Markowitz with regret

Journal of Economic Dynamics & Control 103 (2019) 1–24

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Markowitz with regretR Rainer Baule a,∗, Olaf Korn b, Laura-Chloé Kuntz c a b c

University of Hagen, Universitätsstraße 41, Hagen D-58084, Germany University of Goettingen and Centre for Financial Research Cologne (CFR), Platz der Göttinger Sieben 3, Göttingen D-37073, Germany University of Goettingen, Platz der Göttinger Sieben 3 Göttingen D-37073, Germany

a r t i c l e

i n f o

Article history: Received 13 March 2018 Revised 21 August 2018 Accepted 4 September 2018 Available online 12 April 2019 JEL classification: D81 G11 G40 Keywords: Portfolio selection Regret aversion Regret risk

a b s t r a c t Providing a framework to integrate regret as an additional decision criterion in Markowitz’s model of portfolio selection, we propose two different views on regret: An investor might feel regret with respect to the ex-post best alternative either in terms of return or in terms of preference value. Under both views, regret can be captured by adjusting the vector of expected returns or alternatively by adjusting the return covariance matrix, retaining the tractability of the Markowitz model. The regret model, however, has very different implications for how asset characteristics affect optimal portfolios. While the impact of the skewness of an asset is strengthened, the impact of the variance shrinks. Moreover, we show for a variety of real portfolios that the effects of regret on optimal portfolio weights and the ex-ante return distribution are large. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The portfolio selection approach (Markowitz, 1952; 1987) is a cornerstone of modern finance. It provides a formal yet tractable procedure to find optimal portfolios. This tractability comes at a cost, however. The focus on the mean and variance of returns neglects higher-order return moments like skewness and kurtosis. Moreover, it ignores any aspects beyond final wealth that can be important for investors, in particular emotional and self-expressive benefits. Harry Markowitz himself acknowledges the importance of such benefits. When asked about his personal pension savings in an interview (Zweig, 1998) he admitted, “My intention was to minimize my future regret. So I split my contribution fifty-fifty between bonds and equities.”1 Markowitz’s statement raises interesting questions. How can regret be considered in portfolio selection without losing the formal elegance and tractability of Markowitz’s approach? Within such a new framework, is it really the best choice to split the investment evenly between stocks and bonds? What do optimal portfolios look like and what makes an asset attractive if regret is considered in addition to final wealth? Answers to these questions are important because they can help investors to find better portfolios by allowing for a more realistic preference specification while maintaining the transparency and tractability of Markowitz’s approach. The contribution of this paper is to provide such answers. It proposes a framework of portfolio selection with regret and analyzes R

We thank an anonymous referee and an associate editor for their helpful comments and suggestions. Corresponding author. E-mail addresses: [email protected] (R. Baule), [email protected] (O. Korn), [email protected] (L.-C. Kuntz). 1 This confession is only one example for the importance of regret in investment decisions. In a recent study, Frydman and Camerer (2016) show that neural measures of regret are related to the decision-making of participants in an experimental asset market. ∗

https://doi.org/10.1016/j.jedc.2018.09.012 0165-1889/© 2019 Elsevier B.V. All rights reserved.

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the properties of optimal portfolios. Regret is understood as an emotional feeling caused by the ex-post knowledge that a different decision—other than the one actually taken—would have led to a better outcome. Decision-making under regret aversion has been developed formally in the literature (Bell, 1982; 1983; Loomes and Sugden, 1982) and can rest on an axiomatic foundation (Diecidue and Somasundaram, 2017; Sugden, 1993). In our specific framework an investor potentially cares about two attributes of a portfolio: its return and the regret resulting from holding it. In contrast to the utility-based definition of regret known from the literature, we propose two alternative views on regret that are both compatible with Markowitz’s mean-variance approach: First, an investor might feel regret with respect to the ex-post best performing asset, that is, the asset with the best realized return. Second, given a mean-variance preference function, an investor might feel regret with respect to the ex-post best realized preference value, considering both risk and return. Although our model is primarily meant to be a useful tool to find optimal portfolios for individual investment decisions, the model’s main implications are well in line with observed market characteristics. For example, it is a persistent empirical finding that low volatility stocks tend to earn higher average returns than high volatility stocks (Ang et al., 20 06; 20 09), which seems to contradict the trade-off between risk and return. However, such a pattern can well be in line with the risk-return trade-off in the regret model. Similarly, in our regret model a positive skewness makes an asset more attractive, which is not captured by the Markowitz model but is in line with the empirical observation of skewness premiums (Conrad et al., 2013). Our work belongs to the literature on alternative portfolio theories. As we show, portfolio optimization with regret can be implemented by replacing variance with a different measure of risk, the regret-adjusted variance. In this respect our approach is similar to portfolio models that use alternative risk measures, like lower partial moments (Harlow, 1991; Klebaner et al., 2017; Zhu et al., 2009) or conditional value-at-risk (Quaranta and Zaffaroni, 2008). Moreover, our work belongs to the literature on behavioral portfolio theories, studying portfolio problems with non-standard preferences, like loss aversion and disappointment aversion (Ang et al., 2005; Driessen and Maenhout, 2007; Shi et al., 2015) or different mental accounts (Das et al., 2010; Das and Statman, 2013; Shefrin and Statman, 20 0 0). However, there is a fundamental difference between these approaches and the regret model. In the regret model, utility does not depend on the final outcome of the chosen portfolio alone, but also on the outcome of other feasible portfolios that were not chosen. In this sense the regret model is related to benchmarking (with the ex-post best portfolio as the benchmark) and the analysis of tracking errors (with respect to the ex-post best portfolio). Stated differently, the regret approach does not simply change the view on the risk of an investor’s final wealth but adds a distinctively different investor want to the portfolio problem. To our knowledge only three other papers provide theoretical results on the effects of investor regret on portfolio choice. Muermann et al. (2006) study the asset allocation between a risk-free asset and a risky asset in defined contribution pension schemes. Michenaud and Solnik (2008) study the currency hedging decision of investment managers, a setting that is structurally equivalent to an asset allocation problem with one risk-free and one risky asset.2 The two papers provide important insights that are in line with our findings. Most importantly, when investors consider regret they tend to “hedge away from the extremes”. The works by Muermann et al. (2006) and Michenaud and Solnik (2008), however, are confined to a specific two-asset problem. In contrast, our analysis studies the Markowitz problem of N risky assets and our results equally apply to the case with N − 1 risky and one risk-free asset. As we show, a setting with multiple risky assets has a much richer structure. In particular, the dependence structure between assets changes the problem because it has an impact on the regret risk of a portfolio. Gollier and Salanié (2006) also allow for multiple assets in their framework of a completemarket Arrow-Debreu economy. They study optimal asset allocation as well as equilibrium asset prices when investors are sensitive to regret. Some insights from their model also show up in our results, in particular the importance of skewness for the attractiveness of an asset when investors are regret-averse. The work by Gollier and Salanié focuses on structural properties and equilibrium implications. In contrast, the main goal of this paper is to provide a tractable extension of the Markowitz model which requires few inputs, is easy to understand for investors, and is relatively easy to implement. In particular, in our setting, parameter estimation can proceed in the same way as in the Markowitz model and we are able to obtain optimal portfolio weights in closed form if short-sales constraints are not binding. Thus, if practitioners have specified the perceived importance of regret risk relative to classical risk, they can quite easily incorporate the regret dimension into Markowitz’s mean-variance analysis. The only requirement is a preprocessing of the input data to obtain a regret-adjusted covariance matrix, or, alternatively, a regret-adjusted vector of expected returns, which then can be plugged into an existing Markowitz algorithm. The remainder of the paper is structured as follows: Section 2 introduces our setting and delivers some results on the structure of the portfolio problem and the properties of optimal portfolio weights. In the following Section 3, we present a deeper analysis of the impact of regret on portfolio weights and ask what distributional characteristics make an asset more or less attractive than in the Markowitz model. A comparative static analysis based on analytical results and simulations is presented. Section 4 demonstrates the quantitative importance of regret effects for real portfolios. Some conclusions are provided in Section 5.

2 In Michenaud’s and Solnik’s model there is also another stochastic component beyond exchange-rate risk: the return of the foreign asset in local currency. This second stochastic component cannot be influenced by the investor, however. Therefore, the investment problem is to choose between a risk-free and a risky asset with additional background risk.

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2. The portfolio problem 2.1. Regret in the Markowitz framework We study the one-period portfolio problem where an investor can choose among a set of risky assets or funds. The investor considers two criteria to assess a portfolio. The first is return, as in classical portfolio selection, representing the benefits of wealth and consumption opportunities. The second is the avoidance of regret, representing the emotional and self-expressive benefits of being not too far from the ex-post best choice. In the literature, regret is usually defined as a difference in utility between the chosen portfolio and the ex-post best alternative. Because we work on an extension of the Markowitz model in this paper, our framework is mean-variance analysis instead of expected utility maximization,3 and we are looking for an appropriate regret measure for the mean-variance investor. How might a mean-variance investor evaluate the opportunity set of portfolios from an ex-post (end-of-period) perspective? The answer to this question depends in the way the investor perceives the joint return distribution of all assets based on the end of the period realized returns. A first way to look at the return distribution is that after the realization of returns all uncertainty has disappeared. This view would imply that only the realized returns and not the (ex-ante) risk matter for the perception of regret. It is therefore a natural choice under this view to define regret as the return difference between the ex-post best performing asset and the actually chosen portfolio. We will call this perception of regret the “return-regret” view in the following. A second way to look at the return distribution is that the realization only represents one observation from the joint distribution. This single realization could be used to replace the vector of expected returns in the preference function but adds no information on risk. Consequently, the investor leaves her assessment of ex-ante risk unchanged. This view would imply that both the realized returns and the (ex-ante) variances matter for the perception of regret. Ex ante, the investor has a preference function which relates expected return and variance. For the perception of regret the investor considers the realized preference value of the actually chosen portfolio and compares it to the ex-post optimal preference value among the opportunity set of all portfolios. Basically, an investor feels less regret with respect to an ex-post well performing portfolio if this portfolio wasn’t a good choice ex ante because of a low preference value due to high risk. We will call this perception of regret the “preference-regret” view in the following. To formalize these ideas, we have to define the investment opportunity set and identify the ex-post best alternative within this set. Assume that N risky assets are available at the beginning of the period and denote the random return vector by R := (R1 , R2 , . . . , RN )tr . Let π (μ, σ 2 ) = μ − γ σ 2 denote a preference function with the two arguments expected return and variance, where γ ≥ 0 is the risk aversion coefficient. A portfolio P is characterized by the weight vector ω := (ω1 , ω2 , . . . , ωN )tr with the budget constraint ωtr 1 = 1, so that the portfolio return is given by RP = ωtr R.4 Additionally we introduce short-selling constraints, ωi ≥ 0 ∀i. Restrictions on leverage are required because the ex-post best strategy under the return-regret view would otherwise lead to infinite wealth, which is certainly unrealistic and not a meaningful benchmark for an investor’s regret. The reason is that unlimited leverage would allow an infinite amount of money to be held in the ex-post best-performing asset, financed via short selling the worst-performing asset. Such restrictions would also be required under the utility-based definition of regret used in the literature, because also with this definition, the ex-post best strategy, maximizing realized utility, would imply investing an infinite amount in the ex-post best-performing asset. Under the return-regret view, the ex-post best strategy is to hold the full budget in the asset with the highest realized return. This ex-post best alternative yields a return of Rmax := max(R1 , R2 , . . . , RN ). For any portfolio P, regret is then defined as the difference between this “benchmark return” and the portfolio return RP , i.e., regret is the under-performance in percentage points compared to the ex-post best-performing portfolio that had been feasible ex ante:

RegretPret = Rmax − RP .

