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European Journal of Operational Research 237 (2014) 957–965
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European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Decision Support Equilibrium in an ambiguity-averse mean–variance investors market q ⇑ Mustafa Ç. Pınar
Department of Industrial Engineering, Bilkent University, 06800 Bilkent, Ankara, Turkey
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Article history: In a financial market composed of n risky assets and a riskless asset, where short sales are allowed and Received 2 July 2013 mean–variance investors can be ambiguity averse, i.e., diffident about mean return estimates where con- Accepted 5 February 2014 fidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a Available online 12 February 2014 worst case max–min criterion. Then, in a market where all investors are ambiguity-averse mean–vari- ance investors with access to given mean return and variance–covariance estimates, we investigate con- Keywords: ditions regarding the existence of an equilibrium price system and give an explicit formula for the Robust optimization equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, Mean–variance portfolio theory we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) Ellipsoidal uncertainty Equilibrium price system mean–variance investors’ market has a deflationary effect on equilibrium prices with respect to a pure mean–variance investors’ market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambi- guity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction and background Klibanoff, Marinacci, and Mukerji (2005, 2009). Our study follows the earlier work of Konno and Shirakawa (1994, 1995), and is in A major theme in mathematical finance is the study of inves- particular inspired by the previous work of Deng, Li, and Wang tors’ portfolio decisions using the well-established theory of (2005) where the authors study a similar problem allowing the mean–variance that began with the seminal work of Markowitz mean returns of risky assets to vary over a hyper-rectangle, i.e., (1987). The mean–variance portfolio theory then formed the basis an interval is specified for each mean return estimate and a of the celebrated Capital Asset Pricing Model (CAPM) (Sharpe, max–min approach is used in the portfolio choice as in the present 1964), the most commonly used equilibrium and pricing model paper. We adopt an ellipsoidal uncertainty set for the mean-return in the financial literature. However, it is a well-known fact that vector instead of a hyper-rectangle, and obtain a closed-form port- the investors’ portfolio holdings in the mean–variance portfolio folio rule using a worst-case max–min approach as in the robust theory are very sensitive to the estimated mean returns of the risky optimization framework of Ben-Tal and Nemirovski (1999, 1998). assets; see e.g., Best and Grauer (1991a), Best and Grauer (1991b), In contrast, in Deng et al. (2005) a closed-form portfolio rule is Black and Litterman (1992). The purpose of the present paper is to not possible due to the polyhedral nature of their ambiguity repre- investigate equilibrium relations in a financial market composed of sentation. The ellipsoidal model controls the diffidence of investors n risky assets and a riskless asset using an approach that takes into using a single positive parameter while the interval model of account the imprecision in the mean return estimates. In our mod- Deng et al. (2005) requires the specification of an interval for each el, investors act as mean–variance investors with a degree of diffi- risky asset, and has to resort to numerical solution of a linear pro- dence (or confidence) towards the mean return estimates of risky gramming problem to find a worst-case rate of return vector in the assets. We refer to this attitude of diffidence as ambiguity aversion hyper-rectangle. The linear programming nature of the procedure to distinguish it from risk aversion quantified by a mean–variance may cause several components of the rate of return vector in ques- objective function. Decision making under ambiguity aversion is an tion to assume the lower or upper end values of the interval as a active research area in decision theory and economics; see e.g., by-product of the simplex method (i.e., an extreme point of the hy- per-rectangle will be found). Since the worst case return occurs at
q an extreme point of the hyper-rectangle, it corresponds to an ex- Revision: December 2013. ⇑ Tel.: +90 3122902603. treme scenario where most (or all) risky assets assume their worst E-mail address: [email protected] possible return values, which may translate into an unnecessarily
