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Univariate and multivariate measures of risk aversion and risk premiums with joint normal distribution… Li, Yuming 1987

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UNIVARIATE AND MULTIVARIATE MEASURES OF RISK AVERSION A N D RISK P R E M I U M S WITH JOINT N O R M A L DISTRIBUTION A N D A P P L I C A T I O N S IN P O R T F O L I O S E L E C T I O N M O D E L S by YUMING LI B . S c , Shanghai Jiao Tong University, 1983 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR THE DEGREE OF M A S T E R O F SCIENCE IN BUSINESS  ADMINISTRATION  in T H E F A C U L T Y OF G R A D U A T E STUDIES (The Faculty of Commerce and Business Administration) W e accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH C O L U M B I A September 1987 © Y u m i n g L i , 1987  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his thesis  and  scholarly  or for  her  of  COMMERCE  The University of British C o l u m b i a 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  DE-6(3/81)  October  1,  1987  I  I further  purposes  gain  the  shall  requirements  agree  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  that  the  be  Library  an  advanced  shall  permission for  granted  is  for  by  understood allowed  the  make  extensive  head  that  without  it  of  copying my  my or  written  Abstract  T h i s thesis gives the formal derivations of the so-called Rubinstein's measures of risk aversion and their multivariate generalizations.  T h e applications of these  measures i n portfolio selection models are also presented. A s s u m i n g that a decision maker's preferences can be represented by a unidimensional von N e u m a n n and Morgenstern utility function, we consider a m o d e l w i t h an uninsurable i n i t i a l random wealth and an insurable risk. Under the assumption that the two r a n d o m variables have a bivariate normal distribution, the second-order covariance operator is developed from S t e i n / R u b i n s t e i n first-order covariance operator and is used to derive Rubinstein's measures of risk aversion from the approximations of risk premiums. Rubinstein's measures of risk aversion are proved to be the appropriate generalizations of the A r r o w - P r a t t measures of risk aversion. In a portfolio selection model w i t h two risky investments having a bivariate norm a l distribution, we show that Rubinstein's measures of risk aversion can yield the desirable characterizations of risk aversion and wealth effects on the o p t i m a l portfolio. These properties of Rubinstein's measures of risk aversion are analogous to those of the A r r o w - P r a t t measures of risk aversion i n the portfolio selection model w i t h one riskless and one risky investment. In multi-dimensional decision problems, we assume that a decision maker's pref-  - iii -  erences can be represented by a multivariate utility function. F r o m the model w i t h an uninsurable i n i t i a l wealth vector and insurable risk vector having a joint normal distribution i n the wealth space, we derived the m a t r i x measures of risk aversion w h i c h are the multivariate extension of Rubinstein's measures of risk aversion. T h e derivations are based on the multivariate version of S t e i n / R u b i n s t e i n covariance operator developed by Gassmann and its second-order generalization to be developed i n this thesis. W e finally present an application of the m a t r i x measures of risk aversion i n a portfolio selection model w i t h a multivariate u t i l i t y function and two risky investments. In this model, if we assume that the random returns on the two investments and other r a n d o m variables have a joint normal distribution, the o p t i m a l portfolio can be characterized by the m a t r i x measures of risk aversion.  - iv -  Table of Contents  Abstract  ii  Table of Contents  iv  Acknowledgements  vi  § 1. Introduction  1  § 2. Rubinstein's Measures of Risk A v e r s i o n and R i s k Premiums 2.1 Review of the A r r o w - P r a t t Measures of R i s k Aversion  7  2.2 T h e M a t h e m a t i c a l Properties of T w o Covariance Operators  10  2.3 T h e Derivation of Rubinstein's Measure of Absolute R i s k Aversion  15  2.4 R i s k Aversion w i t h Quadratic U t i l i t y Function  24  2.5 Rubinstein's Measure of Relative R i s k Aversion  29  A p p e n d i x 2.1  32  A p p e n d i x 2.2  34  § 3. A n A p p l i c a t i o n of Rubinstein's Measures i n the Portfolio Theory  38  3.1 F o r m u l a t i o n of the Portfolio M o d e l  40  3.2 Necessary and Sufficient Conditions for Diversification  40  3.3 R i s k Aversion and the Choice between T w o R i s k y Investments  47  3.4 W e a l t h Effect on the Choice between T w o R i s k y Investments  51  - v -  3.5 Remarks § 4. M u l t i v a r i a t e Measures of R i s k Aversion and R i s k P r e m i u m s  55 56  4.1 Introduction to the M u l t i v a r i a t e Measures of R i s k Aversion  58  4.2 T h e M a t h e m e t i c a l Properties of T w o M u l t i v a r i a t e Covariance Operators  60  4.3 T h e Derivation of M a t r i x Measures of R i s k Aversion  64  4.4 Directional M a t r i x Measures of R i s k Aversion  70  A p p e n d i x 4.1  74  § 5. A n A p p l i c a t i o n of M a t r i x Measure of R i s k Aversion i n the Portfolio Theory  77  5.1 Formulation of the M o d e l  78  5.2 R i s k Aversion and the O p t i m a l Portfolio  79  References  82  - vi -  Acknowledgements  I am grateful to my supervisor, Professor W . T . Ziemba for his skilled direction throughout the preparation of this thesis. I would also like to thank the other committee members, Professor S. Brumelle and Professor G . Sick, for their suggestions for this thesis.  -1  -  § 1. I n t r o d u c t i o n  In an uncertain world, a decision maker's preferences for the various consequences of any action can be represented by a utility function. The von Neumann and Morgenstern utility function was derived on the basis of a set of axioms about the decision maker's preferences.  Different sets of axioms that imply the existence of utilities  with the property that expected utility is an appropriate guide for consistent decision making are presented by von Neumann and Morgenstern[51], Savage[47], Luce and Raiffa[2l], Pratt, Raiffa, and Schlaifer [38] and Fishburn[10]. The axioms of utility theory assume that for the possible consequences of any action, either objective probabilities exist, or that the subjective probabilities are known to a decision maker. Under the expected utility hypothesis, if an appropriate utility value is assigned to each consequence and the expected utility is calculated, then the best course of action is the alternative with the highest expected utility. The utility theory is the basis of many mathematical or stochastic models in economics and finance. Consider a portfolio selection problem with two investments. We assume that an investor's preference can be represented by a von Neumann and Morgenstern utility function. He has an initial wealth w to be allocated to the two possibly risky 0  investments with the objective of maximizing the expected utility of final wealth. One formulation of the problem is  max  U(\)  = E[u(w (\(X 0  -Y)  +  Y))]  where u is the investor's utility function, X and Y are (one plus) the returns on the two investments, X and 1 - A are the proportions of the initial wealth to be allocated  - 2-  to the two investments, and E[-] represents the mathematical expectation of a random outcome. In the portfolio selection problem, an interesting question is to determine the relationship between the investor's preference towards risk and his optimal portfolio. Pratt[37] and Arrow[4] developed two measures which attempt to characterize these attitudes: One is the absolute risk aversion measure  r ( i ) = -u"(x)/u'(x),  (1.1)  and the other is the relative risk aversion measure  r*(x) = -xu'\x)/u'{x).  (1.2)  Under the assumption that all of the returns on the different investments are jointly normally distributed or the investor's preference can be represented by a quadratic utility function, Rubinstein[41] employed the alternative measure of risk aversion:  R(W) = -E[u"{W)]/E[u\W)]  (1.3)  where If is a random variable representing an investor's final wealth. The other version of Rubinstein's measure[23] is  R*(w , Z) = -w E[u"{w Z)]/E[u'{w Z)} 0  0  0  Q  (1.4)  where w is the investor's initial wealth and Z is (one plus) the return on the investor's 0  portfolio. If one of the two investments is riskless, the Arrow-Pratt measure of relative (absolute) measure of risk aversion is shown to give the desirable characterization of  - 3 -  risk aversion and wealth effect on the proportional (absolute) allocation of investments in the portfolio[37],[4],[7].  But if both of the investments are risky, as shown by  Cass and Stiglitz[7], Ross[40], and K i r a and Ziemba[30], the Arrow-Pratt measures of risk aversion cannot yield an unambiguous relationship between risk preferences and portfolio allocations without any additional conditions. However, under the assumption of joint normality of returns on different investments, it is shown by Kallberg and Ziemba that the investors with the same Rubinstein's measure of risk aversion (as given by 1.4) have the same optimal portfolio. The different powers of alternative risk aversion measures in determining the risk preferences in portfolio selection problems with different assumptions give rise to the following question: Why are the Arrow-Pratt measures of risk aversion useful only in the portfolio selection problem with one riskless and one risky investments, whereas Rubinstein's measures of risk aversion are powerful in the portfolio selection problem with many risky investments having joint normally distributed returns ? As we know, the Arrow-Pratt measures of risk aversion were derived from the context with only one random outcome[37]. Specifically, let  ir(x,Y)  be the risk pre-  mium a decision maker with non-random initial wealth x is willing to pay to avoid a gamble Y, which satisfies E[u{x + Y)] = u{x + E[Y] - ir(x, Y)).  (1.5)  Then the Arrow-Pratt measure of absolute risk aversion r(x) could be derived from the approximation: ir(x,Y)  where the approximation is of order  «  o(a ). Y  ^a r(x) Y  The corresponding Arrow-Pratt measure  - 4 -  of relative risk aversion was derived in the same fashion. Thus the answer to the first half of the above question maybe comes from the way in which Arrow-Pratt measures of risk aversion were derived. Then the remaining question becomes whether Rubunstein's measures of risk aversion could be derived from the risk premiums in the appropriate contexts. This idea is the motivation of the thesis. One objective of this thesis is to present a formal derivation of Rubinstein's measures of risk aversion and furthur investigate the power of these measures in determining the characteristics of the investor's optimal portfolio. The results in the thesis show that Rubinstein's measures of risk aversion can be justified by the corresponding risk premiums and can be used to yield desirable results for risk aversion and wealth effects on the portfolio with two risky investments having a bivariate normal distribution. Thus they are the appropriate extension of the Arrow-Pratt measures of risk aversion to the situation with two or more jointly normally distributed sources of risks. The other objective of this thesis is to give a multivariate extension of Rubinstein's measure of risk aversion from the corresponding multivariate risk premiums with jointly normally distributed risks. This multivariate extension of risk aversion measures involves a set of directional risk aversion matrices and a combined index of the matrix measures. The matrix measure of risk aversion is applied to a portfolio selection model with multivariate utility function. The outline of this thesis is as follows: in Chapter 2, after a brief review of concepts and properties of Arrow-Pratt measures of risk aversion, we derive the Rubinstein's measures of risk aversion from the approximation of risk premiums with an  -5 insurable and an uninsurable risk, assuming that either the two risks have a bivariate normal distribution or the utility function is quadratic. In Chapter 3, we present the results about the power of Rubinstein's measures of risk aversion in determining the risk aversion and wealth effect on optimal portfolio with two investments having a bivariate normally distributed returns. In Chapter 4, we give a multivariate extension of Rubinstein's measure of risk aversion from the multivariate risk premiums with jointly normally distributed insurable and uninsurable risks. Finally, in Chapter 5, we present an application of the multivariate measures of risk aversion in a portfolio selection model.  J - 6§ 2. T h e R u b i n s t e i n ' s M e a s u r e s of R i s k A v e r s i o n a n d R i s k  Premiums  This chapter is concerned with the study of Rubinstein's measures of risk aversion. If the initial wealth is non-random, the Arrow-Pratt measures of risk aversion have been justified in terms of risk premiums and in the sense of comparative risk aversions and risk premiums[4] [37] . However, if the initial wealth is random, these justifications for the Arrow-Pratt measures of risk aversion no longer hold[40]. In this chapter, we assume that either all random variables are jointly normally distributed or the utility functions are quadratic. Under either of these assumptions, Rubinstein's measures of risk aversion are derived from the risk premiums and are proved to have the corresponding properties of the Arrow-Pratt measures of risk aversion. Thus Rubinstein's measures of risk aversion can be viewed as the appropriate extension of the Arrow-Pratt measures of risk aversion in the random wealth situation when the mean-variance criterion is valid. This chapter is outlined as follows: In section 2.1, we review the basic concepts and properties of the Arrow-Pratt measures of risk aversion. In section 2.2, we present Stein/Rubibstein covariance operator and develop its second-order generalization. In section 2.3, we derive Rubinstein's measure of absolute risk aversion from the risk premium in the context of bivariate normally distributed random wealth and risk. In Section 2.4, we extend the result in 2.3 to the case in which the utility functions are restricted to the quadratic. In Section 2.5, we derive Rubinstein's measure of relative risk aversion under the assumptions made in either Section 2.3 or Section 2.4.  - 7 2.1 Review of Arrow-Pratt Measures of Risk Aversion  In this section, we give a brief review of the basic concepts of the Arrow-Pratt measures of risk aversion, which is the basis of the analyses in the rest of the thesis. These concepts and results can be found in Pratt[37] or Keeney and Raiffa[26]. Suppose that a decision maker's preferences for any particular set of random outcomes can be represented by a von Neumann and Morgenstern utility function. That is, we assume that there exists a mapping («) from the wealth level x into R  1  such that u(x) is the utility for wealth, u is an increasing function. We restrict our analysis to the risk averse utility function, which is defined by D E F I N I T I O N 2.1.1 A decision maker with utility function u is risk averse if  u(E[x + Y])> E[u(x + Y)]  (2.1.1)  for all nondegenerate gambles Y, where x is any initial wealth level. This definition is equivalent to D E F I N I T I O N 2.1.1' A decision maker is risk averse if his utility function u is concave, that is u" < 0. Arrow-Pratt Measure of Absolute Risk Aversion The Arrow-Pratt measure of absolute risk aversion is derived from the following concept: D E F I N I T I O N 2.1.2 The risk premium ir is the amount a decision maker with initial wealth x is willing to pay to avoid a gamble Y. That is,  E[u(x + Y)] = u(x + E[Y] — 7r).  (2.1.2)  Theorem  2.1.1.  A decision maker is risk averse if and only if the risk premium is  positive for all random gambles. Let  r(x) = —u"(x)/u'(x),  Theorem  2.1.2.  then we have  The risk premium of a decision maker with initial wealth  x  for a  fair gamble Y (E[Y] = o) can be approximated by 7r(ar,y) «  where the approximation is of order  ^<T r(x)  (2.1.3)  Y  o(<Xy).  This theorem leads to the following DEFINITION  2.1.3 A decision maker's Arrow-Pratt measure of absolute risk  aversion is  This measure of risk aversion has the following properties: Theorem  2.1.3.  If r is positive for all x, then  u  is concave and the decision maker  is risk averse. Theorem  2.1.4.  If ri(x)  > r (x) 2  for all a;, then  TTI(X,  Y) > ir (x,Y) for all x and Y. 2  Thus the Arrow-Pratt measure of absolute risk aversion gives the same ordering of risk preferences as the corresponding risk premiums. One important concept about this measure of risk aversion is D E F I N I T I O N 2.1.4 A decision maker's Arrow-Pratt measure of risk aversion is decreasing if  dr(x) ^ ^  constant if  dr(x) _ ^  ax  dx  increasing if  dv(x)  > 0.  - 9 -  Arrow-Pratt Measure of Relative Risk Aversion The Arrow-Pratt measure of relative risk aversion was derived from the concept of relative risk premium. D E F I N I T I O N 2.1.2' The relative risk premium TT* is the amount a decision maker with wealth x would be willing to pay as a proportion of his wealth to avoid a risk xY, That is, E[u(x  Let  r*(x)  Theorem  = -xu"(x)/u'(x),  2.1.2'.  + xY)]  = u(x + E[xY]  - xir*)  (2.1.2')  we have  The relative risk premium of a decision maker with wealth  x  for a  fair gamble xY (E[Y] = 0) can be approximated by ir*(x,Y)  where the approximation is of order  « ^a$r*(x)  (2.1.3')  o(oy)-  This theorem leads to D E F I N I T I O N 2.1.3' A decision maker's Arrow-Pratt measure of relative risk aversion is * ^  r  - ^ ¥ -  =  (2.1-4')  One important concept about this relative risk aversion is D E F I N I T I O N 2.1 A'' A decision maker's Arrow-Pratt measure of relative risk aversion is decreasing if  dr (x) ^ ^  ,  — r - ^ - = 0,  , .  constant i l r  increasing if  dx  dr*(x) 1 dx  dr*(x)  ^  > 0.  - 10 2.2 T h e M a t h e m a t i c a l P r o p e r t i e s o f T w o C o v a r i a n c e O p e r a t o r s  The so-called "Covariance Operator" developed by Stein[48] and Rubinstein[4l] seperately has been one of the key tools in the analysis in the following sections. In the rest of this thesis, this operator will be called "First-Order Covariance Operator", which is stated as follows: Lemma 2.1 (First  — Order Covariance  Operator).  Suppose that X , Y have a bivariate normal distribution and g : R —• R is a 1  differentiable function in R . l  1  If E[g'(Y)] exists and  where f(y) is the density of Y , given by  then  Cov(X, g(Y)) = (Cov(X, Y))E[g'(Y)].  Proof  :  (2.2.1)  See Stein[48], or Rubinstein[41].  We now develop a new formula, which will be called "Second-Order Covariance Operator". This operator will also be an important tool in the analyses in the rest of this thesis. Lemma 2.2 (Second — Order Covariance  Operator).  Suppose that X,Y have a bivariate normal distribution and g : R -+ R is twice 1  differentiable. If E[g'(Y)],E[g"(Y)]  exist and l i m (y - u )g(y)f(y) v  y—>-±oo  = 0,  1  -11 -  v  Jim ff'(y)/(y) = o ao  where f(y) is the density of Y , given by  then Cov((X - u )\g{Y))  = (Cov(X, Y)) E[g"(Y)).  (2.2.2)  2  x  Proof :  Cov((X - uxf, 9(Y)) = E[(X - u ) g(Y)] - E[(X -  u ) ]E[g(Y)]  2  2  x  =  x  E[(X- ) g(Y)]-a E[g(Y)}, 2  2  fix  x  Note that the conditional distribution of X given Y is normal with mean  VX\Y =  fiX  + PXY —  (Y  (Ty  - HY),  and variance A  \\Y = C - PXYWX1  From the conditional mean, Px — PX\Y ~ PXY—(Y - p, ).  O~Y  Y  Substituting it into E[(X - fi ) g(Y)], we get 2  X  Cov({X-u )\g{YJ) x  E[g(Y)((X  - U \Y) X  + PXY—(Y  - p )f] Y  - <r E[g(Y)] 2  X  0~Y E[g(Y)(X - u ) } + 2p ^-E[(X  - u \ ){{Y - u )g{Y))]  2  xlY  XY  + P ^E[g(Y)(Y E[g(Y)]E[(X - u ) ] XY  2  xlY  X  Y  - u ) } - a E[g(Y)) +2 -±E[X - p \ ]E[{Y a 2  2  Y  X  P X Y  x Y  Y  „2  <4  + PXY -±E[g(Y)(Y  Y  - u ) ] - <r E[g(Y)}. 2  Y  x  - u )g{Y)} Y  - 12 -  Note that in the first and second terms above, the residual variable (X — UX\Y) is independent of Y and has mean zero and variance  x\Y — (! - PXY^X-  a  Substituting the mean and variance of the residual variable into the first and second terms, and using the result in Appendix 2.1 that  E[(Y - tiy) g(Y)}  = a E[g(Y)]  2  Y  +  a E[g"(Y)} Y  to substitute the third term, yields 2  Cov((X  - u ) \ g(Y)) = E[g(Y)](a x  2 xlY  - a\) + p \ ^-E[g(Y)(Y a  -  Y  uyf]  Y  = - \y^xE[g{Y)]  + PxY^lrB^Y)] y  P  +  cr E[g"(Y)]) Y  = P XY°WYE[ "{Y)] 2  9  =  (Cov(X,Y)) E[g"(Y)}. 2  Q.E.D.  Since the first-order and second-order covariance operators will be applied to u' and u" where u is a utility function, the following definition is given to ease the representations of several results. DEFINITION  2.2.1:  Let  u(x)  be a utility function.  regularity conditions , if u^\x) exist everywhere for  u(x)  i = 1,2,3,  is said to satisfy the  and  lim u'(x)f(x) = 0,  (2.2.3)  lim xu"(x)f(x) = 0,  (2.2.4)  lim u"'(x)f(x) = 0  (2.2.5)  X—•±oo  - 13 -  where f(x) is the density of a normal random variable X, that is  /(*) =  and u(x) is said to satisfy the regularity condition if only (2.2.3) holds. Note that (2.2.4) implies that  lim u"(x)f(x) =  lim -(xu"(x)f(x))  = 0.  (2.2.6)  From (2.2.3) and (2.2.6), u'(x) and u"(x) satisfy the condition for the first-order covariance operator. Note also that (2.2.4) and (2.2.6) imply that  lim (x — ux)u"(x)f(x) = 0.  (2.2.7)  —•dboo  From( 2.2.7) and (2.2.5), u"(x) satisfies the condition for the second-order covariance operator. Conversely, if u'(x) satisfies the condition for the first-order covariance operator, and u"(x) satisfies the conditions for the first-order and the second-order covariance operators, then (2.2.3)- (2.2.5) hold. Thus we have proved the following lemma: Lemma 2.3 : Let u(x) be continuously differentiable of order 4, then u(x) satisfies the regularity conditions of order 3 if and only if u'(x) satisfies the condition for firstorder covariance operator and u"(x) satisfies the conditions for the first-order and the second-order covariance operators.  - 14 The  n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n s f o r u(x) t o s a t i s f y t h e r e g u l a r i t y c o n d i t i o n s  are  w h e r e B = l/(2a ) x  lim  u\x)/e '  lim  xu"(x)/e *' = 0,  Mm  u"\x)/e ' = 0  not  (2.2.3)' (2.2.4)'  B  B  (2.2.5)'  3  s i n c e (2.2.3)' - (2.2.5)' a r e e q u i v a l e n t t o (2.2.3) - (2.2.5),  u(x)  T h u s the regularity conditions for do  = 0,  Bt  a p p r o a c h i n f i n i t y as f a s t as e *  w i l l h o l d as l o n g as  respectively.  