(1)

Under the preference-regret view, the ex-post best strategy is not directly observed. It is the portfolio that leads to the maximum of the preference function. Regret is defined as the difference between this maximum preference value and the actually realized preference value of the portfolio P; i.e., regret is the under-performance in preference value compared to the ex-post best achievable preference value among all portfolios which had been feasible ex ante:5

RegretPpre f = max π (RA , σA2 ) − π (RP , σP2 ), A

(2)

where the maximization is performed over all alternative portfolios A which fulfill the budget and short-selling constraints. After having defined regret, a crucial issue is to specify the (relative) importance of the two criteria return and regret for an investor. To express relative importance, we use the regret-adjusted return ZP of a portfolio P, which is defined as

ZP := (1 − α ) RP − α RegretP ,

(3)

3 Although the mean-variance approach and the expected-utility approach are different concepts, the literature has shown various linkages. For example, Meyer (1987) and Bigelow (1993) provide conditions under which the two approaches lead to the same ranking of decision alternatives. Even if these conditions are not met, restricting choices to the set of mean-variance efficient portfolios often leads to minor utility losses for investors (Kroll et al., 1984; Levy and Markowitz, 1979). 4 The vector 1 denotes a column vector of ones. 5 For ease of notation, we use the terms R both for the ex-ante random returns and the ex-post realized returns.

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with α ∈ [0, 1]. The regret-adjusted return naturally combines the two criteria. It increases with higher portfolio returns and decreases with higher regret. The parameter α governs the relative importance of the two criteria, thereby reflecting the corresponding investor preferences. With α = 0 the investor does not care about regret and we are back at the classical portfolio problem. In the case of the other extreme (α = 1), the investor only considers regret. We will call the former the Markowitz case and the latter the pure regret case. Distinguishing between the two views on regret, the return-regret-adjusted return becomes

ZPret = (1 − α ) RP − α (Rmax − RP ) = RP − α Rmax ,

(4)

while the preference-regret-adjusted return becomes

ZPpre f = (1 − α )RP − α (πmax − (RP − γ = RP − α πmax − α γ

σP2 ))

σP2 ,

(5)

with πmax := maxA π (RA , σA2 ). Because the focus of our paper is to extend Markowitz’s portfolio selection by incorporating regret as an additional criterion, we follow the classical approach and assume that investors evaluate portfolios based on the first two return moments, in our case referring to regret-adjusted returns. In terms of the two components of ZP , such preferences imply that investors like higher expected portfolio returns, dislike higher expected regret and dislike higher risks (variances) both with respect to portfolio returns and regret.6 Under such μ-σ preferences, the investor’s optimization problem in the regret model is finally given by7

max (ZP ) := π (E (ZP ), V ar (ZP )) = E (ZP ) − γ V ar (ZP ), ω

(6)

with the budget and short-selling restrictions on ω as defined above. Due to the single risk-aversion parameter γ in Eq. (6), the model implies that investors have the same risk aversion with respect to classical return risk and regret risk, conditional on their assessment of the relative importance of return and regret, as expressed by the parameter α . 2.2. Structural properties of the portfolio problem Having set up the portfolio problem, we will discuss some structural properties of this problem which have important practical implications and help us to understand the effects of regret on optimal portfolios. In the following, we denote the vector of (classical) expected returns by μ := E[R] and the (classical) covariance matrix by  := (Cov(Ri , R j ))i, j . Furthermore, we introduce the vector of expected regret-adjusted returns, μra := E[Z], and the regret-adjusted covariance matrix via ra := (Cov(Zi , Z j ))i, j . Note that the regret-adjusted return vector and covariance matrix differ under the return-regret view (with regret-adjusted returns Zret ) and the preference-regret view (with regret-adjusted returns Zpref ). Irrespective of the view taken, it can be shown that the matrix  ra is positive definite whenever the covariance matrix  is positive definite.8 Property 1 (Adjustment of Covariance Matrix). Optimal portfolio weights can be found as in standard μ-σ optimization, using the (unadjusted) vector of expected returns and a new covariance matrix. The new covariance matrix is the regret-adjusted covariance matrix under return-regret, ret ˜ ret = ra  ,

(7)

and a combined matrix under preference-regret, pre f ˜ pre f = α  + ra  .

Proof. See the appendix.

(8)



Property 1 is good news for the portfolio model with regret, because in a sense it is not more complicated to work with than with standard μ-σ analysis. The difference is that the risk measure is changed, from the variance of returns to the variance of regret-adjusted returns (or a combined measure under preference-regret). This property makes our approach similar in structure (not necessarily in outcome) to alternative portfolio choice models that replace variance with alternative risk measures, for example lower partial moments. Moreover, once the investor has specified the parameter α , the covariance matrix of regret-adjusted returns can be estimated from historical data in the same way as the covariance matrix of returns.9 Economically, the simple structure of the optimization problem is somewhat surprising, because our analysis did not just propagate an alternative risk measure, but started from the idea of integrating two separate investor wants, namely 6

Bleichrodt et al. (2010) provide empirical evidence that decision makers indeed are averse to regret risk. We use the preference function  for random returns, while the function π explicitly refers to the first two moments. 8 A proof is provided in the appendix. 9 Methods to improve estimation, like shrinkage (Ledoit and Wolf, 2004) or more general norm constraints (DeMiguel et al., 2009), can be used for the covariance matrix of regret-adjusted returns in the same way as for the return covariance matrix. 7

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wealth and avoidance of regret, into portfolio choice. To understand why the resulting structure is so simple, recall that in μ-σ analysis expected returns affect portfolio weights only via return differences between assets, not via the levels of expected returns. Under return-regret the expected regret of investing in a single asset is just the expected return difference between the ex-post maximum-return asset and that specific asset. It therefore becomes clear that the vector of individual assets’ expected regret contains the same information on return differences as the vector of expected returns. The reason is simply that regret is measured according to the same benchmark for all assets, the return of the ex-post maximum-return asset. Also under preference-regret, regret is measured according to the same benchmark for all assets, the preference value of the ex-post best portfolio. Because in both cases expected regret contains redundant information with respect to the determination of portfolio weights, it disappears from the optimization problem. However, it would be equally correct to say that expected returns disappear from the optimization problem as long as information on expected regret is incorporated. One can therefore conclude that both expected returns and expected regret are considered, but simply lead to the same portfolio weights. Property 2 (Adjustment of Expected Returns). Alternatively, optimal portfolio weights can be found by using a new vector of expected returns and the (unadjusted) return covariance matrix of the original returns. Under the preference-regret view, additionally the risk aversion coefficient is scaled by 1 + α . The new (adjusted) vector of expected returns is given by10

μ˜ ret = μ + 2 α γ Cov(R, Rmax )

(9)

under return-regret and by

μ˜ pre f = μ + 2 α γ Cov(R, πmax )

(10)

under preference-regret. Proof. See the appendix.



Property 2 provides a second representation of the optimization problem, which is in a sense the mirror image of the representation from Property 1. Instead of adjusting the covariances and leaving the expected returns unadjusted, we can alternatively use the unadjusted return covariance matrix and adjust the vector of expected returns. We stress that the alternative representations of the optimization problem presented in Properties 2 and 1 are both fully equivalent to the original optimization problem from Equation (6), i.e., all representations will deliver the same portfolio weights. The adjustment that is made for each asset in Property 2 is very intuitive. A first observation is that we require both preference parameters α and γ to be positive for any adjustment to appear. It is obvious that α = 0 leads to no adjustment, because we would then be back in the Markowitz case. But even with a positive α , γ = 0 would seemingly leave us without any regret effect. The reason was already discussed in the context of Property 1. If there is no risk aversion and only expected returns count, using either expected returns or expected regret leads to the same result. What the adjustment does, though, is to account for the effect of regret risk. The regret risk of an asset is expressed via the covariance between the asset’s return and the return of the ex-post best asset (under return-regret) or the preference value of the ex-post best portfolio (under preference-regret). If this covariance is high, the returns of the asset and the benchmark move together and regret risk is low. Therefore, the asset receives a high positive return adjustment. However, again, what matters for the optimal portfolio weights are the differences between the assets’ expected returns, not the levels. Thus, if the adjustments in expected returns were to be the same for all assets, there would be no effect on portfolio weights and it is crucial to look at the differences between the adjustments for different assets. Under preference-regret, there is a second effect which has an impact on optimal portfolio weights: The risk aversion coefficient increases from γ to γ · (1 + α ). The reason for this effect lies in the perception of regret: The investor feels more regret with respect to portfolios with an ex-ante lower variance risk. Given two portfolios with an ex-post identical realized return, the investor feels more regret not having invested in the portfolio with the ex-ante lower variance, because this would have been ex-ante a more reasonable choice according to the preference function. So to reduce regret risk, the investor tends to invest into less risky portfolios, which can be captured by an increased risk aversion coefficient. We believe that both formulations of the optimization problem, as given in Properties 1 and 2, are useful for an understanding of how regret affects optimal portfolios. The mean adjustment (Property 2) provides us with a return measure of the attractiveness of individual assets under regret, which is both simple and intuitive. But it is also important that regret effects can be captured by just changing the risk measure. Investigating the variances and correlations of properly adjusted returns (in comparison to the unadjusted ones) helps us to understand how the characteristics of the whole investment universe affect the risk of individual assets. Property 3 (Weighted Average of Weights). If short-sales constraints are not binding, optimal portfolio weights in the regret model are the weighted average of optimal weights from the Markowitz case and the pure regret case. The optimal weight vector ∗ (α ) in dependence of the regret coefficient α is given by ωra

10

The term Cov (R, Rmax ) denotes a vector with covariances Cov (R j , Rmax ) for j = 1, . . . , N.

