Optimal Portfolio Under State-dependent Expected ...

Optimal Portfolio Under State-dependent Expected Utility Carole Bernard∗, Steven Vanduffel† and Jiang Ye‡

March 14, 2018

Abstract We derive the optimal portfolio for an expected utility maximizer whose utility does not only depend on terminal wealth but also on some random benchmark (state-dependent utility). We then apply this result to obtain the optimal portfolio of a loss-averse investor with a random reference point (extending a result of Berkelaar et al. 2004). Clearly, the optimal portfolio has some joint distribution with the benchmark and we show that it is the cheapest possible in having this distribution. This characterization result allows us to infer the state-dependent utility function that explains the demand for a given (joint) distribution.

Keywords: Optimal portfolio choice, State-dependent utility, Cost-efficiency, Portfolio theory, Expected utility theory, Loss aversion, Prospect theory.

1

Introduction

The Von Neumann & Morgenstern (1947) expected utility theory has been the dominant theory for making decisions under risk. As this theory is not always consistent with observed behavior (e.g., the paradox of Allais 1953; Starmer 2000), alternative decision settings such as the dual theory of choice (Yaari 1987), the rank-dependent utility theory (Quiggin 1992) and the cumulative prospect theory (Tversky & Kahneman 1992) have been proposed. However, all these settings still share a common feature with the expected utility setting in that preferences are increasing (more is better than less) and law-invariant (states in which cash-flows are received do not affect preferences). While it is reasonable to assert that people prefer more dollars to less, the assumption that states of the world have no impact on decision making is more challenging. Rozeff (1994) and ∗

Grenoble Ecole de Management, Department of Accounting, Law and Finance, University Grenoble Alpes, 12 Rue Pierre S´emart, 38000 Grenoble, France. Vrije Universiteit Brussel, Department of Economics and Political Sciences Pleinlaan 2, 1050 Bruxelles, Belgium. (email: [email protected]). C. Bernard gratefully acknowledges support from the Odysseus FWO research grant at Vrije Universiteit Brussel. † Vrije Universiteit Brussel, Department of Economics and Political Sciences Pleinlaan 2, 1050 Bruxelles, Belgium. (email: [email protected]). S. Vanduffel acknowledges the financial support of the Stewardship of Finance Chair at VUB and of FWO. ‡ Corresponding author: Jiang Ye, Vrije Universiteit Brussel, Department of Economics and Political Sciences Pleinlaan 2, 1050 Bruxelles, Belgium. (email: [email protected]). J. Ye acknowledges the financial support from CSC of China. All authors thank three anonymous referees for constructive comments that have been very helpful to improve the paper.

1

Vanduffel et al. (2012) explain that investors who follow a dollar cost averaging strategy, which consists in investing gradually in the stock market rather than at once, do not only pursue this strategy for obtaining a (distribution of) terminal wealth that is positively correlated with the stock market, but also for obtaining protection in bearish markets (especially at the beginning of the investment horizon). A striking example that shows the states matter is the existence of insurance contracts (Bernard & Vanduffel 2014a). To explain this, let us assume that a risk-averse agent is offered the option to choose between an insurance contract to protect his $100,000 property against fire (occurring with 1% probability) or a digital option contract that is long the market and that pays $100,000 with the same probability and zero otherwise. Bernard & Vanduffel (2014a) show that the digital option will typically be cheaper, but might not be preferred by the agent, and this despite the fact that both contracts provide the same distribution (and thus the same expected utility). Another issue with using law-invariant preferences is that all optimal investment portfolios are increasing in the market portfolio, which implies that they yield their worst outcomes when the economy (proxied by the market portfolio) is low (Bernard et al. 2015b). This feature is not acceptable for many investors in that a declining economy may lead to job losses and tax increases, which the investors may not be able to afford. Finally, note that in performance measurement, it is common practice that portfolio returns are compared with those of benchmarks or portfolios of peers. For instance, the performance of a fund manager who invests in the energy sector is typically assessed against an index representing the energy sector. Furthermore, the performance of a fund manager following a volatility strategy should be assessed conditional on a market volatility index. In all these cases, agents do not only consider the expected utility of the portfolio, but also the states in which their portfolio pays off. In this paper, we make the following contributions. To deal with state-dependent preferences, we propose to extend the standard expected utility setting by letting the utility function depend on terminal wealth and on an external random source, i.e., a stochastic benchmark (e.g., a market index), and we derive the optimal portfolio in this setting. Clearly, optimal portfolios have some joint distribution with the benchmark and our second contribution consists in showing that they are the cheapest possible in having this distribution, i.e., optimal solutions to the statedependent expected utility maximization problem are completely characterized by their joint distribution with the stochastic benchmark. Finally, for a given joint distribution of a portfolio with a benchmark, we infer the corresponding state-dependent utility function such that the portfolio is optimal for that utility. In this regard, we extend the work of Bernard et al. (2015a) in a natural way. The rest of the paper is organized as follows. Section 2 describes the market and outlines the problems we want to solve. We derive optimal portfolios under state-dependent utility in Sections 3 and 4. In Section 5, we infer the state-dependent utility function from the joint distribution with the benchmark. Section 6 concludes.