http://dx.doi.org/10.1016/j.ejor.2014.02.016 0377-2217/Ó 2014 Elsevier B.V. All rights reserved. 958 M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965 conservative portfolio. Such extreme behavior does not occur with we have an explicit formula for the equilibrium price system in an ellipsoidal representation of the uncertainty set due to the non- a market of mean–variance and ambiguity-averse investors, linear geometry of the ellipsoid. Besides, the ellipsoidal representa- which reduces to a formula for the equilibrium prices of a mar- tion is also motivated by statistical considerations alluded to in ket of mean–variance investors, Section 2.AsinDeng et al. (2005), in the contributions of Konno we show the deflationary effect of the ambiguity aversion on and Shirakawa (1994, 1995) where short sales are not allowed, risky asset prices, the formula for the equilibrium price vector requires the solution we establish a generalization of the CAPM which reverts to the of an optimization problem as input to the formula whereas we original CAPM when all investors are ambiguity-neutral. have an explicit formula for the equilibrium price. To the best of our knowledge, the present paper is one the few The paper is organized as follows. In Section 2 we examine the studies next to Deng et al. (2005), Wu, Song, Xu, and Liu (2009) to problem of portfolio choice of an ambiguity-averse investor using incorporate ambiguity aversion in asset returns in an equilibrium an ellipsoidal ambiguity set and worst case max–min criterion. framework. However, unlike the present paper, in neither Deng We derive an explicit optimal portfolio rule. In Section 3, we study et al. (2005) nor Wu et al. (2009) there is a truly closed-form result, conditions under which an equilibrium system of prices exist in and furthermore they do not study the impact of ambiguity aver- different capital markets characterized by the presence of ambigu- sion on equilibrium prices. ity-averse or neutral investors, and give an explicit formula for The seminal results on equilibrium in capital markets were equilibrium prices. We illustrate the results with a numerical established in the early works of Lintner (1965), Mossin (1966) example. Section 4 gives some properties of equilibrium. In partic- and Sharpe (1964), which resulted in the celebrated CAPM; see El- ular, separation and proportion properties are shown, as well as a ton and Gruber (1991), Markowitz (1987) for textbook treatments CAPM-like result which reduces to the classical CAPM when inves- of the subject. The theory of equilibrium in capital asset markets tors have full confidence in the estimates of mean rate of return. were later extended in several directions in e.g., Black (1972), Niel- We conclude in Section 5 with a summary and future research sen (1987, 1989, 1990, 1989). In a recent study, Rockafellar, Urya- directions. sev, and Zabarankin (2007) use the so-called diversion measures (an example is Conditional Value at Risk, CVaR) to investigate equilib- 2. Ambiguity-averse mean–variance investor’s portfolio rule rium in capital markets. Balbás, Balbás, and Balbás (2010) use coherent risk measures, expectation bounded risk measures and Let the price per share of asset j be denoted pj, j ¼ 1; 2; ...; n for general deviations in optimal portfolio problems, and study the first n risky assets in the market, we assume the price of the CAPM-like relations. Grechuk and Zabarankin (2012) consider an n þ 1th riskless asset to be equal to one. We denote by x0 the num- optimal risk sharing problem among agents with utility functionals j ber of shares of asset j held initially by the investor while we use xj depending only on the expected value and a deviation measure of to denote the number of shares of asset j held by the investor after an uncertain payoff. They characterize Pareto optimal solutions the transaction, for all j ¼ 1; ...; n þ 1. Unlimited short positions and study the existence of an equilibrium. Kalinchenko, Uryasev, are allowed, i.e., there is no sign restriction on xj. and Rockafellar (2012) use the generalized CAPM based on mixed The n risky assets have random rate of return vector Conditional Value at Risk deviation for calibrating the risk prefer- r ¼ðr1; r2; ...; rnÞ and estimate of mean rate of return vector ences of investors. Hasuike (2010) use fuzzy numbers to represent ^r ¼ð^r1; ^r2; ...; ^rnÞ (that we shall also refer to as the nominal rate investors’ preferences in an extension of the CAPM. Zabarankin, of return) with variance–covariance matrix estimate C which is as- Pavlikov, and Uryasev (2014) uses the Conditional Drawdown-at- sumed positive definite. The ðn þ 1Þth position is reserved for the Risk (CDaR) measure to study optimal portfolio selection and riskless asset with deterministic rate of return equal to R. The CAPM-like equilibrium models. Won and Yannelis (2011) examine investor has a risk aversion coefficient x 2ð0; 1Þ and an initial equilibrium with an application to financial markets without a endowment W0 assumed positive such that riskless asset where uncertainty makes preferences incomplete. Xn They assume a normal distribution for the mean return with an W ¼ p x0 þ x0 : uncertain mean, and adopt a min–max approach using an ellipsoi- 0 j j nþ1 j¼1 dal representation as in the present paper. In the present paper, we investigate the equilibrium implica- Since there are no withdrawals from and injections to the port- tions of ambiguity aversion defined as diffidence vis à vis esti- folio, we still have, after the transaction, mated mean returns. In particular, in a capital market in Xn equilibrium where all investors fully trust estimated mean rates W0 ¼ pjxj þ xnþ1: of return (they are ambiguity-neutral), if one investor decides to j¼1 adopt an ambiguity-averse position, this shift may create a down- Dividing the last equation by W0 and defining the proportions ward pressure on equilibrium prices. In uniform markets where all pjxj yj for j ¼ 1; ...; n þ 1 we have that investors are ambiguity averse, the effect of ambiguity aversion is W0 also deflationary with respect to a fully confident (ambiguity-neu- Xnþ1 tral) investors market. yj ¼ 1: In summary, the contributions of the present are as follows: j¼1