u'(x), xu"(x)  u"'(x)  Therefore,  the  r e g u l a r i t y c o n d i t i o n s are w e a k i n the sense t h a t m o s t c o m m o n u t i l i t y functions  can  satisfy them.  5  2  d o e s as x t e n d s t o i n f i n i t y .  and  - 15 2.3 T h e D e r i v a t i o n o f R u b i n s t e i n ' s M e a s u r e o f A b s o l u t e R i s k  Aversion  In this section, we formally derive Rubinstein's measure of absolute risk aversion from the concept of risk premium in the two-risk situation. Suppose that a decision maker's preference can be represented by a utility function. He has a random initial wealth X and faces some other risk Y. He can insure against risk Y , but not X. Then the risk premium is the amount he is willing to pay to eliminate the insurable risk Y. D E F I N I T I O N 2.3.1 Let X , Y be bivariate random variables, and u(x) be an increasing utility function. The risk premium TT(X, Y) for a decision maker with utility function u(x) is the amount satisfying  E[u(X  + Y)] = E[{u{X +Y-  TT(X, Y)).  (2.3.1)  where Y - E[Y]. The risk premium  TT(X,Y)  can also be interpreted as follows: a decision maker  with random wealth X would be indifferent between receiving a risk Y and receiving a non-random amount E[Y] - Tr(X,Y). In the case that a random wealth Y is added to a non-random initial wealth x, a decision maker is risk averse if and only if  ir(x,Y) > 0.  However, if random wealth  Y  is added to a random initial wealth X, since Y may be negatively correlated with X and then have a hedging effect on the initial random variable, the corresponding risk premium ir(X, Y) may be negative. But if X and Y are bivariate normal random variables, the mean-variance criterion[17] is valid. In this case, we have the following : Theorem 2.3.1 : Suppose that X, Y have a bivariate normal distribution, and TT(X, Y)  - 16 -  is the risk premium for a decision maker with utility function u(x). If the decision maker is risk averse, that is, u" < 0, then (0  ir(X, Y) > 0  (ii)  w(X, Y) = 0  (iii)  x(X, Y)<0  if if if  Var(X + Y)> Var(X); Var(X + Y) = Var(X); Var(X + Y)< Var(X).  Proof : For simplicity, we assume that E[Y] = 0. We only prove (i), because the proofs of (ii) and (iii) can be done similarly. If TT(X,Y) < 0, then E[X + Y]<E[X  -ir].  Suppose that Var(X+ Y)> Var(X),  so that Var(X + Y)> Var(X - TT).  B y Hanoch and Levy[17](Theorem 4), E[u(X + Y)] < E[u(X -  TT)]  for all u" < 0. But this is in contradiction with the definition of ir(X, Y) in (2.3.1). Q.E.D.  This result shows that the sign of the risk premium for random wealth depends only on the difference between the variance of the initial random wealth accompanied by the additional random variation and the variance of the initial wealth alone. If the additional random wealth Y were to increase the total variance, a risk averse individual  - 17 -  with random wealth X plus a random variation Y would be indifferent between the status quo and a reduction in wealth by the amount of risk premium coupled with an elimination of the wealth variation due to Y ; conversely, if the additional random wealth Y were to reduce the total variance, a risk averse individual would have to be compensated by an amount equal to the hedging premium, that is, the money value of the risk premium when the additional wealth were to be withdrawn. Although the sign of the risk premium can be determined by the comparison of variances, its magnitude would be expected to relate to the appropriate measure of a decision maker's risk aversion. Now we derive an approximate expression for the risk premium. We still assume that the initial wealth and risk have a bivariate normal distribution and the utility function is well- behaved. For simplicity, we also assume that the risk is a zero mean random variable. Using the mathematical properties of two covariance operators developed in the last section yields the following Theorem  2.3.2.  Let X , Y have a bivariate normal distribution with  E[Y] = 0 ,  and  u(x) be an increasing utility function. Assume that u(x) is continuously differentiable of order  4,  and satisfies the regularity conditions, and  E[u^\X)},  exist for i =  1,2,3,4.  Then the risk premium TT(X, Y) for the given X can be approximated by 7r(X,Y)«-i(<4 -c4) +y  where the approximation is of the order O(<T  x+y  E[u"(X)]  (2.3.2)  E[u'(X)}  - o- ).  And in particular, for exponential utility function  x  u(x) =  -exp(-Px),  E[u"{X)]  *(x,y) =  -i(<4 y-<4) E[u'(X)} +  •  (2.3.3)  - 18 -  Proof  :  Expanding  u(X - ir) E[u(X  around  X,  and taking expectations, we get  - Tr)] = E[u(X)}  - irE[u'(X)]  + o(ir);  Similarly, expanding u(X + Y) around X, and taking expectations, we get E[u{X  + Y)} = E[u(X)}  +  +  E[Yu'(X)]  -E[Y u"(X)]  1  2  +  o(cr ). Y  Setting the R H S of the above two equations as equal, and solving for ir, we have,  „,XV\~ <  x  '  Y  E[Yu'(X)}  +  (l/2)E[Y u"(X)} 2  W(xj]  )  •  where the approximation is of the order o(a ). Y  Since for any random variables A and B ,  E[AB]  and  - Cov(A,B)  +  E[A]E[B],  E[Y] = 0,  (XY)*  {  '  Cov(Y,u'(X)) >~  lCov(Y ,u"(X)) 2  E[u'(X)] 1  E[u'(X)]  E[u"(X)}  2  2  2  E[u'(X)]  Y  '  = 11 + 12 + 13.  (2.3.4)  Since X , Y have a bivariate normal distribution and u satisfies the regularity conditions, by the First-order Covariance Operator,  Cov(Y,u'(X))  =  Cov(X,Y)E[u"(X)].  Let E[u"(X)} R  {  X  )  -  E[u'(X)}'  ( 2  - 3  5 )  then Cov(X,Y)R(X).  71 =  Let E[uW{X)} E[u"{X)] '  by the Second-order Covariance Operator, i(c (x,y)) g[« (x)] 2  4  OT  2  2  ( C o ( X , y ) j { E[U>(X)]H E[u»(X)}> V  = (c (x,y)) i2(A:) 2  y  OT  = yPxy°X y- Wc r  R  And 13 = i<r i2(X). y  Substituting (2.3.6a)-(2.3.6c) into (2.3.4) yields T T ( X , y ) = -(2Cov{X, Y) + <x )R(X + Y) + (o(a ) +  ^p\ c  l  Y  Y  + ^) + ° K + y - <r )  = j(4+y -  2  x  where the approximation is of order o(a  rr  -I  0 i m  Y  x+Y  - a ) because as a -+ x  Y  - rt"  2  2  ( r ) fpxy<4<4 g  +  S p x y c T A ' f y + Cy  _ f lim 2 ^ 1 = 0, ~ 1l  i  m  ° ^ ) + T ^ W  (In lim, cry -+ 0 is omitted). Substituting from (2.3.5) yields (2.3.2).  if =  0  ,  PXY = 0;  if p y ^ 0 . X  - 20 -  For u(x) = -exp(-Px),  direct calculation (see Freund[12]) yields  E[u(X+Y)] = -exp[-(/3(X) - i / j V ^ y ) ] , E[u(X - TT)] = - p[-(0(X+ ex  Y - TT) - \f<r\)\.  Set E[u(X + Y)] = E[u(X-ir)},  then  Therefore,  Note that u'(x) = j3exp(—j3x), u"(x) - -/3 exp(-l3x), 2  thus E[u"(X)} E[-fex (-p(X))] P  E[pexp(-/3(X))] = 0.  (2.3.7)  Q.E.D.  This theorem suggests the following D E F I N I T I O N 2.3.2. Let X be a random variable and u(x) be an increasing utility function. Assume that u(x) is twice differentiable and E[u'(X)], and E[u"(X)] exist. Rubinstein's measure of absolute risk aversion of utility function u(x) over X is defined as R  {  X  )  - ~ E[u'{X)Y  -'  (2 3 8)  -21 -  Thus the risk aversion R(X) can then be interpreted as twice the risk premium a decision maker requires per unit of incremental variance for infinitesimal risk. Two interesting special cases are as follows: (1) If X is non-random, that is, X is degenerated to any particular constant x, then (1.2.10) becomes *(x,Y) « ^cr r(x) Y  where the approximation is of order o{o~ ). Y  Since this result is identical to that in Theorem 2.1.2, Rubinstein's measure of risk aversion is a generalization of the Arrow-Pratt measure. (2) If X and Y are independent, (2.3.2) becomes  it{X,Y)K -a R{X) l  Y  where the approximation is of order a . Y  This result is the case of risk aversion measure with independent wealth and risk obtained by Kihlstrom, Romer and Williams [29] and Ambarish and Kallberg[3]. For given bivariate normal random variables X and Y , a decision maker with larger Rubinstein's measure of risk aversion than another would require a higher risk premium in the absolute sense if the incremental variance is small. But if two individuals have constant risk aversion then this property holds for any bivariate normal random variables X and Y . Corollary 2.3.2 : Let X , Y be bivariate normal random variables and U{(x) = -e~  PiX  be the exponential utility functions for i =  1,2.  Let Ri(X) and  7r,(X,  Y) be the risk  aversion measures and risk premiums for utility function Ui(x) (i — 1,2), respectively.  - 22 -  If Ri(X) > R (X) > 0, 2  then h n ( X , Y ) | > \ir (X,Y)\. 2  Or, (0  ir (X,Y)>ir (X,Y)>0 1  2  '£  (ii)  TT (X,Y) = IT (X,Y) = 0  11 (T  (in)  *i(X,Y) < TT (X,Y) < 0  11 A  2  1  2  2 2 X+Y= X'> <T  X+Y < ^X-  Proof : This result follows immediately from Theorem 2.3.2. <?.£.£>.  As presented in Section 2.1, , a decision maker is risk averse if and only if r(x) > 0 for all x. If r(x) = p > 0, then u(x) = -e~P . Thus, from (2.3.7), R(X) = p > 0 x  for any random variable X . The more interesting case is that under some regularity conditions on utility function, r(x) > 0 for any x if and only if R(X) > 0 for any normally distributed random variable X . Theorem 2.3.3 Let u(x) be an increasing utility function. Assume that u(x) is twice continuously differentiable and  E[u'(X)]  and  E[u"(X)]  exist. If  l i m ^ i o o u'(x) <  oo, then  a decision maker with utility function u(x) is risk averse, that is, r(x) > 0 for all x, if and only if R(X) > 0 for any normally distributed random variable X . Proof : R(X) > 0 for any normal random variable implies that R(X) > 0 for any degenerated normal random variable X = x. But in this degenerated case, R(X) = r(x). Thus the sufficient condition is proved. The necessary condition is true since u is  -23 i n c r e a s i n g , t h u s r(x) > 0 f o r a n y x i m p l i e s t h e c o n c a v i t y t h e d e f i n i t i o n o f R(X),  R(X)  > 0 forany random  o f u, t h a t i s u" < 0. H e n c e , b y  variable.  For a m o r e delicate proof w i t h o u t t h e use of degenerated n o r m a l r a n d o m variable,  see A p p e n d i x  2.2.  Q.E.D.  2.4 R i s k A v e r s i o n w i t h Q u a d r a t i c U t i l i t y F u n c t i o n s  In this section, we study Rubinstein's measure of risk aversion under the assumption that all the decision maker's utility functions are quadratic. The results in this section are comparable to those in Section 2.3. Since the mean-variance criterion is also valid for the class of quadratic utility functions, the following result holds: Theorem 2.4.1 Let u(x) = x - fix be a quadratic utility function on (-co, jp) where 2  j3 > 0, then for any random variables X and Y on ( - 0 0 ,  the following properties  hold: (»')  v(X, Y) > 0  (ii)  ir(X, Y) = 0  (iii)  n(X, Y) < 0  if if if  Var(X + Y) > Var(X); Var(X + Y)= Var(X); Var(X + Y)< Var(X).  Since this result is a part of Corollary 2.4.2, the proof is omitted. The following result analogous to Theorem 2.3.2 also holds: Theorem 2.4.2 Let X , Y be any bivariate random variables with E[Y] — 0 and u(x) be a quadratic utility function on  (-00,1/2/?)  given by u(x) = x-px where /? > 2  0.  Let R(X)  and n(X,Y) be the risk aversion measure and risk premium of utility function u(x). Assume that X and Y are in the suitable range such that R(X) > 0, and R(X - TT) > 0, then n(X, Y) = R~\X) ((1 + R (X)(a 2  x+Y  - a )) * - l ) , 2  x  or approximately, *{X, Y) » \{o-  2 x+Y  - a\)R(X).  (2.4.1)  -25 where the approximation is of the order o(a  - <? ).  x+Y  x  Proof : u(x) = x — fix , 2  u'(x) = 1 - 2(3x, u"{x) = -2(3. SO  RiX\-  E [ U  "  { X ) ]  2/3  (2.4.2)  1 - 2f3(X) B y the definition of u(x), E[u(X + Y)] = £ [ ( X + Y) - (3(X + Y) } 2  = E[X + Y] - f3E[(X + Y) } 2  = (X)-/3(a  +  x+Y  E[u(X - ir)] = E[(X -  (X) ), 2  /3(X - IT) ] 2  7r) -  = (X-ir)-P(<r  (X-ir) ).  +  x  2  B y the definition of ir(X, Y),  E[u{X + Y)] =  E[u(X-ir)],  so (X)-/3(a  +  2 x+Y  = (X-Tr)-/3(a  2 x  (T) ) 2  +  (X-ir) ). 2  After some rearrangement, we get -2/37T - 2(1 - 2(3{XJ)IT + 2fi(a 2  2  Note that E[uXX)]=l-2f3(X),  E[u"{X)] = -2B,  x+Y  - a ) = 0. 2  x  - 26 -  so E[u"(X)ir  - 2E[u'(X]ir - E[u"(X)(a  2  - <r ) = 0.  2  (2.4.3)  2  x+Y  x  Since R(X) > 0 and 0 > 0, E[u'(X)] > 0,  dividing (2.4.3) by -E[u'(X)] yields R(X)TT + 2TT - R(X)(a 2  - a ) = 0.  2  (2.4.4)  2  x+Y  x  Solving this quadratic equation, we get - a )) * - l ) .  n(X, Y) = R~\X) (±{l + R (X)(a 2  (2.4.5)  2  x+Y  x  Since '  y  2/3 _ 1 - 2(3(X) -  \- 7T  2/?  = R- (X) + ir 1  >0, TT >  -R- (X). X  Hence, in (2.4.5), the feasible solution is *(X, Y) = R~\X) ( ( l + R (X)(a 2  Expanding (l + R (X)(a  7  x+Y  (1 + R (X)(<r 2  x+Y  x+Y  - a )) around a-  2  x  - a )) * = 1 + \R {X){<T 2  x  2  * - l) .  -  2  2 x+Y  2 x+y  Substituting the last equation into (2.4.6), we have  (2.4.6)  - a yields x  - <r ) + o(a 2  x  2  - a ). 2  x+Y  x  -27 Q.E.D.  As shown in Corollary 2.3.2, under the assumption that two random variables have a bivariate normal distribution, and the decision makers have constant risk aversion, one decision maker is more risk averse than the other if and only if the absolute value of the risk premium for the former is larger than that for the latter. Similarly, for the class of quadratic utility function, the above result holds for any random variables with arbitrary joint distribution. Corollary  2.4.2 :  Let X , Y be any two random variables and  u (x),u (x) 1  2  be two  quadratic utility functions given by Ui(x)  = x — PiX  for  2  i =  l,2.  Let R,(X) and -Ki(X,Y) be the risk aversion measures and risk premiums for utility function  (i = 1,2), respectively. Assume that Ri(X) > R (X)  > 0  2  then \*i(X,Y)\  >  \n (X,Y)\. 2  Or, (*')  v (X,Y)>v (X,Y)>0  (»)  7T (X,Y) = 7r (X,Y) = 0  (iii)  TTI(X,  1  1  Proof  :  if  2  <T >(T ; 2  X+Y  if  2  if  Y) < TT (X, Y) < 0 2  a  2 x + Y  X  =a  2 x  ;  <?x+y  From (2.4.6), ic(X, Y) = R~\X)  ((1 + R (X)(a  - a )) 2  2  x+Y  x  * - lY  (2.4.6)  28  (i) if cr  then  > cr ,  x+Y  x  *i(X,Y)  (ii) if <T  > 0;  = °~x> then  x+y  7r (X,y) = 0 ; 1  (iii) if  <T  X  +  Y  <  (T , X  then 7n(X,y)<0.  T a k i n g derivative of ir(X,Y)  (2.4.7)  with respect to R(X), we get  since Ri(X) > R (X) > 0 , 2  (i) if o-  > <r ,  x+Y  x  then dir/dR  >  0,  hence  *i(X,Y)  (ii) if  <T  X  +  Y  = <r , then dir/dR x  > 7r (X,y); 2  hence  — 0,  7r (X,y) = 7r (X,y); 1  (iii) if c r  x + y  < a , then dir/dR x  < 0,  2  hence  T T ^ X . Y ) < 7r (X,y). 2  C o m b i n i n g ( 2 . 4 . 7 ) and  (2.4.8)  yields the desired result.  (2.4.8)  Q.E.D.  - 29 2.5 R u b i n s t e i n ' s M e a s u r e o f R e l a t i v e R i s k  Aversion  So far we have been concerned with assets with given risks. Now let us view everything as a proportion of some fixed value WQ. Specifically, for two random variables X and Y , we consider w X and w Y. The relative risk premium is defined as follows: Q  0  D E F I N I T I O N 2.5.1 Let u(x) be an increasing utility function, and X , Y have a bivariate normal distribution. For a constant  w  0l  the relative risk premium  Tr*(w ; X, Y) 0  is the amount satisfying  E[u(w X  + woY)] = E[u(w X  0  + w (Y - TT*))].  0  0  (2.5.1)  In portfolio selection problems, if we view w as the initial dollar investment, X and 0  Y as the the dollar returns on two risky assets for each dollar's investment, then the the relative risk premium TT* can be interpreted as the reduction in or compensation for the return on the second asset the investor would accept if that asset were to be replaced with a risk-free asset. This concept of relative risk premium yields the following D E F I N I T I O N 2.5.2 Let u(x) be an increasing utility function, and X be a random variable. For a constant  Rubinstein's measure of relative risk aversion  w, 0  is defined to be  Comparing Definition 2.3.1 with Definition 2.5.1, we have  w ir*(w \X,Y) 0  that is  0  = ir(w X, a  w Y), 0  R*(w ,X) 0  -30 TT*(W ;X,Y) =  —n(w X,w Y).  0  0  (2.5.3)  0  Note that if w = 1, then 0  x'(l;X,Y) = x(X,Y).  Similarly, comparing Definition 1.2.2 with Definition 2.5.2, we have (2.5.4)  R*(wo,X) = WOR(WQX).  If w — 1, then 0  R*(1,X) = R(X).  Using Theorem 2.3.2 and Theorem 2.4.2 yields the following result: Theorem 2.5.1 : Let u(x) be an increasing utility function, and X , Y have a bivariate normal distribution with E[Y] = 0. Assume that u(x) is continuously differentiable of order 4, and E[u^(X)] exist for i = 1,2,3,4. If either of the following conditions holds: (i) X and Y have a bivariate normal distribution, and u(x) satisfies the regularity I  conditions ,  (ii) u(x) is a quadratic utility function, then for a constant w , the relative risk premium TT*(W ; X,Y) can be approximated by 0  0  ir*(w ;X, Y) « \{<r 0  where the approximation is of order o(o-  2 x+Y  - a )R*(w ,X)  (2.5.5)  2  x+Y  x  0  - cr ). x  Proof : Based on the given conditons, from Theorem 2.3.2 or Theorem 2.4.2, we have .2  TT(W X,W Y) = -(tr; w ,X+W0Y 0  0  0  ~ <rl x) ( oX R  w  0  + w Y) + 0  = 2™O(°"X+Y - a )R(w X + w Y) + o(a 2  x  0  0  x+  o(a  2 WoX+WoY  -31 -  Dividing the above equation by  w, 0  ir*(w ; X,Y) = \ ( o 0  from  (2.5.3)  and  - o- )R*(w ,X  x + Y  x  (2.5.4),  + Y) + o ( a  0  x + Y  -  a\).  Q.E.D.  From to that of with  (2.5.5), R(X).  (1.1.4),  the interpretation of the relative risk aversion  Note also that if X is non-random  0  ir*(w ;l,Y) 0  (2.5.5)  0  comparing  is similar  (2.5.2), (2.5.3)  we have  R*(w ,l)  then  ( X = 1),  R*(w ,X)  = r*(w  0  + w Y),  = ir*(w ,Y),  R*(w ,X) 0  (2.5.6)  (2.5.7)  0  is recovered to the result in Theorem  of relative risk aversion  0  2.1.2'.  Thus Rubinstein's measure  is a generalization of the Arrow-Pratt measure of  relative risk aversion r*(x), and has the corresponding local property of the latter.  -32 Appendix 2.1  Lemma A2.1 Let Y be a normal random variable with mean n and variance a . Y  Y  Let g : R -+ R be differentiable of order n. If Elg^^Y)] exists for all i = 1,2,... ,n and 1  1  \imjy  =  liyr-W-^Mfiv)  -  for  0  all j = 1,2,...  where f(y) is the density of Y , then T , = E[g{Y)}, 0  p  T  ltg  = <r E[g'(Y)], Y  T , = E[(Y - u ) g(Y)} = <r E[g(Y)} + <r E[g"(Y)}, 2  2  g  T where T  n>9  Y  Y  = (n- l ) c r T _ y  n  Y  + (n - 2)a T ^  2iS  Y  n  gl  + aT- „ Y  n  2tg  for  n> 3  = E[{Y - ti ) g{Y)]. n  n<g  Y  Proof : Since f(y) is the density of the normal random variable Y ,  /(y) =  /'(!/) = /(y) = -  y-  Uy  •f'(y).  (1)  then for n > 3, r„,, = f?[(y - »Y) g{Y)] n  = -*»/ =  (y-uy) - 9(y)f'(y)dy n  l  -<r y((y-u r- g(y)f(y) 2  l  y  f(y)[(n - l)(y - UyY' g{y) + (y - Uy)^ g'{y)]dy) 2  — CO  = trl(A+B + C).  1  - 33 -  B y assumption, A = 0,B =  (n-l)T . . n 2il  Substituting ( l ) into C ; we have +oo  (y-uyr- g'(y)f'(y)dy 2  / =  -oo  -^((y-u r- 9'(y)f(y)\ _Z +  2  y  +oo  f(y)[(n - 2)(y - ^) - g'(y) + (y - n ) ~ g"\y)]dy) n  /  3  n  2  y  -CO  = <r (Al + Bl +  Cl).  2  y  B y assumption, Al = 0 , 5 1 = (n - 2 ) T _ y , C l = T _ n  3  n  2 >  ,».  Therefore, for n > 3,  T,  n a  = (n~  l)a T - , y  n  2 ) ^ T _ , y + <T$Tn-2,a».  + (n -  2  2 g  n  3  It is easy to see that T ,, = 0  T  1>g  T  2t9  =  E[g(Y)],  a E[g'(Y)],  = E[(Y  2  Y  -  u ) g{Y)] 2  Y  = <TyE[g(Y)] +  a E[g"(Y)}. Y  Q.E.D.  -34Appendix  2.2  Proof of Theorem 2.3.3  First note that since u'(x) > 0, r(x) > 0 if and only if u"(x) < 0. So the necessary condition follows immediately from the definition of R(X) in (1.13). To prove the sufficient condition, we assume that R(X) > 0 for any normally distributed random variable X , but there exists at least one x , such that u"(x ) > 0. 0  0  Since u"(x) > 0 is continuous, u"(x ) > 0 implies that there exists a, 6 > 0, such that 0  for all x e[x - 6,xo + 6], u"(x) > 0. 0  Let  f(x;n,(T ) 2  be the density of any normal random variable with mean u and  variance a . If X is a normally distributed random variable with mean x and variance 2  0  c , then 2  = 1 + 11.  (1)  Since u"(x) is continuous and u"(x) > 0 for all x 6 [x - 6,x + 6], 0  =  i ^,-f "( )> \x-x \<6 m i  u  x  0  >  0  Let 0 < i] < S,  I>L  = L(A + B).  )dx  (2)  -35 where  =  / cxp( J\x-x \<r, V2TTO1  { x  ~  X o ) 2  0  2a  )dx 2  f i n / • ,— exp(— —-)dx J\x-x \<r, V2ira 2a 2  -  2  0  ,V (lyexpikl) ) V2lr V " 2 V 2  2  r  v  • +O0  as T) —y 0, a —• 0, and n/a —• 0.  Jrj<\x-x \<6 V21T(T  2a*  0  f  I  S  Jrt<\x-x \<6 V2ira  la*  0  =  _ 2 _ ( ^ ) v^F  -  a  2  ^ P  {  2a ' 2  =(<5-77)V^TT" exp(f5 /2cr ) 2  2  • 0  as  77 —*  0,a —• 0.  Hence, b y (2),  J > L(A + B) — + 0 0  as 77 —»• 0, a —>• 0 a n d 77/cr —* 0.  (3)  -36  Since for \x — XQ\ > 6, 2\  tt  («-»o) \  (  1  2  /(«;«o.O=^e*p( £ 5 - )  </(*;0,i) 1  , <5 , 2  = X  since df(6;0,a)/do- > And  0  for  cr  e [0,1].  f  r+00  rx — 6 0  /  J\x-x \>&  u"{x)dx= I  u"(x)dx + / u"(x)da  J-ca  0  J Xn + i  = w'(a;o - <5) - w'(-oo) + M'(+OO) - u'(x + 6) 0  < OO  by the assumption on u(x). so  u"(x)Kdx < 00.  /  '\x-x \>5 0  Hence /  u"(x)f(x; x , <r )dx 2  '\x-x \>6 J\x—Xn\>6  0  0  converges uniformly o m r e  B y Apostol (Theorem 14.22),  [0,1].  lim /  „ « ( , ) / ( , ; x ,* )dx 2  0  ^° J\x-x \>6  <7  0  u"(x) lim f(x; xo, a )dx  = /  2  J\x-x \>6 0  = 0.  Hence, II —>0  as  cr —»• 0.  (4)  Combining  (1 ),(3),  and  (4),  E[u"{X)] + 0 0  as  a - f 0, n  0  and  77/cr  0.  Therefore, for a sufficiently small a, E[u"(X)} > 0. Then  ^K(x)] for X with mean x and the sufficiently small variance a . But this is in contradiction 2  0  with the condition that R(X) > 0 for any normally distributed random variable X . Q.E.D.  -38 § 3. A n A p p l i c a t i o n o f R u b i n s t e i n ' s M e a s u r e s i n t h e P o r t f o l i o T h e o r y  In this chapter, we study the power of Rubinstein's measures of risk aversion in determining the investor's preference towards risk in the portfolio problem with two risky investments. Assume that the investor's goal is to maximize the expected utility of his terminal wealth. If one of the investments is risk-free, the Arrow-Pratt measures of risk aversion derived from the context of one random outcome have been used to successfully characterize the risk aversion and wealth effect on the investor's portfolio. However, if both of the investments are risky, those measures derived from that context cannot give a desirable characterization of the investor's preference towards risk. Under the assumption that the returns on the two investments have a bivariate normal distribution, we show that Rubinstein's measure of relative (or absolute) risk aversion derived from the context of two random outcomes can yield the exact results about investor's portfolio if the portfolio is stated in terms of proportional (or absolute) allocation between the two risky investments. The conditions for diversification between two investments are established. The results show that analogous to the case of one risk-free and one risky investment, the investor with larger Rubinstein's measures of risk aversion will take a less risky portfolio position. Furthermore, the proportion of the portfolio allocated to the higher mean investment is an increasing, constant, or decreasing function of wealth as the investor's Rubinstein measure of relative risk aversion is decreasing, constant, or increasing at the initial wealth. This chapter is outlined as follows: in Section 3.1, we formulate the portfolio problem and in Section 3.2, we establish the conditions for diversification, that is, a non-trivial allocation between the two investments. In Section 3.3, we prove that the  - 39 -  proportional allocation of an investor's portfolio in the higher return investment is larger if his Rubinstein's measure of relative risk aversion is smaller. In Section 3.4, we show that the proportion of the portfolio allocated to the more risky investment is an increasing, constant or decreasing function of initial wealth as the relative risk aversion is decreasing, constant or increasing at the initial wealth, respectively.  -40 3.1 F o r m u l a t i o n o f t h e P o r t f o l i o M o d e l  Now we consider that an investor is to allocate his initial wealth w between two 0  risky investments with random returns X and Y. We assume that the investor has an increasing and concave utility function u(x). His objective is to maximize the expected utility of his terminal wealth at the end of his planning horizon. If the proportion of his investment to the investment with return X is A, his decision problem is max <7(A) = E[u(w {\{X - Y).+ Y))] 0  (3.1.1)  where u' > 0, u" < 0.  3.2.  Necessary  a n d Sufficient C o n d i t i o n s for Diversification  We first establish the necessary and sufficient conditions for a non-trivial allocation. Lemma 3.1  Let u(x) be a utility function which is twice differentiable with u' >  0,u" < 0 and satisfies the regularity conditions, and X,Y have a bivariate normal distribution. Assuming that all expectations are finite, then 0 < A* < 1 if and only if R*(w ,Y)Cov(Y,X 0  -Y)  < E[X]-E[Y]  < R* (w , X)Cov(X, X - Y); 0  (3.2.1)  A* = 0 if the left-hand side of the inequity is violated; A* = 1 if the right-hand side of the inequity is violated, where A* is the optimal allocation for (3.1.1).  1  Proof : Let W = w (\(X - Y) + Y). If u" < 0, 0  <7"(A) = E[u"{W)wl(X - Y) ] < 0, 2  (3.2.2)  - 41 w h i c h i m p l i e s t h a t t/(A) i s c o n c a v e . H e n c e A* = 0 i f a n d o n l y i f C7'(0) < 0, a n d A* = 1 i f a n d o n l y i f U'(l)  > 0.  Using covariance  identity  E[AB] = Cov(A, B) + E[A]E[B]  and then the First-order Covariance  Operator sequentially,we have  U'{0) = E[u'(w Y)w {X - Y)] 0  0  = Cov(u'(w Y),w (X - Y)) + E[u'(w Y)]E[w (X - Y)] 0  0  0  0  = E[u"(w Y)]Cov(w Y, w (X - Y)) + E[u'(w Y)]E[w (X - Y)] 0  0  0  0  0  = E[u'(Y)]{-R(w Y)wlCov(Y, X - Y) + E[w (X - Y)]} 0  0  = w E[u'(w Y)]{-R*{w ,Y)Cov(Y,X-Y)+(E[X]-E[Y])};  (3.2.3)  = w E[u'(w X)]{-R*(w X)Cov(Y,X-Y) + (E[X] - E[Y])};  (3.2.4)  0  0  0  Similarly,  U\l)  0  a  0  (3.2.1) f o l l o w s s i n c e  0 < A* < 1  ifa n d only if  C7'(0) > 0  and  U'(l)  < 0.  Q.E.D.  -42 -  Assume that P(X = Y) ^ 1, then  Cou(X,  X-Y)>  Cov(Y, X-Y)  (3.2.5)  because Cov(X, X-Y)-  Cov(Y, X —Y) = Cov(X — Y,X — Y) = Var(X - Y) > 0.  By the symmetry of X and Y in (3.2.1), we further assume that E[X] > E[Y], and discuss (3.2.1) in detail. (1) E[X] = E[Y\. In this case, since for any risk-averse investor, R*(w Y) > 0, and R*(w X) > 0, 0  0  (3.2.1) holds if and only if  Cov(Y, X - Y) < 0  and  Cov(X, X - Y) > 0  which is equivalent to Cov(X, Y) < mm{Var(X), Var(Y)}.  (3.2.6)  In fact, under this condition, diversification by any one of risk-averse investor is equivalent to diversification by all risk-averse investors. Brumelle[6] has given the necessary and sufficient conditions for diversification by all risk-averse investors without any assumptions about the bivariate distribution of the two investments.  He defined X and Y as negatively interdependent if that  condition is satisfied. For X and Y having a bivariate normal distribution, we give the following definition.  - 43 -  D E F I N I T I O N 3.2.1 X,Y having a bivariate normal distribution are negatively interdependent if E[X] = E[Y]  and Cov(X,Y) < min{Var(X),Var(Y)}.  Thus a sufficient condition for X,Y having bivariate normal distribution to be negatively interdependent is that X, Y have a common mean and be independent or negatively correlated. The above discussions also lead to the following result. Theorem  3.2.1.  Let X, Y have a bivariate normal distribution. Then all risk-averse  investors with increasing and concave utility functions which satisfy the regularity conditions will diversify between two investments with returns X and Y if and only if X and Y are negatively interdependent. Proof. The sufficient condition follows directly from the above discussion. For the necessary condition, notice that if (3.2.1) holds for any arbitrary R*, it must hold for any arbitrary small R*. This implies that E[X] = E[Y]. From this condition, the other condition for diversification also follows from the previous discussion. Q.E.D. (2) E[X] > E[Y]. In this case, if Cov(X, X-Y) < 0, then no risk-averse investors will diversify because the right-hand side of (3.2.1) is violated, which implies that A* = 1. But from (3.2.5), Cov(X,X - Y) < 0 gives  Cov{Y, X-Y)<  Cov{X, X-Y)<0,  which is equivalent to that  Var(X) < Cov(X,Y) < Var(Y)  Thus any risk-averse individual will invest only in the investment with the expected return no less than the other but with the smaller variance when two investments have some positive correlation. Then a necessary condition for diversification between the two investments X, Y is that Cov(X, X-Y)>0  which is same as Cov(X, Y) < Var(X)  or  This necessary condition can be divided into two cases for discussions: (a)  Var(X) > Cov(X,Y) > Var(Y).  In this case, Cov(X, X-Y)>  Cov(Y, X-Y)>0.  (3.2.1) can be written as  L(X,Y) < R*(w ,X) 0  where  and  R*(w ,Y) < U(X,Y) 0  (3.2.7)  - 45 -  Thus an investor will diversify between the two risky investments if and only if his relative risk aversion is greater than the lower bound L(X,Y), and less than the upper bound U(X, Y). If his relative risk aversion is less than the lower bound, the investor will hold the investment with higher expected return only; if his relative risk aversion is greater than the upper bound, the investor will hold the investment with lower expected return only. (b)  Cov(X, Y) < Var(X), and Cov(X,Y) < Var(Y).  This case is true if and only if Cov(Y, X - Y) < 0, but Cov(X, X - Y) > 0.  Since Cov(Y,X- Y) < 0, the left-hand side of (3.2.1) will always hold, that is, A* > 0. This implies that when the correlation between X and Y is small, any riskaverse investor will always include the investment with the higher expected return in his portfolio. This is obviously obtained when X and Y are independent or negatively correlated.  This result for the independent investment case has been obtained by  Samuelson[45] and later generalized by McEntire[33]. The additional condition for diversification is that L(X,Y) < R*(w X). 0  To summarize the results from the above discussions, we give the following theorem. Theorem 3.2.2. Let u(x) be an increasing and concave utility function which sat-  isfies the regularity conditions, and X,Y have a bivariate normal distribution with E[X) > E[Y]. Then an investor with utility function u(x) will diversify between the two investments with returns X and Y , that is, 0 < A* < 1, if and only if either of the following two conditions holds:  -46 -  (a)  Var(X) > Cov(X,Y) > Var(Y),  and  L(X,Y) < R*(w X), R*{w Y) < U(X,Y);  (b)  Cov(X, Y) < Var(X),Cov(X, Y) < Var(Y),  and  0  0  L(X,Y) < R*(w X). 0  where L(X,Y) and U(X, Y) are given by (3.2.8).  \  i  \  -47 3.3 R i s k A v e r s i o n a n d t h e C h o i c e b e t w e e n T w o R i s k y I n v e s t m e n t s  Now we establish the first-order condition for the decision problem  Lemma  3.2.  Under the hypothesis of Theorem  optimal solution for problem  (3.2.1)  if and only if  where  R*(w ,Z*)  = -w E[u"(w ,  Z* = X*(X-Y)  + Y,  and  0  0  3.2.2,  that is,  Z*)}/E[u'(w ,  0  0  (3.2.1).  0 < A* < 1, A*  Z*)}  is the  is Rubinstein's  measure of relative risk aversion.  Proof  :  By  ( 3 . 2 . 2 ) , (7(A)  is concave, thus  0 < A* < 1  is an optimal solution to  (3.1.1)  if and only if  U(X*) = E[u'(W*)w (X  -Y)] = 0  0  where  W* = w Z* 0  and  Z*  (3.3.2)  is given by the above.  Since X , Y have a bivariate normal distribution, and  W*, X-Y  also have a bivariate normal distribution. Given that u(x) satisfies the  regurality condition, applying the Covariance Identity  E[AB]  = Cov(A,B)  +  E[A]E[B],  - 48 -  and the First-order Covariance Operator to (3.2.4) yields Cov(u'(W*),X -Y)+ E[u'(W*)}(E[X] - E[Y}) = 0 E[u"{W*)}Cov{W*,X-Y) + E[u\W*)](E[X] - E[Y)) = 0, <^=> Cov(W* ,X-Y) = (E[X] - E[Y)) (-  w [X*Var(X - Y) + Cov(Y, X — Y)] = 0  (NoteW* = w (X*(X -Y) + Y) 0  E[X] - E[Y] <^ X*Var(X - Y) + Cov(Y, X — Y) = R*(w ,Z*) 0  The proof is then complete.  Q.E.D.  The above lemma explicitly links an investor's optimal allocation of his investments to Rubinstein's measure of relative risk aversion. From this lemma, it is clear that any investors with the same Rubinstein's measures of relative risk aversion at the terminal wealth will have the same optimal portfolio. This result has been obtained by Kellberg and Ziemba [23] for many risky investment case. But in the two risk investment case, much stronger results can be obtained. In fact, the orderings of the optimal investments in the higher return investment can be compared with the orderings of Rubinstein's measures of relative risk aversion. Under the assumption that the returns on the two investments have a bivariate normal distribution, the risk and return characteristics of the portfolios can be completely described by the their means and variances. Now we use Lemma 3.2 to establish Theorem  3.3.1.  Let Ui(x) be a utility function which is continuously differentiable  with u'i > 0, u'l < 0, and satisfies the regularity condition for i = 1,2, and let X , Y have  - 49 -  a bivariate normal distribution. Consider the following two portfolio problems  m a x E[ui(wi(\(X -Y) + Y ) ) l 0<A<1  '  where w refers to investor i's initial wealth for i = 1,2. If E[X] > E[Y], and 0 < A* < 1, t  then the following four statements are equivalent: (a)  R\{w^Zl)<R* {w ,Z*), 2  2  (b)  A*>A;,  (c)  E[Z{] > E[Z* ), 2  (d)  Var(Zl) > Var(Z^).  where A* is the optimal solution ,Z* = \"-(X - Y ) + Y and R*(wi,Z*) is the relative risk aversion of investor i for i = 1,2.  Proof : We first prove that (a) is equivalent to (b). B y Lemma 3.2,  X*Var(X — Y) + Cov{Y, X - Y) =  E  [  * J  E  }J\  where i = 1,2. Since E[X] - E[Y] > 0, (a) is then equivalent to (b). Now we prove that (b) is equivalent to (c). This is the case since  E[Z*] = X*(E[X]-E[Y]) + E[Y]  (3.3.3)  -50 -  Lastly, we prove that (b) is equivalent to (d). B y definition, Var{Z{) - Var(Z^) = [(Kf - (X*2) ]Var(X -Y) + 2(AJ - X* )Cov(X - Y, Y) 2  2  = ( * i - 2)[( A  =( I A  A  I + A ; ) ^ a r ( X — Y ) + 2 C o v ( X - Y, 7 ) ]  - A;)[(A*Far(X - Y) +  Cov(X - Y, Y))  + (\ Var(X — Y) + Cov(X - Y, Y ) ) ] 2  _ { A l  r  / W ~ E[Y] R\{w ,Z{)  A 2 ) l  x  +  E[X]-E[Y] R* (w ,Z* ) 2  2  J  2  ( S y (3.3.3))  >0.  