2

Preliminaries

We assume an arbitrage-free and frictionless financial market (Ω, F, P) with a fixed investment horizon of T > 0. Let ξT be the pricing kernel that is agreed upon by all agents. We assume that it has a positive density on R+ \{0}. The value ω0 at time 0 of a consumption XT at T is then computed as ω0 = E[ξT XT ].

2

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We consider only terminal consumptions XT such that ω0 is finite. Let AT be a reference benchmark in the market (AT is a random vector) and let (a, b) ⊆ R. A state-dependent utility function U maps x ∈ (a, b) into U (AT , x). Equalities and inequalities among random variables are understood to hold almost surely. At first, we additionally impose that it is sufficiently regular (smooth). Definition 1 (Smooth state-dependent utility function). A state-dependent utility function mapping x ∈ (a, b) 7→ U (AT , x) is called smooth when it is continuously differentiable and ∂ strictly increasing in x (conditionally on AT ). Furthermore, writing ∂x U (AT , x) = U 0 (AT , x), 0 it is also required that U (AT , x) is strictly decreasing in x (so that the investor is risk averse, conditionally on AT ), U (AT , c) = 0 for some c ∈ (a, b), U 0 (AT , a) := limx&a U 0 (AT , x) = +∞, and U 0 (AT , b) := limx%b U 0 (AT , x) = 0. It is however also interesting to consider the case in which utility functions have a kink and lack differentiability. In order to deal with this situation we introduce the notion of generalized e (AT , ·) : R → R ∪ {−∞} in the same spirit as the generalized state-dependent utility functions U utility function considered in Bernard et al. (2015a). e (AT , ·) : R → Definition 2 (Generalized state-dependent utility function). Let (a, b) ⊆ R. U R ∪ {−∞} is a generalized utility function if it is of the form  U (AT , x) for x ∈ (a, b),    −∞ for x < a, e (AT , x) = U (2) lim U (A , x) for x = a,  T x&a   limx%b U (AT , x) for x > b, where U (AT , ·) : (a, b) → R is strictly increasing and strictly concave on (a, b) for each realization e 0 (AT , ·) on R as follows: of AT . We then define U    −1 U e (AT , x + ε) − U e (AT , x)  lim ε for x ∈ (a, b),  ε→0     +∞ for x < a, e 0 (AT , ·) = 0 (A , x) U (3) lim U for x = a, T x&a   0  limx%b U (AT , x) for x = b,    0 for x > b. In this paper we first solve the optimal expected utility investment problem and discuss the solution in some examples (Section 3 in the case of a smooth state-dependent utility function and Section 4 for a generalized state-dependent utility function). We then derive the correspondence between the cheapest payoff XT with given joint cumulative distribution function (AT , XT ) ∼ G and a maximum expected utility problem with state-dependent utility function U (AT , x) and we use this correspondence to infer the state-dependent utility function that explains the demand for XT given its behavior with respect to a benchmark AT (Section 5). We deal thus with the following problems: Problem 1 (Maximization of smooth state-dependent expected utility). Let U be a smooth state-dependent utility function. We consider the problem max E[U (AT , XT )], XT |E [ξT XT ] = ω0 where ω0 > 0 is the initial budget available for investing. 3