If we denote the true (unknown) mean rates of return by rj for we use an ellipsoidal representation of the ambiguity in mean j ¼ 1; ...; n the mean rate of return of portfolio x (with proportions returns which avoids extreme scenarios, and thus alleviates yj) is equal to the overly conservative nature of the resulting portfolios, Xn our ellipsoidal ambiguity model allows for a truly closed-form r y þ Ry portfolio rule, j j nþ1 j¼1 we establish a sufficient condition for a unique equilibrium P P n n T price vector in financial markets with ambiguity averse inves- with variance equal to j¼1 k¼1Cjkyjyk ¼ y Cy where y denotes the tors, as well as a necessary and sufficient condition for existence vector with components ðy1; ...; ynÞ. Note that the random end-of- of non-negative equilibrium prices, period wealth W1 is given as M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965 959 "# ! Xn W 1 x H W ¼ W r y þ Ry : x ¼ 0 ðC 1l^Þ ; j ¼ 1; ...; n: 1 0 i i nþ1 j p 2x H j i¼1 j The investor is also ambiguity averse with ambiguity aversion If P H, then it is optimal for an AAMVP investor to keep all initial coefficient such that his/her confidence in the mean rate of return wealth in the riskless asset. vector estimate is expressed as a belief that the true mean rate of return lies in the ellipsoidal set Proof. The function is strictly concave. The first-order necessary 1=2 6 – Ur ¼frjkC ðr ^rÞk2 g; and sufficient conditions (assuming a solution y 0) yields the can- didate solution: that is, an n-dimensional ellipsoid centered at ^r (the estimated mean return vector) with radius . The idea is that the decisions ð1 xÞr 1 y ¼ C l^; of an ambiguity averse investor are made by considering the worst ð1 xÞ þ 2rx case occurrences of the true mean rate of return r within the set pffiffiffiffiffiffiffiffiffiffiffiffi where we defined r yT Cy. Using the definition of r we obtain Ur. Therefore, more conservative portfolio choices are made when 2 2 2 the volume of the ellipsoid is larger, i.e. for greater values of , ð1 xÞ H ¼ðð1 xÞ þ 2rxÞ . Developing the parentheses on while an ambiguity-neutral investor with no doubt about errors the right side we obtain a quadratic equation in r in the estimated values sets equal to zero. The differences be- 2 2 2 2 2 4x r þ 4xð1 xÞ r þð1 xÞ ð H Þ¼0 tween the true mean rate of return r and its forecast ^r depend ð1 xÞðH Þ on the variance of the returns, hence they are scaled by the inverse with the positive root rþ ¼ 2x provided that < H. Then the of the covariance matrix. To quote Fabozzi, Kolm, Pachamanova, result follows by simple algebra. If P H then our supposition that and Focardi (2007): ‘‘The parameter corresponds to the overall a non-zero solution exists has been falsified, in which case we re- amount of scaled deviations of the realized returns from the vert to the origin as the optimal solution. h forecasts against which the investor would like to be protected’’. Garlappi, Uppal, and Wang (2007) show that the ellipsoidal repre- Notice that when the investor is not ambiguity averse, i.e., sentation of the ambiguity of estimates may also lead to more ¼ 0, one recovers the optimal portfolio rule of a mean–variance stable portfolio strategies, delivering a higher out-of-sample investor, namely, Sharpe ratio compared to the classical Markowitz portfolios. It is 1 x also well-known (see Johnson & Wichern (1997, p. 212)), that y ¼ C 1l^: the random variable 2x H T 1 The factor < 1 in the optimal portfolio of a diffident investor ðr ^rÞ C ðr ^rÞ; H whose diffidence is bounded above by the slope of the Capital Mar- has a known distribution (F-distribution under standard assump- ket Line (we shall refer to such investors as mildly diffident, we shall tions on the time series of returns), and this fact can be exploited also be using the terms bounded diffidence or limited diffidence in using a quantile framework to set meaningful values for in prac- the same context), tends to curtail both long and short positions tical computation with return data, c.f. Garlappi et al. (2007). with respect to the portfolio of a fully confident (i.e., ambiguity- The ambiguity-averse mean–variance investor is interested in neutral) investor. choosing his/her optimal portfolio according to the solution of An alternative proof would proceed by exchanging the max and the following problem the min as in Deng et al. (2005). Solving the max problem first for fixed r, one finds the point maxminð1 xÞðrT y þð1 eT yÞRÞ xyT Cy y r2U r 1 x y ¼ C 1ðr ReÞ: ð1Þ where e represents an n-vector of ones and the scalar x 2ð0; 1Þ rep- 2x resents the degree of risk aversion of the investor. The larger the va- Then minimizing the resulting maximum lue of x, the more risk averse (in the sense of aversion to variance of portfolio return) the investor. Processing the inner min we obtain as ð1 xÞ2 ð1 xÞR þ ðr ReÞT C 1ðr ReÞ usual the problem: 4x T T 1=2 T maxð1 xÞð^r y þð1 e yÞR kC yk Þ xy Cy over the set U one finds the worst case rate of return r as the unique y 2 r minimizer of the above function (this is missing in the analysis of that is referred to as AAMVP (abbreviation of Ambiguity Averse Deng et al. (2005)): Mean–Variance Portfolio). Let l^ ¼ ^r Re. Hence we can re-write H R AAMVP as r ¼ ^r þ e; ð2Þ H H T 1=2 T maxð1 xÞðl^ y þ R kC yk2Þ xy Cy: y which when plugged into (1) for r results in the solution we have 2 1 obtained in Proposition 1. Let us define the market optimal Sharpe ratio as H ¼ l^ T C l^.
