From stochastic dominance to meanrisk models: Semideviations as risk measures 1


 Stephanie McDonald
 2 years ago
 Views:
Transcription
1 European Journal of Operational Research 116 (1999) 33±50 Theory and Methodoloy From stochastic dominance to meanrisk models: Semideviations as risk measures 1 Wøodzimierz Oryczak a, *, Andrzej Ruszczynski b a Department of Mathematics and Computer Science, Warsaw University, Banacha 2, Warsaw, Poland b Department of Manaement Science and Information Systems, Ruters University, 94 Rockafeller Rd, Piscataway, NJ 08854, USA Received 1 November 1997; accepted 1 April 1998 Abstract Two methods are frequently used for modelin the choice amon uncertain outcomes: stochastic dominance and meanrisk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quanti es the problem in a lucid form of two criteria with possible tradeo analysis, but cannot model all riskaverse preferences. In particular, if variance is used as a measure of risk, the resultin mean±variance (Markowitz) model is, in eneral, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the meanrisk model consistent with the second deree stochastic dominance, provided that the tradeo coe cient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome±Risk (O±R) diaram, which appears to be particularly useful for comparin uncertain outcomes. Ó 1999 Elsevier Science B.V. All rihts reserved. Keywords: Decisions under risk; Portfolio optimization; Stochastic dominance; Meanrisk model 1. Introduction * Correspondin author. 1 This work was partially supported by the International Institute for Applied Systems Analysis, Laxenbur, Austria, and distributed online as the IIASA Report IR (June 1997). Comparin uncertain outcomes is one of fundamental interests of decision theory. Our objective is to analyze relations between the existin approaches and to provide some tools to facilitate the analysis. We consider decisions with realvalued outcomes, such as return, net pro t or number of lives saved. A leadin example, oriinatin from  nance, is the problem of choice amon investment opportunities or portfolios havin uncertain returns. Althouh we discuss the consequences of our analysis in the portfolio selection context, we do not assume any speci city related to this or /99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rihts reserved. PII: S ( 9 8 )
2 34 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 another application. We consider the eneral problem of comparin realvalued random variables (distributions), assumin that larer outcomes are preferred. We describe a random variable X by the probability measure P X induced by it on the real line R. It is a eneral framework: the random variables considered may be discrete, continuous, or mixed (Pratt et al., 1995). Owin to that, our analysis covers a variety of problems of choosin amon uncertain prospects that occur in economics and manaement. Two methods are frequently used for modelin choice amon uncertain prospects: stochastic dominance (Whitmore and Findlay, 1978; Levy, 1992), and meanrisk analysis (Markowitz, 1987). The former is based on an axiomatic model of riskaverse preferences: it leads to conclusions which are consistent with the axioms. Unfortunately, the stochastic dominance approach does not provide us with a simple computational recipe ± it is, in fact, a multiple criteria model with a continuum of criteria. The meanrisk approach quanti es the problem in a lucid form of only two criteria: the mean, representin the expected outcome, and the risk: a scalar measure of the variability of outcomes. The meanrisk model is appealin to decision makers and allows a simple tradeo analysis, analytical or eometrical. On the other hand, meanrisk approaches are not capable of modelin the entire amut of riskaverse preferences. Moreover, for typical dispersion statistics used as risk measures, the meanrisk approach may lead to inferior conclusions. The seminal Markowitz (1952) portfolio optimization model uses the variance as the risk measure in the meanrisk analysis. Since then many authors have pointed out that the mean± variance model is, in eneral, not consistent with stochastic dominance rules. The use of the semivariance rather than variance as the risk measure was already suested by Markowitz (1959) himself. Porter (1974) showed that the meanrisk model usin a xedtaret semivariance as the risk measure is consistent with stochastic dominance. This approach was extended by Fishburn (1977) to more eneral risk measures associated with outcomes below some xed taret. There are many aruments for the use of xed tarets. On the other hand, when one of performance measures is the expected return, the risk measure should take into account all possible outcomes below the mean. Therefore, we focus our analysis on central semimoments which measure the expected value of deviations below the mean. To be more precise, we consider the absolute semideviation (from the mean) d X ˆ Z l X ˆ 1 2 l X n P X dn jn l X j P X dn and the standard semideviation 0 r X Z l X 1 l X n 2 P X dn A 1=2 1 ; 2 where l X ˆ EfX. We show that meanrisk models usin standard or absolute semideviations as risk measures are consistent with the stochastic dominance, if a bounded set of meanrisk tradeo s is considered. In the portfolio selection context these models correspond to the Markowitz (1959, 1987) mean±semivariance model and the Konno and Yamazaki (1991) MAD model with absolute deviation. The paper is oranized as follows. In Section 3 we recall the basics of the stochastic dominance and meanrisk approaches. We also specify what we mean by consistency of these approaches. In Section 3 we introduce a convenient raphical tool for the stochastic dominance methodoloy: the Outcome±Risk (O±R) diaram, and we examine various risk measures within the diaram. We further use the O±R diaram to establish consistency of meanrisk models usin the absolute semideviation (Section 4) and the standard semideviation (Section 5), respectively, with the second deree stochastic dominance rules. In a similar way we reexamine the standard deviation as a possible risk measure in Section 6. Owin to the use of the O±R diaram all proofs are easy, nevertheless, riorous.
3 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33± Stochastic dominance and meanrisk models Stochastic dominance is based on an axiomatic model of riskaverse preferences (Fishburn, 1964). It oriinated in the majorization theory (Hardy et al., 1934) for the discrete case and was later extended to eneral distributions (Hanoch and Levy, 1969; Rothschild and Stilitz, 1970). Since that time it has been widely used in economics and  nance (see Bawa (1982) and Levy (1992) for numerous references). In the stochastic dominance approach random variables are compared by pointwise comparison of some performance functions constructed from their distribution functions. Let X be a random variable with the probability is de ned as the rihtcontinuous cumulative distribution function itself: measure P X. The rst performance function F 1 X F 1 X ˆ F X ˆ PfX 6 for 2 R: The weak relation of the rst deree stochastic dominance (FSD) is de ned as follows: X FSD Y () F X 6 F Y for all 2 R: 3 The second performance function X is iven by areas below the distribution function F X : X ˆ F X n dn for 2 R; and de nes the weak relation of the second deree stochastic dominance (SSD): X SSD Y () X 6 Y for all 2 R: 4 The correspondin strict dominance relations FSD and SSD are de ned by the standard rule X Y () X Y and Y ² X : 5 Thus, we say that X dominates Y under the FSD rules (X FSD Y ), if F X 6 F Y for all 2 R, where at least one strict inequality holds. Similarly, we say that X dominates Y under the SSD rules (X SSD Y ), if X 6 Y for all 2 R, with at least one inequality strict. Certainly, X FSD Y implies X SSD Y and X FSD Y implies X SSD Y. Note that F X expresses the probability of underachievement for a iven taret value. Thus the rst deree stochastic dominance is based on the multidimensional (continuumdimensional) objective de ned by the probabilities of underachievement for all taret values. The FSD is the most eneral relation. If X FSD Y, then X is preferred to Y within all models preferrin larer outcomes, no matter how riskaverse or riskseekin they are. For decision makin under risk most important is the second deree stochastic dominance relation,. If X SSD Y, then X is preferred to Y within all riskaverse preference models that prefer larer outcomes. It is therefore a matter of primary importance that an approach to the comparison of random outcomes be consistent with the second deree stochastic dominance relation. Our paper focuses on the consistency of meanrisk approaches with SSD. Meanrisk approaches are based on comparin two scalar characteristics (summary statistics), the rst of which ± denoted l ± represents the expected outcome (reward), and the second ± denoted r ± is some measure of risk. The weak relation of meanrisk dominance is de ned as follows: associated with the function X X l=r Y () l X P l Y and r X 6 r Y : The correspondin strict dominance relation l=r is de ned in the standard way, as in Eq. (5). We say that X dominates Y under the l=r rules (X l=r Y ), if l X P l Y and r X 6 r Y, and at least one of these inequalities is strict. Note that random variables X and Y such that l X ˆ l Y and r X ˆ r Y are indi erent under the l=r rules (X l=r Y and Y l=r X ). We say then that X and Y enerate a tie in the l=r model. An important advantae of meanrisk approaches is the possibility of a pictorial tradeo analysis. Havin assumed a tradeo coe cient k between the risk and the mean, one may directly compare real values of l X kr X and l Y kr Y. Indeed, the followin implication holds: X l=r Y ) l X kr X P l Y kr Y for all k > 0: We say that the tradeo approach is consistent with the meanrisk dominance.
