A Robust Capital Asset Pricing Model

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Robust Capital Asset Pricing Model Doriana Ruffino 2014-01 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A Robust Capital Asset Pricing Model (cid:3) Doriana Ru¢ no y December 17, 2013 Abstract We build a market equilibrium theory of asset prices under Knightian uncertainty. Adopting the mean-variance decisionmaking model of Maccheroni, Marinacci, and Ru¢ no (2013a), we derive explicit demands for assets and formulate a robust version of the two-fund separation theorem. Upon market clearing, all investors hold ambiguous assets in the same relative proportions as the assets(cid:146)market values. The resulting uncertainty-return tradeo⁄is a robust security market line in which the ambiguous return on an asset is measured by its beta (systematic ambiguity). A simple example on portfolio performance measurement illustrates the importance of writing ambitious, robust asset-pricing models. The views in this paper are solely the responsibility of the authors and should not be interpreted (cid:3) as re(cid:135)ecting the views of the Board of Governors of the Federal Reserve System or of any other person associated with the Federal Reserve System. Board of Governors of the Federal Reserve System. E-mail address: doriana.ru¢ no@frb.gov. y 1

1 Introduction The Great Recession of 2008 has prompted calls for a microeconomic theory to predict the behavior of capital markets under Knightian uncertainty.1 Although many insights can be drawn from the capital asset pricing model (CAPM) under risk, and from its many variations2, thusfarfewstudieshaveexploredthee⁄ectsonequilibriumassetpricesofuncertainty in investors(cid:146)preferences. We develop a robust capital asset pricing model (RCAPM) that o⁄ers powerful predictions about how to measure the uncertainty-return tradeo⁄. Also, the model(cid:146)s analytical tractability renders it immediately applicable to capital-budgeting estimations, to the evaluation of professionally managed portfolios, and so on. Since the CAPM of Treynor (1962), Sharpe (1964), Lintner (1965), and Mossin (1966) is foundedontheformalquantitativetheoryforoptimalportfolioselectionofMarkowitz(1952) andTobin(1958),we(cid:133)rstproposeageneralsolutionfortheportfolio-selectionproblemunder uncertainty. Our objective function features the second-order approximation of the certainty equivalentofMaccheroni, Marinacci, andRu¢ no(2013a), whichistheanalogeoftheArrow- Prattapproximationundermodeluncertainty. Italsointroducesanambiguitypremiumthat captures variations in returns due to model uncertainty. First, we (cid:133)nd a mean-variance e¢ cient portfolio that depends on the investor(cid:146)s tastes (cid:150)his aversion to risk and uncertainty (cid:150)as well as his beliefs over expected returns. Next, we use the optimal solution to derive a robust version of the two-fund separation theorem: here, all mean-variance e¢ cient portfolios come from combining the riskless asset with the mean-variance e¢ cient portfolio made of ambiguous assets only. Last, assuming that all investors are mean-variance optimizers who make decisions according to the same normative model, we derive the set of prices at which everyone(cid:146)s demand is satis(cid:133)ed in equilibrium. The resulting relationship between asset returns and uncertainty resembles the CAPM securitymarketlineunderrisk. Inaddition,itdisplaysarobustbetacoe¢ cientthatmeasures an asset(cid:146)s systematic ambiguity. Given the results of our model, we discuss the case in which 1We use the terms Knightian uncertainty (from Knight, 1921) and ambiguity interchangeably. 2For example, Merton (1973) proposes an intertemporal model for the capital market. Lucas (1978), Breeden (1979), and Grossman and Shiller (1981) derive consumption-based asset-pricing models. 2