if and only if AJ > \ . 2  The proof is then complete.  Q.E.D.  This result suggests that an individual with a smaller Rubinstein's measure of relative risk aversion will invest a larger proportion in the investment with a higher mean return, and thus will have a more risky position in his investments. Therefore, Rubinstein's measure of relative risk aversion gives the desirable orderings in the comparison between different investors' portfolios and their risk-bearing behaviours when the returns on investments have a joint normal distribution.  -51 3.4 W e a l t h E f f e c t o n t h e C h o i c e b e t w e e n T w o R i s k y I n v e s t m e n t s  In the one risk-free, one risky investment case, Cass and Stiglitz[7] showed that the proportion of an investor's portfolio allocated to the risky investment is an increasing, constant or decreasing function of wealth if the investor has an decreasing, constant, or increasing Arrow-Pratt measure of relative risk aversion. In the two-risk investment case, much attention has been given to the relationship between the wealth effect on the portfolio and Arrow- Pratt measure of relative risk aversion or absolute risk aversion. Cass and Stiglitz[7] showed that this relationship is not certain in general. But a sufficient condition for the desirable relationship between wealth effect on the portfolio and Arrow-Pratt measure of relative risk aversion was given by Kira and Ziemba[30], under some additional assumptions about the bivariable distribution of returns on the two investments. In this section, we study the relationship between the wealth effect on the portfolio and Rubinstein's measure of relative risk aversion. D E F I N I T I O N 3.4.1 Rubinstein's measure of relative risk aversion is said to be i  •  .  decreasing at ,  ,  i  w  0  constant at  w  increasing at  w  ' r dR*(w ,Z) ^ < 0; dwo  it  0  n  0  it  0  dR*(w ,Z) ^ = 0; dwo 0  n  ., dR*(w ,Z) „ ^ > 0, dwo 0  where Z = Z(w ) is a random variable. 0  B y using Lemma 3.2, we obtain Theorem 3.4.1 Under the hypothesis of Lemma 3.2, the optimal proportion of the portfolio allocated to X , that is, A*, is an increasing, constant, or decreasing function  of initial wealth u; , as Rubinstein's measure of relative risk aversion 0  R*(w ,Z*) Q  of the  utility function u(x) is decreasing, constant, or increasing at w , respectively. 0  That is,  (a)  d\* -— > 0 duiQ  as  (6)  £1 = 0  as  n  dR*(w ,Z*) < 0; dwo 0  =  dwo  dA*  (c)  -— < 0  as  n  dwQ  Proof :  0  ;  dwo  dR*(w ,Z*)  >  0  dwo  0-  Taking the derivative of (3.3.1) with respect to w yields 0  d\* _  Since E[X - Y] > 0, and  JS[X-Y]_  (  R*(w , Z*) > 0, 0  ^-2dR(w ,Z*) 0  the result is immediate.  The proof is then complete.  Q.E.D.  This theorem leads to Theorem 3.4.2. Under the hypothesis of Theorem 3.4.1, the mean, and variance  of the return of the optimal portfolio per dollar invested are increasing, constant , or decreasing functions of the initial wealth as the investor's Rubinstein measure of relative risk aversion is decreasing, constant, or increasing at the initial wealth. That is,  (a)  JL-^fz.]) >  0)  dwo  (») () C  -fL(Var(Z*)) > 0 dwo  £<si*T) = o.  £(V**-)) =o  J-( [z-])<0,  jL(Vor(Z-))< 0 dw  E  dw  0  0  as  *^,Z*)  dR  .  < Q  dwo  as  ^ d  R  M =»  ;  ' ^ "> > 0. Z  dw  0  -53 -  Proof: Taking the derivative of E[Z*} with respect to w$ yields  Taking the derivative of Var(Z*) with respect to wo yields  J-Var(Z*) = 2^-{\*Var(X - Y) + Cov(X - Y, Y)) =  2  d A * E[X]-E[Y] dw R*(w ,Z*)~' 0  0  Since E[X] > E[Y], and R*(w ,Z*) > 0, by the Theorem 3.4.1, the results in the 0  theorem are proved.  Q.E.D.  Example : For the exponential utility function u(x) = -e~$ {fi > 0), we have x  V  R*(w ,Z*) = w /3, 0  0  thus du>o  From Theorem 3.4.1 and Theorem 3.4.2, if an investor has an exponential utility function, as his initial wealth increases the optimal allocation of his portfolio with two risky investments having a bivariate normal distribution to the higher mean return investment will decrease; thus the mean return and variance of his portfolio will fall. For general utility frunctions, it is usually difficult to determine the increasing, constant or decreasing property of Rubinstein's measure of relative risk aversion at initial wealth. Fortunately, however, as Kallberg and Ziemba[23] illustrated, if the variance of a random variable is small, Rubinstein's measure of relative risk aversion can be resonably approximated by the Arrow-Pratt measure of risk aversion. Thus in  the small-variance case, Rubinstein's measure of relative risk aversion can be expected to have the same increasing, constant or decreasing property with respect to initial wealth as the Arrow-Pratt measure of relative risk aversion for any particular utility function. For example, as the logarithmic utility function u(x) = log(x) has constant ArrowPratt measure of relative risk aversion at any wealth level, Rubinstein's measure of relative risk aversion will also remain constant as wealth changes if the variance of the random variable associated the measure is small. Therefore, the wealth effect on the proportional allocations of a portfolio consisting of two risky investments with returns having a bivariate normal distribution will be small if the variance of the portfolio is small(e.g. daily returns ).  - 55 3.5  Remarks  In one risk-free and one risky investment case, Cass and Stiglitz[7] have shown that, if the portfolio is characterized by proportional holdings in different investments, then the Arrow-Pratt measure of relative risk aversion will be associated with the proportional allocation of investments and some statistical properties of the portfolio; but if the portfolio is characterized by absolute holdings, then the Arrow-Pratt measure of absolute risk aversion will be associated with the corresponding absolute allocation and the statistical properties. In the case of two risky investments with returns having a bivariate normal distribution, we have established theorems concerning the relationship between Rubinstein's measure of relative risk aversion and proportional allocations in the two investments and some statistical properties. The corresponding relationship between Rubinstein's measure of absolute risk aversion and absolute allocations in the two investments can also be established.  - 56 § 4 M u l t i v a r i a t e Measures of R i s k A v e r s i o n a n d R i s k  Premiums  This chapter develops the characterizations of the risk premiums and measures of risk aversion for multi-dimensional problems.  Suppose that a decision maker's  preferences can be represented by a multivariate utility function. From the model with an insurable risk and an uninsurable risk with a joint normal distribution in the wealth space, we derive the matrix measure of risk aversion and directional matrix measures of risk aversion from the approximations of the corresponding risk premiums. The first-order multivariate covariance operator generalized by Gassmann[l4] from Stein/Rubinstein first-order covariance operator and the second-order multivariate covariance operator to be developed in this chapter from the second-order covariance operator developed in Chapter 2 provide the basic tools for the derivation. The matrix measure of risk aversion is a multivariate generalization of Rubinstein's measure[4l].  This matrix measure is also a generalization of the matrix measure  developed by Duncan[9] with non-random initial wealth vector. A n example of the application of the matrix measures to the portfolio selection model with multivariate utility function is given. This chapter is outlined as follows. In Section 4.1, we give an introduction to the multivariate measures of risk aversion. In Section 4.2, we give the mathematical properties of two multivariate covariance operators.  In Section 4.3, we use these  covariance operators to derive the risk aversion matrix from the approximation of the corresponding risk premiums for the problems involving multi-dimensional random wealth and risks. In Section 4.4, we give a set of directional risk aversion matrices from the directional premiums. The risk aversion matrix is shown to be a combination  -57-  of the set of directional risk aversion matrices.  -58 4.1  -  I n t r o d u c t i o n t o the M u l t i v a r i a t e M e a s u r e s of R i s k  Aversion  In multi-dimensional decision problems, a decision maker's preferences can be represented by a multivariate utility function. In these situations, a multivariate representation of preferences towards risks is often necessary. The multivariate measures of risk aversion have been developed from the unidimensional risk theory for both non-random and random wealth cases. In the non-random initial wealth case, the Arrow-Pratt measure of absolute risk aversion has been extended to multivariate situations in many different ways. The first is to require the formulation to be reducible to one direction, for example, through requiring the risk to lie along one direction [27].  The second approach to multivariate theory[28][36], is to assume that all indi-  viduals have the same preference orderings. Ambarish and Kallberg[2] have shown how restrictive this assumption is. The third one [9] is to use a generalization of the Taylor series approach employed by Pratt[37] in the univariate case to derive the risk aversion matrix from the corresponding risk premiums. In dealing with multivariate risk aversion measures with random wealth levels, Karni[25] used the indirect utility function to reduce the problem to one direction, and Ambarish and Kallberg[3] used a model with an insurable and an uninsurable risk which are quasi-independent to develop the risk premiums through the Taylor series approach. The results from these approaches have some limitations in the applications. For example, Kami's results from indirect utility functions are ineffective when a multi-dimensional representation of preferences is necessary; and Ambarish and Kallberg's assumption about independence of risks is restrictive since different risks are usually correlated in real-world decision problems.  /  Thus it seems to be necessary to develop a measure of risk aversion with direct multivariate utility function and correlated risks. We adopt the model developed by Ambarish and Kallberg with insurable and uninsurable risks but we assume that these risks are correlated and have a joint normal distribution. We then derive the approximations of risk premiums and give the matrix measures of risk aversion. The next section presents the key tools for the development of these multivariate measures of risk aversion.  - 60 4.2 T h e M a t h e m a t i c a l P r o p e r t i e s  of T w o Multivariate Covariance Operators  The First-Order Covariance Operator developed by Stein[48] and Rubinstein[4l] has been generalized to a multivariate version by Gassmann[14] is given below: Lemma 4.1 (First — Order Multivariate Covariance Operator)  Let X , Y be multivariate random vectors on R  m  and R , respectively, with joint n  distribution  Let g : R i — • R be differentiable with Jacobian matrix 3 of dimension k x n. Assume n  k  g  that Cov(X,g(Y)) and E[3 (Y)] exist. If G  t  j i m ^ g(y + ty)f (y 0  Y  0  + ty) = 0  (4.2.1)  for all y,y 6 R , where / Y is the probability density of Y , given by n  0  /  v  (  y  )  =  ( 2 ^ S Y ] ^  E  *  P  (  ~ ^  (  Y  " " ) Y  T  V (y - ^v)),  S  (4.2.2)  then Cov(X, g(Y)) = Cov(X, Y)E[3 (Y)f. G  (4.2.3)  Proof : See Gassmann[14]. Now we develop a new formula which will be called "Second-Order Multivariate Covariance Operator".  This operator is the multivariate version of Second-Order  Covariance Operator in the univariate case developed in Chapter 2. Lemma 4.2 (Second — Order Multivariate Covariance Operator)  Let X , Y be multivariate random vectors on R , with joint distribution n  -61 -  Let its  H :R  i—• R  n  ith  row  H  8  be twice continuously differentiable with the Jacobian matrix of  n x n  being  of dimension  J[Hj(j/)]  lim ( y + <y)H(y 0  If  n x n.  + * y ) / ( y o + t y ) = 0,  0  (4.2.4)  Y  t—t-XOO  lim J [ H i ( y  0  + ty)]/ (y Y  + ty) = 0,  0  (4.2.5)  t—»-±00  for i = 1,2, • • •, n, and yo, y 6 R , where / n  E[(X -  Proof  :  Y  is the density of Y , then  ) H ( Y ) ( X - / i ) ] = tr(£ £[H(Y)]) + O (*r(£ )).  (4.2.6)  T  M  X  x  x  XY  The conditional distribution of X given  is known to be a normal with  Y  mean vector /*X|Y  = /lX + S x Y S Y ( Y - A i Y ) ,  (4.2.7)  1  and covariance matrix Sx|Y  Let  .ffx-Y = X  given by  —  A*X|Y  =  £  -  X  (4.2.8)  S X Y S Y S Y X .  be the residual vector with mean zero and covariance matrix  (4.2.8).  Then £[(X-/ ) H(Y)(X-/.x)] T  J x  =E[((X  T  xlY  -  =£[(X  +  - u ) + (/i | - /i )) H(Y)((X - ^ | ) + ( Y  | ) H(Y)(X T  / J X  Y  E[(U \Y - ^ X  =A +A +A 1  X  2  3  X  X  X  /JXIY)] +  )H(Y)(X-  ^  X  E[(X  | Y ) ] +  -  ^  X  Y  | Y )  E[(u  xlY  T  M X  H ( Y ) (  |  M  -  Y  X  |  Y  ))]  M x  -  - u ) H(Y)(u  A*X)]  T  x  + A.  x]Y  - /a )] x  (4.2.9)  4  From the identity E[X QX\ T  = E[X] QE[X] T  + tr(QE ) x  - 62 -  for any random vector X and constant matrix Q, where tr(-) means the trace, the sum of the diagonal elements of a square matrix, we have  A  x  = E  [f? [(X - / / | ) H ( Y ) ( X -  ^ | )]|Y]  T  X  Y  X  Y  X  = E [E [(X - u ) }H(Y)E[(X  - p , )] +  T  Y  X  Y  x]Y  x  = £ [(X - /. |Y) ]^Y[H(Y)]^ [X - u ] X  X  X  Y  + J5[tr(H(Y)S |Y)]  T  X  * r ( H ( Y ) £ , ) | Y]  Y  X  x]Y  = tr(S[H(Y)]E | ) X  (4.2.10a)  Y  where the first term disappears because the mean of the residual - R - Y = X - n \ is X  X  Y  zero. Substituting (4.2.7) into the second term of (4.2.9) yields  A  2  = E[(X - ^  X  |  Y  ) H ( Y ) E  X  Y  E  ( Y -  1 Y  u )} Y  = £ [ ( X - u )]E[H(Y)X Z \Y xlY  XY  - u )}  Y  Y  = 0  (4.2.106)  where the second equality follows due to the independence between X - ^ | X  Y  and Y  and the third is due to zero mean of X - u \ . x  Y  Thus, A  = A% = 0.  3  (4.2.10c)  Substituting from (4.2.7) also yields  A  4  Let A =  SxySy-  = E[(Y - P Y )  T  S Y  1  E  Y X H ( Y ) E  X  Y S  1 Y  (  From Appendix 4.1, we have A = E[(Y - u ) A H(Y)A(Y T  4  Y  T  - u )] Y  Y  ~ MY)]-  - 63 = tr (A AEY E[H(Y)]) + 0 ( t r ( £ T  X Y  ))  = ^ ( E X Y E ^ E Y X ^ P C Y ) ] ) + 0(tr(E  X Y  )).  (4.2.10d)  Substituting (4.2.10a)-(4.2.10d) into (4.2.9) and then using (4.2.8) gives E[(X -  ) H ( Y ) ( X - u )] T  / J X  =<r((E | + E X Y  x  E E 1  X Y  + 0(tr(£  Y  X Y  Y X  )^[H(Y)])  ))  =7>(E £[H(Y)])+0(fr(E Y))x  The  proof is then complete.  X  Q.E.D.  -644.3 T h e D e r i v a t i o n o f M a t r i x M e a s u r e s o f R i s k  Aversion  Suppose that the preferences of a decision maker can be represented by a multivariate utility function u(x) with x  T  = (xi, x , • • •,x )- Here we may interpret x as a 2  n  wealth vector. We can either say that the wealth vector represents a decison maker's entire wealth measured at different points of time, or say that it represents the wealth hold in different assets at the same point of time. We can also allow the dimension of x to include the factors which affect the real wealth, thus making x be the measure of real wealth. The utility functions are assumed to be increasing and concave. Consider that a decision maker has two sources of risks in his wealth holdings. He has an initial wealth X which is a random vector. He also faces some other risk Y which affects the wealth levels in the wealth space. These two risks may be correlated. Assume that risk Y can be insured but the initial wealth risk is uninsurable. We are interested in determining the amount of risk premiums that a decision maker is willing to pay to make him indifferent between having the other risk with random wealth X + Y and eliminating that risk by paying the premium n with remaining wealth X - n . This model was proposed by Ambariah and Kallberg[3] and was developed from Ross[40]'s partially insurable lottery and from the random initial wealth model of Kihlstrom, Romer and Williams[29]. Now we formally state the assumptions of our model. A decision maker's preferences for wealth are represented by a von Neumann and Morgenstern utility function which is a mapping from the wealth space into the real line. The wealth variables are allowed to be negative. u: R  n  i — •  R.  - 65 -  ( A l ) The gradient of u at x , V M(X) T  =  (du(x)/dxi, • • •, du(x)/dx ) n  satisfies du(x)/dxi > 0 for all x e R  n  and all i.  (A2) The Hessian of u at x, H ( x ) is an n x n symmetric, negative semidefinite matrix with (i, j)th element d u(x.)/dxidxj. 2  (A3) u is continuously differentiable on R of order 4 and all of the integrals are finite. n  D E F I N I T I O N 4.3.1 Let X , Y be two random vectors on R , and u(x) be a utility n  function satisfying (A1)-(A3). The risk premiums I I are any constant vector on R  n  satisfying E[u(X + Y)] =  £[u(X+£[Y]-II)].  (4.3.1)  The left-hand side of (4.3.1) represents the expected utility of wealth when the decision maker have two types of risks; and the right-hand side represents the expected utility after one of the risks is eliminated. The following lemma is a result of Ambariah and Kallberg[3] with minor different assumptions. Lemma 4.3 Given that (A1)-(A3) are valid and E[Y] — 0, then £[V u(X)]II T  =  -E[V u(X)Y] T  (1/2)£[Y H(X)Y] + T  o(*r£ ). x  (4.3.2)  This result is derived from the Taylor's expansion of the both sides of (4.3.1). The proof will be omitted here. To derive the measure of risk aversion with the correlated random wealth vector and risk vector, we make the following crucial assumption:  - 66 -  (A4) X, Y are multivariate joint normal random vectors on R , with joint distribution n  And for simplicity, assume that E[Y] = 0. D E F I N I T I O N 4.3.2 Given that (A1)-(A4) are valid, u is said to satisfy the regularity condition if the gradient of u, V-u(x) satisfies condition (4.2.1); and u is said to satisfy the regularity conditions if u satisfies the regularity condition and the Hessian of u, H(x) satisfies conditions (4.2.4) and (4.2.5) where all of the variables are replaced by x. (A5) u satisfies the regularity conditions . Theorem 4.3.1 Given that (A1)-(A5) are valid, an approximation of order o(<r(E  x+Y  - Ex)) is H « -i(^ V«(X)) tr((E -Ex)£[H(X)]). T  x+Y  where (E Vu(X)) T  (4.3.3)  is a generalized inverse[39] of E[Vu(X)].  Proof. From Lemma 4.1, the first term in (4.3.2) ( Here we assumed that E[Y] — 0.) E[V u(X)Y] T  •tr(E[Vu(X)Y ]) T  tr(Cov(Y, V«(X)) •tr(Cov(X,Y)E[H(X)]) :tr(E Y^[H(X)]). X  Applying Lemma 4.2 to the second term in (4.3.2) gives £[Y H(X)Y] T  (4.3.4)  =*r(E £[H(X)]) +  0(<r(E  Y  X Y  )).  (4.3.5)  Substituting (4.3.4), (4.3.5) into (4.3.2) yields  E[V u(X)]U T  =  - i * r ( ( 2 £ Y + £ )£[H(X)]) +  o(*r(E )) +  =  -i*r((E  o(tr(E  X  x + Y  Y  -E )jE7[H(X)]) x  +  0(*r(E  x  x  +  Y  X Y  ))  - E ))  (4.3.6)  x  where the last term of approximation can be verified directly from the definition. Premultiplying (4.3.6) by  yields (4.3.3).  (£[Vu(X)])~  Q.E.D.  Note that for any m x n matrix A , the n x m matrix A  -  is a generalized inverse if  A A ~ A = A . Thus the solutions for n are not unique. This is to say, the risk premiums can be paid in different ways to induce the same expected utility. Let the ijth element of  E  x  +  Y  - E  be aY,-, and  x  u  t  and  = du/dxi,  Uij  = H . t J  Written  in summation form, (4.3.6) is then n  £[«,-(x)]n,t=i I " * -  2 ^ 1  dijE[uij{X)]  n  =  n  diji-EluijiXWEluiiX)]))  ([Eui(X)]^2  i=l  (4.3.7)  j=l  Let  ^ = '^)i-(-W)  then (4.3.7) can be written as  £[V «(X)](I1 T  |dta  f f  ((E +Y-Sx)R(X)) x  where diag(A) is a diagonal vector of the matrix  A.  «  0.  (4.3.8)  -  From  68 -  (4.3.8), II  0  irfia ((Ex+Y - Sx)R(X))  a  (4.3.9)  f f  is a typical approximation of the risk premium. A n d the other approximations are located on a hyperplane passing through it and orthogonal to E[Vu(X)]. D E F I N I T I O N 4.3.3 Given that (A1)-(A5) are valid. The absolute risk aversion matrix is EMM)  K(X)-(  If the dimension of the wealth vector is one, then this matrix measure is the same as Rubinstein's measure and we have the same approximated relationship between risk aversion and risk premium as obtained in Chapter 2. Thus this matrix measure is a multivariate generalization of Rubinstein's measure of risk aversion. If the wealth vector  X  and risk vector  IT  0  ta  Y  are independent, from  id«aff(E R(X)).  (4.3.9),  .  Y  (4.3.10)  This relationship has actually been implied by Ambarish and Kallberg [3]. In particular, if X is non-random, that is X = x, then  This is the matrix measure of risk aversion for non-random wealth developed by Duncan [9]. The approximated risk premium in  (4.3.9)  suggests that the multivariate formu-  lation of risk aversion is not important if ( E X + Y - S X ) or  R(X)  is diagonal. In the first  case, the correlations between different dimensions of wealth would be unchanged if  - 69 -  the risk premium were paid. In the second case, the utility function is additive, thus the risks in different dimensions of wealth are mutually independent. The meanings of the matrix measure of risk aversion will be further understood after the directional measures of risk aversion are introduced in the next section.  -70 4.4 D i r e c t i o n a l M a t r i x M e a s u r e s o f R i s k  Aversion  In this section, we assume that the risk premiums can be paid only in one of the directions of the wealth vector. First of all, we give the following D E F I N I T I O N 4.4.1 The risk premium in jfcth direction DJW is the fcth element of the risk premium vector n = (IL;) satisfying  -  l i j  for any  \0,  if j * * .  k = 1, • • •, K  From  (4.3.6),  the directional risk premiums can be approximated by i n  (*)  » -|<r((E Y-Ex)£[H(X)]/f?[«ib]).  (4.4.1)  x+  Thus we make the following D E F I N I T I O N 4.4.2 The matrix measure of risk aversion in kth. direction is  (*)(x) =  R  (_ MM.) E  E[u (X)]. k  for  k — 1,2, • • •, n.  The directional matrix measures of risk aversion are positive definite since is negative definite and  E[u (X)]  definition with Definition  k  4.3.2,  > 0  for all of k. Furthermore, by comparing this  we have  R<*> = Diag(E[ }/E[u ], Ul  where  Diag(A))  E[H(X)]  k  - • • ,E[u ]/E[u ])K(X). n  k  (4.4.2)  for a vector A means the matrix with diagonal elements of A .  