4 36 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Suppose now that the meanrisk model is consistent with the SSD model by the implication X SSD Y ) X l=r Y : Then meanrisk and tradeo approaches lead to uaranteed results: X l=r Y ) Y ² SSD X ; l X kr X > l Y kr Y for some k > 0 ) Y ² SSD X : In other words, they cannot strictly prefer an inferior decision. In this paper we show that some meanrisk models are consistent with the SSD model in the followin sense: there exists a positive constant a such that for all X and Y X SSD Y ) l X P l Y and l X a r X P l Y a r Y : 6 In particular, for the risk measure r de ned as the absolute semideviation (1) or standard semideviation (2), the constant a turns out to be equal to 1. Yitzhaki (1982) showed a similar result for the risk measure de ned as the RGini's R mean (absolute) di erence r X ˆ C X ˆ 1 2 jn j PX dn P X d. Relation (6) directly expresses the consistency with SSD of the model usin only two criteria: l and l a r. Both, however, are de ned by l and r, and we have l X P l Y and l X a r X P l Y a r Y ) l X kr X P l Y kr Y for 0 < k 6 a: Consequently, Eq. (6) may be interpreted as the consistency with SSD of the meanrisk model, provided that the tradeo coe cient is bounded from above by a. Namely, Eq. (6) uarantees that l X kr X > l Y kr Y ) Y ² SSD X : for some 0 < k 6 a Comparison of random variables is usually related to the problem of choice amon risky alternatives in a iven feasible set Q. For instance, in the simplest problem of portfolio selection (Markowitz, 1987) the feasible set of random variables is de ned as all convex combinations (weihted averaes with nonneative weihts totalin 1) of a iven number of investment opportunities (securities). A feasible random variable X 2 Q is called e cient under the relation if there is no Y 2 Q such that Y X. Consistency (6) leads to the followin result. Proposition 1. If the meanrisk model satis es (6), then except for random variables with identical l and r, every random variable that is maximal by l kr with 0 < k < a is e cient under the SSD rules. Proof. Let 0 < k < a and X 2 Q be maximal by l kr. This means that l X kr X P l Y kr Y for all Y 2 Q. Suppose that there exists Z 2 Q such that Z SSD X. Then, from Eq. (6), l Z P l X and l Z ar Z P l X ar X : 7 Addin these inequalities multiplied by 1 k=a and k=a, respectively, we obtain 1 k=a l Z k=a l Z ar Z P 1 k=a l X k=a l X ar X ; which after simpli cation reads: l Z kr Z P l X kr X : But X is maximal, so we must have l Z kr Z ˆ l X kr X ; 8 that is, equality in Eq. (8) holds. This combined with Eq. (7) implies l Z ˆ l X and r Z ˆ r X : Proposition 1 justi es the results of the meanrisk tradeo analysis for 0 < k < a. This can be extended to k ˆ a provided that the inequality l X a r X P l Y a r Y turns into equality only in the case of l X ˆ l Y. Corollary 1. If the meanrisk model satis es (6) as well as X SSD Y and l X > l Y ) l X a r X > l Y a r Y 9
5 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 37 then, except for random variables with identical l and r, every random variable that is maximal by l kr with 0 < k 6 a is e cient under the SSD rules. Proof. Due to Proposition 1, we only need to prove the case of k ˆ a. Let X 2 Q be maximal by l ar. Suppose that there exists Z 2 Q such that Z SSD X. Hence, by Eq. (6), l Z P l X. If l Z > l X, then Eq. (9) yields l X ar X < l Z ar Z, which contradicts the maximality of X. Thus, l Z ˆ l X and, by Eq. (6) and the maximality of X, one has l X ar X ˆ l Z ar Z. Hence, l Z ˆ l X and r Z ˆ r X : It follows from Proposition 1 that for meanrisk models satisfyin Eq. (6) the optimal solution of the problem maxfl X k r X : X 2 Q 10 with 0 < k < a, if it is unique, is e cient under the SSD rules. However, in the case of nonunique optimal solutions, we only know that the optimal set of Problem (10) contains a solution which is e cient under the SSD rules. The optimal set may contain, however, also some SSDdominated solutions. Exactly, due to Proposition 1, an optimal solution X 2 Q can be SSD dominated only by another optimal solution Y 2 Q which enerates a l=r tie with X (i.e., l Y ˆ l X and r Y ˆ r X ). A question arises whether it is possible to additionally reularize (re ne) problem (10) in order to select those optimal solutions that are e cient under the SSD rules. We resolve this question durin the analysis of speci c risk measures. In many applications, especially in the portfolio selection problem, the meanrisk model is analyzed with the socalled critical line alorithm (Markowitz, 1987). This is a technique for identifyin the l=r e cient frontier by parametric optimization (10) for varyin k > 0. Proposition 1 uarantees that the part of the e cient frontier (in the l=r imae space) correspondin to tradeo coe cients 0 < k < a is also e cient under the SSD rules. 3. The O±R diaram The second deree stochastic dominance is based on the pointwise dominance of functions. Therefore, properties of the function are important for the analysis of relations between the second deree stochastic dominance and the meanrisk models. The followin proposition summarizes the basic properties which we use in our analysis. Proposition 2. If EfjX j < 1, then the function X is well de ned for all 2 R and has the followin properties: P1. X is continuous, convex, nonneative and nondecreasin. P2. If F X 0 > 0, then X is strictly increasin for all P 0. P3. X ˆ R n df X n ˆ R n P X dn ˆ PfX 6 Ef X jx 6. P4. lim! X ˆ 0. P5. X l X ˆ R 1 n df X n ˆ R 1 n P X dn ˆ PfX P EfX jx P. P6. X l X is a continuous, convex, nonneative and nonincreasin function of. P7. lim!1 X l X Š ˆ 0. P8. For any iven 0 2 R X P X 0 0 supff X n j n < 0 P X 0 0 for all < 0 ; X 6 X 0 0 supff X n j n < 6 X 0 0 for all > 0 : Properties P1±P4 are rather commonly known but frequently not expressed in such a riorous form for eneral random variables. Properties P5± P8 seem to be less known or at least not widely used in the stochastic dominance literature. In Appendix A we ive a formal proof of Proposition 2. From now on, we assume that all random variables under consideration are interable in the sense that EfjX j < 1. Therefore, we are allowed us to use all the properties P1±P8 in our analysis.