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R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24 ∗ ∗ ∗ ∗ ωra (α ) = ωra (0 ) + α (ωra (1 ) − ωra (0 )) ∗ ∗ = (1 − α ) ωra (0 ) + α ωra (1 )

(11)

under return-regret and by

2α ∗ ∗ (ωra (1 ) − ωra (0 )) 1+α 2α ∗ (0 ) + ωra ω ∗ (1 ) 1 + α ra

∗ ∗ ωra (α ) = ωra (0 ) +

=

1−α 1+α

(12)

under preference-regret. Proof. See the appendix.



Property 3 provides a considerable simplification for the understanding of regret effects on portfolio composition. If short-sales constraints are not binding, an analysis of such effects can concentrate on just the differences between the Markowitz case and the pure regret case, because all other cases are simple combinations of these two. Even with binding short-sales constraints, where the result does not necessarily hold exactly, these two cases are important reference points. In summary, Properties 1–3 provide important information for implementing the regret model and understanding the effects of regret. First, the analysis of the pure regret case is often sufficient for an understanding of how the regret model changes the portfolio composition compared to the Markowitz model. Second, the specification of the expected return component is as simple (or better to say as difficult) as in the Markowitz approach, because there is no need to estimate expected regret. Third, regret risk effects can be incorporated in two different ways, either by using the full covariance matrix of regret-adjusted returns or by using the covariances between the individual assets’ returns and the returns (return-regret) or preference values (preference-regret) of the ex-post best performing alternative. All required quantities referring to the joint distribution of asset returns can be estimated from historical data as in the Markowitz case. Another important aspect of the model’s practical implementation is the required information on preferences. In comparison to the Markowitz model, there are two additional aspects. The first one is the estimation of the importance of regret risk relative to classical risk, as represented by the parameter α . The second one is the choice between the return-regret model and the preference-regret model. While the development and detailed discussion of appropriate estimation procedures lies beyond the scope of this paper, the general idea is to adapt methods from the literature on the identification of preferences. For example, Bleichrodt et al. (2010) develop an iterative procedure of binary choices between simple alternatives to infer the regret aversion of individuals. In the same spirit, appropriate sequences of choices between alternatives need to be developed to assess the relative importance of regret risk and classical risk for a specific investor. Similarly, one could try to find out if an investor’s view on regret confirms more with return-regret or preference-regret. 3. Optimal portfolios In this section, we investigate in some detail how regret affects optimal portfolios. We ask which characteristics of an asset make it more or less attractive than in the Markowitz world11 and look at the dependence on investor preferences, as represented by the model parameters γ and α . The previous section already presented two properties that are very useful for our analysis. First, it is regret risk, not expected regret, that leads to changes in portfolio composition as opposed to the Markowitz model. Second, an asset’s covariance with the ex-post best alternative is crucial for its attractiveness. These two properties can guide us in our understanding of optimal portfolios. Moreover, the second property stresses a fundamental characteristic of the regret model: Changes in the distribution of any individual asset usually change the distribution of the ex-post best alternative—the common benchmark—and thereby change the regret risk of all other assets too. Similarly, the addition or deletion of assets from the investment opportunity set potentially changes the risk of all assets. Therefore, a proper definition of the investor’s investment opportunities is especially important in the regret model. A natural starting point for an analysis of the effects of asset characteristics on optimal portfolios is the case of an investor choosing between two assets (asset classes, funds). As our base scenario, we consider two homogeneous assets, i.e., assets with the same distribution. With homogeneous assets the optimal policy is to invest half of the budget in each asset, both in the Markowitz case and the pure regret case. Such an equal weighting of assets therefore defines our reference point portfolio composition. The concrete specification of the base scenario uses a bivariate normal distribution with expected returns of 10%, standard deviations of 25%, and a correlation of 0.5. The risk-aversion parameter is γ = 3. Since normally distributed returns provide an ideal setting for the Markowitz model, this assumption allows us to identify regret effects without being hampered by other limitations of the model. In the comparative static analyses to follow, we calculate the vector of mean returns and the covariance matrix of (regret-adjusted) returns for each parameter combination. The latter is obtained as the sample covariance matrix from two million simulated (regret-adjusted) returns. With these inputs, we obtain optimal portfolio weights either numerically or via the analytical formulas given in the appendix.

11 For the Markowitz model the effects of different input parameters on portfolio characteristics are well documented in the literature (Best and Grauer, 1991; Chopra and Ziemba, 1993). We can therefore concentrate on the differences between the Markowitz model and the regret model.

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Fig. 1. Effects of a shift in expected return. This figure shows the effects of a shift in the expected return of an asset (Asset 1) on its optimal portfolio weight (Part A) and on the covariance with the ex-post best portfolio (Part B). In Part A the solid line refers to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret. In Part B the solid line refers to the return covariance of Asset 1 with the ex-post best portfolio (Cov (R1 , Ropt )) and the dotted line to the covariance of Asset 2 with the ex-post best portfolio (Cov (R2 , Ropt )). These covariances are numerically almost indistinguishable for the return-regret model (covariance with the ex-post best asset) and the preference-regret model (covariance with the preference value of the ex-post best portfolio), i.e., the depicted lines refer to both of them. The base scenario uses two assets with a bivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and a correlation of 0.5. The investor’s risk aversion is γ = 3.

Expected return. The effects of changing the expected return of one asset (Asset 1) are presented in Fig. 1. The upper part of the figure (Part A) shows the optimal portfolio weights of Asset 1 for different cases. The first one is the Markowitz case without any regret effects (α = 0), the second one is the pure regret case (α = 1) under return-regret, and the third one is the pure regret case (α = 1) under preference-regret. We can restrict our attention to the extreme cases with α equal to zero or one because for any in between value of α the resulting portfolio weights are just the weighted average weights according to Property 3. In all cases, the portfolio weight of Asset 1 grows with higher expected returns, as expected. The effect is strongest in the pure regret case under return-regret, which implies that return-regret increases the sensitivity of portfolio weights with respect to expected returns compared to the Markowitz model.12 The lower part of the figure (Part B) provides an intuition for this result. It shows the return covariance of both assets with the ex-post best asset. According to Property 2, this

12 This result also holds generally for N assets. As long as short-sales constraints are not binding, the sensitivity is always positive in the return-regret model and larger than in the Markowitz model. This result is not restricted to the multivariate normal distribution of returns but holds for any elliptical distribution. A proof is given in the appendix.

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quantity is crucial for the optimal portfolio weight. The covariance of Asset 1 increases with its expected return, reflecting the fact that a higher expected return makes it more likely to become the ex-post best asset. Hence the regret risk of Asset 1 shrinks. Importantly, changes in the expected return of Asset 1 also affect Asset 2, which can be seen from its decreasing return covariance with the ex-post best asset. The covariance decreases because Asset 2 is less likely to be the ex-post best asset when the expected return of Asset 1 grows. In the pure regret case under preference-regret, the impact of a changing expected return is smaller than under returnregret and even smaller than in the Markowitz case. According to Property 2, we have to consider two effects now. The first one is represented by the covariances of asset returns with the preference value of the ex-post best portfolio. These covariances are very close to the covariances under return-regret, as given in Part B of the figure. That is why we do not include them in the figure. The second effect of preference-regret, however, is an additional adjustment of the risk aversion parameter (from γ to (1 + α )γ ), i.e., the investor considers risk to be relatively more important. Therefore, the investor is less sensitive to changes in expected returns than the investor under return-regret. There is a trade-off between being closer to the ex-post best portfolio and being less diversified. Choosing a less diversified portfolio (higher weight of Asset 1) is more likely to be close to the ex-post best portfolio. However, if the ex-post best portfolio turns out to be a more diversified one, regret would be relatively high, compared to the opposite case of a chosen portfolio being well diversified and the expost best performing portfolio being less diversified.

