 1/3 | 0 |

US Corp. Bond | 0 | 0 | 0 | 0 | 1 |

Estimated returns | 0.11% | 0.05% | 0.11% | 0.05% | 0.09% |

Historical | Equilibrium | View Adjusted | |||||
---|---|---|---|---|---|---|---|

Mean | Std. Dev. | Mean | Std. Dev. | ||||

S&P500 | 0.0000 | 0.0019 | 0.0017 | 0.0006 | 0.0014 | ||

S&P500 Growth | 0.0003 | 0.0023 | 0.0019 | 0.0002 | 0.0022 | ||

S&P500 Value | 0.0014 | 0.0026 | 0.0029 | 0.0015 | 0.0017 | ||

S&P600 | 0.0014 | 0.0023 | 0.0020 | 0.0009 | 0.0017 | ||

S&P600 Growth | 0.0017 | 0.0025 | 0.0020 | 0.0005 | 0.0020 | ||

S&P600 Value | 0.0020 | 0.0026 | 0.0027 | 0.0015 | 0.0017 | ||

S&P400 | 0.0014 | 0.0024 | 0.0020 | 0.0011 | 0.0017 | ||

S&P400 Growth | 0.0018 | 0.0025 | 0.0021 | 0.0010 | 0.0018 | ||

S&P400 Value | 0.0019 | 0.0029 | 0.0030 | 0.0019 | 0.0020 | ||

US Corp. Bond | 0.0013 | −0.0001 | 0.0001 | 0.0005 | 0.0002 | ||

In order to test the coherence of the risk measure over time, we construct 12 optimal portfolios, i.e., for 3 levels of risk and for 4 different perceptions of risk, every 4 weeks, starting on June 29th, 2001, until November 5th, 2010, and we observe the evolution of their ERA.

The observation of these average values already hints over the complexity induced by the various notions of risk perceived by the investors. Indeed, a protective investor is not always the one with the highest allocation in bonds. Taking a look at the average allocations of the portfolios (right part of

The last column of

ERA | Risk Measure | Investors | Weights | Average Std. Dev. | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MVaR | Protect | Stable | Markow | S&P500 | S&P500 Growth | S&P500 Value | S&P400 | S&P400 Growth | S&P400 Value | S&P600 | S&P600 Growth | S&P600 Value | Corp. Bond | |||

Defensive (ERA = 50%) | Var. | 73% | 46% | 50% | 50% | 0.04 | 0.04 | 0.12 | 0.05 | 0.05 | 0.12 | 0.12 | 0.05 | 0.15 | 0.27 | 8.2% |

Bell_{C=5} | 72% | 45% | 50% | 50% | 0.03 | 0.00 | 0.19 | 0.003 | 0.00 | 0.18 | 0.09 | 0.02 | 0.20 | 0.29 | 3.7% | |

Bell_{C=34} | 76% | 50% | 55% | 55% | 0.03 | 0.00 | 0.17 | 0.003 | 0.00 | 0.17 | 0.12 | 0.04 | 0.20 | 0.27 | 4.2% | |

MVaR | 50% | 19% | 23% | 23% | 0.01 | 0.01 | 0.10 | 0.02 | 0.03 | 0.08 | 0.07 | 0.04 | 0.13 | 0.51 | 7.0% | |

Median (ERA = 75%) | Var. | 89% | 74% | 76% | 75% | 0.05 | 0.04 | 0.13 | 0.07 | 0.06 | 0.13 | 0.16 | 0.09 | 0.15 | 0.12 | 8.4% |

Bell_{C=5} | 88% | 72% | 75% | 74% | 0.04 | 0.00 | 0.20 | 0.02 | 0.00 | 0.18 | 0.17 | 0.04 | 0.20 | 0.15 | 4.2% | |

Bell_{C=34} | 89% | 75% | 78% | 77% | 0.03 | 0.00 | 0.18 | 0.02 | 0.00 | 0.18 | 0.18 | 0.07 | 0.20 | 0.14 | 4.4% | |

MVaR | 75% | 48% | 54% | 53% | 0.03 | 0.02 | 0.12 | 0.04 | 0.05 | 0.12 | 0.13 | 0.07 | 0.16 | 0.25 | 7.9% | |

Aggressive (ERA = 100%) | Var. | 101% | 103% | 101% | 100% | 0.02 | 0.05 | 0.17 | 0.05 | 0.04 | 0.15 | 0.16 | 0.14 | 0.19 | 0.04 | 6.5% |

Bell_{C=5} | 100% | 100% | 100% | 99% | 0.00 | 0.02 | 0.20 | 0.01 | 0.04 | 0.19 | 0.16 | 0.14 | 0.20 | 0.05 | 3.9% | |

Bell_{C=34} | 100% | 100% | 99% | 97% | 0.01 | 0.001 | 0.18 | 0.02 | 0.01 | 0.19 | 0.18 | 0.16 | 0.20 | 0.05 | 3.4% | |

MVaR | 100% | 99% | 98% | 96% | 0.03 | 0.02 | 0.15 | 0.08 | 0.06 | 0.16 | 0.17 | 0.13 | 0.17 | 0.03 | 6.9% |

The portfolios are optimized every 4 weeks on the basis of the preceding 78 weekly returns (from June 29th, 2001 to November 5th, 2010) of 10 indices, with 4 different risk measures. The indices are S&P500, S&P500 Growth, S&P500 Value, S&P400, S&P400 Growth, S&P400 Value, S&P600, S&P600 Growth, S&P600 Value and JPM US Corp Bond index. The risk measures are the variance, the Bell’s Risk Measure (Bell) with a perception parameter C equal to 5, and equal to 34, and the Modified Value-at-Risk with a 97.5% level of confidence. Under each risk measure, we construct three portfolios: a defensive (ERA = 50%), a median-risk (ERA = 75%) and an aggressive (ERA = 100%) portfolio. The ERA is the value of the risk measure of the optimal portfolio, divided by the value of the same risk measure for the benchmark. The benchmark chosen is an equally weighted portfolio of the 9 equity indices.

We then analyze the evolution of the ERA of each portfolio over time. Each portfolio’s ERA is computed every week after the optimization until 3 years later. The procedure is reproduced over a rolling window of 18 months. ^{8}, the evolution of the averages, the 5% and 95% quantiles of these ERAs, according to the number of weeks elapsed since the allocation date. The dashed lines represent the target value of the portfolios optimizations (75% in this example).

The value taken by the ERAs after three years is on average equal to respectively 82%, 85%, 88% and 92% for resp. the MVaR, Bell_{C=34}, Bell_{C=5} and variance, which means that the portfolios optimized with the traditional variance approach are the least likely to remain close to their risk target.

Although it provides informative outputs, the previous exercise is not very close to the reality of asset managers. Hence, in the following application, we do not consider the different allocations of each investor separately, but we study the dynamics of the 12 portfolios reallocated every four weeks.

Next, we study the evolution of the risk taken by the investors through time. _{t}) and the average ERA four weeks later, just before the next allocation (ERA_{t+4}). The average differences between both ERA (bias) and the Root Mean Square Errors (RMSE), expressed in percentage of the ERA_{t}, are also reported. The shaded sections highlight the measures when the right method is used for the right investor.

MVaR investor Protective investor

Investor with Risk Aversion Such as ERA = 50% | |||||
---|---|---|---|---|---|

MVaR | Protective | Stable | Markowitz | ||

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 | 0.2% | 0.6% | 0.4% | 0.4% | |

RMSE | 3.3% | 5.7% | 3.5% | 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 | 0.5% | 1.1% | 0.8% | 0.8% |
---|---|---|---|---|

RMSE | 3.2% | 6.7% | 3.6% | 3.3% |

Panel C―Optimization of Bell Utility Function with C = 34 | ||||

ERAt | 75.8% | 50.0% | 55.3% | 54.9% |

ERAt+4 | 76.2% | 50.6% | 55.7% | 55.3% |

Bias | 0.5% | 1.2% | 0.8% | 0.7% |

RMSE | 3.0% | 7.0% | 3.5% | 3.1% |

Panel D―Optimization with MVaR | ||||

ERAt | 50.0% | 18.9% | 23.2% | 23.3% |

ERAt+4 | 50.2% | 19.0% | 23.3% | 23.4% |

Bias | 0.4% | 0.9% | 0.5% | 0.4% |

RMSE | 4.5% | 7.0% | 4.7% | 4.6% |

Average Equivalent Risky Allocation of 4 portfolios rebalanced every 4 weeks on the day of the allocation, ERAt, and 4 weeks after, just before the next allocation (ERAt+4), for 4 different investors (Markowitz, Stable, Protector and MVaR). The portfolios are optimal portfolios under the constraint that the ERA is set to 50% for 4 different risk measures (variance, Bell’s Risk Measure with perception parameter C = 5 and 34 and MVaR). The bias and root mean squared errors (RMSE) are computed as a percentage of the average of ERAt.

Investor with Risk Aversion Such as ERA = 75% | |||||
---|---|---|---|---|---|

MVaR | Protective | Stable | Markowitz | ||

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 | 0.1% | 0.2% | 0.2% | 0.3% | |

RMSE | 1.8% | 3.4% | 2.3% | 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 | 0.3% | 0.9% | 0.6% | 0.6% | |

RMSE | 2.2% | 5.2% | 2.7% | 2.4% | |

Panel C―Optimization of Bell Utility Function with C = 34 | |||||

ERAt | 89.4% | 75.0% | 77.9% | 77.1% | |

ERAt+4 | 89.7% | 75.6% | 78.4% | 77.5% | |

Bias | 0.3% | 0.8% | 0.6% | 0.5% | |

RMSE | 1.8% | 4.6% | 2.5% | 2.2% | |

Panel D―Optimization with MVaR | |||||

ERAt | 75.0% | 48.4% | 53.7% | 53.3% | |

ERAt+4 | 75.3% | 48.8% | 53.9% | 53.5% | |

Bias | 0.3% | 0.8% | 0.5% | 0.5% | |

RMSE | 2.7% | 5.2% | 3.0% | 2.9% | |

Average Equivalent Risky Allocation of 4 portfolios rebalanced every 4 weeks on the day of the allocation, ERAt, and 4 weeks after, just before the next allocation (ERAt+4), for 4 different investors (Markowitz, Stable, Protector and MVaR). The portfolios are optimal portfolios under the constraint that the ERA is set to 75% for 4 different risk measures (variance, Bell’s Risk Measure with perception parameter C = 5 and 34 and MVaR). The bias and root mean squared errors (RMSE) are computed as a percentage of the average of ERAt.

MVaR investor Protective investor

and median-risk portfolios of the protective investors tend to be perceived as riskier than expected by the other investors.

The portfolios of the MVaR investors look less risky than targeted for the other investors. Thus even if the MVaR measure seems more stable in a passive strategy, this measure is totally inaccurate for investors using variance or Bell Risk Measure to take their decisions.

The bias levels show that the allocation processes perform well in regards to their objective, as all biases are lower than 1%, except for the Bell_{34} investor where a wrong measure can reach a 3.6% bias. It nonetheless does not outreach the 1.1% threshold when the correct measure is used.

If we look at the RSME, we clearly see that, if one does not know the investor’s profile, the MVaR risk measure is most likely to be misleading. It can produce ERA more than 20% far away from its objective. This tends to improve for higher degree of risk. For the Bell Risk Measure, even if one misestimates the perception parameter, the highest RMSE is half the MVaR’s, and the method even improves comparatively when the level of risk rises, with a RMSE of maximum 6.3% for an ERA of 100%.

After having checked for the time consistency of the risk measure, an ex-post validation is necessary. The investor must be able to verify afterwards whether the portfolio held during the whole investment horizon has realized its objectives in term of risk.

To test the portfolio consistency of our 12 portfolios, we rebuild the historical of the rebalanced portfolios and compute their ERA on the last day of the holding period, that is, on November 5th, 2010.

Results in

The relative increase of the ERA is lower, the higher the expected ERA. This might be due to the S&P600 index and, to a smaller extent, to S&P500 and 400 indices, which have experienced a reduction in variance and kurtosis over the tested period. The spread between the ERAs of different investor types for a same portfolio

Risk Measure | ERA | Investors | Max Spread | |||
---|---|---|---|---|---|---|

MVaR | Protective | Stable | Markowitz | |||

Variance | Defensive | 0.84 | 0.44 | 0.51 | 0.51 | 40% |

Median | 0.89 | 0.67 | 0.74 | 0.74 | 22% | |

Aggressive | 0.95 | 0.99 | 1.00 | 1.01 | 5% | |

Bell_5 | Defensive | 0.83 | 0.46 | 0.54 | 0.55 | 38% |

Median | 0.91 | 0.74 | 0.79 | 0.80 | 17% | |

Aggressive | 0.96 | 1.00 | 1.02 | 1.03 | 7% | |

Bell_34 | Defensive | 0.91 | 0.57 | 0.64 | 0.64 | 34% |

Median | 0.96 | 0.81 | 0.85 | 0.86 | 14% | |

Aggressive | 1.00 | 1.04 | 1.04 | 1.05 | 4% | |

MVaR | Defensive | 0.65 | 0.19 | 0.25 | 0.25 | 45% |

Median | 0.76 | 0.50 | 0.56 | 0.56 | 26% | |

Aggressive | 0.86 | 0.97 | 0.99 | 0.99 | 13% |

Equivalent Risky Allocation of 12 portfolios rebalanced every 4 weeks, for 4 different investors (Markowitz, Stable, Protector and MVaR). The portfolios are optimized every 4 weeks on the basis of the preceding 78 weekly returns (from June 29th, 2001 to November 5th, 2010) of 10 indices, with 4 different risk measures. The indices are S&P500, S&P500 Growth, S&P500 Value, S&P400, S&P400 Growth, S&P400 Value, S&P600, S&P600 Growth, S&P600 Value and JPM US Corp Bond index. The risk measures are the variance, the Bell’s Risk Measure (Bell) with a perception parameter C equal to 5, and equal to 34, and the Modified Value-at-Risk with a 97.5% level of confidence. Under each risk measure, we construct three portfolios: a defensive (ERA = 50%), a median-risk (ERA = 75%) and an aggressive (ERA = 100%) portfolio. The ERA is the value of the risk measure of the optimal portfolio, divided by the value of the same risk measure for the benchmark. The benchmark chosen is an equally weighted portfolio of the 9 equity indices.

illustrates the consequence of a wrong profiling. A misunderstanding of the investor’s perception of risk could make her endure a risk up to 45% away from her objective. Besides, using the wrong risk measure for an investor appears to be more damageable for defensive portfolios, as the maximum spread is higher for those portfolios.

Our framework happens to yield significant implications for portfolio management at three levels: i) by introducing an operationally affordable tool to account for risk perception in investor profiling; ii) by showing how actual portfolio allocations are sensitive to the proper adequacy between asset return properties and two-di- mensional investor profiles; iii) by summarizing the risk measure in an intuitive and easily interpretable fashion which is especially relevant in the European MiFID regulations context.

On the one hand, the traditional notion of risk aversion refers to the rational expected utility theory. On the other hand, the dimension of risk perception, which involves a personal assessment of the nature of risk, more closely relates to behavioral finance. Thanks to the linex utility function, these two dimensions are reconciled in a single framework that summarizes the investor with two parameters. Unlike Kahneman and Tversky’s prospect theory [

Does the explicit account for risk perception really lead to improving the quality of portfolio advice and the outcomes of the asset allocation process? Our tests clearly show that, even for relatively homogenous equity investments, an inadequate investor profile that ignores the heterogeneity of investors regarding their risk perception is likely to lead to significant errors, both in static allocations and in dynamic portfolio properties. The attribution of “one-type-fits-it-all” portfolios optimized with the wrong underlying utility function yields potential investor dissatisfaction because the portfolio behavior does not correspond to her profile. Furthermore, the dynamic properties of such misspecified portfolios induce a risk drifting that aggravates the initial mistake. Ignoring risk perception, while this is a major determinant of an investor’s attitudes towards risk, has considerable implications in expected utility maximization.

Finally, the introduction of ERA as a way to express risk extends its applicability beyond the use of Bell’s risk measure, but is particularly suitable in this context. It meets the challenge of expressing the risk of a financial instrument when this risk is not homogeneously perceived by investors. Our original framework has addressed several important issues in risk perception analysis and delivers in the end several meaningful insights for either practitioners or national and international regulatory authorities.

This study has demonstrated the relevance of the perception of risk, defined as the subjective judgment of an investor about the characteristics and severity of a risk, for portfolio allocation. Although confirmed several times in experimental finance (see [

The Bell Risk Measure allows us to adequately address the request of an increasing number of researchers and practitioners advocating for the introduction of higher moments of the distribution of returns when optimizing portfolios. For that purpose, we also compare our results with an optimization using MVaR, another risk measure taking into account higher moments of distribution.

The risk measure retrieved from the linex function is much more complicated to interpret than the mere variance or a VaR. As a first contribution, we have proposed a standardization technique that simplifies its interpretation but keeps its individualization specificity: the Equivalent Risky Allocation (ERA). This measure simply expresses the risk measure of an asset in terms of a percentage of the wealth to invest in a selected benchmark to obtain the same risk value.

Through an application of portfolio optimization using nine equity indices and a bond index, we test the time and portfolio consistency of the measure and compare it with the variance of Markowitz and the MVaR which we also express in terms of ERA. Although the variance is a particular case of the Bell Risk Measure with its risk perception C that tends to zero, our research shows the relevance of the recognition of the risk perception in portfolio allocation. Moreover, we demonstrate the consequences of a misspecification of the risk profiles of the investors who are brought to encounter a too high risk or to miss potentially higher returns. Finally, we demonstrate the weakness of MVaR which tends to produce too defensive portfolios for investors using other risk measure as reference.

The limit of this study remains, as in all quantitative approaches, in the need of sufficient historical data. Introducing non classical assets in the portfolio, such as structured products or derivative products, would require using additional techniques in order to “translate” those assets into time series. This methodology also could be tested on larger portfolios and other time periods. Finally, whereas the authors have shown the consequences of a wrong risk profile, the value of the risk perception C of the utility function may be difficult to parametrize.

This paper has benefitted from comments by Yves Crama, André Farber, Olesya Grishchenko, Olivier Lecourtois, Aline Muller and François Quittard-Pinon, as well as EM Lyon seminar participants and the 2010 French Finance Association Conference (St-Malo) attendees. We thank Laurent Bodson and Gaël Minon for research support. Georges Hübner acknowledges financial support from Deloitte. Séverine Plunus acknowledges financial support from KBL Private Bankers.