6 38 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Note that, due to property P3, X ˆ PfX 6 Ef X j X 6 thus expressin the expected shortae for each taret outcome. So, in addition to bein the most eneral dominance relation for all riskaverse preferences, SSD is a rather intuitive multidimensional (continuumdimensional) risk measure. Therefore, we will refer to the raph of X as to the O±R diaram for the random variable X (Fi. 1). The raph of the function X has two asymptotes which intersect at the point l X ; 0. Speci cally, the axis is the left asymptote (property P4) and the line l X is the riht asymptote (property P7). In the case of a deterministic outcome (X ˆ l X ), the raph of X coincides with the asymptotes, whereas any uncertain outcome with the same expected value l X yields a raph above (precisely, not below) the asymptotes. Hence, the space between the curve ; X, 2 R, and its asymptotes represents the dispersion (and thereby the riskiness) of X in comparison to the deterministic outcome of l X. We shall call it the dispersion space. Both size and shape of the dispersion space are important for complete description of the riskiness. Nevertheless, it is quite natural to consider some size parameters as summary characteristics of riskiness. As the simplest size parameter one may consider the maximal vertical diameter. By properties P1 and P6, it is equal to X l X. Moreover, property P3 yields the followin corollary. Corollary 2. If EfjX j < 1, then X l X ˆ d X. The absolute semideviation d X turns out to be a linear measure of the dispersion space. There are many aruments (see, e.., Markowitz, 1959) that only the dispersion related to underachievements should be considered as a measure of riskiness. In such a case we should rather focus on the downside dispersion space, that is, to the left of l X. Note that d X is the larest vertical diameter for both the entire dispersion space and the downside dispersion space. Thus d X seems to be a quite reasonable linear measure of the risk related to the representation of a random variable X by its expected value l X. Moreover, the absolute deviation d X ˆ jn l X j P X dn 11 is symmetric in the sense that d X ˆ 2 d X for any (possible nonsymmetric) random variable X. Thus absolute mean d also can be considered a linear measure of riskiness. A better measure of the dispersion space should be iven by its area. To evaluate it one needs to calculate the correspondin interals. The followin proposition ives these results. Proposition 3. If EfX 2 < 1, then X f df ˆ 1 2 n 2 P X dn ˆ 1 2 PfX 6 Ef X 2 jx 6 ; 12 Fi. 1. The O±R diaram.
7 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 39 X f f l X Š df ˆ 1 2 n 2 P X dn ˆ 1 2 PfX P Ef X 2 jx P : 13 Formula (12) was shown by Porter (1974) for continuous random variables. The second formula seems to be new in the SSD literature. In the Appendix A we ive a formal proof of both formulas for eneral random variables. Corollary 3. If EfX 2 < 1, then Z lx r 2 X ˆ 2 X f df; 14 Z lx r 2 X ˆ 2 X f df 2 X f f l X Š df: l X 15 Hereafter, whenever considerin variance r 2 or semivariance r 2 (standard deviation r or standard semideviation r) we will assume that EfX 2 < 1. Therefore, we are eliible to use formulas (14) and (15) in our analysis. By Corollary 3, the variance r 2 X represents the doubled area of the dispersion space of the random variable X, whereas the semivariance r 2 X is the doubled area of the downside dispersion space. Thus the semimoments d and r 2, as well as the absolute moments d and r 2, can be rearded as some risk characteristics and they are well depicted in the O±R diaram (Fis. 2 and 3). In further sections we will use the O±R diaram to prove that the meanrisk model usin the semideviations d and r is consistent with the second deree stochastic dominance. Geometrical relations in the O±R diaram make the proofs easy. However, as the eometrical relations are the consequences of Propositions 2 and 3, the proofs are riorous. To conclude this section we derive some additional consequences of Propositions 2 and 3. Let us observe that in the O±R diaram the diaonal line X 0 0 is parallel to the riht as Fi. 2. The O±R diaram and the semimoments. Fi. 3. The O±R diaram and the absolute moments.
8 40 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 ymptote l X and intersects the raph of X at the point 0 ; X 0. Therefore, property P8 can be interpreted as follows. If a diaonal line (parallel to the riht asymptote) intersects the raph of X at ˆ 0, then for < 0, X is bounded from below by the line, and for > 0, X is bounded from above by the line. Moreover, the boundin is strict except in the case of supff X n j n < 0 ˆ 1 or F X 0 ˆ 1, respectively. Settin 0 ˆ l X we obtain the followin proposition (Fi. 4). Proposition 4. If EfX 2 < 1, then r X P d X and this inequality is strict except in the case r X ˆ d X ˆ 0. Proof. From P8 in Proposition 2, X > X l X l X for all < l X ; since supff X n j n < l X < 1. Hence, in the case of X l X > 0, one has 1 2 r2 X > 1 d 2 2 X and r X > d X. Otherwise r X ˆ d X ˆ 0: Recall that, due to the Lyapunov inequality for absolute moments (Kendall and Stuart, 1958), the standard deviation and the absolute deviation satisfy the followin inequality: r X P d X : 16 Proposition 4 is its analoue for absolute and standard semideviations. While considerin two random variables X and Y in the common O±R diaram one may easily notice that, if l X < l Y, then the riht asymptote of X (the diaonal line l X ) must intersect the raph of Y at some 0. By property P8, X P l X P Y for P 0 : Moreover, since l Y is the riht asymptote of (property P7), there exists 1 > 0 such that Y X > Y for P 1 : Thus, from the O±R diaram one can easily derive the followin, commonly known, necessary condition for the second deree stochastic dominance (Fishburn, 1980; Levy, 1992). Proposition 5. If X SSD Y, then l X P l Y. While considerin in the common O±R diaram two random variables X and Y with equal expected values l X ˆ l Y, one may easily notice that the functions X and Y have the same asymptotes. It leads us to the followin commonly known result (Fishburn, 1980; Levy, 1992). Proposition 6. For random variables X and Y with equal means l X ˆ l Y ; X SSD Y ) r 2 X 6 r2 Y ; 17 X SSD Y ) r 2 X < r2 Y : Absolute deviation as risk measure In this section we analyze the meanrisk model with the risk de ned by the absolute semideviation d iven by Eq. (1). Recall that dx ˆ X l X (Corollary 2) and it represents the larest vertical diameter of the (downside) dispersion space. Fi r2 X P 1 2 d 2 X.
9 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 41 Hence, d is a well de ned eometrical characteristic in the O±R diaram. Consider two random variables X and Y in the common O±R diaram (Fi. 5). If X SSD Y, then, is bounded from above by Y, and, by Proposition 5, l X P l Y. For P l Y, Y is bounded from above by d Y l Y (second inequality of P8 in Proposition 2). Hence, by the de nition of SSD, X d X ˆ X l X 6 Y l X 6 d Y l X l Y : This simple analysis of the O±R diaram allows us to derive the followin necessary condition for the second deree stochastic dominance. Proposition 7. If X SSD Y, then l X P l Y and l X d X P l Y d Y, where the second inequality is strict whenever l X > l Y. Proof. Due to the considerations precedin the proposition, we only need to prove that l X d X > l Y d Y whenever X SSD Y and l X > l Y. Note that from the second inequality of P8 ( ˆ l X, 0 ˆ l Y ), in such a case, d X ˆ X l X 6 X l Y l X l Y supff X n j n < l X < d Y l X l Y : Proposition 7 says that the l= d meanrisk model is consistent with the second deree stochastic dominance by the rule (6) with a ˆ 1. Therefore, a l= d comparison leads to uaranteed results in the sense that l X k d X > l Y k d Y for some 0 < k 6 1 ) Y ² SSD X : For problems of choice amon risky alternatives in a iven feasible set, due to Corollary 1, the followin observation can be made. Corollary 4. Except for random variables with identical mean and absolute semideviation, every random variable X 2 Q that is maximal by l X k d X with 0 < k 6 1 is e cient under the SSD rules. The upper bound on the tradeo coe cient k in Corollary 4 cannot be increased for eneral distributions. For any e > 0 there exist random variables X SSD Y such that l X > l Y and l X 1 e d X ˆ l Y 1 e d Y : As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 1= 1 e, PfX ˆ 1 ˆ e= 1 e ; and Y de ned as PfY ˆ 0 ˆ 1. Konno and Yamazaki (1991) introduced the portfolio selection model based on the l=d meanrisk model. The model is very attractive computationally, since (for discrete random variables) it leads to linear prorammin problems. Therefore, it is recently applied to various nance problems (e.., Zenios and Kan, 1993). Note that the absolute deviation d is a symmetric measure and the absolute semideviation d is its half. Hence, Proposition 7 is also valid (with factor 1/2) for the l=d meanrisk model. Thus, for the l=d model there exists a bound on the tradeo s such Fi. 5. SSD and the absolute semideviations: X SSD Y ) d X 6 d Y l X l Y.