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e be a generProblem 2 (Maximization of generalized state-dependent expected utility). Let U alized state-dependent utility function. We consider the problem i h e (AT , XT ) , (5) max E U XT |E[ξT XT ]=ω0

where ω0 > 0 is the initial budget available for investing. Problem 3 (Conditional cost-efficiency). Let AT be a n-dimensional random vector. Let G be a n + 1 dimensional multivariate distribution. We consider the problem min E[ξT XT ]. XT |(AT , XT ) ∼ G

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We refer to Problem 3 as a “conditional cost-efficiency” problem because the standard costefficiency problem (Dybvig 1988a, Bernard et al. 2014) consists in minimizing the cost of a strategy for a given distribution of terminal wealth. Here, since AT is a given random variable, the constraint in Problem 3 is equivalent to fixing the conditional distribution, conditionally on AT . We will illustrate our theoretical results in a one-dimensional Black-Scholes model with one risky asset ST . In this case, the pricing kernel ξT is unique and can be expressed explicitly in terms of the stock price ST as follows  −β ST ξT = α , (7) S0
with α = exp β(µ − σ 2 /2)T − (r + θ2 /2)T , β = θ/σ and θ = (µ−r)/σ. In Table 1, we present the parameters that will be used in all examples of the paper. Note that, in practice, µ > r typically holds and in this case ξT is strictly decreasing in ST . Table 1: Market Parameters and Budget

ω0 100

S0 100

µ 0.05

r 0.02

σ 0.2

T 1

ω0 denotes the initial budget of investor, S0 is the initial stock price, µ is the drift parameter of the stock return, r is risk-free rate, σ is the stock price volatility, and T is the investment horizon.

3

Optimal portfolio under smooth concave state-dependent expected utility

The following proposition finds the payoff for an expected utility maximizer who uses a smooth state-dependent utility function to evaluate his terminal wealth. Proposition 1. Assume that (AT , ξT ) has joint density with respect to the Lebesgue measure and that Fξ−1 (1) = +∞ and Fξ−1 (0) = 0. Consider a smooth state-dependent utility function T |AT T |AT U (AT , XT ) and assume that ω0 ∈ (E[ξT a], E[ξT b]). The unique optimal solution XT∗ to the expected utility maximization problem 1 is given by XT∗ := [U 0 (AT , ·)]−1 (λ∗ ξT ) 4

(8)

where λ∗ > 0 is such that E[ξT XT∗ ] = ω0 . Furthermore, XT∗ has a continuous conditional distribution FXT∗ |AT , which is strictly increasing on (a, b) with limx&a FXT∗ |AT (x) = 0, limx%b FXT∗ |AT (x) = 1. Proof. Given ω ∈ Ω, consider the following auxiliary problem max {U (AT (ω), x) − λξT (ω)x}

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x∈(a,b)

with λ > 0. This is an optimization over the interval (a, b) of a concave function. The first-order conditions imply that the optimum x∗ is at U 0 (AT (ω), x) − λξT (ω) = 0, i.e., x∗ = [U 0 (AT (ω), ·)]−1 (λξT (ω)). For each ω ∈ Ω, define the random variable Xλ∗ by Xλ∗ (ω) = [U 0 (AT (ω), ·)]−1 (λξT (ω)).

(10)

Choose λ∗ > 0 such that E[ξT Xλ∗∗ ] = ω0 . The existence of λ∗ is ensured by the conditions imposed on x 7→ U (AT , x), by continuity of λ 7→ E[ξT Xλ∗ ] and the fact that the budget ω0 ∈ (E[ξT a], E[ξT b]). For every final wealth XT that satisfies the budget constraint (E[ξT XT ] = ω0 ), we have by construction U (AT (ω), XT (ω)) − λ∗ ξT (ω)XT (ω) 6 U (AT (ω), Xλ∗∗ (ω)) − λ∗ ξT (ω)Xλ∗∗ (ω) ∀ ω ∈ Ω,