the existence of superior-performance assets (that is, assets whose robust alpha coe¢ cient is greater than zero) allows for the creation of (cid:147)super-e¢ cient(cid:148)portfolios. Our work is related to recent research on optimal portfolio-selection theory under risk and uncertainty. Maccheroni, Marinacci, and Ru¢ no (2013a) and Gollier (2011) study the e⁄ects of higher ambiguity aversion on optimal portfolio rebalancing. They also set the conditions under which ambiguity reinforces (or mitigates) risk. Izhakian and Benninga (2008) (cid:133)nd similar results on the separation of risk and ambiguity. Garlappi, Uppal, and Wang (2007) and Chen, Ju, and Miao (2013) contrast the optimal portfolio allocation from the Bayesian and the ambiguity models. Portfolio optimization under uncertainty also helps to explain some puzzling investor behaviors. For example, Epstein and Miao (2003) and Boyle, Garlappi, Uppal, and Wang (2012) provide a rational justi(cid:133)cation for holding (cid:147)familiar(cid:148) assets. Easley and O(cid:146)Hara (2009) and Illeditsch (2011) tackle poor investor participation in the stock market. Wesharesomefeatureswiththecapital asset-pricingmodelsof ChenandEpstein(2002); Collard, Mukerji, Sheppard, and Tallon (2011); Ju and Miao (2012); and Izhakian (2012). In Chen and Epstein(cid:146)s (2002) model, excess returns also re(cid:135)ect a compensation for risk and a separate compensation for ambiguity. In Collard and others (2011) and Ju and Miao(cid:146)s (2012)models, theinvestor(cid:146)spessimisticbehavioristiedtoavarietyofdynamicasset-pricing phenomena (equity premium, risk-free rate, and so forth). In Izhakian(cid:146)s (2012) model, equilibrium prices contain a systematic beta similar to ours. We di⁄er in that his derivation is founded on shadow probability theory, while ours is based on the smooth preferences of Klibano⁄, Marinacci, and Mukerji (2005). The rest of the paper is organized as follows. In Section 2 we introduce the mathematical setup,inSection3wederivethemean-variancee¢ cientportfolioandpresentarobustversion of the two-fund separation theorem, in Section 4 we obtain the RCAPM, and in Section 5 we conclude. 3

2 Theoretical decision framework The sure amount of money that a von Neumann-Morgenstern expected-utility maximizer with utility u and wealth w considers equivalent to a risky investment h is given by c(w+h;P) = u 1(E (u(w+h))); (1) (cid:0) P where P is the probabilistic model that describes the stochastic nature of the problem. The classic approximation of (1) by Arrow (1971) and Pratt (1964) 1 c(w+h;P) w+E (h) (cid:21) (w)(cid:27)2 (h); (2) (cid:25) P (cid:0) 2 u P where (cid:21) = u =u denotes the decisionmaker(cid:146)s risk attitude, is widely used in models of u 00 0 (cid:0) investment because it ties the risk premium associated with h to its variance, (cid:27)2 (h). But P if a decisionmaker is uncertain about the true probabilistic model P and instead adopts alternative models Q, then c(w+h;Q) becomes a variable amount of money that depends on Q. The smooth characterization of (1) under ambiguity is the certainty equivalent of Klibano⁄, Marinacci