W. Ogryczak and A. Ruszczyski. From stochastic dominance to mean-risk models: Semideviations as risk measures. Working paper IR97027, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....ffl subadditive ffl not concave 11 ffl generalized convolution concave ffl not homogeneous unless if h(u) u fl ffl isotonic w.r.t. OE SD(2) if ffi 1 Isotonicity w.r.t. OE Sd(2) for the special case of h(u) u and h(u) u 2 was shown by Ruszczynski and Ogryczak ([15]) We show here a more general result: 2.1) Proposition. If h is monotone, convex, differentiable and u 7 h 0 ffi h Gamma1 (u) is concave, then S given by (1) is isotonic w.r.t. OE SD(2) for 0 ffi 1. Proof. We begin with showing that for all random variables Z and b 0 h Gamma1 ....

Ruszcynski A., Ogryczak W. (1997): From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures. IR-97-027, IIASA, Laxenburg, Austria

.... constraints [30] General references for stochastic programming models and solution techniques are [7] 14] 22] 33] Different applications of stochastic programming to optimal asset al..location including analysis of different risk measures were considered in [9] 10] 12] 13] 15] 17] 26] [27], 34] 2 Properties of safety measures A safety measure assigns a numerical value to a random distribution of wealth. Safety measures can be compared on the basis of their properties. In this section, some of these properties are discussed. First, we collect some de#nitions of properties of ....

W. Ogryczak and A. Ruszczynski. From stochastic dominance to mean-risk models: Semideviations as risk measures. European Journal of Operational Research, 116:33--50, 1999.

....mean 1 2 mean absolute deviation mean lower semi standard deviation 5 quantile Then we specify the investor s objective as max E (W ) E [jW E (W )j] where measures the degree of risk aversion. The mean absolute deviation is a good measure of the risk associated with the decision (see [18, 1, 29, 28, 31] for a review of properties of mean absolute deviation as a risk measure) The risk aversion factor allows to adapt the objective to the speci c views of the decision maker. Moreover, its preserves linearity: if the terminal wealth W is linear in the decision variables x, then the objective is a ....

W. Ogryczak and A. Ruszczyski. From stochastic dominance to mean-risk models: Semideviations as risk measures. Working paper IR97027, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1997.

....the degree of risk aversion. The mean absolute deviation E (jW E (W )j) is a good measure of the risk associated with the decision. It is consistent with the second order stochastic dominance (SSD) in a sense that a unique optimal solution of a problem with such objective is SSD e cient (see [OR99] for this and related results) Also, the upper bound on can be increased and the above result can be substantially strengthened in case of symmetric distributions. The risk aversion factor allows to adapt the objective to the speci c needs of the decision maker. Moreover, MAD preserves ....

W. Ogryczak and A. Ruszczyski. From stochastic dominance to mean-risk models: Semideviations as risk measures. European J. of Operations Research, 116:3350, 1999. Preliminary version available as IIASA Interim Report number IR97027 at http://www.iiasa.ac.at/Publications/Documents /IR-97-027.ps.

....of risk aversion. The mean absolute deviation E (jW GammaE (W )j) is a good measure of the risk associated with the decision. It is consistent with the second order stochastic dominance (SSD) in a sense that a unique optimal solution of a problem with such objective is SSD eOEcient 3 (see [OR97] for this and related results) The risk aversion factor ae allows to adapt the objective to the speci c needs of the decision maker. Moreover, MAD preserves linearity: if the terminal wealth W is linear in the decision variables x, then the objective is also linear in x. ffl lower semivariance ....

W. Ogryczak and A. Ruszczy#ski. From stochastic dominance to mean-risk models: Semideviations as risk measures. Working paper IR97027, International Institute for Applied Systems Analysis, 1997. PostScript URL: !http: //www.iiasa.ac.at/Publications/Documents/IR-97-027.ps?.

....under risk because it does not rely on a relation of stochastic dominance (c.f. Whitmore and Findlay, 1978; Levy, 1992) However, the MAD model is consistent with the second degree stochastic dominance, provided that the trade o# coe#cient between risk and return is bounded by a certain constant (Ogryczak and Ruszczynski, 1997). The proposed extension of the MAD model retains consistency with the stochastic dominance. The paper is organized as follows. In the next section we discuss the original MAD model. Section 3 deals with the proposed extension of MAD, enabling to incorporate (downside) risk aversion of an ....

....type of return distributions, what facilitated its application to portfolio optimization for mortgage backed securities (Zenios and Kang, 1993) and other classes of investments where distribution of rate of return is known to be not symmetric. Recently, the MAD model was further validated by Ogryczak and Ruszczynski (1997) who demonstrated that if the trade o# coe#cient # is bounded by 1, then the model is partially consistent with the second degree stochastic dominance (Whitmore and Findlay, 1978) Origins of a stochastic dominance are in an axiomatic model of risk averse preferences (Fishburn, 1964; Hanoch and ....

[Article contains additional citation context not shown here]

Ogryczak, W., Ruszczynski, A. (1997), "From Stochastic Dominance to Mean-- Risk Models: Semideviations as Risk Measures", Interim Report IR--97--027, IIASA, Laxenburg (European Journal of Operational Research, in print).

....with stochastic dominance relations. The classical Markowitz [14] model uses the variance as the risk measure in the mean risk analysis. Since then many authors have pointed out that the mean variance model is, in general, not consistent with stochastic dominance rules. In our preceding paper [18] we have proved that the standard semideviation (square root of the semivariance) or the mean absolute deviation (from the mean) as the risk measures make the corresponding mean risk models consistent with the second degree stochastic dominance, provided that the trade off coefficient is bounded ....

.... the discrete case and was later extended to general distributions [8, 23] Since that time it has been widely used in economics and finance (see [3, 12] for numerous references) Detailed and comprehensive discussion of a stochastic dominance and its relation to downside risk measures is given in [18, 19]. In the stochastic dominance approach random variables are compared by pointwise comparison of some performance functions constructed from their distribution functions. For a real random variable X, its first performance function F (1) X : R [0; 1] is defined as the right continuous ....

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W. Ogryczak and A. Ruszczy' nski, From stochastic dominance to mean--risk models: semideviations as risk measures, European Journal of Operational Research, 116 (1999), pp. 33--50.

....x 1 2 oe 2 y 1 2 oe 2 x Figure 3: Second necessary condition: x (2) y ) 1 2 oe 2 x 1 2 oe 2 y 1 2 ( x Gamma y ) ffi x ffi y ) 8 proof of Corollary 2 for k = 1 and m = 2. For more details on the properties of the O R diagram the reader is referred to (Ogryczak and Ruszczy nski, 1997). A careful analysis of the proof of Theorem 1 reveals that its assertion can be slightly strengthened. Indeed, estimating in (13) the quantities Pfx y g and Pfx x g from above by some constant ae, Pfx x g ae 1; we obtain the necessary condition ae 1=k x Gamma ffi (k) x ....

Ogryczak, W., A. Ruszczy'nski (1997). From Stochastic Dominance to Mean--Risk Models: Semideviations as Risk Measures , Interim Report 97-027, International Institute for Applied Systems Analysis, Laxenburg, Austria.

....Y ) X ; 0) and ( X ; ffi X ) Employing the Lyapunov inequalities ffi X oe X and ffi Y oe Y , we obtain a graphical proof of Corollary 2 for k = 1 and m = 2. For more details on the properties of the O R diagram in the case of second degree dominance, the reader is referred to (Ogryczak and Ruszczy nski, 1997). For a higher degree k 1 it is more convenient to analyze the graph of the function G (k) X (j) i k F (k 1) X (j) j 1=k = k max(0; j Gamma X)k k ; 17) instead of F (k 1) X itself. It has the following properties. Proposition 6 Let k 1 and X 2 L k . Then (i) lim j Gamma1 G ....

Ogryczak, W., A. Ruszczy'nski (1997). From Stochastic Dominance to Mean--Risk Models: Semideviations as Risk Measures, Interim Report 97-027, International Institute for Applied Systems Analysis, Laxenburg, Austria.

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W. Ogryczak and A. Ruszczyski. From stochastic dominance to mean-risk models: Semideviations as risk measures. Working paper IR97027, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1997.

No context found.

Ogryczak, W., Ruszczynski, From stochastic dominance to mean-risk models: semideviation as risk measures, European Journal of Operational Research, 116(1999), 33-50.

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