10 42 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 that for smaller tradeo s the model is consistent with the SSD rules. Speci cally, due to Corollary 4, the followin observation can be made. Corollary 5. Except for random variables with identical mean and absolute deviation, every random variable X 2 Q that is maximal by l X kd X with 0 < k 6 1=2 is e cient under the SSD rules. The upper bound on the tradeo coe cient k in Corollary 5 can be substantially increased for symmetric distributions. Proposition 8. For symmetric random variables X and Y, X SSD Y ) l X P l Y and l X d X P l Y d Y : Proof. If X SSD Y then, due to Proposition 5, l X P l Y. From the second inequality of P8 in Proposition 2, 1 2 d X ˆ X l X 6 X l Y l X l Y supff X n j n < l X d Y 1 2 l X l Y ; since for symmetric random variables supff X n j n < l X 6 1=2: Hence, l X d X P l Y d Y : For problems of choice amon risky alternatives in a iven feasible set, Propositions 1 and 8 imply the followin result. Corollary 6. Within the class of symmetric random variables, except for random variables with identical mean and absolute deviation, every random variable X 2 Q that is maximal by l X kd X with 0 < k < 1, is e cient under the SSD rules. The bound on the tradeo coe cient k in Corollary 6 cannot be increased. There exist symmetric random variables X SSD Y such that l X > l Y and l X d X ˆ l Y d Y. As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 0:5, PfX ˆ 4 ˆ 0:5; and Y de ned as PfY ˆ 0 ˆ 0:5, PfY ˆ 2 ˆ 0:5. It follows from Corollary 4 that the optimal solution of the problem maxfl X k d X : X 2 Q; 0 < k 6 1; 19 is e cient under the SSD rules, if it is unique. In the case of multiple optimal solutions, the optimal set of problem (19) contains a solution which is e cient under SSD rules but it may contain also some SSD dominated solutions. Exactly, due to Corollary 4, an optimal solution X 2 Q can be SSD dominated only by another optimal solution Y 2 Q which enerates a l= d tie with X (i.e., l Y ˆ l X and d Y ˆ d X ). A question arises how di erent can the random variables be that enerate a tie (are indi erent) in the l= d meanrisk model. Absolute semideviation is a linear measure of the dispersion space and therefore many di erent distributions may tie in the l= d comparison. Note that two random variables X and Y with the same expected value l X ˆ l Y are l= d indi erent if X l X ˆ Y l X, independently of values of X and Y for all other 6ˆ l X. Consider two nite random variables: X de ned as PfX ˆ 20 ˆ 0:5, PfX ˆ 20 ˆ 0:5; and Y de ned as PfY ˆ 000 ˆ 0:01, PfY ˆ 0 ˆ 0:98, PfY ˆ 1000 ˆ 0:01. They are l= d indi erent, because l X ˆ l Y ˆ 0 and d X ˆ d Y ˆ 10. Nevertheless, X SSD Y and X < Y for all 0 < jj < As an extreme one may consider the case when X < Y for all 2 R except for ˆ l X ˆ l Y, and despite that X and Y are l= d indi erent (Fi. 6). Therefore, the l= d model, althouh consistent with the second deree stochastic dominance for bounded tradeo s, dramatically needs some additional reularization to resolve ties in comparisons. Ties in the l= d model can be resolved with additional comparisons of standard deviations or variances. By its de nition, a tie in the l= d model may occur only in the case of equal means, and this is exactly the case when Proposition 6 applies. We can simply select from X and Y the one that has a smaller standard deviation. It can be for
11 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 43 malized as the followin lexicoraphic comparison: l X k d X ; r X P lex l Y k d Y ; r Y () l X k d X > l Y k d Y or l X k d X ˆ l Y k d Y and r X P r Y : The lexicoraphic relation de nes a linear order. Hence, for problems of choice amon risky alternatives in a iven feasible set, the lexicoraphic maximization of l k d; r is well de ned. It has two phases: the maximization of l k d within the feasible set, and the selection of the optimal solution that has the smallest standard deviation r. Owin to Eq. (18), such a selection results in SSD e cient solutions (even in the case of multiple optimal solutions). Corollary 7. Every random variable X 2 Q that is lexicoraphically maximal by l X k d X ; r X with 0 < k 6 1 is e cient under the SSD rules. For the l=d portfolio selection model (Konno and Yamazaki, 1991) the results of our analysis can be summarized as follows. While identifyin the l=d e cient frontier by parametric optimization maxfl X k d X : X 2 Q 20 for tradeo k varyin in the interval 0; 0:5Š the correspondin imae in the l=d space represents SSD e cient solutions. Thus it can be used as the meanrisk map to seek a satisfactory l=d compromise. It does not mean, however, that the solutions enerated durin the parametric optimization (20) are SSD e cient. Therefore, havin decided on some values of l and d one should apply the reularization technique (minimization of standard deviation) to select a speci c portfolio which is SSD e cient. 5. Standard semideviation as risk measure In this section we analyze the meanrisk model with the risk de ned by the standard semideviation r iven by Eq. (2). Recall that the standard semideviation is the square root of the semivariance which equals to the doubled area of the downside dispersion space (Corollary 3). Hence, r is a well de ned eometrical characteristic in the O±R diaram. Consider two random variables X and Y in the common O±R diaram (Fi. 7). If X SSD Y, then, is bounded from above by Y, and, by Proposition 5, l X P l Y. Due to the convexity of X, the downside dispersion space of X is no reater than the downside dispersion space of Y plus the area of the trapezoid with the vertices: l Y ; 0, l X ; 0, l X ; X l X and l Y ; Y l Y. Formally, by the de nition of SSD, X 1 2 r2 X r2 Y 1 2 l X l Y d X d Y : 21 This inequality allows us to derive new necessary conditions for the consistency with SSD of the bicriteria meanrisk model usin standard semideviation as the risk measure. Fi. 7. SSD and the semivariances: X SSD Y ) 1 2 r2 X r2 Y 1 2 l X l Y d X d Y.