Proposition 1. If < H then AAMVP admits the unique optimal 3. Existence of an equilibrium price system solution In this section we shall analyze the existence of an equilibrium 1 x H y ¼ C 1l^; y price system in capital markets where investors adopt or relin- 2x H nþ1 quish an ambiguity-averse attitude. First, we shall look at markets Xn 1 x H where all investors are either ambiguity-averse or ambiguity-neu- ¼ 1 ðC 1l^Þ 2x H j tral. We refer to such markets as uniform markets. Then, we inves- j¼1 tigate the effect on equilibrium prices of introducing an ambiguity- i.e., an ambiguity-averse mean–variance investor with limited diffi- averse investor in a market of ambiguity-neutral investors. We dence ( < H) makes the optimal portfolio choice in the risky assets shall refer to such markets as mixed. 960 M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965
We denote the price system by the vector ðp1; p2; ...; pnÞ for the Proposition 2. In an ambiguity-averse mean–variance investors’ n risky assets. The price of the riskless asset is assumed to be equal market where every investor has limited diffidence (i.e., i < H for to one. We make the following assumptions: all i ¼ 1; ...; m) if a–1, then there exists a unique solution p to the equilibrium system (11) given by 0 1. The total number of shares of asset j is xj , j ¼ 1; 2; ...; n þ 1. Xm 1 f 1 x H 2. Investors i ¼ 1; ...; m make their static portfolio choices p ¼ j i i x0 ; j ¼ 1; ...; n: ð12Þ j 1 a x0 2x H inþ1 according to the ambiguity-averse mean–variance portfolio j i¼1 i P model AAMVP of the previous section; they all agree on m 1 xi H i 0 If i 1 x P 0; j ¼ 1; 2; ...; n þ 1, and no investor is the nominal excess return vector l^ (i.e., they all agree on ¼ 2xi H ij the same nominal rate of return vector ^r and the same short on risky assets, i.e., fj P 0 for all j ¼ 1; ...; n, then the market riskless rate R) and positive-definite variance–covariance admits a unique non-negative equilibrium price vector p if and only matrix C. if a < 1. 0 3. InvestorÀÁi invests an initial wealth Wi in an initial portfolio x0 ; x0 ; ...; x0 . i1 i2 inþ1 Proof. The proof is almost identical to the proof of Theorem 4.1 in 4. Investor i has risk aversion coefficient x and ambiguity aver- i Deng et al. (2005) with minor modifications. Let sion coefficient (diffidence level) . i Xm 1 x H c ¼ i i x0; j ¼ 1; 2; ...; n þ 1; We have j 2x H ij i¼1 i Xm 0 0 xij ¼ xj ; j ¼ 1; 2; ...; n þ 1; ð3Þ and i¼1 0 Xn dj ¼ fj=xj ;;j ¼ 1; 2; ...; n: 0 0 0 pjxij þ xinþ1 ¼ Wi : ð4Þ j¼1 Let c be the vector with components ðc1; ...; cnÞ and d the vector T with components ðd1; ...; dnÞ. Then we can express a as a ¼ c d. The system (11) can now be re-written as 3.1. Uniform markets Xm Xn Xm fj 1 xi H i 0 fj 1 xi H i 0 pj ¼ 0 pkxik þ 0 xinþ1 Using the result from the previous section we have that each x 2xi H x 2xi H j i¼1 k¼1 j i¼1 investor i holds the percentage portfolio Xn Xm 1 xi H i 0 ¼ dj pk xik þ djcnþ1 1 xi H i 1 2x H k¼1 i¼1 i yij ¼ ðC l^Þj; j ¼ 1; 2; ...; n; ð5Þ 2xi H Xn Xn Xn 1 x H ¼ ckpk þ djcnþ1; j ¼ 1; 2; ...; n: y ¼ 1 y ¼ 1 i i ðC 1l^Þ ð6Þ inþ1 ij 2x H j k¼1 j¼1 i j¼1 In vector form we have the equation under the assumption that each investor i operates under limited T p ¼ dðc pÞþcnþ1d; diffidence, i.e., i < H; i ¼ 1; ...; m. Passing to the corresponding as- set portfolio holdings (shares) x we have ij or, equivalently 0 0 W y W 1 x H T x ¼ i ij ¼ i i i ðC 1l^Þ ; j ¼ 1; 2; ...; n; ð7Þ ðI dc Þp ¼ c d: ij p p 2x H j nþ1 j j i ! ! Xn Xn 1 x H Then, when a–1 the system has the unique solution x ¼ W0y ¼ W0 1 y ¼ W0 1 i i ðC 1l^Þ : ð8Þ inþ1 i inþ1 i ij i 2x H j ! j¼1 i j¼1 T 1 dc c p ¼ c ðI dcT Þ d ¼ c I þ d ¼ nþ1 d; ð13Þ The market clearing condition requires the following equation nþ1 nþ1 1 a 1 a to hold:
Xm where the second equality follows from the Sherman–Morrison– 0 Woodbury formula.1 The rest of the proof consists of applying Farkas xij ¼ xj ; j ¼ 1; 2; ...; n þ 1; ð9Þ i¼1 Lemma (c.f. chapter 2 of Mangasarian (1994)) to the system i.e., we have T ðI dc Þp ¼ cnþ1d; p P 0; Xm 0 W 1 x H i i i ðC 1l^Þ ¼ x0; j ¼ 1; 2; ...; n þ 1; ð10Þ and its alternative p 2x H j j i¼1 j i ðI cdT Þy 6 0; dT y > 0; Re-arranging this equation and recalling (4) we have the equation system with n equations and n unknowns: under the conditions c P 0; fj P 0 for all j ¼ 1; ...; n and a < 1. If ! a < 1, then the unique solution in (13) is non-negative. If a P 1, Xm Xn 1 xi H i then y ¼ c satisfies the alternative system, hence no non-negative p x0 ¼ðC 1l^Þ p x0 þ x0 ; j j j 2x H k ik inþ1 equilibrium prices exist. h i¼1 i k¼1