and Mukerji (2005): C(w+h) = v 1(E (v(c(w+h)))) (3) (cid:0) (cid:22) = v 1 E v u 1(E(u(w+h))) ; (cid:0) (cid:22) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1)(cid:1)(cid:1) where (cid:22) denotes the decisionmaker(cid:146)s priorprobabilityonthe space of possible models Q, and v is his attitude toward model uncertainty. In Maccheroni, Marinacci, and Ru¢ no (2013a) we derive a second-order approximation of (3): 1 1 C(w+h) w+E (h) (cid:21) (w)(cid:27)2 (h) ((cid:21) (w) (cid:21) (w))(cid:27)2 (E(h)); (4) (cid:25) Q(cid:22) (cid:0) 2 u Q(cid:22) (cid:0) 2 v (cid:0) u (cid:22) (cid:22) where (cid:21) = v =v is the decisionmaker(cid:146)s ambiguity attitude, Q is the reduced probability v 00 0 (cid:0) induced by the prior (cid:22), and E(h) is the random variable that associates the expected value 4

E (h) to each model Q. Q The last term in (4) (cid:150)the ambiguity premium (cid:150)is new relative to (2).3 Speci(cid:133)cally, the ambiguitypremiumchanges the certaintyequivalent throughthe decisionmaker(cid:146)s aversionto ambiguity (cid:21) as well as the variance of the return E(h). Hence, this parsimonious extension v of the mean-variance model under risk is fully determined by three parameters: (cid:21) , (cid:21) , u v and (cid:22). Higher values of (cid:21) and (cid:21) indicate stronger negative attitudes toward risk and u v ambiguity, respectively. Higher values of (cid:27)2 (E(h)) indicate poorer information on outcomes (cid:22) andmodels. Inthe special case of (cid:27)2 (E(h)) = 0 (that is, where (cid:22) is a trivial measure), there (cid:22) is no source of model uncertainty and (4) reduces to (2). Last, the variance decomposition between state and model uncertainty, (cid:27)2 (h) = E (cid:27)2(h) +(cid:27)2 (E(h)); (5) Q(cid:22) (cid:22) (cid:22) (cid:0) (cid:1) allows us to rearrange (4) by the Arrow-Pratt coe¢ cients of u and v as follows: (cid:21) (w) (cid:21) (w) C(w+h) w+E (h) u E (cid:27)2(h) v (cid:27)2 (E(h)):4 (6) (cid:25) Q(cid:22) (cid:0) 2 (cid:22) (cid:0) 2 (cid:22) (cid:0) (cid:1) In (6), risk aversion and ambiguity aversion determine the decisionmaker(cid:146)s response to the average variance, E ((cid:27)2(h)), and the variance of averages, (cid:27)2 (E(h)), respectively. (cid:22) (cid:22) Next, we set (cid:21) (w) = (cid:21) and (cid:21) (w) (cid:21) (w) = (cid:18). Then, a decisionmaker is risk averse u v u (cid:0) when (cid:21) > 0 and ambiguity averse when (cid:18) > 0.5 Last, we assume that The ratio of (cid:18) to (cid:21) is equal for all investors. (cid:15) Q (cid:22) is equal to the baseline probability P.6 (cid:15) 3Nau (2006), Izhakian and Benninga (2008), and Jewitt and Mukerji (2011) obtain approximations for the ambiguity premium on the basis of special assumptions. 4This formulation shows that, when the indexes u and v are su¢ ciently smooth, both state and model uncertainty have at most a second order e⁄ect on the evaluation. In Maccheroni, Marinacci, and Ru¢ no (2013b)westudyindetailordersofriskaversionandofmodeluncertaintyaversioninthesmoothambiguity model. 5Ambiguity neutrality is modeled as (cid:18) =0. 6Under (ii), the certainty equivalent (4) is always (cid:133)nite. 5

In Section 3 we apply the (cid:147)enhanced(cid:148)Arrow-Pratt approximation to the mean-variance model of optimal portfolio-selection theory. 3 Mean-variance portfolio theory We allow for an arbitrary number of investors who make portfolio decisions based on their prior probability (cid:22) on the space of possible probabilistic models of their end-of-period wealth.7 For the normative results that follow, investors need not agree on the prospects of the various investments; thus, in general, beliefs are not homogenous. The market is formed of n ambiguous assets ((cid:27)2 (E(r )) > 0, i = 1;:::;n) with expected rate of return r and a (cid:22) i i risk-less asset, whose return r is known with certainty. Denote by r the vector of returns f on the (cid:133)rst n assets and by w the mean-variance e¢ cient portfolio with expected return b r = r +w (r r 1); (7) w f f (cid:1) (cid:0) b b where 1 is the n-dimensional unit vector. We assume a friction-less market environment in which assets are traded in the absence of transaction costs, of spreads between the borrowing andthelendingrates, andofshortsalerestrictions. From(4)and(7), w mustbethesolution to the portfolio problem: b (cid:21) (cid:18) maxC(r ) = max E (r ) (cid:27)2 (r ) (cid:27)2 (E(r )) : (8) w Rn w w Rn P w (cid:0) 2 P w (cid:0) 2 (cid:22) w 2 2 (cid:18) (cid:19) To deliver the optimality condition, set E [r r 1] = [E (r r );:::;E (r r )]|; P f P 1 f P n f (cid:0) (cid:0) (cid:0) (cid:6) [r] = (cid:27) (r ;r )n ; P P i j i;j=1 (cid:6) [E[r]] = (cid:27) (E(r );E(r ))n ; (cid:22) (cid:22) i j i;j=1 (cid:4) = (cid:21)(cid:6) [r]+(cid:18)(cid:6) [E[r]]: P (cid:22) 7Investorsaremyopicinthesensethattheyfocusontheirwealthonlyonedecision-periodaheadoftheir current decision. 6

Hence, from (8), we have that 1 maxC(r ) = max w E [r r 1] w|(cid:4)w : w P f w Rn w Rn (cid:1) (cid:0) (cid:0) 2 2 2 (cid:18) (cid:19) The (cid:133)rst-order condition for a maximum is: E [r r 1] (cid:4)w= 0; P f (cid:0) (cid:0) b which can be solved by matrix inversion assuming that (cid:4) is positive-de(cid:133)nite. The meanvariance e¢ cient portfolio is: w = [(cid:21)(cid:6) [r]+(cid:18)(cid:6) [E[r]]] 1E [r r 1]; (9) P (cid:22) (cid:0) P f (cid:0) where (cid:6) [r] and (cid:6) [E[br]] are the variance-covariance matrixes of returns and expected P (cid:22) excess returns under P and (cid:22), respectively, and E [r r 1] is the vector of expected excess P f (cid:0) returns under P. The optimal solution has the merit to naturally adjust the classic risk model to re- (cid:135)ect investors(cid:146)uncertainty over expected returns, (cid:6) [E[r]]. In fact, if investors are either (cid:22) ambiguity-neutral ((cid:18) = 0) or approximately unambiguous ((cid:6) [E[r]] = 0), the Markowitz- (cid:22) Tobin derivation readily obtains.8 The form of (9) is especially convenient because it allows for a direct application of the ample research on mean-variance preferences developed for problems involving risk to the analysis of problems involving ambiguity. In particular, provided that information on r , E [r], (cid:6) [r], and (cid:6) [E[r]] is available, all mean-variance f P P (cid:22) investors use the normative model (8) to select the optimal combination of ambiguous assets withthe risk-free asset. The resulting allocationis a smoothfunctionof the taste parameters (cid:21) and (cid:18), which is particularly well suited for comparative statics analysis. Maccheroni, Marinacci and Ru¢ no (2013a) exhaustively map the conditions under which higher ambiguity aversion (or higher ambiguity in expectations) lowers an investor(cid:146)s optimal exposures to the 8Approximately unambiguous prospects are de(cid:133)ned by Maccheroni, Marinacci, and Ru¢ no (2013a), p. 1086. 7

ambiguous assets, spurring severe (cid:147)(cid:135)ights-to-quality(cid:148)and investments (cid:147)in the familiar.(cid:148)9;10 Now assume that investors share the same prior probability (cid:22) on the space of possible probabilistic models. Then, for E [r r 1] > 0, the allocations to ambiguous assets in P f (cid:0) (9) have the same relative proportions, independent of wealth, risk aversion, or ambiguity aversion, as long as (cid:18) is a (cid:133)xed proportion of (cid:21) equal for all investors. We have that w n (cid:24) E (r r ) i = l=1 l;i P l (cid:0) f ; i;j = 1;:::;n; w n (cid:24) E (r r ) 8 j Pl=1 l;j P l (cid:0) f b P where (cid:24) is de(cid:133)ned ascthe i;j element of the inverse of (cid:4). That is, (cid:4) 1 = (cid:24) n . i;j (cid:0) i;j i;j=1 We de(cid:133)ne the optimal combination of ambiguous assets (OCAA) as th(cid:2)e m(cid:3)ean-variance e¢ cientportfoliothatcontainsambiguousassetsonly. Labelingw\OCAA thefractionofOCAA i made up by i, we have that: w\OCAA solves (8). (cid:15) i n w\OCAA = 1. (cid:15) i=1 i P Thus, n (cid:24) E (r r ) w\OCAA = l=1 l;i P l (cid:0) f ; i = 1;:::;n: (10) i n n (cid:24) E (r r ) 8 iP=1 l=1 l;i P l (cid:0) f Theorem 1 Denote by (cid:25) thePfractPion of an investor(cid:146)s mean-variance e¢ cient portfolio (9) that is allocated to the OCAA portfolio. From (10), it follows that the fraction of the investor(cid:146)s portfolio allocated to asset i is w = (cid:25)w\OCAA. i i In other words, the mean-variance e¢ cbient portfolio with expected return r w can be constructed from mixing the optimal combination of ambiguous assets with the risk-less asset (cid:150) b a robust two-fund separation theorem. In particular r = r +(cid:25) r r 1 ; (11) w f (cid:1) w\OCAA (cid:0) f (cid:16) (cid:17) 9Allocation strategies driven by anbinvestor(cid:146)s geographical or professional proximity to a particular stock are generally conceptualized in the term familiarity. See, among others, Huberman (2001). 10Gollier (2011) makes a similar point in a static two-asset portfolio problem with one safe asset and one uncertain one. 8

where r is the expected return on OCAA. w\OCAA 4 Equilibrium prices of capital assets In this section we propose a positive asset-pricing theory that formalizes the relationship between asset returns and uncertainty, assuming that investors follow the mean-variance norm (9). First, we derive the set of prices that clears the market. Then, we discuss superior portfolio performance measurement when the RCAPM fails. Theorem (1) characterizes an investor(cid:146)s demand function with respect to the meanvariancee¢ cientportfoliothatcontainsambiguousassetsonly. Thus, therelativeproportion of i to j is given by w w\OCAA i = i ; i;j = 1;:::;n: (12) w j w\OCAA 8 j b The equilibrium implication of (12) is summarized in Theorem (2). c w\OCAA Theorem 2 In equilibrium the relative proportions i , i;j = 1;:::;n must equal the w\OCAA 8 j relative proportions of the asset values in the market. That is, only if the market portfolio is mean-variance e¢ cient (equal to the optimal combination of ambiguous assets) is it feasible for all investors to hold ambiguous assets in the same relative proportions as OCAA. Denoting by r the return on the market portfolio, M the equilibrium expected return on asset i is given by E (r ) = r +(cid:12)AE (r r );11 (13) P i f i P M (cid:0) f with (cid:21)(cid:27) (r ;r )+(cid:18)(cid:27) (E(r );E(r )) (cid:12)A = P i M (cid:22) i M : (14) i (cid:21)(cid:27)2 (r )+(cid:18)(cid:27)2 (E(r )) P M (cid:22) M 11TheequilibriumexpectedreturnE (r )obtainsfrom(9),bycombiningassetiwiththemarketportfolio P i (of which i is a part) and requiring a pure investment in M. 9