12 44 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Proposition 9. If X SSD Y, then l X P l Y and l X r X P l Y r Y, where the second inequality is strict whenever l X > l Y. Proof. If X SSD Y then, due to Proposition 5, l X P l Y. Moreover, inequality (21) is valid. From Proposition 4 we have r X P d X and r Y P d Y. Usin these inequalities in (21) we et r 2 X r2 Y 6 l X l Y r X r Y : Hence, r X r Y 6 l X l Y, and nally l X r X P l Y r Y. Moreover, from Proposition 4, r X ˆ d X and r Y ˆ d Y can occur only if r X ˆ r Y ˆ 0. Hence, X SSD Y and l X > l Y ) l X r X > l Y r Y ; which completes the proof. The messae of Proposition 9 is that the l=r meanrisk model is consistent with the second deree stochastic dominance by the rule (6) with a ˆ 1. Therefore, l=r comparisons lead to uaranteed results in the sense that l X kr X > l Y kr Y for some 0 < k 6 1 ) Y ² SSD X : For problems of choice amon risky alternatives in a iven feasible set, Corollary 1 results in the followin observation. Corollary 8. Except for random variables with identical mean and standard semideviation, every random variable X 2 Q that is maximal by l X kr X with 0 < k 6 1 is e cient under the SSD rules. The upper bound on the tradeo coe cient k in Corollary 8 cannot be increased for eneral distributions. For any e > 0 there exist random variables X SSD Y such that l X > l Y and l X 1 e r X ˆ l Y 1 e r Y. As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 1 e 2, PfX ˆ 1 ˆ 1 1 e 2 ; and Y ˆ 0. It follows from Corollary 8 that the optimal solution of the problem maxfl X k r X : X 2 Q; 0 < k is e cient under the SSD rules, if it is unique. In the case of nonunique optimal solutions, however, we only know that the optimal set of problem (22) contains a solution which is e cient under SSD rules. Thus, similar to the l= d model, the l=r model may enerate ties (Fi. 8) and the optimal set of problem (22) may contain also some SSD dominated solutions. However, two random variables that enerate a tie (are indi erent) in the l=r meanrisk model cannot be so much di erent as in the l= d model. Standard semideviation r X is an area measure of the downside dispersion space and therefore it takes into account all values of X for 6 l X. Note that, if two random variables X and Y enerate a tie in the l=r model, then l X ˆ l Y and Z l X Z l Y X f df ˆ Y f df: Fi. 8. A tie in the l=r model: l X ˆ l Y, r X ˆ r Y and X SSD Y.
13 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 45 Functions are continuous and nonneative. Hence, if X SSD Y enerate a l=r tie, then X ˆ Y for all 6 l X. Thus a tie in the l=r model may happen for X SSD Y but the second deree stochastic dominance X over Y is then related to overperformances rather than the underperformances. Summin up, the l=r model needs some additional reularization to resolve ties in comparisons, but it is not such a dramatic need as in the l= d model. Similar to the l= d model, ties in the l=r model can be resolved by additional comparisons of standard deviations or variances. In the case when l X kr X ˆ l Y kr Y, one may select from X and Y the one that has a smaller standard deviation. It can be formalized as the followin lexicoraphic comparison: l X kr X ; r X P lex l Y kr Y ; r Y () l X kr X > l Y kr Y or l X kr X ˆ l Y kr Y and r X P r Y : For problems of choice amon risky alternatives in a iven feasible set, the lexicoraphic maximization of l kr; r has two phases aain: maximization of l kr on the feasible set, and selection of the optimal solution that has the smallest standard deviation r, if many optimal solutions occur. Due to Eq. (18), such a selection results in SSD e cient solutions (even in the case of multiple optimal solutions). Corollary 9. Every random variable X 2 Q that is lexicoraphically maximal by l X kr X ; r X with 0 < k 6 1 is e cient under the SSD rules. The mean±semivariance optimization approach was proposed by Markowitz (1959). It is quite an intuitive modi cation of the mean±variance model, since an investor worries about underperformance rather than overperformance. Nevertheless, it is less used in portfolio optimization. One reason is that it is more di cult to compute the mean± semivariance e cient frontier that for the mean± variance model. Still, Markowitz et al. (1993) have developed a critical line alorithm for the mean± semivariance e cient frontier. The use of semivariance r 2 or standard semideviation r in the meanrisk analysis may be considered to be equivalent, with the former easier to implement. In fact, both de ne exactly the same e cient set, since standard deviation is nonneative and the square function is strictly increasin for nonneative aruments. However, our result that the l=r model with tradeo s bounded by one is consistent with the SSD rules cannot be directly applied to the mean±semivariance model. Note that X 2 Q that is maximal by l kr 2 may be not maximal by l kr X r, in eneral. Consider two random variables X and Y with l X ˆ 0, r X ˆ 1 and l Y ˆ 1, r Y ˆ 2, respectively. For k ˆ 0:4, l X 0:4r 2 X ˆ 0:4 > 0:6 ˆ l Y 0:4r 2 Y but l X 0:4r X ˆ 0:4 < 0:2 ˆ l Y 0:4r Y. While comparin two random variables X and Y by the mean±semivariance tradeo analysis the followin relationship is valid: l X kr 2 X P l Y kr 2 Y () l X k r X r Y Šr X P l Y k r X r Y Šr Y : Thus for problems where the l=r 2 e cient set is bounded (like typical portfolio selection problems), there exists an upper bound on the tradeo coe cients which uarantees that for smaller tradeo s the correspondin meanrisk e cient solutions are also e cient under the SSD rules. It explains the hih number of SSD e cient solutions included in the l=r 2 e cient set observed in experiments with reallife portfolio selection problems (Porter, 1974). The upper bound, thouh, may be very tiht. 6. Standard deviation as risk measure After the work of Markowitz (1952) the variance (or the standard deviation) is the most frequently used risk measure in meanrisk models for portfolio selection. The O±R diaram (Fi. 3) shows the variance as a natural area measure of dispersion. Comparison of random variables with equal means leads to uaranteed results (Proposition 6). However, for eneral random variables X and Y with unequal means, no relation involvin their standard deviations is known to be necessary
14 46 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Fi. 9. SSD and the variances: X SSD Y ) 1 2 r2 X r2 Y 1 2 l X l Y M X l X M Y l Y Š. for the second deree stochastic dominance of X over Y. Inp the case of symmetric distributions one has r ˆ 2 r, and r in Proposition 9 can be replaced p with standard deviation r multiplied by factor 2 =2. It turns out, however, that for symmetric distributions the relation between d and r can be described in more detail. Proposition 10. For symmetric random variables X and Y, X SSD Y ) l X P l Y and l X r X P l Y r Y : 23 Proof. If X SSD Y then, by Proposition 5, l X P l Y. Moreover, inequality (21) is valid. Lyapunov inequality (16) for symmetric variables yields r X P 2 d X and r Y P 2 d Y. Usin these inequalities in (21) we et r 2 X r2 Y 6 l X l Y r X r Y : Hence, r X r Y 6 l X l Y, and nally l X r X P l Y r Y which completes the proof. For problems of choice amon risky alternatives in a iven feasible set, Propositions 1 and 6 imply the followin result. Corollary 10. Within the class of symmetric random variables, every random variable X 2 Q that is maximal by l X kr X with 0 < k < 1, is e cient under the SSD rules. The bound on the tradeo coe cient in Corollary 10 is the best in the sense that there exist symmetric random variables X SSD Y such that l X r X ˆ l Y r Y. As an example one may consider two nite random variables: X de ned as PfX ˆ 0 ˆ 0:5, PfX ˆ 4 ˆ 0:5; and Y de ned as PfY ˆ 0 ˆ 0:5, PfY ˆ 2 ˆ 0:5. Therefore, the upper bound on the tradeo coe cient k in Corollary 10 cannot be increased. In the eneral case of nonsymmetric random variables, standard deviation is not a symmetric measure and there is no direct analoue of Proposition 9 for the standard deviation. Some similar, but much weaker, necessary conditions for the SSD dominance can be derived for distributions bounded from above (random variables with an upper bounded support). Note that, if X is upper bounded by a real number M X (i.e. PfX > M X ˆ 0), then M X P l X and for P M X, F X ˆ 1. Thus for P M X the function X coincides with its riht asymptote ( X ˆ l X ). Hence