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since Xλ∗∗ (ω) is the optimal solution to (9). Now take the expectation on both sides of the above inequality, we obtain E[U (AT , XT )] 6 E[U (AT , Xλ∗∗ )],

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which ends the proof that Xλ∗∗ is optimal for Problem 1. Since y 7→ [U 0 (AT (ω), ·)]−1 (y) is strictly decreasing on (a, b) for given ω ∈ Ω, and ξT |AT has a continuous distribution, it is clear that XT∗ |AT has a continuous cumulative distribution function that is strictly increasing on (a, b) with limx&a FXT∗ |AT (x) = 0 and with limx%b FXT∗ |AT (x) = 1. Hereafter, we present some examples that illustrate Proposition 1. From this proposition we observe that optimal payoffs will be decreasing in ξT conditionally on AT . Since in a BlackScholes market ξT is monotonic in ST , we can express the optimal portfolios also as a function of ST and AT , which may facilitate their interpretation. We will use such representation in the various graphical illustrations. Example 1 (Beating a benchmark - CRRA utility). Consider an investor with CRRA utility, i.e., ( log(w) η=1 u(w) = (13) 1 1−η η 6= 1, η > 0, 1−η w in which η is the so-called constant coefficient of relative risk aversion, i.e., R(w) := −

wu00 (w) = η. u0 (w)

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The optimal portfolio for an investor who maximizes E[u(XT )] subject to the available budget ω0 is given as XT∗ = (λ1 ξT ) 5

− η1

,

(15)

in which λ1 follows from the budget constraint ω0 (Merton 1971). We obtain that   θ2 exp (1 − η)(r + 2η )T λ1 = . ω0η

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Next, we assume that the investor faces a benchmark AT that he wants to beat in that he aims at maximizing E[u(XT /AT )]. Merely for mathematical convenience, we further choose as a benchmark AT = Stc , 0 < t < T. In this case,   c− ηc − 1 w Stcη−c 0 −1 = , [u ] (y) = S y η, u0 (17) t c St wη and from Proposition 1 we obtain that the optimal strategy is now given as c− ηc

YT∗ = St

− η1

(λ2 (c)ξT )

,

(18)

in which λ2 (c) is determined by the budget ω0 . After some calculations we obtain   2 2 2 2 c(η−1) S0 exp (η − 1)(cµt − θ2ηT + (c η−2βcη−c2η+2βc−cη)σ t − rT ) λ2 (c) = . ω0η

(19)

Note that when c tends to 0 the benchmark AT becomes equal to 1 and we expect to recover the results from standard expected utility theory. Indeed, limc→0 λ2 (c) = λ1 and thus YT∗ converges to XT∗ when c goes to 0. It is of interest to compare the optimal portfolio XT∗ of the traditional expected utility maximizer (maximizing nominal terminal wealth) with the optimal choice YT∗ of the investor who aims at outperforming the benchmark (maximizing terminal wealth expressed in units of the benchmark). We identify the following three cases: Case 1: η < 1. From the expressions for XT∗ and YT∗ , YT∗

>

XT∗

 if St 6

λ2 (c) λ1



1 cη−c

.

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In other words, YT∗ dominates XT∗ under scenarios in which the benchmark is low. To further illustrate this case, we take η = 0.5 and the market parameters displayed in Table 1. We obtain that YT∗ > XT∗ when St 6 101.005 and P(YT∗ > XT∗ ) = 0.4859. The optimal payoffs are shown in Figure 1 for t = 1/2 and c = 2. Case 2: η > 1. In this case we find that YT∗

>

XT∗

 if St >

λ2 (c) λ1



1 cη−c

.

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Hence, YT∗ dominates XT∗ under scenarios in which the benchmark is high. To illustrate this case, we take η = 1.5 and the parameters from Table 1, and we obtain that YT∗ > XT∗ when St > 101.6806 and P(YT∗ > XT∗ ) = 0.4953. The optimal payoffs are illustrated in Figure 2 for t = 1/2 and c = 2. In this case, the terminal payoff is lower when the benchmark is lower. Case 3: η = 1. In this case, u(XT /St ) = log(XT ) − log(St ). Hence, maximization of E[u(XT )] and E[u(XT /St )] yield the same solutions, i.e. XT∗ = YT∗ .