j ¼ 1; 2; ...; n: ð11Þ The scalar a plays an important role in the existence of equilib- 1 rium results (see also Deng et al. (2005), Konno & Shirakawa (1995) Now, define for convenience fj ¼ðC l^Þj for j ¼ 1; 2; ...; n and and the scalar c in Corollary 1 below). However, a financial inter- pretation of the condition involving a is missing from the litera- Xm Xn 0 1 x H x ture. Note that the double summation in a, considered without a ¼ i i ij f : 2x H x0 j i¼1 j¼1 i j 1 T 1 1 A 1 uvT A 1 ðA þ uv Þ ¼ A 1 . 1þvT A u M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965 961
x0 Corollary 1. In a mean–variance investors’ market (with no ambi- the ratio term ij (which represents the investor i’s initial fraction of x0 j guity aversion) if c–1, then there exists a unique solution p to the shares of asset j) would give the total of fraction portfolio holdings equilibrium system (11) given by (yij) in the market, summed over all investors and all risky assets. f Xm Thus, the scalar a gives a measure of the weighted total of fraction mv 1 j 1 xi 0 pj ¼ xinþ1; j ¼ 1; ...; n: ð14Þ 0 portfolio holdings where each y is weighted by the corresponding 1 c xj 2xi ij i¼1 x0 P ij m 1 xi 0 ratio . If this weighted total is strictly less than 1, an equilibrium If x P 0; j ¼ 1; 2; ...; n þ 1, and fj P 0 for all x0 i¼1 2xi ij j price exists as is shown in the proposition above. The condition is j ¼ 1; ...; n, then the market admits a unique non-negative equilib- also necessary. The condition rium price vector p if and only if c < 1. Xm 1 x H As in Proposition 2 the scalar c gives a measure of the weighted i i 0 xij P 0; j ¼ 1; 2; ...; n þ 1 total of fraction portfolio holdings where each yij is weighted by 2xi H i¼1 x0 ij the corresponding ratio x0. also represents a weighted total of initial portfolio holdings over all j 1 xi H i An interesting case is when all ambiguity-averse investors agree investors in the market. The weight 2x H encodes the risk i on the same level of limited diffidence, i.e., i ¼ < H for all aversion and ambiguity aversion attitudes of the investor. i ¼ 1; ...; m. In that case, the equilibrium price vector p has a sim- The existence of strictly positive prices is a harder question that plified expression: is rarely addressed (with the exception of Rockafellar et al. (2007)) Xm H f 1 x although zero prices would hardly make economic sense in prac- pH ¼ j i x0 ; j ¼ 1; ...; n: ð15Þ j Hð1 aÞ x0 2x inþ1 tice. Interestingly, we can also prove the following negative result j i¼1 i on the existence of a strictly positive system of equilibrium prices. H ðH Þð1 cÞ mv P Obviously, the above expression implies pj ¼ H ðH Þc pj . Now, m 1 x H If the condition of Proposition 2 i i x0 P 0; j ¼ 1; since we have i¼1 2xi H ij 2; ...; n partially holds (only for the risky assets), i.e., a weighted ðH Þð1 cÞ H Hc þ c 0 < ¼ < 1 total of initial portfolio holdings of risky assets over all investors H ðH Þc H Hc þ c in the market is non-negative, while this total is negative for the riskless asset then it is not possible to have positive equilibrium as c < 1 in equilibrium, and H > > 0. Therefore, in a homoge- prices in the market. neously and mildly diffident ambiguity-averse mean–variance investors’ market (where diffidence is bounded above by the slope P m 1 xi H i 0 of the Capital Market Line), equilibrium prices are under downward Proposition 3. If i¼1 2x H xij P 0; j ¼ 1; 2; ...; n; cnþ1 < 0, i pressure with respect to a purely confident mean–variance inves- no investor is short on risky assets, i.e., f P 0 for all j ¼ 1; ...; n, and j tors’ market. We summarize these observations below. a 2ð0; 1Þ then a strictly positive equilibrium price system does not exist in an ambiguity-averse mean–variance investors’ market where Proposition 4. In a homogeneously and mildly diffident (where all every investor has limited diffidence (i.e., < H for all i ¼ 1; ...; m). i investors have the same < H) ambiguity-averse mean–variance investors’ market in equilibrium prices are smaller than the equilib- Proof. We shall invoke the non-homogeneous Stiemke theorem rium prices in a pure mean–variance investors’ market. (Stiemke, 1915) for the system: Another interesting observation is the following. Assume no 0 investor has an initial liability, i.e., xij > 0 for all i ¼ 1; ...; m and T ðI dc Þp ¼ cnþ1d; p > 0; fj > 0 for all j ¼ 1; ...; n. Then we have the immediate conse- mv quence that a < c. This implies straightforwardly that pj < pj , The alternative of the above system according to Stiemke’s theo- for all j ¼ 1; ...; n. In other words, in an ambiguity-averse mean–var- rem2 is the system ! ! iance investors market with bounded diffidence, if all investors have T T long initial positions, then equilibrium leads to smaller prices compared I cd I cd x P 0; x – 0; to the equilibrium prices of purely mean–variance investors’ market, c d c d nþ1 nþ1 everything else being equal. Hence, the introduction of ambiguity aver- If x ¼ c then sion or diffidence in rate of return estimates into a market with all po- sitive initial positions creates a deflationary pressure on equilibrium I cdT cð1 aÞ x ¼ : prices. cnþ1d cnþ1a A Numerical Example. For illustration we consider an example P m 1 xi H i 0 with three investors and three assets (two risky assets and one Since by assumption we have i 1 x P ¼ 2xi H ij riskless asset). The relevant data for the risky assets are specified 0; j ¼ 1; 2; ...; n, we have c P 0. Due to the hypotheses that as follows a 2ð0; 1Þ and cnþ1 < 0 we have x ¼ c that satisfies the alternative T system. h l^ ¼ð0:1287 0:1096Þ If the market consists of fully confident (in the mean rate of 0:4218 0:0530 return estimates) investors (i.e., ambiguity-neutral), we have the C ¼ : 0:0530 0:2230 following equilibrium result in a mean–variance capital market. Let us define for convenience We assume x0 ¼ 10 for all three assets j ¼ 1; 2; 3, and the initial j Xm Xn 0 portfolio holdings 1 x x c ¼ i ij f : 2x x0 j T T T i¼1 j¼1 i j ½433 ; ½622 ; ½334
2 T T for each asset respectively, e.g., investor 1 holds initially 4 shares of Stiemke’s Theorem: Either A y ¼ b; y > 0 has a solution or Ax P 0; b x P 0; Ax asset 1, 6 shares of asset 2 and 3 units of the riskless asset. We have – 0 has a solution, but never both, c.f. Chapter 6 of Panik (1993). T bT x H ¼ 0:2822 and f ¼ C 1l^ ¼½0:2509 0:4319 .InFig. 1 we plot the 962 M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965
uniformly ambiguity−averse investors turn more risk−averse at epsilon=0.01 al investors turn increasingly more ambiguity−averse at omega=0.5 50 0.35
45 0.3 40 0.25 35
30 0.2
25 price
price 0.15 20
15 0.1
10 0.05 5 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 epsilon omega Fig. 3. Effect of increasing ambiguity aversion equally across the board with Fig. 1. Effect of increasing risk aversion coefficient x when all investors are equally xi ¼ 0:5 for i ¼ 1; 2; 3. ambiguity averse with i ¼ 0:01 for i ¼ 1; 2; 3. evolution of the prices of the two risky assets in a uniformly ambi- coefficient < H and risk aversion coefficient x2. All other guity-averse investors’ market with i ¼ 0:01 for i ¼ 1; 2; 3. Increas- assumptions about the assets traded in the market are still valid. ing x, i.e., increasing the risk aversion of investors (expressed as an The ambiguity-neutral investor makes the portfolio choice increasing emphasis on a smaller variance of portfolio return) ! 0 Xn equally for all investors while ambiguity aversion remains fixed W 1 x 1 x x ¼ 1 1 f ; j ¼ 1; ...; n; x ¼ W0 1 1 f ; across the board has a sharp deflationary effect on asset prices. In 1j p 2x j 1nþ1 1 2x j j 1 1 j¼1 Figs. 2 and 3 we show the impact of increasing ambiguity aversion equally for all investors at two different levels of risk aversion, while the ambiguity-averse investor makes the choice x ¼ 0:25 and x ¼ 0:5, respectively. Both figures show clearly the W0 ð1 x ÞðH Þ deflationary effect on asset prices of increasing ambiguity aversion x ¼ 2 2 f ; j ¼ 1; ...; n; 2j p 2Hx j at both levels of risk aversion. The decrease in prices in response to j 2 ! Xn an increase in ambiguity aversion is much more pronounced when ð1 x ÞðH Þ x ¼ W0 1 2 f : the investors are less risk-averse at x ¼ 0:25. 2nþ1 2 2Hx j 2 j¼1
3.2. Mixed markets As in the proof of Proposition 2 we define