15 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 47 Z lx Z M X r 2 X ˆ 2 X f df 2 X f f l X Š df: l X Consider two random variables X and Y such that PfX > M X ˆ 0 and PfY > M Y ˆ 0 in the common O±R diaram (Fi. 9). If X SSD Y, then, by the de nition of SSD, X is bounded from above by Y and, by Proposition 5, one has l X P l Y. Due to the convexity of X, the area between this function and its asymptotes cannot be reater than the area between Y and its asymptotes plus the area of the trapezoid de ned by the vertices: l Y ; 0, l X ; 0, M X ; M X l X and M Y ; M Y l Y. This is valid for M X > M Y (like in Fi. 9), as well as for M X 6 M Y. Formally, 1 2 r2 X r2 Y 1 2 l X l Y M X l X M Y l Y Š: 24 This inequality is similar but stroner than the inequality derived by Levy (1992), Theorem 9, p. 570, which reads: r 2 X r2 Y 6 l X l Y 2 maxfm X ; M Y l X l Y : Inequality (24) allows for the formulation of the followin necessary conditions for the bicriteria meanrisk model with the standard deviation used as the risk measure. Proposition 11. Suppose that a common upper bound h > 0 is known for X l X =r X and Y l Y =r Y. Then X SSD Y ) l X P l Y and l X 1 h r X P l Y 1 h r Y : Proof. If X SSD Y, then due to Proposition 5, l X P l Y. Further, note that inequality (24) can be rewritten as r 2 X r2 Y 6 h r X r Y l X l Y : This immediately yields the required result. Corollary 11. Within the class of random variables such that PfX > l X hr X ˆ 0, every random variable that is maximal by l X kr X with 0 < k < 1=h, is e cient under the SSD rules. Note that Proposition 11 is applicable to any pair of nite random variables. Similarly, Corollary 11 shows that in the case of a portfolio selection problem with nite random variables (for example de ned by historical data), there exists a positive bound on the tradeo coe cient for the standard deviation which uarantees that for smaller tradeo s the correspondin meanrisk e cient solutions are also e cient under the SSD rules. Analoously to the case of the semivariance discussed in the previous section, an upper bound on the variance tradeo coe cients exists which uarantees that l=r 2 e cient solutions are also e cient under the SSD rules. However, the upper bound may be very tiht. Corollary 11 provides a theoretical explanation for the results of numerous experimental comparisons of mean±variance and SSD e cient sets on reallife portfolio selection problems (Porter, 1974, and references therein). Most of them, like that performed by Porter and Gaumnitz (1972) on over 900 portfolios of securities randomly selected from the Chicao Price Relative File, provided some support for the idea that mean±variance and SSD choices are empirically similar. The main di erence was the tendency of the mean±variance e cient set to include some low mean, low variance portfolios that were eliminated by the SSD rules. Althouh e cient in the mean±variance analysis, they obviously correspond to lare tradeo coe cients for the variance. 7. Concludin remarks The second deree stochastic dominance relation is based on an axiomatic model of riskaverse preferences, but does not provide us with a simple computational recipe. The meanrisk approach quanti es the problem in only two criteria: the mean, representin the expected outcome, and the risk: a scalar measure of the variability of outcomes (usually, a central moment or the correspondin deviation). This is appealin to decision makers and allows a simple tradeo analysis, analytical or eometrical. In the paper we have analyzed the consistency of these two approaches. We have shown that standard semideviation (the square root of the
16 48 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 semivariance) as the risk measure makes the meanrisk model consistent with the second deree stochastic dominance, provided that the tradeo coe cient is bounded by one. Similar results have been obtained for the absolute semideviation as the risk measure. These results are valid for all (possibly nonsymmetric) random variables for which these moments are wellde ned. In the case of symmetric random variables the same results are valid for the standard and absolute deviations, respectively. In many applications, especially in portfolio selection problems, the meanrisk model is analyzed by the critical line alorithm. This is a technique for identifyin the meanrisk e cient frontier via parametric optimization with a varyin tradeo coe cient. Our results uarantee that when risk is measured by the standard or the absolute semideviation (the standard or the absolute deviation in the case of symmetric distributions), the part of the e cient frontier (in the meanrisk imae space) correspondin to tradeo coe cients smaller than one is also e cient under the SSD rules. In some way our analysis justi es the critical line methodoloy for typical risk measures, provided that it is not extended too far in terms of the tradeo coe cient. It also explains some results of experimental comparisons of the SSD and meanrisk e cient sets for portfolio selection problems. In the analysis we have used a new raphical tool, the O±R diaram, which appears to be particularly convenient for comparin uncertain outcomes and examinin second deree stochastic dominance. Typical dispersion statistics, commonly used as risk measures (absolute deviation and semideviation, variance and semivariance) are well depicted in the O±R diaram, and it may be useful for various types of comparisons of uncertain outcomes, especially in computerized decision support systems. Appendix A A.1. Proof of Proposition 2 P1. Simple consequence from de nition of X. P2. Simple consequence from de nition of X. P3. Chanin the order of interation by Fubini's theorem we obtain X ˆ P lim ˆ ˆ ˆ F X f df Z f n P X dn df df P X dn n P X dn ˆ PfX 6 Ef X jx 6 : X! 6 lim! ˆ lim! jnj P X dn ˆ 0; n P X dn because n 6 jnj for n 6 < 0, and EfjX j < 1. P5. X l X ˆ ˆ ˆ n P X dn l X P X dn n P X dn n P X dn ˆ PfX P EfX jx P : P6. Simple consequence from P5. P7.
17 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33± lim X l X Š!1 ˆ lim!1 Z1 6 lim!1 Z1 n P X dn n P X dn ˆ 0; similarly to P4. P8. If < 0, then X 0 X ˆ 0 F X n dn 6 0 supff X n j n < : Interchanin and 0 we obtain the second inequality. A.2. Proof of Proposition 3 By Fubini's theorem, X f df ˆ 2 Z f 4 f n P X dn 5 df f 6 n 6 f 3 Z Z ˆ f n P X dn df ˆ ˆ n 3 7 f n df5 P X dn n 2 P X dn ˆ 1 2 PfX 6 Ef X 2 jx 6 : Analoously, notin that f l X ˆ we obtain f n P X dn ; X f f l X Š df ˆ n f P X dn 5 df f 6 f f 6 n 3 Z Z ˆ n f P X dn df ˆ ˆ Z n 3 7 n f df5 P X dn n 2 P X dn ˆ 1 2 PfX P Ef X 2 jx P : References Bawa, V.S., Stochastic dominance: A research biblioraphy. Manaement Science 28, 698±712. Fishburn, P.C., Decision and Value Theory, Wiley, New York. Fishburn, P.C., Meanrisk analysis with risk associated with below taret returns. The American Economic Review 67, 116±126. Fishburn, P.C., Stochastic dominance and moments of distributions. Mathematics of Operations Research 5, 94± 100. Hanoch, G., Levy, H., The e ciency analysis of choices involvin risk. Review of Economic Studies 36, 335±346. Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities, Cambride University Press, Cambride, MA. Kendall, M.G., Stuart, A., The Advanced Theory of Statistics, vol. 1: Distribution Theory, Gri n, London. Konno, H., Yamazaki, H., Meanabsolute deviation portfolio optimization model and its application to Tokyo stock market. Manaement Science 37, 519±531. Levy, H., Stochastic dominance and expected utility: Survey and analysis. Manaement Science 38, 555±593. Markowitz, H.M., Portfolio selection. Journal of Finance 7, 77±91. Markowitz, H.M., Portfolio Selection. Wiley, New York. Markowitz, H.M., Mean±Variance Analysis in Portfolio Choice and Capital Markets. Blackwell, Oxford. Markowitz, H.M., Todd, P., Xu, G., Yamane, Y., Computation of mean±semivariance e cient sets by the critical line alorithm. Annals of Operations Research 45, 307±317.