6

350 *

XT 300

T T/2

1200

Y*T|ST/2=100

1000

Y*T|ST/2=110

800 T

200

Y*

Terminal Payoff

250

Y* |S =90

600 400

150

200

100

0 200

50

200

150 150 100

0 40

60

80

100

120

140

160

180

200

220

S

100 50

ST

50

ST/2

T

Panel A

Panel B

Figure 1: Panel A: We plot the terminal payoff XT∗ as a function of the underlying stock price ST . We also plot the terminal payoff YT∗ as a function of ST for three possible realizations of the benchmark, namely St = 90, St = 100 and St = 110. Panel B: We show a three-dimensional graph representing the optimal wealth YT∗ as a function of the benchmark AT = St and the underlying asset ST . Market parameters are displayed in Table 1, t = 1/2 and c = 2.

180 X*T 160

Y*T|ST/2=90 Y*T|ST/2=100

250

Y*T|ST/2=110

200

T

150 Y*

Terminal Payoff

140

120

100 50

100 0 200 80

200

150 150 100

60 0

50

100

150

200

250

ST

ST

Panel A

100 50

50

ST/2

Panel B

Figure 2: Panel A: We plot the terminal payoff XT∗ as a function of the underlying stock price ST . We also plot the terminal payoff YT∗ as a function of ST for three possible realizations of the benchmark, namely St = 90, St = 100 and St = 110. Panel B: We show a three-dimensional graph representing the optimal wealth YT∗ as a function of the benchmark AT = St and the underlying asset ST . Market parameters are displayed in Table 1, t = 1/2 and c = 2.

Example 2 (Random Risk Aversion - Exponential Utility). We assume an investor with exponential utility, i.e., u(w) = −δ −1 exp(−δw), with δ > 0. In this case the parameter δ reflects the so-called constant coefficient of absolute risk aversion, i.e., −u00 (w)/u0 (w) = δ. Since u0 (w) = exp(−δw), [u0 ]−1 (y) =

ln y , −δ

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we find from Proposition 1 that the optimal portfolio XT∗ (as a solution to maximize E[u(XT )])

7

is given as XT∗ = −

ln(λ1 ξT ) , δ

(23)


in which λ1 = exp rT − θ2 T /2 − δω0 erT . It seems reasonable to assume that risk aversion is decreasing in bull markets (corresponding to optimism of the investor) and increasing in bear markets (corresponding to pessimism). We introduce randomness by defining a state-dependent risk aversion δ 0 = δ/(St /S0 )c (where δ is constant). Hence, u(St , w) = −(δ 0 )−1 exp(−δ 0 w) and thus S c ln y u0 (St , w) = exp(−δ 0 w), [u0 ]−1 (y) = − t c . (24) δS0 From Proposition 1, we obtain that the optimal strategy is now given as YT∗ = −

ln(λ2 (c)ξT )Stc , δS0c

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where  λ2 (c) = exp 

2

−rT +c(µ− σ2 )t+

e

e

(c2 −2βc)σ 2 t 2

(rT +

βcσ 2 t

−

θ2 T 2 )

− δω0

2 (c2 −2βc)σ 2 t −rT +c(µ− σ2 )t+ 2

 .

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In a similar way as in Example 1, we observe that YT∗ converges to XT∗ when c tends to zero (i.e., limc→0 λ2 (c) = λ1 ). We illustrate this example for two specific choices of the absolute risk aversion coefficient δ, namely δ = 0.005 and δ = 0.015. Note that these choices give rise to relative risk aversion coefficients that are equal to respectively 0.5 and 1.5 when measured at the level of initial budget equal to 100. We also take the market parameters from Table 1. The optimal payoffs are displayed in Figure 3 (Panel A: δ = 0.005, Panel B: δ = 0.015). 250

160 X*T

200

X*T

Y*T|ST/2=90

140

Y*T|ST/2=100 Y*T|ST/2=110

120 Terminal Payoff

Terminal Payoff

Y*T|ST/2=100 Y*T|ST/2=110

150

100

100

50

80

0

60

−50 50

Y*T|ST/2=90

100

150

200

250

40 40

ST

60

80

100

120

140

160

180

200

220

ST

Panel A: δ = 0.005

Panel B: δ = 0.015

Figure 3: In each panel, we plot the terminal payoff XT∗ as a function of the underlying stock price ST , and the terminal payoff YT∗ as a function of the underlying stock price ST for three possible realizations of the benchmark, namely ST /2 = 90, ST /2 = 100 and ST /2 = 110. Market parameters are displayed in Table 1 and c = 2.