1 1 x1 0 Consider now a uniform market with ambiguity-neutral inves- cj ¼ x1j j ¼ 1; ...; n þ 1; 2x1 tors where an investor decides to adopt an ambiguity-averse posi- tion. For simplicity we shall examine the case where we have two for investor 1, and investors. Investor indexed 1 is ambiguity-neutral with risk aver- 1 x H ~2 2 0 sion coefficient x1, investor indexed 2 is ambiguity-averse with cj ¼ x2j j ¼ 1; ...; n þ 1; 2x2 H all investors turn increasingly more ambiguity=averse at omega=0.25 for investor 2, and d ¼ f =x0 for j ¼ 1; ...; n. Then we can express the 50 j j j equilibrium price system for the mixed market as 45 c1 þ ~c2 pm ¼ nþ1 nþ1 d; 40 1 am 35 where am ¼ dT ðc1 þ ~c2Þ, and we assume that the conditions guaran- teeing the non-negativity of pm as expressed in Proposition 2 hold. 30 Now, we compare the equilibrium price system pm to the equi- 25 librium price system of a uniform ambiguity-neutral investors
price market. I.e., if investor 2 were to be ambiguity-neutral as well, 20 we would have the following price system pp: 15 c1 þ c2 pp ¼ nþ1 nþ1 d; 10 1 ap T 5 where ap ¼ d ðc1 þ c2Þ with
0 2 1 x2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 cj ¼ x2j j ¼ 1; ...; n þ 1: 2x2 epsilon We assume again the conditions guaranteeing the non-negativ- p Fig. 2. Effect of increasing ambiguity aversion equally across the board with ity of p expressed in Corollary 1 hold. Now, it is a simple exercise xi ¼ 0:25 for i ¼ 1; 2; 3. to see that M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965 963
H ~2 2 (i) a portfolio composed of the riskless asset andÀÁ a non-negative cj ¼ cj ; j ¼ 1; ...; n þ 1 0 0 0 0 H multiple ki of the initialP total holdings x ¼ x1; x2; ...; xn of m and risky assets, where i¼1ki ¼ 1 and ð1 aÞW0ð1 x ÞðH Þ m p H T 2 i i i ki ¼ P a ¼ a þ 1 d c : m 1 xk 0 H xi ðH kÞx k¼1 xk knþ1 These observations imply that for i ¼ 1; ...; m;
1 H 2 (ii) a percentage portfolio which is a linear (ni; 1 ni) combination m cnþ1 þ cnþ1 p ¼ ÀÁH d: of the percentage riskless portfolio ð0; 0; ...; 0; 1Þ and the (aug- p H T 2 ÀÁ 1 a þ 1 H d c mented) master fund z1; z2; ...; zn; 0 consisting only of risky T 1 xi H 2 2 m p assets where ni ¼ ðB RAÞ. Therefore, if cnþ1 > 0 and d c > 0, we have p < p , i.e., if an xi H investor with positive initial holdings moves from ambiguity-neu- tral position to (bounded) ambiguity-averse position, this change has a deflationary effect on equilibrium prices. We summarize this Proof. Recall that in equilibrium each investor i holds the optimal result below. We define portfolio 0 0 i 1 xi 0 Wi ð1 xiÞðH iÞfj Wi ð1 xiÞðH iÞfj c ¼ x j ¼ 1; ...; n þ 1; i ¼ 1; ...; m x ¼ ¼ P j ij ij 1 fj m 1 x 0 2xi 2xiHpj k 2xi 1 a x0 k¼1 2x ðH Þ xknþ1 j k k ÀÁfor every investor i, and refer to the n-vector with components 0 i i i Wi ð1 xiÞðH iÞð1 aÞ 0 c1; ...; cn as c . ¼ P x : m 1 xk 0 j xi ðH kÞx k¼1 xk knþ1 Proposition 5. In a uniform market of m ambiguity-neutral mean– P Since we have m x ¼ xj , we infer immediately that variance investors in equilibrium, assume investor m adopts an P i¼1 ij 0 m k ¼ 1. This proves part (i). ambiguity-averse attitude with coefficient < H. Then the following i¼1 i ð1 xiÞðH iÞ For part (ii), recall that y ¼ fj and statements hold: ij 2xiH ð1 x ÞðH Þ y ¼ 1 i i eT f. Since eT f ¼ B RA, we can re-write inþ1 2xiH ð1 x ÞðH Þ ð1 x ÞðH Þ 1. A non-negative equilibrium price system y ¼ i i ðB RAÞz and y ¼ 1 i i ðB RAÞ. ij 2xiH j inþ1 2xiH P h m 1ci þ ~cm Hence, the result follows. pm ¼ i¼1 nþ1 nþ1 d 1 am We note that the weight ni in part (ii) of the previous result is smaller m m exists, if and only if a < 1 where a is defined as than the corresponding weight that would result if were taken equal to ! i Xm 1 zero, i.e., the investor were ambiguity-neutral. This observation implies T am ¼ d ci þ ~cm : that ambiguity aversion leads to giving less weight to master fund z . i¼1 Let the vector yM and zM be defined with components
0 0 2. If the initial holdings xmj for all j ¼ 1; ...; n þ 1 of investor m are x p M j j positive, the equilibrium prices of the mixed market are smaller y ¼ P ; j ¼ 1; 2; ...; n þ 1; j nþ1x0p than the equilibrium prices of the uniform market. j¼1 j j and The above result is not surprising if one bears in mind that an 0 x pj ambiguity-averse investor holds smaller long positions in risky as- zM ¼ P j ; j ¼ 1; 2; ...; n; sets compared to an ambiguity-neutral investor, which leads to a j nþ1 0 j¼1 xj pj decreased demand for risky assets, and hence exerts a downward pressure on equilibrium prices. called, respectively, the market portfolio of all assets and the mar- A similar analysis can be made when the ambiguity aversion of ket portfolio of risky assets in Deng et al. (2005). We also have one investor is not classified as mildly diffident, but rather, signif- the