The Security Market Line (13) features a robust beta, (cid:12)A, that measures the marginal contrii butionofassetitotheambiguityoftheoptimalportfolioM. Borrowingfromtheterminology of the CAPM under risk, we say that (cid:12)A measures the systematic ambiguity of asset i. i Observe that if there exists an asset j whose expected return violates (13), then the market portfolio is not the optimal combination of ambiguous assets. Instead, one can create OCAA by combining j with the market portfolio. De(cid:133)ne (cid:11)A to be the deviation of j asset j from the expected return pro(cid:133)le (13). We decompose the excess return on asset j by means of the ordinary least square coe¢ cients to obtain (cid:27) (r ;r ) (cid:11)A = E (r r ) P j M E (r r ):12 (15) j P j (cid:0) f (cid:0) (cid:27)2 (r ) P M (cid:0) f P M Here, (cid:11)A is the expected value of the residual for the regression of (r r ) on (r r ) (cid:150) j j (cid:0) f M (cid:0) f that is, the expected value of the portion of (r r ) that is uncorrelated with (r r ). j f M f (cid:0) (cid:0) Then, from (13) and (14), it follows that (cid:27) (r ;r ) (cid:11)A = (cid:12)AE (r r ) P j M E (r r ) (16) j j P M (cid:0) f (cid:0) (cid:27)2 (r ) P M (cid:0) f P M (cid:18) (cid:27) (E(r );E(r ))(cid:27)2 (r ) (cid:27) (r ;r )(cid:27)2 (E(r )) = (cid:22) j M P M (cid:0) P j M (cid:22) M E (r r ): (cid:27)2 (r ) (cid:21)(cid:27)2 (r )+(cid:18)(cid:27)2 (E(r )) P M (cid:0) f (cid:2) P M P M (cid:22) M (cid:3) (cid:2) (cid:3) It is easy to show that the sign of (cid:11)A is: j (cid:27) (E(r );E(r )) (cid:27) (r ;r ) sgn(cid:11)A = sgn (cid:22) j M P j M ; (17) j (cid:27)2 (E(r )) (cid:0) (cid:27)2 (r ) (cid:20) (cid:22) M P M (cid:21) where (cid:27)(cid:22)(E(rj);E(rM)) and (cid:27)P(rj;rM) are the (cid:147)pure(cid:148)marginal contributions of holding asset j (cid:27)2 (cid:22) (E(rM)) (cid:27)2 P (rM) to the risk and the ambiguity of the market portfolio, respectively. In brief, asset j is: Underpriced if (cid:27)(cid:22)(E(rj);E(rM)) > (cid:27)P(rj;rM). (cid:15) (cid:27)2 (cid:22) (E(rM)) (cid:27)2 P (rM) Overpriced if (cid:27)(cid:22)(E(rj);E(rM)) < (cid:27)P(rj;rM). (cid:15) (cid:27)2 (cid:22) (E(rM)) (cid:27)2 P (rM) 12The ratio (cid:27)P(rj;rM) on the right-hand side of (15) measures the systematic risk of asset j. (cid:27)2 P (rM) 10

Fairly priced if (cid:27)(cid:22)(E(rj);E(rM)) = (cid:27)P(rj;rM). (cid:15) (cid:27)2 (cid:22) (E(rM)) (cid:27)2 P (rM) Finally, weremarkthat theoriginal capital assetpricingmodel underriskisnestedinour equilibriummodelofthecapitalmarketunderuncertainty. Infact,setting(cid:27) (E(r );E(r )) = (cid:22) i j 0, i;j = 1;:::;n, we have that E (r ) = r + (cid:27)P(rj;rM)E (r r ) and (cid:11)A = 0. Figure 1 8 P j f (cid:27)2 P (rM) P M (cid:0) f j summarizes our results. Figure 1 Classic and Robust CAPM Notes: (1) (cid:12) = (cid:27)P (rj;rM ) j =1;:::;n. j (cid:27)2 P (rM) 8 (2) (cid:12) = (cid:27)P(ri;rM) =1 (cid:18)[(cid:27)(cid:22)(E(ri);E(rM))(cid:27)2 P(rM) (cid:0) (cid:27)P(ri;rM)(cid:27)2 (cid:22)(E(rM))] . i (cid:27)2 P (rM) (cid:0) (cid:27)2 P (rM)[(cid:21)(cid:27)2 P (rM)+(cid:18)(cid:27)2 (cid:22)(E(rM))] (3) RSML has been drawn assuming (cid:27)(cid:22)(E(ri);E(rM)) > (cid:27)P(ri;rM) i=1;:::;n. (cid:27)2 (cid:22)(E(rM)) (cid:27)2 P (rM) 8 The menu of equilibrium expected returns if the CAPM holds is given by the Security Market Line (SML). If the RCAPM holds, and ambiguity reinforces risk, the robust Security MarketLine(RSML)issteeperthanSML,re(cid:135)ectinghigherreturnsduetomodeluncertainty. Instead, asset j is created to violate both SML and RSML. Note that if an investor makes decisions according to the CAPM, he identi(cid:133)es j as underpriced ((cid:11) > 0) when in fact, j properly accounting for model uncertainty, j is overpriced ((cid:11)A < 0). Then, the portfolio j constructed by combining j with the market portfolio is not a superior performer as the investor believes, but an inferior one. 11

Finally, if ambiguity mitigates risk, RSML is (cid:135)atter than SML and assets command lower returnsforthesamelevelofrisk. Thisresultisimportantinlightofcompellingevidencethat theU.S.stockmarketSMLismuch(cid:135)atterthantheCAPM-impliedSML.Forexample,Black, Jensen, and Scholes (1972) analyze the returns of portfolios formed by ranking U.S. stocks on (cid:12) values. They (cid:133)nd that between 1926 and 1966 low-beta stocks earned higher average returns than the CAPM predicts, and vice versa. Even after accounting for measurement errors, the CAPM relation between expected returns and (cid:12) is so weak that the model must berejected. Bycontrast, thejointe⁄ectofriskanduncertaintyonindividualdecisionmaking allows the RCAPM to predict a proportional relation between expected returns and (cid:12) (cid:150)one that is consistent with the above-mentioned empirical evidence. 5 Conclusions We extend the capital asset pricing model by including Knightian uncertainty about asset returns. First, we derive the mean-variance e¢ cient portfolio using the certainty equivalent approximation of Maccheroni, Marinacci, and Ru¢ no (2013a). The optimal portfolio holdings decrease with ambiguity aversion and perceived ambiguity (cid:150)a phenomenon usually referred to as (cid:135)ight to quality. Next, we de(cid:133)ne the optimal combination of ambiguous assets and propose a robust version of the two-fund separation theorem: we show that all meanvariance e¢ cient portfolios come from combining the optimal combination of ambiguous assets with the risk-free asset. Last, we (cid:133)nd the robust security market line and compare it with the security market line under risk. In particular, we show that the interaction of risk and ambiguity can predict a robust security market line whose (cid:135)atter slope (cid:133)ts the data. We argue that this result is important to address known empirical failures of the capital asset pricing model. 12

References [1] Arrow, K. Essays in the Theory of Risk Bearing. Markham Publishing Co., Chicago, IL (1971). [2] Black, F., Jensen, M., and Scholes, M. (cid:147)The Capital Asset Pricing Model: Some Empirical Tests.(cid:148)In Studies in the Theory of Capital Markets, Michael C. Jensen, ed., New York (1972), pp. 79-121. [3] Boyle, P., Garlappi, L., Uppal, R. and Wang, T. (cid:147)Keynes Meets Markowitz: The Tradeo⁄ Between Familiarity and Diversi(cid:133)cation.(cid:148)Management Science, Vol. 58 (2012), pp. 253-272. [4] Breeden, D. (cid:147)An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities.(cid:148)Journal of Financial Economics, Vol. 7 (1979), pp. 265- 296. [5] Chen, Z., and Epstein, L. (cid:147)Ambiguity, Risk, and Asset Returns in Continuous Time.(cid:148) Econometrica, Vol. 70 (2002), pp. 1403-1443. [6] Chen, H., Ju, N., and Miao, J. (cid:147)Dynamic Asset Allocation with Ambiguous Return Predictability.(cid:148)Unpublished manuscript, (2013). [7] Collard, S. Mukerji, S., Sheppard, K., and Tallon, J. (cid:147)Ambiguity and the historical equity premium.