From Figure 3, we observe that state-dependent exponential utility maximizers get an optimal portfolio that yields a terminal wealth that is very low when the benchmark is at the low level. 8

This feature is at odds with the intuition (belief ) that a more risk averse individual would be willing to purchase more protection and therefore to use a larger part of his budget for obtaining protection against the worst states of the market. In fact, this intuition is the reason why we model stochastic risk aversion as a decreasing function of the benchmark. Although this link between risk aversion coefficient and the state of the market appears natural, the resulting optimal behavior of the investor is counter-intuitive. We also note that for high risk aversion (i.e., when δ is high) the optimal payoff is flatter than the opposite case. This feature is intuitive, as with infinite risk aversion the optimal portfolio consists of a risk-free asset.

4

Optimal portfolio under non-smooth concave state-dependent utility functions

Proposition 1 makes it possible to derive optimal portfolios for state-dependent expected utility maximizers, but requires differentiability of the utility function. Since every non-differentiable utility function can be closely approximated by a differentiable one, one may argue that nondifferentiability is not an important issue (investors may not be able to make a distinction between the original function and the approximation). It turns out, however, that non-differentiability might be crucial in modeling investors’ behavior in that for instance kinked utility functions appear as necessary in modeling preferences of agents and in explaining observed behavior (Kahneman & Tversky 1979, Berkelaar et al. 2004). Hereafter, we extend Proposition 1 to deal with non-differentiability. e (AT , ·) on (a, b) ⊂ Proposition 2. Consider a generalized state-dependent utility function U ∗ R and let ω0 ∈ (E[ξT a], E[ξT b]). The optimal solution XT to the generalized state-dependent expected utility maximization Problem 2 exists, is unique and is given by e 0 (AT , ·)]−1 (λ∗ ξT ), XT∗ := [U

(27)

e 0 (AT , ·) is introduced in Definition 2, where λ∗ > 0 is such that E[ξT X ∗ ] = ω0 and where where U T a pseudo inverse is defined as n o e 0 (AT , ·)]−1 (y) := inf x ∈ (a, b)|U e 0 (AT , x) 6 y , [U (28) with the convention that inf {∅} = b. Furthermore, XT∗ may have mass points. Proof. Note that U (AT , ·) is strictly concave and strictly increasing (for almost surely all realizations of the variable AT ), and (a, b) ⊂ R. The proof consists of three steps: Step 1: Given λ > 0 and ω ∈ Ω, we first prove that the unique optimum of n o e (AT (ω), y) − λξT (ω)y max U

(29)

y∈[a,b]

e 0 (AT (ω), ·)]−1 (λξT (ω)) ∈ R, where the inverse of U e 0 (AT , ·) is defined by is equal to Yλ∗ (ω) := [U (28). For λ > 0, we have that λξT (ω) > 0. Denote by h i−1 e 0 (AT (ω), ·) x∗ := U (λξT (ω)).

9

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It is clear that x∗ ∈ [a, b]. Note that x∗ ∈ R. Indeed, if a = −∞, then x∗ > a because e 0 (AT , a) = ∞. Similarly, if b = +∞, then x∗ < b because U e 0 (AT , b) = 0. Thus, x∗ ∈ R in U e (AT (ω), x) is continuous and all cases. Let us prove that x∗ is an optimum. We observe that U e (AT (ω), ·) strictly increasing on (a, b) (almost surely). Hence, the left and right derivatives of U exist at each point of (a, b) and are equal to each other, except in a countable number of points e (AT , ·) is not differentiable. We can then conclude that U e 0 (AT (ω), ·) is left-continuous where U and decreasing on (a, ∞) with discontinuities in a set (xi )i∈I with xi ∈ (a, b] and I ⊂ N. As e 0 (AT (ω), ·) is left-continuous, from (30), U e 0 (AT (ω), x∗ ) > λξT (ω) and we consider the following U two cases: e 0 (AT (ω), x∗ ) = λξT (ω). Then for x ∈ (a, x∗ ), Case 1: U e (AT (ω), x∗ ) − λξT (ω)x∗ − (U e (AT (ω), x) − λξT (ω)x) U " # e (AT (ω), x∗ ) − U e (AT (ω), x) U = − λξT (ω) (x∗ − x) x∗ − x