following proportion property. icantly diffident, i.e., with P H, in which case this investor would put all his/her initial wealth into the riskless asset. It can be shown Proposition 7. Let the capital market be in equilibrium. Then the again that such behavior leads to a drop in equilibrium prices. This following hold: is left as an exercise. (i) the market portfolio yM is proportional to the market portfolio zM of risky assets; 4. Properties of the equilibrium price system (ii) the market portfolio zM of risky assets is identical to z . We devote this section to the study of some interesting proper- ties of portfolios in equilibrium. More precisely, we follow the ref- Proof. Using the definition of yM we have erences Deng et al. (2005), Konno and Shirakawa (1994), Konno P and Shirakawa (1995) to examine the properties of the portfolios m i¼1xijpj in equilibrium in a market of mildly diffident mean–variance yM ¼ P P P j n m x p þ m x investors. Define the master fund z ¼ f =eT f for all j ¼ 1; ...; n. j¼1 i¼1 ij j i¼1 inþ1 j j hiP We begin with the following two-fund separation property. Let us m ð1 xiÞðH iÞ 0 i¼1 2x H Wi fj define A ¼ eT C 1e and B ¼ eT C 1^r. ¼ P hiP i P n m ð1 xiÞðH iÞ 0 m 0 ð1 xiÞðH iÞ t j¼1 i¼1 2x H Wi fj þ i¼1Wi 1 2x H e f hii hii Proposition 6. Let the price system in the mildly diffident mean– P P m ð1 xiÞðH iÞ 0 m ð1 xiÞðH iÞ 0 W fj ðB RAÞ W variance investors’ market be as defined in (12). Then, after the i¼1 2xiH i i¼1 2xiH i ¼ P ¼ P z : transaction, each investor i holds m 0 m 0 j i¼1Wi i¼1Wi 964 M.Ç. Pınar / European Journal of Operational Research 237 (2014) 957–965
For the second part we have H have the beta that is scaled by the ratio H which is a number hiP P m 1 H 0 larger than one when we have 0 < < H. Therefore, the constant m ð xiÞð iÞ x p i¼1 2x H Wi fj M P iP¼1 ij j hii of proportionality and hence the new beta which relates in our case z ¼ n m ¼ P P ¼ z : j n m ð1 x ÞðH Þ 0 j j 1 i 1x pj i i the nominal excess return to the total worst case return of the mar- ¼ ¼ ij j¼1 i¼1 2x H Wi fj i ket is larger than the beta of the classical CAPM. Let the random (uncertain) rate of return of the market portfolio be denoted by 5. Conclusions
Xn In this paper, we analyzed existence of equilibrium in a finan- r r zM; M ¼ j j cial market composed of risky assets and a riskless asset, where j¼1 mean–variance investors can display aversion to ambiguity, i.e., with the worst-case value aversion to imprecision in the estimated mean rates of return of
Xn risky assets. We first derived a closed-form optimal portfolio rule M for a mean–variance investor with aversion to ambiguity modeled rM ¼ E½rM ¼ rj zj : j¼1 using an ellipsoidal uncertainty set, borrowing the concept from robust optimization. The optimal portfolio rule reduces to the port- where r is as defined in (2). It is the rate of return where the max- folio choice of a mean–variance investor when the investor is imum in the min max portfolio selection model AAMVP of Sec- ambiguity-neutral. We examined conditions under which an equi- tion 2 is attained. Then, we have the following CAPM-like librium exists in a market of ambiguity-averse investors as well as property which expresses the nominal excess rate of return of risky conditions that lead to deflationary pressure on equilibrium prices asset j as proportional to the worst-case excess rate of return of the with respect to a pure mean–variance investors’ (i.e., ambiguity- market portfolio of risky assets. In addition to the terms that are neutral) market. A CAPM-like result is derived, which reduces to encountered in the classical CAPM, the proportionality also depends the usual CAPM in the absence of ambiguity aversion. on the square root of the market optimal Sharpe ratio H2 and the Future research can address equilibrium in the absence of the ambiguity aversion coefficient . riskless asset, limitations or exclusion of short sales, equilibrium with other risk measures such as robust CVaR or expected shortfall Proposition 8. Let a capital market of homogeneously diffident under mean return ambiguity, and equilibrium under transaction investors with common < H be in equilibrium. Then the excess costs. nominal rate of return on each risky asset is proportional to the excess worst-case rate of return on the market portfolio of risky assets; i.e., the following holds References
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