Standard deviation. Fig. 2 presents the effects of a change in the standard deviation of Asset 1. Again, Part A of the figure depicts the asset’s portfolio weight. From the three cases, the Markowitz case and the pure regret case under return-regret show the most different reactions. Whereas the weight of Asset 1 strongly decreases with growing standard deviation in the Markowitz case, it does not change at all under return-regret. This finding implies that differences in standard deviation between the two assets do not translate into similar differences in regret risk. Actually, irrespective of the standard deviation of Asset 1, the relation between the two assets is completely symmetric with respect to return-regret. The distribution of regret-adjusted returns is always identical for both assets, given that the normal distribution is a symmetric distribution and mean returns are the same. Therefore, the two assets always have an identical standard deviation of regret-adjusted returns, regardless of any differences in (unadjusted) return standard deviations. Interestingly, the result that each asset obtains a portfolio weight of 0.5 in the pure regret case still holds if the standard deviation of Asset 1 drops to zero, making it a risk-free asset in the sense that returns are certain. However, in terms of return-regret risk this “risk-free asset” is still as risky as Asset 2. The effects of a change in standard deviation under preference-regret are in between the Markowitz model and returnregret model. The reason is that variance affects the relative regret risk of the two assets under preference-regret. With growing variance, the expected weight of Asset 1 in the ex-post optimal portfolio shrinks, which is the benchmark with respect to regret risk. Therefore, the investor should also reduce the weight of Asset 1 in her ex-ante optimal portfolio to keep regret-risk low. Part B of the figure shows the return covariances of the two assets with the return of the ex-post best asset (returnregret) or the ex-post best preference value (preference-regret). These covariances increase for both assets under both regret specifications. The reason is that the variance of the ex-post best strategy grows if the standard deviation of one of the assets becomes larger. However, the covariance increases much more for Asset 1 than for Asset 2 under return-regret, finally leaving portfolio weights unaffected. Under preference-regret the covariance of Asset 1 also increases more strongly than the one of Asset 2, however, less so than under return-regret. Our results on the effects of changing mean returns and return standard deviations have interesting implications for the perceived risk-return trade-off of assets, which can be quite different for a classical Markowitz investor and an investor considering regret. Recall the situation as shown in Fig. 2 with two homogeneous assets. Under return-regret, if we lower the standard deviation of Asset 1, the relative regret risk of the two assets is unchanged, i.e., an investor considering regret risk only would judge the two assets still as equally risky. If we additionally lower the expected return of Asset 1, we increase its relative regret risk as compared to Asset 2, because Asset 1 is less likely to be the ex-post best asset. In summary, Asset 1 has now both a lower expected return and a lower standard deviation than Asset 2, which is the natural risk-return profile for a Markowitz investor. However, in terms of regret risk Asset 1 is more risky than Asset 2. As a numerical example, consider μ1 = 0.05 and σ1 = 0.1 for Asset 1 and the base case parameters for Asset 2 (μ2 = 0.1, σ2 = 0.25). The return correlation is still 0.5. Then the regret-adjusted standard deviation of Asset 1 is about 0.144 and the one of Asset 2 about 0.11, i.e., the asset with the lower standard deviation has the higher regret risk. As this example shows, an investor considering regret risk may well require a higher risk compensation for low volatility stocks than for high volatility stocks, which would be consistent with empirical findings on the low volatility anomaly (Ang et al., 2006; 2009). The dependency between expected return and risk in the regret model has also an effect on the way optimal portfolio weights change with changing standard deviation. Actually, the optimal portfolio weights might even increase with standard deviation under return-regret if expected returns are different. Fig. 3 provides an illustration for the setting of our previous example (μ1 = 0.05, μ2 = 0.1, σ2 = 0.25, correlation=0.5). As we see from the figure, increasing the standard deviation of Asset 1 leads to a higher portfolio weight under return-regret. The reason is that the standard deviation affects the probability of becoming the ex-post best asset. Because Asset 1 has a disadvantage in terms of expected returns, a higher standard deviation raises this probability and reduces its relative regret risk, as compared to Asset 2.

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Fig. 2. Effects of a shift in standard deviation. This figure shows the effects of a shift in the standard deviation of an asset (Asset 1) on its optimal portfolio weight (Part A) and on the covariance with the ex-post best portfolio (Part B). In Part A the solid line refers to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret. In Part B the solid lines refers to the return covariance of Asset 1 with the ex-post best portfolio (Cov (R1 , Ropt )) and the dotted lines to the covariance of Asset 2 with the ex-post best portfolio (Cov (R2 , Ropt )). The thick lines refer to the return-regret case (covariance with the ex-post best asset) and the thin lines to the preference-regret case (covariance with the preference value of the ex-post best portfolio). The base scenario uses two assets with a bivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and a correlation of 0.5. The investor’s risk aversion is γ = 3.

Skewness. We have already seen that the expected return of an asset changes its risk in the regret world. Given the intuition behind this result, i.e., the impact of expected returns on the probability of becoming the ex-post best asset and on the risk of deviating from this benchmark, it is plausible that the degree of asymmetry of the return distribution has also an effect on regret risk. We now examine this issue and ask to what extent optimal portfolio weights can be affected by the asymmetry of the return distribution in our model of portfolio choice. For such an analysis we have to drop the assumption of normally distributed returns. Instead, we use the Gram-Charlier density, which allows for a direct modeling of skewness.13 The GramCharlier density can be seen as an extension of the normal distribution with different higher moments. If skewness and excess kurtosis are zero, the density coincides with the normal density. Our comparative static analysis leaves the setting of the base case untouched, but varies the skewness of Asset 1 between −1 and +1. Part A of Fig. 4 shows the corresponding effects on portfolio weights. As expected, the weight of Asset 1 increases with skewness in the pure regret case both under return-regret and preference-regret. For left-skewed distributions the portfolio weights are below 0.5 and for right-skewed distributions they are above 0.5. For highly asymmetric distributions, the quantitative effects are quite substantial, leading to weights of only about 0.35 for negative skewness and about 0.65 for positive skewness under return-regret. In the preference-regret model the effects are weaker, because we have a 13

See, e.g., Jondeau and Rockinger (2001).

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Fig. 3. Effects of a shift in standard deviation with different expected returns. This figure shows the effects of a shift in the standard deviation of an asset (Asset 1) on its optimal portfolio weight in a base scenario with different expected returns of the assets. The scenario uses two assets with a bivariate normal return distribution, identical standard deviations of 25%, and a correlation of 0.5. The expected return of Asset 1 equals 5% and the expected return of Asset 2 equals 10%. The investor’s risk aversion is γ = 3. The solid line refers to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret.

similar trade-off as already discussed for a change in expected return. On the one hand, taking a higher weight in the asset with the higher skewness is good because it reduces regret risk. On the other hand, however, deviating too much from a well diversified portfolio increases the risk of a high regret, because the variance enters into the preference function under preference-regret. Part B of the figure stresses the intuition behind the skewness effects. The higher the skewness of Asset 1 is, the higher the covariance with the ex-post best asset and the higher the relative advantage as compared to Asset 2. In conclusion, a positive skewness is beneficial in the regret model because of its impact on the covariance matrix of regret-adjusted returns, making an asset with a positively skewed return distribution less risky than an asset with a negatively skewed distribution. This property is in sharp contrast to the Markowitz case, where skewness has no effect on portfolio weights. Correlation. We next look at changes in correlations. In the two-asset case a comparative-static change in the correlation has no effect on portfolios weights if the assets have identical expected returns and standard deviations, leaving them fully homogeneous irrespective of their correlation. Only the multi-asset case allows us to introduce some heterogeneity in correlations that potentially has an impact on portfolio weights. To investigate correlation effects, the new base scenario consists of three (ten) assets with a multivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and identical correlations of 0.5. We then vary the correlation between Asset 1 and all other assets. Fig. 5 provides the corresponding results. Part A of the figure refers to the three-asset case and Part B to the ten-asset case. Both parts of the figure show the same tendency: the higher the correlation, the lower the portfolio weight. This result holds for the Markowitz case, the pure regret case under return-regret, and the pure regret case under preference-regret. The well-known intuition for the Markowitz case states that an asset that is highly correlated with other assets offers fewer diversification benefits. The same intuition works with respect to regret risk. Think of an asset with a low (even negative) correlation. This asset imposes a high regret risk on the other assets if it happens to be the ex-post best asset. To reduce the regret risk of the portfolio, one should therefore overweight this asset in comparison to a weight of 1/N. The correlation effects in the pure regret case, however, are less pronounced than in the Markowitz case, suggesting that the introduction of regret into portfolio selection still mitigates correlation effects to some degree. In line with the results for a variation of the standard deviation, we find that the weights according to the preference-regret model fall in between the weights resulting from the Markowitz model and the return-regret model. Risk aversion. We now analyze the effects of changing preference parameters and start with the risk aversion γ . If assets were homogenous, we would obtain portfolio weights of 1/N, irrespective of the degree of risk aversion and irrespective of using the Markowitz model or the regret model. Therefore, we have to introduce some heterogeneity in the return distribution to study the effects of a varying γ . We look at two assets with a bivariate normal distribution. Asset 1 has an expected return of 5% and a standard deviation of 10%, which are the lowest values we consider in our previous analyses in Figs. 1 and 2. Asset 2 has an expected return of 15% and a standard deviation of 40%, which are the highest values we consider in our previous analysis. The return correlation equals 0.5.

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Fig. 4. Effects of a shift in skewness. This figure shows the effects of a changing return skewness of an asset (Asset 1) on its optimal portfolio weight (Part A) and on the covariance with the ex-post best portfolio (Part B). In Part A the solid line refers to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret. In Part B the solid line refers to the return covariance of Asset 1 with the ex-post best portfolio (Cov (R1 , Ropt )) and the dotted line to the covariance of Asset 2 with the ex-post best portfolio (Cov (R2 , Ropt )). These covariances are numerically almost indistinguishable for the return-regret case (covariance with the ex-post best asset) and the preference-regret case (covariance with the preference value of the ex-post best portfolio), i.e., the depicted lines refer to both cases. The base scenario uses two assets with a bivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and a correlation of 0.5. For Asset 1 a Gram-Charlier density with varying skewness is assumed, while the return distribution of Asset 2 stays normal with zero skewness. The investor’s risk aversion is γ = 3.

Fig. 6 shows the optimal weight of Asset 1 for different levels of γ . In the Markowitz case, a risk-neutral investor would invest entirely in the asset with the higher expected return (Asset 2). With higher risk aversion, portfolio variance becomes more and more important, and the optimal portfolio eventually converges to the minimum variance portfolio, which is a full investment in Asset 1 in our setting. In the pure regret case, a risk-neutral investor would chose the asset with the lowest expected regret, which is again the asset with the highest expected return. This statement is true both under return-regret and preference-regret, because the preference functions coincide for γ = 0. According to our previous analysis, differences between expected returns are more important and differences between standard deviations less important under returnregret than under the Markowitz model. This is the reason why the weight of Asset 1 does not increase so much. Even if regret aversion becomes greater and finally the minimum variance portfolio is chosen, this minimum variance portfolio (with respect to the variance of regret) still has a high weight in Asset 2. Under preference-regret, the effects of both expected return and standard deviation are weaker than in the Markowitz case, leaving the overall effect unclear. As we see from the figure, the weight of the low-risk-low-return asset (Asset 1) is higher than in the Markowitz case for low levels of risk aversion and higher for higher levels of risk aversion.

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Fig. 5. Effects of a shift in correlation. This figure shows the effects of a shift in the return correlation of an asset (Asset 1) with all other assets on its optimal portfolio weight in an investment universe with three assets (Part A) and ten assets (Part B). The base scenario uses three (ten) assets with a multivariate normal return distribution, identical expected returns of 10%, identical standard deviations of 25%, and identical correlations of 0.5. The investor’s risk aversion is γ = 3. The solid lines refer to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret.

Relative importance of investor wants. Our model contains a second preference parameter, α , describing the relative importance of the two investor wants: return and regret. Our analysis of portfolio weights concentrated on the extreme cases α = 0 and α = 1, because Property 3 provides information about the intermediate cases. However, portfolio weights are not the only important aspect to look at. Another one is the characterization of the investment opportunity set and the efficient frontier in terms of ex-ante return and risk. Fig. 7 plots attainable risk-return combinations for different values of α , using the same two-asset universe that underlies Fig. 6.14 A first observation is that (regret-adjusted) expected returns fall with growing α , because α times the expected regret is subtracted from the (unadjusted) expected returns. The difference in expected returns between Asset 1 and Asset 2, however, does not change with α , because the same benchmark return is subtracted from the returns of any portfolio. A second observation refers to the (regret-adjusted) standard deviation. The lowest attainable value is achieved for α = 0.5. Intuitively, for α = 0.5, diversification effects between regret variance and return variance are most pronounced in this case. Finally, we look at the set of efficient portfolios. For the Markowitz case (α = 0), all combinations of Asset 1 and Asset 2 are efficient. With growing α portfolios with a high weight in Asset 1 become inefficient. For α = 1 even the minimum variance

14 Fig. 7 concentrates on return-regret because the concept of an efficient frontier does not make sense under preference-regret in our view. Under preference-regret both expectation and variance of every portfolio’s regret-adjusted returns already depend on the risk aversion of the investor, i.e., there would be a different “efficient frontier” for each value of γ . However, if γ is specified, there is already a unique optimal portfolio.

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Fig. 6. Effects of a shift in risk aversion on the portfolio weight of Asset 1. This figure shows the effects of a changing risk aversion (γ ) on the optimal portfolio weight of Asset 1. The solid line refers to the Markowitz case (α = 0), the dashed line to the pure regret case (α = 1) under return-regret, and the dotted line to the pure regret case (α = 1) under preference-regret. The return scenario uses two assets with a bivariate normal return distribution. Asset 1 has an expected return of 5% and a standard deviation of 10%. Asset 2 has an expected return of 15% and a standard deviation of 40%. The return correlation is 0.5.

Fig. 7. Attainable risk-return-combinations for different values of α . This figure shows attainable combinations of (regret-adjusted) expected return and return standard deviation for different values of α . Regret is measured as return-regret. The return scenario uses two assets with a bivariate normal distribution of (unadjusted) returns. Asset 1 has an expected return of 5% and a standard deviation of 10%. Asset 2 has an expected return of 15% and a standard deviation of 40%. The return correlation is 0.5.

portfolio invests only 40% in Asset 1. This finding shows again that regret risk can be very different from standard deviation, and higher standard deviation does not necessarily translate into higher regret risk. Summary. What makes an asset more or less attractive in the regret model than in the Markowitz model? In terms of its return moments, we obtain the following answers: (i) High expected returns are more important than in the Markowitz model under return-regret but less important under preference-regret. (ii) Standard deviation and correlation have a lower impact on portfolio weights in the regret model (both under return-regret and preference-regret) than in the Markowitz model. (iii) Return skewness does not enter the Markowitz model but has an impact on the regret standard deviation. The higher its skewness, the more attractive an asset becomes in the regret model. These properties translate into the ex-ante return distribution of an optimal portfolio. Under return-regret, we obtain an ex-ante return distribution of the optimal portfolio that tends to have a higher expected return, a higher standard deviation, and a higher skewness than in the Markowitz case. Under preference-regret, it is unclear if an optimal portfolio’s expectation and variance of (unadjusted) returns will be above or below the corresponding moments in the Markowitz case. As we have seen in Fig. 6, these properties may well depend on the investor’s risk aversion.

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R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24 Table 1 Description of investment universes. Universe

Constituents

Period

Source

Specification

Stocks-Bonds

Stock index Bond index

Jan 1988–Feb 2017

Datastream

Russell 10 0 0 as broad US stock market index and Bloomberg Barclays US aggregated bond index

Investment Styles

Market Size Value Momentum

Nov 1926–Dec 2016

French Library

Calculation based on CRSP universe

Regions

Latin America North America Asia Europe Pacific Basin

Jul 1994–Feb 2017

Datastream

Specification based on Datastream’s regional indices

This table provides information about the asset universes and data sets used to obtain optimal portfolios. It shows the constituents of the universes, the data periods used to estimate input parameters, the data sources, and how the individual constituents (funds) are specified.

4. Regret effects for real portfolios 4.1. Empirical setup and data To study the impact of regret on real portfolios we use three different investment universes with a varying number of assets (funds). The universes represent different problems of asset allocation. These are concerned with the choice between stocks and bonds (two constituents), different investment styles (four constituents) and different regions (five constituents). More detailed information on the universes and the data that we use to obtain the necessary input parameters is given in Table 1. We determine optimal portfolios both for the Markowitz case (α = 0) and the pure regret case (α = 1) under both returnregret and preference-regret. As for the simulated portfolios, we use a risk-aversion parameter of γ = 3 for μ-σ portfolio optimization.15 In addition, the minimum-variance portfolios based on the (unadjusted) return variance and the regretadjusted variance, respectively, are determined. The reason is that the global minimum variance portfolio is the only efficient portfolio that can be formed without knowledge of expected returns. Because expected returns are difficult to estimate (Merton, 1980), there is an ongoing interest in risk-minimizing strategies. All necessary input parameters are obtained from the corresponding sample moments using a daily data frequency.16 For each universe we report in the respective Panel A of Tables 2–4 the mean return, the standard deviation, the skewness, the regret adjustment of the expected return (covariance between the fund’s return and the return of the ex-post best fund or covariance between the fund’s return and the ex-post best preference value) according to Property 2, and the regret-adjusted standard deviation of each fund. Under preference-regret the benchmark depends of γ , so there are different benchmarks for the μ-σ case and the minimum-variance case, resulting in different μ-adjustments and regret standard deviations. Means and standard deviations are presented as annualized values. Because the relative attractiveness of an asset within a specific universe depends on how its return characteristics deviate from those of the other assets, we also present the deviations of individual funds’ return characteristics from the average values of all funds (in parentheses). Panels B of Tables 2–4 show the optimal portfolio weights resulting from μ-σ optimization and variance minimization both for the Markowitz case and the pure regret case (return-regret and preference-regret). Moreover, Panels C of Tables 2–4 report information on the ex-ante return distribution of optimal portfolios for the Markowitz case, the pure regret case (both views) and for an intermediate case with a regret parameter α = 0.5. The characteristics of the ex-ante return distribution are obtained via simulations from the empirical distribution, using optimal portfolio weights. Finally, the impact of regret on portfolio characteristics is studied graphically based on the efficient frontier in the Markowitz case as a benchmark (Fig. 8). Relative to this benchmark, the positions of six different portfolios in μ-σ space are presented: the μ-σ -optimal portfolio in the Markowitz case and the two pure regret cases as well as the minimum-variance portfolio for all three cases.

15 16

Other levels of risk aversion have been studied, but lead to the same conclusions. Estimation with weekly and monthly data has also been performed, but changes the results only marginally.

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Table 2 Results for the stocks-bonds universe. Panel A: return characteristics

Mean return Std. deviation Skewness

Stock index

Bond index

0.1120 (0.0257) 0.1696 (0.0497) −0.1411 (−0.0287)

0.0606 (−0.0257) 0.0702 (−0.0497) −0.0838 (0.0287)

0.0381 (0.0166) 0.1252 (0.0035)

0.0049 (−0.0166) 0.1182 (−0.0035)

0.0128 (0.0055) 0.1261 (0.0043) 0.0017 (−0.0011) 0.1714 (0.0729)

0.0017 (−0.0055) 0.1176 (−0.0043) 0.0040 (0.0011) 0.0256 (−0.0729)

Pure regret: return view Regret adjustment μ Regret-adj. std. dev. Pure regret: preference view

μ-σ (γ =3)

Regret adjustment μ Regret-adj. std. dev. Regret adjustment μ

Min-Var

Regret-adj. std. dev. Panel B: portfolio weights

μ-σ (γ =3)

Markowitz

Min-Var Pure regret:

μ-σ (γ =3)

Return view

Min-Var

Pure regret:

μ-σ (γ =3)

Preference view

Min-Var

Stock index

Bond index

0.6374 (0.1374) 0.1839 (−0.3161) 0.9312 (0.4312) 0.4776 (−0.0224) 0.5476 (0.0476) 0.1515 (−0.3485)

0.3626 (−0.1374) 0.8161 (0.3161) 0.0688 (−0.4312) 0.5224 (0.0224) 0.4524 (−0.0476) 0.8485 (0.3485)

Panel C: ex-ante return distribution

μ-σ (γ =3)

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95) ES (α = 0.99)

Markowitz

Return view

(α = 0)

Regret (α = 0.5)

Pure regret (α = 1)

Preference view Regret (α = 0.5)

Pure regret (α = 1)

0.0966 0.0989 0.6680 0.2292 −0.0947 −0.1422

0.1044 0.1204 0.6135 0.2392 −0.1275 −0.1843

0.1122 0.1436 0.5686 0.2356 −0.1647 −0.2327

0.0939 0.0923 0.6875 0.2204 −0.0851 −0.1301

0.0926 0.0892 0.6971 0.2147 −0.0807 −0.1245

Markowitz

Return view

(α = 0)

Regret ( α = 0.5 )

Pure regret (α = 1)

Regret ( α = 0.5 )

Pure regret (α = 1)

0.0735 0.0646 0.6662 0.1420 −0.0549 −0.0903

0.0813 0.0684 0.7434 0.1393 −0.0548 −0.0922

0.0891 0.0813 0.7214 0.1945 −0.0699 −0.1111

0.0724 0.0649 0.6471 0.1495 −0.0563 −0.0916

0.0719 0.0651 0.6369 0.1537 −0.0571 −0.0924

Min-Var

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95) ES (α = 0.99)

Preference view

This table shows the return characteristics (Panel A) and optimal portfolio weights (Panel B) of each asset as well as some characteristics of the ex-ante return distribution (Panel C) of optimized portfolios for the stocks-bonds universe. The universe consists of two assets: one fund with stocks (stock index) and one fund with bonds (bond index). Mean return, standard deviation, and skewness are calculated for the unadjusted returns of each asset. The regret-adjustment for μ is the covariance between the asset’s return and the return of the ex-post best alternative, which differs between return-regret and preference-regret. The adjustment can be transferred into any investor-specific adjustment by multiplying it with 2αγ . The regret-adjusted standard deviation is the standard deviation of regret-adjusted returns. Panel B displays the optimal weights for a Markowitz investor, a pure return-regret investor and a pure preference-regret investor. The optimal portfolio weights under μ-σ optimization (γ =3) and variance minimization are shown. In Panels A and B, for all presented values the deviations from their mean are given in parentheses below each value. Panel C provides some properties of the ex-ante distribution of each optimized portfolio: mean return, standard deviation, Sharpe ratio, skewness and expected shortfall (ES). All distributional values are annualized.