(31)

> (λξT (ω) − λξT (ω))(x∗ − x) = 0, e (AT (ω), ·). For x ∈ (x∗ , b), using the concavity of U e (AT (ω), x) where we use the concavity of U again, we have that e (AT (ω), x∗ ) − λξT (ω)x∗ − (U e (AT (ω), x) − λξT (ω)x) U e (AT (ω), x) − U e (AT (ω), x∗ ) U = [− + λξT (ω)](x − x∗ ) x − x∗ > (−λξT (ω) + λξT (ω))(x − x∗ ) = 0.

(32)

e (AT (ω), x∗ ) − λξT (ω)x∗ − (U e (AT (ω), x) − λξT (ω)x) > 0. For all x ∈ (a, b) we have proved that U e 0 (AT (ω), x∗ ) > λξT (ω). This implies that x∗ is at some point of non-differentiability Case 2: U e (AT (ω), ·) in xi is strictly larger than its right derivative, xi , x∗ = xi , and the left derivative of U (r) e 0 e 0 (AT (ω), x∗ ) > λξT (ω) > a(r) . Observe that when which we denote by ai . U (AT (ω), xi ) = U i e (AT (ω), ·), a < x < xi , by concavity of U e (AT (ω), xi ) − λξT (ω)xi − (U e (AT (ω), x) − λξT (ω)x) U e (AT (ω), xi ) − U e (AT (ω), x) U =[ − λξT (ω)](xi − x) xi − x e 0 (AT (ω), xi ) − λξT (ω))(xi − x) > 0. > (U

(33)

e (AT (ω), ·) again, When b > x > xi , using the concavity of U e (AT (ω), xi ) − λξT (ω)xi − (U e (AT (ω), x) − λξT (ω)x) U e (AT (ω), x) − U e (AT (ω), xi ) U = [− + λξT (ω)](x − xi ) xi − x

(34)

(r)

> (−ai + λξT (ω))(x − xi ) > 0. e (AT (ω), x∗ ) − λξT (ω)x∗ − For all x ∈ (a, b), and thus for all x ∈ [a, b], we have proved that U e (AT (ω), x) − λξT (ω)x) > 0. (U

10

Step 2: The optimum of n o e max U (AT (ω), y) − λξT (ω)y y∈R

(35)

is then also equal to Yλ∗ (ω) because e (AT (ω), z) − λξT (ω)z < U e (AT (ω), b) − λξT (ω)b ∀z > b, U e (AT (ω), z) − λξT (ω)z < U e (AT (ω), a) − λξT (ω)a ∀z 6 a, U

(36)

e (AT (ω), b) = U e (AT (ω), z), and z < a, U e (AT (ω), z) − λξT (ω)z = −∞ and Since for z > b, U λ > 0. We can thus conclude that x∗ is an optimum. It is also unique for the slope of y 7→ e (AT (ω), y) − λξT (ω)y can never be 0 on an interval that is not reduced to a point. U Step 3: If there exists λ∗ > 0 such that the pathwise optimum Yλ∗∗ satisfies the budget constraint E[ξT Yλ∗∗ ] = ω0 , then Yλ∗∗ solves Problem 2. The existence of λ∗ is guaranteed thanks to the budget constraint ω0 ∈ (E[ξT a], E[ξT b]), the properties that the cost as a function of λ is continuous on (0, ∞), that if λ → +∞ the limit of the pseudo inverse of the left derivative is a, and that if λ → 0, the limit of the pseudo inverse of the left derivative is b. Based on these three steps, