WWW.ABSTRACT.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Abstract, dissertation, book
 
<< HOME
CONTACTS



Pages:     | 1 ||

«1 Introduction A commonly used risk metrics is the standard deviation. For examples mean-variance portfolio selection maximises the expected utility ...»

-- [ Page 2 ] --

−1 gα (u) = (u) + α, (3) where is the Gaussian cumulative distribution. In other words he applies the same perspective of preference to quantify the risk associated to gain and risk. Thus, a risk manager evaluates the risk associated to the upside and downside risks with the same function g implying a symmetric consideration for the two effects due to the distortion. Moreover it induces the same confidence level for the losses and the gain which implies the same level of risk aversion associated to the losses and the gains.

In Fig. 2 we illustrate the impact of the Wang (2000) distortion function introduced in Eq. (3) on the logistic distribution provided in Table 1. We can remark that the distorted distribution is always symmetrical under this kind of distortion function, and we observe a shift of the mode of the initial distribution towards the left.

To avoid the problem of symmetry in the previous distorsion, Sereda et al. (2010) propose to use two different functions issued from the same polynomial with different

coefficients, say:

–  –  –

Fig. 2 Distortion of logistic distribution with mean 0 using a Wang distortion function with confidence level 0.65. It illustrates the effect of distortion with gi (u) = u + ki u − u2 for ki ∈ ]0, 1] et ∀i ∈ {1, 2}. With this approach one models loss and gains differently relatively to the values of the parameters ki, i = 1, 2.

Thus upside and downside risks are modeled in different ways. Nevertheless the calibration of the parameters ki, i = 1, 2 remains an open problem.

To create bimodal or multi-modal distributions we have to impose other properties to the distortion function g. Indeed, transforming an unimodal distribution into a bimodal one provides different approaches to the risk aversion of losses and gains.

This will allow us to introduce a new coherent risk measure in that latter case.

3.2 A New Coherent Risk Measure

We begin to discuss the choice of the function g to obtain a bimodal distribution. To do so we need to use a function g which creates saddle points. The saddle point generates a second hump in the new distribution which allows us to take into account different patterns located in the tails. The distortion function g fulfilling this objective is an inverse S-shaped polynomial function of degree 3 given by the following equation

and characterized by two parameters δ and β:

–  –  –

Fig. 3 Curves of the distortion function gδ introduced in Eq. (5) for several value of δ and fixed values of β = 0.001 In Fig. 3, the value of the level of the discrimination of an event is given by β = 0.001 then we plot the function gδ for different values of δ. This parameter β illustrates the fact that some events are discriminating more than others. Figure 3 shows the location of the saddle point creating convex and concave parts inside the domain [0, 1]. The convex part can be associated to the negative values of the returns associated to the losses and the concave part is associated to positive returns. We observe in this picture that for high values of δ the concave part diminishes and then the effect of saddle point decreases.

Variations in β in Fig. 4 exhibit different patterns for a fixed value of δ.

To understand the influence of the parameter β on the shape of the distortion function we use three graphs in Fig. 4. The two left graphs correspond to the same value of the parameters. The middle figure zooms on the x-axis from [0, 1] to [−4, 4].

We show that the function g may not have a saddle point on ]0, 1[ depending on the values of β. The right graph provides different representations of the distorsion function for several values of β. We observe that if β tends to 1 then the distortion function g tends to the identity mapping and when β tends to 0 the curve is more important and the effect of g on the distribution will be more important.

Figure 5 illustrates the effect of distortion of the Gaussian distribution for several values of β and fixed δ = 0.50. We observe the same effects as in Fig. 4. For small values of the parameter β (0.00005 or 0.005) the distortion function has two distinct parts, one convex part for x ∈ ]0, 0.5[ and one concave part for x ∈ ]0.5, 1[.

Moreover when β is close to 1 then the distorted cumulative distribution tends to the initial Gaussian variable.

80 D. Guégan and B. Hassani Fig. 4 The effect of β on the distortion function for a level of security δ = 0.75 showing that if β tends to 1, the distortion function tends to the identity function Fig. 5 The effect of β on the cumulative Gaussian distribution for δ = 0.50 Figure 6 points out the effect of distortion on the density of the Gaussian distribution using the same values of the parameters than those used in Fig. 5. Again we generate a new distribution with two humps. Making both parameters varying permits to solve one of our objective: to create a asymmetrical distribution with more than one hump.

It is important to notice that the function gδ creates a distorted density function which associates a small probability in the centre of the distribution and put greater weight in the tails. This phenomenon is illustrated in Fig. 7 where the derivative of g (density) indicates how weights on the tails can be increased.

Such discrimination is also illustrated in Fig. 8 which exhibits the particular effect of parameter β when δ is fixed to 0.75 for the creation of humps. From a Gaussian distribution, applying gδ defined in (5), with δ = 0.75 and β = 0.48 we create a distribution for which the probability of occurrences of the extremes in the right part is bigger than the probability of occurrence of the extremes in the left part which can be counter-intuitive for risk management but interesting from a theoretical point of view.

http://www.springer.com/978-3-319-07523-5



Pages:     | 1 ||


Similar works:

«PERCEPTIONS OF THE SERPENT IN THE ANCIENT NEAR EAST: ITS BRONZE AGE ROLE IN APOTROPAIC MAGIC, HEALING AND PROTECTION by WENDY REBECCA JENNIFER GOLDING submitted in accordance with the requirements for the degree of MASTER OF ARTS in the subject ANCIENT NEAR EASTERN STUDIES at the UNIVERSITY OF SOUTH AFRICA SUPERVISOR: PROFESSOR M LE ROUX November 2013 Snake I am The Beginning and the End, The Protector and the Healer, The Primordial Creator, Wisdom, all-knowing, Duality, Life, yet the terror in...»

«Audubon Jr.-Sr. High School Clubs and Student Activities Math Honor Society – Mr. Marino 7th Grade Government – Mr. Webb Mini Bridge – Ms. Graham 8th Grade GovernmentMs. Stack National Honor Society (election) Mrs. 9 Grade Government Ms. McGuire th D’Aprile 10th Grade Government – Ms. Bulskis National Jr. Honor Society (by election) – 11th Grade Government – Mr. Niglio Mrs. Van Fossen 12th Grade Government – Mr. Tomasetti/ Office Aides – Mrs. Clune Ms. Bulskis The Parrot –...»

«Tapirskrift Rasmus Fleischer Detta verk är licensierat enligt Creative Commons Erkännande-Ickekommersiell-IngaBearbetningar 2.5 Sverige licens. För att visa licensen, besök http://creativecommons.org/licenses/by-nc-nd/2.5/ se/ eller skicka ett brev till Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA. Grafisk form: Konst & Teknik Typsnitt omslag: Okapi av Gro Janarv Typsnitt inlaga: Indigo Antiqua Pro Text, Johan Ström Korrektur: Erik Lindman-Mata Axl...»

«ACADEMIC CATALOG 2015 – 2016 COE COLLEGE CATALOG (2013-2014) PREFACE Non-Discrimination Coe College does not discriminate on the basis of race, color, ethnicity, age, religion, national origin, sexual orientation, gender identity, sex, marital status, disability, or status as a U.S. Veteran. All students have equal access to the facilities, financial aid, and programs of the College. Higher Education Opportunity Act The college complies with Readmission Requirements for Service Members as...»

«Fit 2 Love “We all want to be loved. JJ provides help for everyone who wants to love themselves, their bodies, and to attract more love from others.” – DR. CHÉRIE CARTER-SCOTT #1 New York Times Bestselling author If Life is a Game, These are the Rules If Love is a Game, These are the Rules, If Success is a Game, These are the Rules “Just when you think you’ve read everything about attracting THE relationship, BOOM, along comes JJ Flizanes with new information and a magnetizing...»

«KEARNEY LOUGHLIN, ET * NO. 2013-CA-1285 AL. * VERSUS COURT OF APPEAL * UNITED SERVICES FOURTH CIRCUIT AUTOMOBILE ASSOCIATION * STATE OF LOUISIANA ******* APPEAL FROM CIVIL DISTRICT COURT, ORLEANS PARISH NO. 2012-10478, DIVISION “D-16” Honorable Lloyd J. Medley, Judge ****** Judge Madeleine M. Landrieu ****** (Court composed of Chief Judge James F. McKay, III, Judge Dennis R. Bagneris, Sr., Judge Daniel L. Dysart, Judge Madeleine M. Landrieu, Judge Sandra C. Jenkins) JENKINS, J., DISSENTS...»





 
<<  HOME   |    CONTACTS
2016 www.abstract.xlibx.info - Free e-library - Abstract, dissertation, book

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.