(cid:148)Oxford University Economics Dept. WP 550 (2011). [8] Easley, D,.andO(cid:146)Hara, M.(cid:147)AmbiguityandNonparticipation: TheRoleofRegulation.(cid:148) Review of Financial Studies, Vol. 29 (2009), pp. 1817-1843. [9] Epstein, L., and J. Miao, J., (cid:147)A Two-person Dynamic Equilibrium Under Ambiguity.(cid:148) Journal of Economic Dynamics and Control, Vol. 27 (2003), pp. 1253-1288. [10] Garlappi, L., Uppal, R. and Wang, T. (cid:147)Portfolio Selection with Parameter and Model Uncertainty: A Multi-prior Approach.(cid:148)Review of Financial Studies, Vol. 20 (2007), pp. 41-81. [11] Gollier, C. (cid:147)Portfolio Choices and Asset Prices: The Comparative Statics of Ambiguity Aversion.(cid:148)Review of Economic Studies, Vol. 78 (2011), pp. 1329-1344. [12] Grossman, S., and Shiller R. (cid:147)The Determinants of the Variability of Stock Market Prices.(cid:148)American Economic Review, Vol. 71 (1981), pp. 222-227. [13] Huberman, G. (cid:147)Familiarity Breeds Investment.(cid:148)Review of Financial Studies, Vol. 14 (2001), pp. 659-680. [14] Illeditsch, P. (cid:147)Ambiguous Information, Risk Aversion and Asset Pricing.(cid:148)Journal of Finance, Vol. 66 (2012), pp. 2213-2247. [15] Izhakian, Y. (cid:147)Capital Asset Pricing Under Ambiguity.(cid:148)NYU WP 2451/31464 (2012). [16] Izhakian, Y. and Benninga S. (cid:147)The Uncertainty Premium in An Ambiguous Economy.(cid:148) Unpublished manuscript, (2008). 13

[17] Jewitt, I. and Mukerji, S. (cid:147)Ordering Ambiguous Acts.(cid:148)Oxford University Economics Dept. WP 553 (2011). [18] Ju, N., and Miao, J. (cid:147)Ambiguity, Learning and Asset Returns.(cid:148)Econometrica, Vol. 80 (2012), pp. 559-591. [19] Klibano⁄, P., Marinacci, M., and Mukerji, S. (cid:147)A Smooth Model of Decision Making Under Ambiguity.(cid:148)Econometrica, Vol. 73 (2005), pp. 1849-1892. [20] Knight, F. Risk, Uncertainty, and Pro(cid:133)t. Hart, Scha⁄ner & Marx; Houghton Mi› in Co., Boston, MA (1921). [21] Lintner, J. (cid:147)The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.(cid:148)Review of Economics and Statistics, Vol. 47 (1965), pp. 13-37. [22] Lucas, R. (cid:147)Asset Prices in an Exchange Economy.(cid:148)Econometrica, Vol. 46 (1978), pp. 1429-1446. [23] Maccheroni, F., Marinacci, M., and Ru¢ no D. (cid:147)Alpha as Ambiguity: Robust Mean- Variance Portfolio Analysis.(cid:148)Econometrica, Vol. 81 (2013a), pp. 1075-1113. [24] Maccheroni, F., Marinacci, M., and Ru¢ no D. (cid:147)Does Uncertainty Vanish in the Small? The Smooth Ambiguity Case.(cid:148)IGIER WP391 (2013b), pp. 1085-1088. [25] Merton, R. (cid:147)An Intertemporal Capital Asset Pricing Model.(cid:148)Econometrica, Vol. 41 (1973), pp. 867-887. [26] Mossin, J. (cid:147)Equilibrium in a Capital Asset Market.(cid:148)Econometrica, Vol. 35 (1966), pp. 768-783. [27] Nau, R. (cid:147)Uncertainty Aversion With Second-order Utilities and Probabilities.(cid:148)Management Science, Vol. 52 (2006), pp. 136-145, 2006. [28] Pratt, J. (cid:147)Risk Aversion in the Small and in the Large.(cid:148)Econometrica, Vol. 32 (1964), pp. 122-136 [29] Sharpe, W. (cid:147)Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.(cid:148)Journal of Finance, Vol. 19 (1964), pp. 425-442. [30] Treynor,J.(cid:147)TowardaTheoryofMarketValueofRislyAssets.(cid:148)Unpublishedmanuscript (1962). Final version in Asset Pricing and Portfolio Performance, Robert Korajcyzk, ed., London: Risk Books (1999), pp. 15-22. 14