18 50 W. Oryczak, A. Ruszczynski / European Journal of Operational Research 116 (1999) 33±50 Porter, R.B., Semivariance and stochastic dominance: A comparison. The American Economic Review 64, 200±204. Porter, R.B., Gaumnitz, J.E., Stochastic dominance versus mean±variance portfolio analysis. The American Economic Review 62, 438±446. Pratt, J.W., Rai a, H., Schlaifer, R., Introduction to Statistical Decision Theory, MIT Press, Cambride, MA. Rothschild, M., Stilitz, J.E., Increasin risk: I. A de nition. Journal of Economic Theory 2, 225±243. Whitmore, G.A, Findlay, M.C. (Eds.), Stochastic Dominance: An Approach to DecisionMakin Under Risk, Heath, Lexinton, MA. Yitzhaki, S., Stochastic dominance, mean variance, and Gini's mean di erence. The American Economic Review 72, 178±185. Zenios, S.A., Kan, P., Meanabsolute deviation portfolio optimization for mortaebacked securities. Annals of Operations Research 45, 433±450.
Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationPortfolio selection based on upper and lower exponential possibility distributions
European Journal of Operational Research 114 (1999) 115±126 Theory and Methodology Portfolio selection based on upper and lower exponential possibility distributions Hideo Tanaka *, Peijun Guo Department
More informationA Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu March 2009 Abstract This paper takes the
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationIntroduction. and set F = F ib(f) and G = F ib(g). If the total fiber T of the square is connected, then T >
A NEW ELLULAR VERSION OF LAKERSMASSEY WOJIEH HAHÓLSKI AND JÉRÔME SHERER Abstract. onsider a pushout diaram of spaces A, construct the homotopy pushout, and then the homotopy pullback of the diaram one
More informationExact Nonparametric Tests for Comparing Means  A Personal Summary
Exact Nonparametric Tests for Comparing Means  A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationAdvanced Microeconomics
Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationIntroduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by
Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes
More informationTarget Strategy: a practical application to ETFs and ETCs
Target Strategy: a practical application to ETFs and ETCs Abstract During the last 20 years, many asset/fund managers proposed different absolute return strategies to gain a positive return in any financial
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationAEROSOL STATISTICS LOGNORMAL DISTRIBUTIONS AND dn/dlogd p
Concentration (particle/cm3) [e5] Concentration (particle/cm3) [e5] AEROSOL STATISTICS LOGNORMAL DISTRIBUTIONS AND dn/dlod p APPLICATION NOTE PR001 Lonormal Distributions Standard statistics based on
More information14.451 Lecture Notes 10
14.451 Lecture Notes 1 Guido Lorenzoni Fall 29 1 Continuous time: nite horizon Time goes from to T. Instantaneous payo : f (t; x (t) ; y (t)) ; (the time dependence includes discounting), where x (t) 2
More informationMeasuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices  Not for Publication
Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices  Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationThe E ect of Trading Commissions on Analysts Forecast Bias
The E ect of Trading Commissions on Analysts Forecast Bias Anne Beyer and Ilan Guttman Stanford University September 2007 Abstract The paper models the interaction between a sellside analyst and a riskaverse
More informationPricing Cloud Computing: Inelasticity and Demand Discovery
Pricing Cloud Computing: Inelasticity and Demand Discovery June 7, 203 Abstract The recent growth of the cloud computing market has convinced many businesses and policy makers that cloudbased technologies
More information5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0
a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a
More informationChapter 3: The Multiple Linear Regression Model
Chapter 3: The Multiple Linear Regression Model Advanced Econometrics  HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics
More informationLecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model Recall that the price of an option is equal to
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationA Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu May 2010 Abstract This paper takes the
More informationEconS 503  Advanced Microeconomics II Handout on Cheap Talk
EconS 53  Advanced Microeconomics II Handout on Cheap Talk. Cheap talk with Stockbrokers (From Tadelis, Ch. 8, Exercise 8.) A stockbroker can give his client one of three recommendations regarding a certain
More information5. Convergence of sequences of random variables
5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,
More information1 Portfolio mean and variance
Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a oneperiod investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring
More informationUnraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets
Unraveling versus Unraveling: A Memo on Competitive Equilibriums and Trade in Insurance Markets Nathaniel Hendren January, 2014 Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that
More informationChapter 2 Portfolio Management and the Capital Asset Pricing Model
Chapter 2 Portfolio Management and the Capital Asset Pricing Model In this chapter, we explore the issue of risk management in a portfolio of assets. The main issue is how to balance a portfolio, that
More informationA Portfolio Model of Insurance Demand. April 2005. Kalamazoo, MI 49008 East Lansing, MI 48824
A Portfolio Model of Insurance Demand April 2005 Donald J. Meyer Jack Meyer Department of Economics Department of Economics Western Michigan University Michigan State University Kalamazoo, MI 49008 East
More informationChapter 3: Section 33 Solutions of Linear Programming Problems
Chapter 3: Section 33 Solutions of Linear Programming Problems D. S. Malik Creighton University, Omaha, NE D. S. Malik Creighton University, Omaha, NE Chapter () 3: Section 33 Solutions of Linear Programming
More informationOptimal investment and consumption models with nonlinear stock dynamics
Math Meth Oper Res (1999) 50: 271±296 999 Optimal investment and consumption models with nonlinear stock dynamics Thaleia Zariphopoulou* Department of Mathematics and School of Business, University of
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More information1. The Classical Linear Regression Model: The Bivariate Case
Business School, Brunel University MSc. EC5501/5509 Modelling Financial Decisions and Markets/Introduction to Quantitative Methods Prof. Menelaos Karanasos (Room SS69, Tel. 018956584) Lecture Notes 3 1.
More informationVoluntary Voting: Costs and Bene ts
Voluntary Voting: Costs and Bene ts Vijay Krishna y and John Morgan z November 7, 2008 Abstract We study strategic voting in a Condorcet type model in which voters have identical preferences but di erential
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationMidterm March 2015. (a) Consumer i s budget constraint is. c i 0 12 + b i c i H 12 (1 + r)b i c i L 12 (1 + r)b i ;
Masters in EconomicsUC3M Microeconomics II Midterm March 015 Exercise 1. In an economy that extends over two periods, today and tomorrow, there are two consumers, A and B; and a single perishable good,
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationOligopoly. Chapter 10. 10.1 Overview
Chapter 10 Oligopoly 10.1 Overview Oligopoly is the study of interactions between multiple rms. Because the actions of any one rm may depend on the actions of others, oligopoly is the rst topic which requires
More informationA goal programming approach for solving random interval linear programming problem
A goal programming approach for solving random interval linear programming problem Ziba ARJMANDZADEH, Mohammadreza SAFI 1 1,2 Department of Mathematics, Semnan University Email: z.arjmand@students.semnan.ac.ir,
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationDirected search and job rotation
MPRA Munich Personal RePEc Archive Directed search and job rotation Fei Li and Can Tian University of Pennsylvania 4. October 2011 Online at http://mpra.ub.unimuenchen.de/33875/ MPRA Paper No. 33875,
More informationNotes on Chapter 1, Section 2 Arithmetic and Divisibility
Notes on Chapter 1, Section 2 Arithmetic and Divisibility August 16, 2006 1 Arithmetic Properties of the Integers Recall that the set of integers is the set Z = f0; 1; 1; 2; 2; 3; 3; : : :g. The integers
More informationOptimal insurance contracts with adverse selection and comonotonic background risk
Optimal insurance contracts with adverse selection and comonotonic background risk Alary D. Bien F. TSE (LERNA) University Paris Dauphine Abstract In this note, we consider an adverse selection problem
More informationElectric Company Portfolio Optimization Under Interval Stochastic Dominance Constraints
4th International Symposium on Imprecise Probabilities and Their Applications, Pittsburgh, Pennsylvania, 2005 Electric Company Portfolio Optimization Under Interval Stochastic Dominance Constraints D.