Since the elements in the diagonal matrix are the marginal rates of substitution of utility in one direction for the other, the directional matrix measures of risk aversion  - 71 -  are determined once the the matrix measure of risk aversion R ( X ) and these marginal rates of substitution are known. On the other hand, since the kth row of R ( X ) is identical to that of R W , the matrix measure of risk aversion is a combination index of the set of directional matrix measures of risk aversion.  Let the approximation of risk premium in kth. direction be nW, from (4.4.1),  fiW  =  i<r((S  x  +  Y  -S )R  (  ,  :  )  x  (X)).  (4.4.3)  We then have Theorem  4.4.1  Given that (Al)-(A5) are valid,  > 0  if  S +Y - E x  (b)  I I « = 0  if  E +Y =  (c)  n  if  SX+Y -  (a)  w  <  0  is positive definite;  X  E  X  S  ;  X  is negative definite.  X  for all jfe= 1,2, •••,K. Proof.  We only prove (a), since the rest can be proved in the same way.  If E + Y - S x X  is positive definite, it can be decomposed into  SX+Y - E  = U DU T  X  where U is an orthogonal matrix such that U U = I, and D is a diagonal matrix T  D = Diag(\ • • •, A ) u  with Xi > 0 for all i = 1, 2, • • •, n.  n  Thus II« = i*r((E  =  x +  Y - Sx)R  (fc)  )  ^r((U DU)R«) T  = i<r((U U)DR( )) T  fc  = i<r(lDR(»))  =  ^r(DRW) 1 "  > o  because A* > 0,  < 0 and u > 0 for all i and k. k  Q.E.D.  It is well known that if all individuals' preferences are represented by univariate concave utility functions, and the random variables have a joint normal distribution, the risks of any random variables can be completely represented by the variances. With the assumption that all individuals have multivariate concave utility functions, and the random vectors have a joint multivariate normal distribution, the risks of any random vectors can analogously be represented by the covariance matrices. One random vector can be said to be more(less) risky than the other if the difference between the first covariance matrix and second covariance matrix is positive(negative) definite. Thus the results in the last theorem can be interpreted as follows: the directional risk premium is positive if the risk with two components of random wealth is higher than that with only one component; the directional risk premium is negative if the risk with two components of random wealth is lower than that with only one component. Therefore, the directional risk aversion matrices can yield the desirable results with  multivariate utility functions and joint normally distributed wealth variables.  -74A p p e n d i x 4.1  Lemma  A4.1 Under  the hypothesis of Lemma 4.2,  E[Y A H(Y)AY] T  Proof.  = tr(A AE E[H(Y)])  T  + 0(ir(A AE )).  T  T  Y  From the definition of  Y  the gradient of /  f {y), Y  Y  is given by  V / ( y ) = - E ^ y - AiY)/v(y).  (1)  -E v/ (y).  (2)  Y  Thus (y - P Y ) / ( y ) = Y  Y  G = E[(Y - u ) A H(Y)A(Y T  Y  =  j  ( y - u ) A H(y)A  =  /  (y-/i ) A  T  T  T J  u )] Y  ((y - /i )/ (y)) d y  T  Y  Y  Y  Y  -  T  Y  r 7 ( y ) A ( - E V / ( y ) ) dy Y  Y  ^ R"  (substituting from (2))  where  N (y) T  =  [  -((y-^ ) A F(y)AS )V/ (y)  =  /  - N ( y ) V / ( y ) dy  T  Y  Y  dy  T  Y  = (y- /i ) A J7(y)AE T  T  Y  T  Y  Y  with the ith element  Ni(y).  That is,  = t  I  - ^ ( y )dyi^ ^ y -  Integrating by parts yields 9  N  i  f  «—/Y oyi  .  /  A  dy  n  M N ) / Y c(y - ^ J  A  »  (3)  -75 -  where J N is the Jacobian matrix of N(y) and A  H  =  / —(Nifc) JR» dyi f /  =  dy  f °° (/  d •z—(N f-Y)dy ) dyx • • • dyt-idyi+i • • • dy  +  i  i  n  where the inner integral is f + CO Q / =  d  -Z-WY)  lim  Nif -  yi  lim  Y  y;-» + oo  Nf {  Y  y;-»-oo  = 0  by th regularity condition (4.2.4).  Thus the second term in (3) vanishes. Let M(y) = A i J ( y ) A £ with ith column vector M,-, then by definition, T  Y  N ( y ) = (y -  u ) M(y).  T  Applying  T  Y  the product rule to N(y) yields / (y- /*Y) JM,  \  t  J  = M(y) J _  N  ( y  M y )  + \(y-^Y)  =  T j  M„/  M(y) + R(y)  (4)  where J( _ ) is equal to the identity matrix. y  MY  Substituting (4) into (3) yields  G=  <r(M(y))/  f  = /  Y  dy + f  tr(A #(y)A£ )/ T  Y  Y  tr(R(y))/ (y) dy Y  dy  J Ft"  + /  *r(R(y))/  dy  Y  J R."  = 7>(A AE T  + i J R"  Y  /  H(y)f  Y  dy)  7>(R(y))/ dy Y  -76 =  tr(A AX E[H(y)}) T  Y  + f  <r(R(y))/y dy.  (5)  JR."  Similar to univariate case, it can be shown that /  t r ( R ( y ) ) / dy Y  =0(ir(A A£ )). T  Y  Thus G = ir ( A A £ £ ' [ J 7 ( Y ) ] ) + 0 ( i r ( A A £ ) ) . T  T  Y  This is what we required.  Y  Q.E.D.  -77 § 5.  A n Application of M a t r i x Measures i n the Portfolio Theory  In this chapter, we give an application of the matrix measure of risk aversion in a portfolio selection model. This model is a generalization of the portfolio selection model in Chapter 3 and is developed from the model by Losq and Chateau[20]. Suppose that an investor's portfolio consists of two risky investments.  The investor is  concerned not only about the nominal value of his portfolio, but also about some uncontrollable factors which affect the real value of the wealth. Assume that the investor's preferences can be represented by a multivariate utility function which is a function of the nominal value of his portfolio and these relevant factors. The investor's objective is to maximize the expected utility of the real value of his portfolio at the end of the investment horizon. In Section 5.1, we formally formulate this portfolio selection model. From this model, in Section 5.2, we show that if the returns and factors have a joint normal distribution, the optimal allocation of his portfolio depends on a vector of the components of the risk aversion matrix in the wealth dimension. A sufficient condition for any investors to have the same optimal portfolio is that they have these same components in their risk aversion matrices.  -78 5.1 Formulation of the M o d e l  We start with a discription of our model. Assume that there are two risky investments with nominal returns X and Y with a bivariate normal distribution, and there are K factors F  T  = (F ,F , x  2  • • •,F ) K  with a joint normal distribution, which affect the  real value of wealth. The two returns and K factors are also assumed to have a joint normal distribution. Consider that an investor is to allocate his initial wealth w be0  tween the two investments. We assume that in the existence of the uncertain factors, the investor's preference for the real value of wealth can be represented by a multivarir ate increasing and concave utility function u(w,F), where w represents the nominal value of the final wealth, and F is the vector of factors affecting the real value of wealth. Also assume that the investor's objective is to maximize the expected utility of the real value of his portfolio at the end of his investment horizon. Let a be the amount of dollar allocation of the initial wealth w to the investment 0  with (one plus) rate of return X. Thus his decision problem is max  V(a) = E[u(a(X-Y)+w Y,F)]. 0  (5.1.1)  -79 5.2 R i s k A v e r s i o n a n d t h e O p t i m a l P o r t f o l i o  Suppose that the optimal allocation of the portfolio is a non-trivial allocation of investments. Differentiating (5.1.1) with respect to a yields the first-order condition V'(a) = E[u (W*,F)(X-Y)] = 0  (5.2.1)  w  where W* = a*(X — Y) + w Y is the value of the optimal portfolio, and u = du(w,F)/dw 0  w  is the derivative of u with respect to w. By the covariance identity,  Cov(A, B) = E[AB] - E[A]E[B]  for any random variable A and B, (5.2.1) can be written as Cov(u (W*,F),X - Y) = -E[u (W*,F)]E[X - Y}. w  (5.2.2)  w  We now present the following Theorem 5.2.1 Assume that the bivariate random variables X, Y, and the K dimensional random vector F have a joint normal distribution. Also assume that u{w,F) : R  K  +  1  i — • R is an increasing and concave utility function which is twice 1  continuously differentiable and satisfies the regularity condition. If 0 < a* < io , then 0  the optimal allocation of the portfolio a* satisfies  Y^X  V  -  + ».C„„(y, X-Y)  =  ,JWB",F) x - Y) ^ ; ;  E[X-Y]  j - L ^ l j - £  v  V  r  (5.2.3)  where ( R w , R w F i , • • •, R w F ) is the first row of the absolute matrix measure of risk W  K  aversion K  ' '  K  E[ui{W,F)]>  '  J  '  U  '' K  - 80 -  Proof. Applying the first-order covariance operator to (5.2.2) yields K  Cov{W*, X - Y)E[u {W*,Z)] + J2  ( k.  Cov  ww  F  x  ~ Y)E[u (W*, F)] wFk  *=i = - E[u (W*,F)]E[X - Y] w  Substituting from W* = a*(X - Y) + w Y gives 0  (Var{X - Y)a* + w Cov(Y, X - Y))E[u (W*, Z)} 0  ww  K  + J2 Cov(F ,X-Y)E[u (W*,F)] k=l k  wFk  = - E[u (W*,F))E[X -Y] w  Dividing both sides by E[u (W*, Z)} and rearranging terms gives ww  E[X - Y) Var(X - Y)a* + w Cov(Y, X-Y) = (-E[u (W*, F)]/E[u (W*, F)]) 0  ww  w  Y V W F X w-E[u (W*,F)]/E[u (W*,F)}, -^Cov{F X-Y){ _ * * )• Y  k)  wFk  w  E [ U w w { w  > F ) ] / E [ U w { w  t F ) ]  Substituting from the notations D R  m  -  * , - E[u (W*,F)] ' = ~ E[u (W*,F)} 7  {  ww  W  Z )  w  _ E[u (W*,F)] = ~ E[uUW*,F)} wFk  R  ^  {  W  '  F )  for k = 1,2, •••,!< yields (5.2.3).  Q.E.D.  This result links the optimal allocation of the portfolio to the absolute matrix measure of risk aversion defined in § 4.3. From (5.2.3), if the second term in the right-hand side is zero, the optimal portfolio is analogous to what we obtained in § 3.3, except that the portfolio allocation and measure of risk aversion are expressed in proportional terms there. A sufficient condition for this case is that Cov(F ,X-Y) = 0, k  or  R (W*,F) = 0 wFk  for all k- 1,2, Thus the multivariate representation of utility function and the multivariate measure of risk aversion are not important if the difference in returns or the marginal utility of wealth u are independent of all of the factors. w  -81 -  However, in general case, in the absence of the independences, the optimal portfolio depends on the vector  (R ,R F WW  W  1  , • - •, R F )- And the effect of each of these factors W  K  on the optimal allocation depends on the signs of  Cov(F ,X k  - Y)  and  R F (W*, W  k  F)  for  any particular k. For any two given investments, if two investors have the same subjective joint probability distributions of the returns and the factors, and have the same vectors in the risk aversion matrices, they will have the same optimal portfolio.  - 82 References  1. R.B.Ash,  Real  Analysis  and  Academic Press, New York, 1972.  Probability,  2. R.Ambarish and J.G.Kallberg, 'Risk Aversion and Minimum Concave Utility Functions', New York University, Working Paper, 1983. 3.  , 'Multivariate Risk Premiums', Theory  and  Decision  22,  4. K.J.Arrow,  Essays  5. R.Bellman,  Introduction  in the  1987.  Theory  of  to Matrix  Risk — Bearing,  Analysis,  Markham, New York, 1971.  McGraw-Hill, New York, 1959.  6. S.L.Brumelle, 'When Does Diversification between Two Investments Pay?' JFQA  9,  June 1974.  7. D.Cass, and J.E.Stiglitz, 'Risk Aversion and Wealth Effects on Portfolio with Many Assets',  R.E.  Studies  39,  July 1972.  8. 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Management  Optimization  Science  Models  21,  Oct. 1974.  in  Finance,  

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