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Table 3 Results for the investment styles universe. Panel A: return characteristics

Mean return Std. deviation Skewness

Market

SMB

HML

MOM

0.0726 (0.0234) 0.1697 (0.0511) −0.1124 (0.3256)

0.014 (−0.0352) 0.0931 (−0.0255) −0.7698 (−0.3318)

0.0431 (−0.0061) 0.0930 (−0.0256) 0.7451 (1.1831)

0.0671 (0.0179) 0.1191 (0.0 0 05) −1.6148 (−1.1768)

0.0190 (0.0161) 0.1575 (0.0102)

−0.0054 (−0.0083) 0.1450 (−0.0023)

0.0012 (−0.0017) 0.1286 (−0.0187)

−0.0033 (−0.0062) 0.1582 (0.0109)

0.0080 (0.0052) 0.1583 (0.0113) 0.0028 (0.0 0 03) 0.1597 (0.0534)

0.0 0 01 (−0.0027) 0.1441 (−0.0029) 0.0025 (−0.0 0 01) 0.0785 (−0.0279)

0.0023 (−0.0 0 05) 0.1280 (−0.0190) 0.0023 (−0.0 0 02) 0.0804 (−0.0259)

0.0 0 08 (−0.0020) 0.1575 (0.0106) 0.0025 (0.0 0 0 0) 0.1067 (0.0 0 04)

Pure regret: return view Regret adjustment

μ

Regret-adj. std. dev. Pure regret: preference view

μ-σ (γ =3)

Regret adjustment

μ

Regret-adj. std. dev. Min-Var

Regret adjustment

μ

Regret-adj. std. dev. Panel B: portfolio weights

Markowitz

μ-σ (γ =3) Min-Var

Pure regret:

μ-σ (γ =3)

Return view

Min-Var

Pure regret:

μ-σ (γ =3)

Preference view

Min-Var

Market

SMB

HML

MOM

0.3857 (0.1357) 0.1107 (−0.1393) 0.5402 (0.2902) 0.3177 (0.0677) 0.3591 (0.1091) 0.1177 (−0.1323)

0.0 0 0 0 (−0.2500) 0.3366 (0.0866) 0.0 0 0 0 (−0.2500) 0.1975 (−0.0525) 0.0 0 0 0 (−0.2500) 0.3380 (0.0880)

0.0 0 0 0 (−0.2500) 0.3259 (0.0759) 0.0 0 0 0 (−0.2500) 0.2943 (0.0443) 0.1872 (−0.0628) 0.3151 (0.0651)

0.6143 (0.3643) 0.2268 (−0.0232) 0.4598 (0.2098) 0.1905 (−0.0595) 0.4538 (0.2038) 0.2292 (−0.0208)

Panel C: ex-ante return distribution

μ-σ (γ =3) Markowitz

Return view

( α = 0)

Regret ( α = 0. 5)

Pure regret (α = 1)

Preference view Regret ( α = 0. 5)

Pure regret (α = 1)

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95)

0.0867 0.1098 0.4823 0.0955 −0.1372

0.0883 0.1097 0.4958 0.1053 −0.1349

0.0898 0.1143 0.4883 0.1592 −0.1395

0.0832 0.0965 0.5113 0.0830 −0.1141

0.0809 0.0904 0.5198 0.0787 −0.1046

ES (α = 0.99)

−0.2093

−0.2064

−0.2128

−0.1772

−0.1652

Markowitz

Return view

( α = 0)

Regret ( α = 0. 5)

Pure regret (α = 1)

Regret ( α = 0. 5)

Pure regret (α = 1)

0.0551 0.0681 0.3096 0.3253 −0.0762 −0.1310

0.0610 0.0740 0.3651 0.3168 −0.0823 −0.1466

0.0670 0.0846 0.3900 0.3555 −0.0943 −0.1629

0.0553 0.0682 0.3129 0.3228 −0.0763 −0.1318

0.0554 0.0683 0.3145 0.3216 −0.0763 −0.1322

Min-Var

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95) ES (α = 0.99)

Preference view

This table shows the return characteristics (Panel A) and optimal portfolio weights (Panel B) of each asset as well as some characteristics of the ex-ante return distribution (Panel C) of optimized portfolios for the styles universe. The universe consists of four assets: one fund representing the market portfolio (Market), one fund representing the size factor (SMB), one fund representing the value factor (HML), and one fund representing the momentum factor (MOM). Mean return, standard deviation, and skewness are calculated for the unadjusted returns of each asset. The regret-adjustment for μ is the covariance between the asset’s return and the return of the ex-post best alternative, which differs between return-regret and preference-regret. The adjustment can be transferred into any investor-specific adjustment by multiplying it with 2αγ . The regret-adjusted standard deviation is the standard deviation of regret-adjusted returns. Panel B displays the optimal weights for a Markowitz investor, a pure return-regret investor and a pure preference-regret investor. The optimal portfolio weights under μ-σ optimization (γ =3) and variance minimization are shown. In Panels A and B, for all presented values the deviations from their mean are given in parentheses below each value. Panel C provides some properties of the ex-ante distribution of each optimized portfolio: mean return, standard deviation, Sharpe ratio, skewness and expected shortfall (ES). All distributional values are annualized.

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

17

Table 4 Results for the regions universe. Panel A: return characteristics Mean return Std. deviation Skewness

LA

NA

AS

EU

PB

0.0976 (0.0199) 0.2296 (0.0379) −0.3689 (−0.1881)

0.1078 (0.0301) 0.1812 (−0.0105) −0.1733 (0.0075)

0.0439 (−0.0338) 0.1788 (−0.0129) −0.0759 (0.1049)

0.0922 (0.0145) 0.1894 (−0.0023) −0.1815 (−0.0 0 07)

0.0468 (−0.0309) 0.1794 (−0.0123) −0.1043 (0.0765)

0.0831 (0.0226) 0.1649 (0.0014)

0.0573 (−0.0032) 0.1571 (−0.0064)

0.0486 (−0.0119) 0.1721 (0.0086)

0.0642 (0.0037) 0.1523 (−0.0112)

0.0495 (−0.0110) 0.1711 (0.0076)

0.0275 (0.0074) 0.1659 (0.0025) 0.0210 (0.0027) 0.1686 (0.0352)

0.0191 (−0.0010) 0.1567 (−0.0067) 0.0187 (0.0 0 04) 0.1155 (−0.0179)

0.0162 (−0.0039) 0.1717 (0.0084) 0.0152 (−0.0031) 0.1395 (0.0060)

0.0214 (0.0012) 0.1519 (−0.0115) 0.0213 (0.0029) 0.1057 (−0.0278)

0.0165 (−0.0036) 0.1707 (0.0073) 0.0155 (−0.0028) 0.1379 (0.0045)

Pure regret: return view Regret adjustment μ Regret-adj. std. dev. Pure regret: preference view

μ-σ (γ =3)

Regret adjustment μ Regret-adj. std. dev.

Min-Var

Regret adjustment μ Regret-adj. std. dev.

Panel B: portfolio weights

Markowitz

μ-σ (γ =3) Min-Var

Pure regret:

μ-σ (γ =3)

Return view

Min-Var

Pure regret:

μ-σ (γ =3)

Preference view

Min-Var

LA

NA

AS

EU

PB

0.0 0 0 0 (−0.20 0 0) 0.0 0 0 0 (−0.20 0 0) 0.2696 (0.0696) 0.2533 (0.0533) 0.1185 (−0.0815) 0.0 0 0 0 (−0.20 0 0)

0.7078 (0.5078) 0.4186 (0.2186) 0.5108 (0.3108) 0.2662 (0.0662) 0.4923 (0.2923) 0.3982 (0.1982)

0.0 0 0 0 (−0.20 0 0) 0.4565 (0.2565) 0.0 0 0 0 (−0.20 0 0) 0.1588 (−0.0412) 0.0 0 0 0 (−0.20 0 0) 0.3028 (0.1028)

0.2704 (0.0704) 0.1249 (−0.0751) 0.2195 (0.0195) 0.1627 (−0.0373) 0.2182 (0.0182) 0.2134 (0.0134)

0.0218 (−0.1782) 0.0 0 0 0 (−0.20 0 0) 0.0 0 0 0 (−0.20 0 0) 0.1589 (−0.0411) 0.1710 (−0.0290) 0.0856 (−0.1144)

Panel C: ex-ante return distribution

μ-σ (γ =3)

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95) ES (α = 0.99)

Markowitz

Case 1

(α = 0)

Regret (α = 0.5)

Pure regret (α = 1)

Regret (α = 0.5)

Case 2 Pure regret (α = 1)

0.1016 0.162 0.4764 0.1534 −0.2236 −0.3111

0.1045 0.1727 0.4639 0.1699 −0.2360 −0.3274

0.1049 0.1836 0.4381 0.2221 −0.2524 −0.3446

0.0975 0.1665 0.4389 0.1553 −0.2319 −0.321

0.0948 0.1686 0.417 0.1688 −0.2378 −0.3272

Markowitz

Case 1

(α = 0)

Regret (α = 0.5)

Pure regret (α = 1)

Regret (α = 0.5)

Pure regret (α = 1)

0.0812 0.1558 0.3336 0.1988 −0.2351 −0.3165

0.0818 0.1674 0.3422 0.2062 −0.2449 −0.3299

0.0862 0.1808 0.3413 0.2541 −0.2621 −0.348

0.0813 0.1606 0.3456 0.1857 −0.235 −0.3179

0.0818 0.1674 0.3422 0.2062 −0.2449 −0.3299

Min-Var

Mean return Std. deviation Sharpe ratio Skewness ES (α = 0.95) ES (α = 0.99)

Case 2

This table shows the return characteristics (Panel A) and optimal portfolio weights (Panel B) of each asset as well as some characteristics of the ex-ante return distribution (Panel C) of optimized portfolios for the regions universe. The universe consists of five assets: one fund with stocks from Latin America (LA), one fund with stocks from North America (NA), one fund with stocks from (Asia), one fund with stocks from Europe (EU), and one fund with stocks from the Pacific Basin (PB). Mean return, standard deviation, and skewness are calculated for the unadjusted returns of each asset. The regret-adjustment for μ is the covariance between the asset’s return and the return of the ex-post best alternative, which differs between return-regret and preference-regret. The adjustment can be transferred into any investor-specific adjustment by multiplying it with 2αγ . The regret-adjusted standard deviation is the standard deviation of regret-adjusted returns. Panel B displays the optimal weights for a Markowitz investor, a pure return-regret investor and a pure preference-regret investor. The optimal portfolio weights under μ-σ optimization (γ =3) and variance minimization are shown. In Panels A and B, for all presented values the deviations from their mean are given in parentheses below each value. Panel C provides some properties of the ex-ante distribution of each optimized portfolio: mean return, standard deviation, Sharpe ratio, skewness and expected shortfall (ES). All distributional values are annualized.

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R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

Fig. 8. Efficient frontier and optimal portfolios. This figure shows the efficient frontier in the Markowitz case for the stocks-bonds universe (Part A), the styles universe (Part B) and the regions universe (Part C). All efficient portfolios in the Markowitz case with short sales constraints are plotted as solid lines. In addition, the optimal portfolios for a μ-σ investor with γ = 3 as well as a Min-Var investor are depicted for the Markowitz case, the pure regret case under return-regret, and the pure regret case under preference-regret.

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

19

4.2. Results Stocks-bonds universe Table 2 presents the asset characteristics, optimal portfolio weights and information on the ex-ante return distribution for the stocks-bonds universe. Because the universe contains only two assets, it is a convenient starting point. Moreover, it directly relates to Harry Markowitz’s statement in the introduction. If the investor’s wealth is split up equally between stocks and bonds, does this really minimize regret? As Table 2 shows, the mean returns, standard deviations, and skewnesses of the two funds clearly differ. The mean return and the standard deviation are higher for the stock index, as one would expect. In the Markowitz case the stock index is more attractive and receives a clearly higher weight than the bond index under μ-σ optimization, while variance minimization leads to an overweighting of the bond index. The pure regret case under return-regret shows a very different picture. The higher expected return and only slightly lower skewness make the stock index even more attractive. Moreover, the role of the (unadjusted) standard deviation is diminished. These effects show up in the relatively high regret adjustment for μ, leading to a strong overweighting of the stock index. Actually, in terms of regret risk the two funds are almost equally risky, which translates into an about fifty-fifty allocation for the (regret) variance minimization. Under preferenceregret, the μ-adjustment for the stock index is smaller than under return-regret. Together with the scaled risk aversion coefficient, this property translates into higher portfolio weights for the bond index, making the outcome more similar to the Markowitz case. The ex-ante return distribution of optimal portfolios confirms the observation that the Markowitz case and the preference-regret view are closer together. In contrast, return-regret leads to an optimal portfolio with higher expected return, higher risk (both standard deviation and expected shortfall) and higher skewness. This is in line with our previous results for simulated data in Section 3. The results allow us to give an answer to the question relating to Markowitz’s investment problem. The answer depends on the relative importance of expected regret and regret risk as well as the view on regret (return-regret versus preferenceregret): Under return-regret, splitting the wealth equally between stocks and bonds is not even close to the optimal solution for the pure regret case if expected regret plays a significant role, i.e., if γ is not very large. Stocks should then be strongly overweighted as compared to bonds. If the goal is to minimize regret risk (meaning a risk aversion parameter γ tending to infinity), however, a fifty-fifty rule is nearly the optimal allocation. Under preference-regret, for the avoidance of regret risk, bonds should be clearly overweighted, even more than in the Markowitz case. In contrast, if expected regret plays a significant role, the optimal weighting of stocks and bonds can become close to the fifty-fifty rule. Investment styles universe The results for the investment styles universe are presented in Table 3. We clearly observe that the effects of high expected returns and high standard deviations are very different in the Markowitz case, the pure-regret case under returnregret, and the pure-regret case under preference-regret. Based on μ-σ optimization, the highest weight of the market fund—the asset with the highest expected return—is achieved under return-regret, while preference-regret leads to the lowest value. In contrast, the standard deviation of returns is least important under return-regret: Considering variance minimization, the two funds with the lowest standard deviations (SMB, HML) receive relatively low weights under return-regret as compared to the Markowitz model and the preference-regret model. The characteristics of the ex-ante return distribution confirm our previous results. The μ-σ optimal portfolio under return-regret has the highest expected return, the highest standard deviation, the highest expected shortfall, and the highest skewness. The optimal portfolios in the Markowitz case and under preference-regret are again rather similar in their return characteristics. Regions universe Table 4 shows the results for our last example, the regions universe consisting of five assets. From this universe we can learn different things. First, we see again that the effects of a move from the Markowitz case to the pure regret case on optimal portfolio weights can be very large, in particular under return-regret. If we consider the LA fund, the Markowitz investor would not invest at all, whereas the regret-averse investor would buy (preference-regret view) and even overweight (return-regret view) this fund. The reason is the very high standard deviation of the fund that makes it unattractive for the Markowitz investor. Second, it can well be that two assets with apparently similar return characteristics react quite differently to the introduction of regret. An example are the AS and PB funds, which have very similar (univariate) return characteristics. The weight of the AS fund is reduced substantially under variance minimization as compared to the Markowitz case, while the weight of the PB fund increases from zero to a noteworthy amount. This observation holds true for both views on regret. This result emphasizes the importance of the co-movement of the assets’ returns, which can well be different for the (unadjusted) returns and the regret-adjusted returns. Third, an above-average regret-adjustment for μ is not generally an indication of an increasing portfolio weight when moving to the pure regret case (see the EU fund). Instead, the specific structure of the adjustments of all other assets also plays a role. Fourth, the regret effects can be substantially different for μ-σ optimization than for variance minimization. Portfolio weights might well increase in one case and decrease in the other, as can be seen for the PB fund. Finally, the ex-ante return distributions confirm the results from the previous examples that return-regret leads to optimal portfolios with relatively high expected returns, high variance, high shortfall risk, and high skewness.

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Efficient frontiers Our examples show that optimal portfolios under regret aversion can look quite differently than optimal portfolios in the classical Markowitz case. To illustrate these differences graphically, we take a look at the (unadjusted) means and standard deviations of optimal portfolios under regret compared to the classical benchmark. Fig. 8 displays the Markowitz efficient frontier for each universe (Part A: Stocks-Bonds, Part B: Styles, Part C: Regions). Within this (unadjusted) μ-σ space, additionally to the optimal portfolios in the Markowitz case (μ-σ optimization and variance minimization), also the optimal portfolios in the pure regret case under both views on regret are shown. Besides the illustration of some stylized facts already discussed above, Fig. 8 reveals that optimal portfolios under regret may be well below the Markowitz efficient frontier. This observation holds true for both views on regret, but it is more pronounced for the return-regret view. Furthermore, the distance to the efficient frontier can become larger with a growing number of assets. (With only two assets as in the stocks-bonds universe, all portfolios lie on the same line, which is efficient above the minimum-variance portfolios.17 ) So we can confirm that introducing regret cannot simply be captured by changing the risk aversion parameter, but is substantially different from classical Markowitz optimization. Summary In summary, the examples of problems in real asset allocation have demonstrated that the quantitative effects of regret risk on optimal portfolios can be very large. Moreover, the intuition from Section 3 about which return characteristics make an asset more or less attractive in the regret world as compared to the Markowitz world are broadly confirmed. However, our analysis also shows that it is not always easy to anticipate the effects of regret based on some simple univariate indicators, which highlights the importance of a formal yet tractable optimization procedure. 5. Conclusions This paper has extended Markowitz’s portfolio selection by considering investors’ regret as an additional decision criterion beyond final wealth. We have introduced two views on regret which are both compatible with mean-variance analysis. Under both views, the additional criterion does not destroy the simplicity and tractability of the Markowitz model. As has been shown, regret affects optimal portfolios as compared to the Markowitz model only because it leads to a different risk measure. However, this risk measure does not only depend on the portfolio’s return variance but also on the expected return and skewness. These results demonstrate that the regret-averse investor, seeking to be close to the ex-post best portfolio (either in terms of return or of preference value), has a distinctively different view on risk than the traditional mean-variance investor. This different view can have a strong impact on optimal portfolio weights and the ex-ante return distribution of optimal portfolios, which has been shown for different examples of real portfolios. Moreover, the notion of a risk-free asset is a different one in the regret model, because no asset exists that is free of regret risk. Therefore, we do not have to distinguish between risk-free and risky assets in our approach. It was our intention to stay as close as possible to the Markowitz model in our analysis. However, different extension could be considered. A first one would go beyond the mean and variance of regret-adjusted returns in the preference function (6) and consider higher-order moments like skewness and kurtosis. Formally, such an extension is straightforward. However, it would increase the number of input parameters considerably, complicate analytical results, and make the interpretation of results more difficult. Furthermore, the limitation to a single risk aversion coefficient for both regret risk and classical risk could be abandoned. While this limitation comes with the advantage that no additional parameter has to be estimated, there might be investors with different attitudes to classical risk and regret risk. Introducing an additional risk parameter would be a natural way to allow for this aspect of investor preferences. Another natural variation of the preference function would be to replace the variance of regret-adjusted returns by some measure of downside risk, like lower-partial moments. Moreover, one could consider extending the investment universe to non-linear instruments. As suggested by the results of Muermann et al. (2006) on return guarantees in defined contribution pension plans and Korn and Rieger (2019) on optimal hedging instruments for corporate hedging under regret, such non-linear instruments could be valuable for investors who are concerned about regret. Finally, a significant enhancement would be a multi-period or time-continuous setting in contrast to Markowitz’s one-period model. Although such approaches are discussed in portfolio optimization for decades, they would require a completely new paradigm for regret theory because there are many ways to capture regret in such a dynamic setting. Appendix Positive definiteness of covariance matrix of regret-adjusted returns: The covariance matrix of the (regret-adjusted) random variables is positive definite if and only if there are no two linear combinations of the random variables, represented by weight vectors ω1 and ω2 , which are perfectly (positively or negatively) correlated. Consider the vector of regret-adjusted returns under return-regret, Z ret := (R1 − α Rmax , R2 − α Rmax , . . . , RN − α Rmax )tr . Looking at any two linear combinations ω1tr Z ret and ω2tr Z ret , we obtain: 17

Note that the styles universe essentially collapses to a two-asset universe under μ-σ optimization.

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

ω1tr Z ret = ω1tr R − α Rmax ,

21

(13)

ω2tr Z ret = ω2tr R − α Rmax .

(14)

ω1tr

ω2tr

If the original covariance matrix  is positive definite, the linear combinations R and R are not perfectly correlated. Then ω1tr Z ret and ω2tr Z ret will also not be perfectly correlated, because subtracting the same random variable α Rmax from the random variables ω1tr R and ω2tr R cannot lead to perfect correlation. Under preference regret, we can make the same argument. Subtracting the random variable α π max from two random variables that are not perfectly correlated cannot introduce perfect correlation. Therefore, the covariance matrix of regretadjusted returns is positive definite if and only if the covariance matrix of the original returns is positive definite. Proof of Property 1. To show that Property 1 holds true, we first consider return-regret. Writing the investor’s preference function from Eq. (6) as a function of the portfolio weights ω, we obtain tr ret (ZPret ) = ωtr μret ra − γ (ω ra ω ), ret = ωtr μ − α E[Rmax ] − γ (ωtr ra ω ),

(15)

because portfolio weights add up to one. The term α E[Rmax ] is independent of ω, i.e., it cannot be changed by choosing a different portfolio. Thus, for the determination of optimal portfolio weights we can ignore this term and work with μ instead of μret ra . Under preference-regret, the preference function to be optimized reads

(ZPpre f ) = μP − α γ σP2 − α E (πmax ) − γ V ar (RP − α πmax ) = μP − γ v − α E (πmax )

(16)

with an adjusted variance term

v := V ar (RP − α πmax ) + α V ar (RP ).

(17)

Also using the portfolio weight vector ω, the variance term becomes

v = ωtr (rapre f + α ) ω. With the same argument as for return-regret, we can ignore the term α E(π max ), which shows that the property also holds under preference-regret.  Proof of Property 2. To show Property 2 under return-regret, we start by noting that the (i, j) entry of the covariance matrix of regret-adjusted returns is ret (ra )i, j = Cov(Ri − α Rmax , R j − α Rmax )

(18)

= Cov(Ri , R j ) − α Cov(Ri , Rmax ) − α Cov(R j , Rmax ) + α 2 V ar (Rmax ). It follows:

ωtr raret ω = ωtr  ω − 2 α



ωi

i

=



ω j Cov(R j , Rmax ) + α 2V ar (Rmax )

j

ωtr  ω − 2 α ωtrCov(R, Rmax ) + α 2V ar (Rmax ).

(19)

Using the alternative expression for ωtr  ra ω from the right hand side of Eq. (19) and inserting it into Eq. (15) yields

(ZPret ) = ωtr (μ + 2 α γ Cov(R, Rmax ) ) − γ (ωtr  ω ) − c , (20) with a constant c = α E[Rmax ] + γ α 2 V ar (Rmax ). This constant, however, is irrelevant for the determination of the optimal portfolio weights. Quite analogously, under the preference-regret view

(rapre f )i, j = Cov(Ri − α πmax , R j − α πmax )

(21)

= Cov(Ri , R j ) − α Cov(Ri , πmax ) − α Cov(R j , πmax ) + α V ar (πmax ). 2

It follows

ωtr (rapre f + α ) ω = (1 + α )ωtr  ω − 2 α ωtr Cov(R, πmax ) + α 2 V ar (πmax ).

(22)

So we can write

(ZPpre f ) = ωtr (μ + 2 α γ Cov(R, πmax )) − γ (1 + α ) ωtr  ω − c

(23)

with a constant

c = α E (πmax ) + γ

α 2 V ar (πmax ).

(24) 

Proof of Property 3. To show Property 3, note that according to Property 2 we can use a vector of adjusted expected returns μ ˜ and the unadjusted return covariance matrix to find optimal portfolio weights. When short-sales constraints are

22

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

not binding, the following closed-form solution for the optimal weights is available:18 ∗ ωra =

1 −1  2γ˜

  1tr  −1 μ ˜ − 2γ˜ μ˜ − 1 . 1tr  −1 1

(25)

(Under return-regret, γ˜ = γ , while under preference-regret, γ˜ = γ (1 + α ).) As under return-regret γ˜ does not depend on α , we can verify directly that a weighted average of the solutions for the Markowitz case (with weight 1 − α ) and the pure regret case (with weight α ) delivers the optimal solution in Eq. (25). To see this note that we can concentrate on μ ˜ , because no other terms depend on whether we are in the Markowitz case or in the pure regret case. The decomposition

μ˜ = (1 − α ) μ + α [μ + 2 γ Cov(R, Rmax )]

(26)

delivers the required result. For preference-regret, things are a little bit more complicated, as γ˜ depends on α . Inserting μ ˜ = μ + α ν with ν := 2 γ Cov(R, πmax ) and γ˜ = γ (1 + α ) in Eq. (25) yields

  1tr  −1 (μ + α ν ) − 2γ (1 + α )  −1 ω (α ) = 1 . μ+αν − 2γ ( 1 + α ) 1tr  −1 1 ∗ ra

(27)

∗ ( α ) − ω ∗ ( 0 ): Now calculate the difference ωra ra ∗ ∗ ωra (α ) − ωra (0 )  tr −1     −1 μ + α ν 1  (μ + α ν ) − 2γ (1 + α ) 1tr  −1 μ − 2γ = −μ− − 1 2γ 1+α 1tr  −1 1 (1 + α ) 1tr  −1 1   1tr  −1 (ν − μ ) α  −1 = · (ν − μ ) − 1 , 1 + α 2γ 1tr  −1 1

(28)

using

μ+αν α −μ= ( ν − μ ). 1+α 1+α

(29)

Note that

  1tr  −1 (ν − μ )  −1 (ν − μ ) − 1 2γ 1tr  −1 1 is a constant vector, independent of α . So we can insert α = 1 and get 2α ∗ ∗ ∗ ∗ ωra (α ) = ωra (0 ) + (ωra (1 ) − ωra (0 )). 1+α

(30)

(31) 

Optimal portfolio weights: The classical optimal portfolio is obtained by maximizing the preference function

(RP ) = ωtr μ − γ ωtr  ω subject to the budget constraint

ωtr 1 = 1. Differentiating the Lagrange function

L(ω ) = ωtr μ − γ

ωtr  ω − λ(ωtr 1 − 1 )

with respect to ω yields

∂ L (ω ) = μ − 2γ  ω − λ1, ∂ω thus

ω∗ =

1 −1  (μ − λ1 ). 2γ

The Lagrange parameter λ is obtained by differentiating L with respect to λ:

∂ L (ω ) = ωtr 1 − 1 ∂λ =

18

1 2γ

μtr  −1 1 −

A derivation is provided below.

λ

2γ

1tr  −1 1 − 1,

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

23

yielding

λ=

1tr  −1 μ − 2γ 1tr  −1 1

Thus

ω

∗

1 −1 =  2γ

 μ−

.

1tr  −1 μ − 2γ 1tr  −1 1

 1 .

For obtaining the optimal portfolio under the return-regret model, we can use the representation of the preference function from Eq. (20)

(ZP ) = ωtr μra − γ ωtr  ω − c with

μra = μ + 2 α γ Cov(R, Rmax ). Analogously to the classical case we get ∗ ωra =

1 −1  2γ

  1tr  −1 μra − 2γ 1 . μra − 1tr  −1 1

The same reasoning can be used for the preference-regret case. Impact of a change in expected returns on optimal portfolio weights: We look at the sensitivity of the optimal portfolio weights with respect to a change in the mean vector μ. In the classical model we can calculate the first derivative as

∂ω∗ = ∂μ



∂ωi∗ ∂μ j



i, j

1 −1 1 −1 1 1tr  −1 =  −  2γ 2γ 1tr  −1 1





1 1tr  −1 1 −1 =  I − tr , 2γ 1  −1 1 where I is the identity matrix. This value of the derivative is a constant, independent of μ. Thus, the optimal weight of an asset increases linearly with its mean in the classical model. In the return-regret model we have ∗ ∗ ∂ωra,i ∂ωra,i ∂μra, j = · , ∂μ j ∂μra, j ∂μ j

with ∗ ∂ωra,i ∂ωi∗ = ∂μra, j ∂μ j

as above. Furthermore,

 ∂μra, j ∂  ∂ = Cov(R j , Rmax ). μ j + 2 α γ Cov(R j , Rmax ) = 1 + 2 α γ ∂μ j μj ∂μ j To analyze the derivative of the covariance term, let us assume that the return vector R has a joint elliptical distribution with density f( · ; μ,  ). Then,

Cov(R j , Rmax ) = E[R j · Rmax ] − μ j · E[Rmax ]    = ··· R j · max{Rk } − μ j · max{Rk } · f (R; μ, ) dRn · · · dR1 k k    = ··· (R j + μ j ) · max{Rk + μk } − μ j · max{Rk + μk } · k



 =

···

k

f (R; 0, ) dRn · · · dR1 R j · max{Rk + μk } · f (R; 0, ) dRn · · · dR1 k

24

R. Baule, O. Korn and L.-C. Kuntz / Journal of Economic Dynamics & Control 103 (2019) 1–24

with substitution R˜ = R − μ. Differentiating under the integral, we obtain

∂ Cov(R j , Rmax ) = ∂μ j





··· 



=

··· 

 =

···

∂ R · max{Rk + μk } · f (R; 0, ) dRn · · · dR1 ∂μ j j k R j · 1R j =maxk {Rk +μk } · f (R; 0, ) dRn · · · dR1

(R j − μ j ) · 1R j =maxk {Rk } · f (R; μ, ) dRn · · · dR1

= E[(R j − μ j ) · 1R j =maxk {Rk } ] = E[R j − μ j |R j = max{Rk }] · P (R j = max{Rk } ) k

k

with re-substitution R = R˜ + μ. As E[(R j − μ j )] = 0, the expectation conditional on R j = maxk {Rk } is greater than zero. Hence, the sensitivity of the portfolio weight with respect to the expectation μ in the return-regret model,

∗ ∂ωra,i ∂ωi∗  = · 1 + 2αγ E[R j − μ j |R j = max{Rk }] P (R j = max{Rk } ) , ∂μ j ∂μ j k k

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