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationIDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011
IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION
More informationBiinterpretability up to double jump in the degrees
Biinterpretability up to double jump in the degrees below 0 0 Richard A. Shore Department of Mathematics Cornell University Ithaca NY 14853 July 29, 2013 Abstract We prove that, for every z 0 0 with z
More informationComputer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR
Computer Handholders Investment Software Research Paper Series TAILORING ASSET ALLOCATION TO THE INDIVIDUAL INVESTOR David N. Nawrocki  Villanova University ABSTRACT Asset allocation has typically used
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationPortfolio Allocation and Asset Demand with MeanVariance Preferences
Portfolio Allocation and Asset Demand with MeanVariance Preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany.
More informationCorporate Income Taxation
Corporate Income Taxation We have stressed that tax incidence must be traced to people, since corporations cannot bear the burden of a tax. Why then tax corporations at all? There are several possible
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationPanel Data Econometrics
Panel Data Econometrics Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans University of Orléans January 2010 De nition A longitudinal, or panel, data set is
More informationNumerical Summarization of Data OPRE 6301
Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting
More information14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model
14.452 Economic Growth: Lectures 2 and 3: The Solow Growth Model Daron Acemoglu MIT November 1 and 3, 2011. Daron Acemoglu (MIT) Economic Growth Lectures 2 and 3 November 1 and 3, 2011. 1 / 96 Solow Growth
More informationLesson 5. Risky assets
Lesson 5. Risky assets Prof. Beatriz de Blas May 2006 5. Risky assets 2 Introduction How stock markets serve to allocate risk. Plan of the lesson: 8 >< >: 1. Risk and risk aversion 2. Portfolio risk 3.
More information1.2 Solving a System of Linear Equations
1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems  Basic De nitions As noticed above, the general form of a linear system of m equations in n variables
More informationFE670 Algorithmic Trading Strategies. Stevens Institute of Technology
FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 MeanVariance Analysis: Overview 2 Classical
More informationLab M3: The Physical Pendulum
M3.1 Lab M3: The Physical Pendulum Another pendulum lab? Not really. This lab introduces anular motion. For the ordinary pendulum, we use Newton's second law, F = ma to describe the motion. For the physical
More informationIntrinsic Geometry. 2 + g 22. du dt. 2 du. Thus, very short distances ds on the surface can be approximated by
Intrinsic Geometry The Fundamental Form of a Surface Properties of a curve or surface which depend on the coordinate space that curve or surface is embedded in are called extrinsic properties of the curve.
More information1 Capital Asset Pricing Model (CAPM)
Copyright c 2005 by Karl Sigman 1 Capital Asset Pricing Model (CAPM) We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available
More informationA NOTE ON SEMIVARIANCE
Mathematical Finance, Vol. 16, No. 1 (January 2006), 53 61 A NOTE ON SEMIVARIANCE HANQING JIN Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin,
More informationTheorem 5. The composition of any two symmetries in a point is a translation. More precisely, S B S A = T 2
Isometries. Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a mapping which preserves distances. The word metric is a synonym to
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino email: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationST 371 (IV): Discrete Random Variables
ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible
More information1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easytocheck
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationXBRL Generation, as easy as a database. Christelle BERNHARD  christelle.bernhard@addactis.com Consultant & Deputy Product Manager of ADDACTIS Pillar3
XBRL Generation, as easy as a database. Christelle BERNHARD  christelle.bernhard@addactis.com Consultant & Deputy Product Manaer of ADDACTIS Pillar3 Copyriht 2015 ADDACTIS Worldwide. All Rihts Reserved
More informationSolving Equations and Inequalities Graphically
4.4 Solvin Equations and Inequalities Graphicall 4.4 OBJECTIVES 1. Solve linear equations raphicall 2. Solve linear inequalities raphicall In Chapter 2, we solved linear equations and inequalities. In
More informationWater Quality and Environmental Treatment Facilities
Geum Soo Kim, Youn Jae Chan, David S. Kelleher1 Paper presented April 2009 and at the Teachin APPAMKDI Methods International (Seoul, June Seminar 1113, on 2009) Environmental Policy direct, twostae
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More information2. Meanvariance portfolio theory
2. Meanvariance portfolio theory (2.1) Markowitz s meanvariance formulation (2.2) Twofund theorem (2.3) Inclusion of the riskfree asset 1 2.1 Markowitz meanvariance formulation Suppose there are N
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationSimilarityBased Mistakes in Choice
INSTITUTO TECNOLOGICO AUTONOMO DE MEXICO CENTRO DE INVESTIGACIÓN ECONÓMICA Working Papers Series SimilarityBased Mistakes in Choice Fernando Payro Centro de Investigación Económica, ITAM Levent Ülkü Centro
More informationUsing Generalized Forecasts for Online Currency Conversion
Using Generalized Forecasts for Online Currency Conversion Kazuo Iwama and Kouki Yonezawa School of Informatics Kyoto University Kyoto 6068501, Japan {iwama,yonezawa}@kuis.kyotou.ac.jp Abstract. ElYaniv
More informationThe degree, size and chromatic index of a uniform hypergraph
The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a kuniform hypergraph in which no two edges share more than t common vertices, and let D denote the
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond marketmaker would deltahedge, we first need to specify how bonds behave. Suppose we try to model a zerocoupon
More informationCooperation with Network Monitoring
Cooperation with Network Monitoring Alexander Wolitzky Microsoft Research and Stanford University July 20 Abstract This paper studies the maximum level of cooperation that can be sustained in sequential
More informationRegret and Rejoicing Effects on Mixed Insurance *
Regret and Rejoicing Effects on Mixed Insurance * Yoichiro Fujii, Osaka Sangyo University Mahito Okura, Doshisha Women s College of Liberal Arts Yusuke Osaki, Osaka Sangyo University + Abstract This papers
More informationPortfolio Optimization within Mixture of Distributions
Portfolio Optimization within Mixture of Distributions Rania Hentati Kaffel, JeanLuc Prigent To cite this version: Rania Hentati Kaffel, JeanLuc Prigent. Portfolio Optimization within Mixture of Distributions.
More informationAn Introduction to Portfolio Theory
An Introduction to Portfolio Theory John Norstad jnorstad@northwestern.edu http://www.norstad.org April 10, 1999 Updated: November 3, 2011 Abstract We introduce the basic concepts of portfolio theory,
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationMULTIPLECRITERIA INTERACTIVE PROCEDURE IN PROJECT MANAGEMENT
UTIPECRITERIA INTERACTIVE PROCEDURE IN PROJECT ANAGEENT Tomasz Błaszczyk The Karol Adamiecki University of Economics ul. 1 aja 50, 40287 Katowice Email: blaszczyk@ae.katowice.pl aciej Nowak The Karol
More information1 Another method of estimation: least squares
1 Another method of estimation: least squares erm: estim.tex, Dec8, 009: 6 p.m. (draft  typos/writos likely exist) Corrections, comments, suggestions welcome. 1.1 Least squares in general Assume Y i
More informationDynamics and Stability of Political Systems
Dynamics and Stability of Political Systems Daron Acemoglu MIT February 11, 2009 Daron Acemoglu (MIT) Marshall Lectures 2 February 11, 2009 1 / 52 Introduction Motivation Towards a general framework? One
More informationCharting Income Inequality
Charting Inequality Resources for policy making Module 000 Charting Inequality Resources for policy making Charting Inequality by Lorenzo Giovanni Bellù, Agricultural Policy Support Service, Policy Assistance
More information10. FixedIncome Securities. Basic Concepts
0. FixedIncome Securities Fixedincome securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationTiered and Valuebased Health Care Networks
Tiered and Valuebased Health Care Networks Chingto Albert Ma Henry Y. Mak Department of Economics Department of Economics Boston Univeristy Indiana University Purdue University Indianapolis 270 Bay State
More informationChapter 2. Dynamic panel data models
Chapter 2. Dynamic panel data models Master of Science in Economics  University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans April 2010 Introduction De nition We now consider
More informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More information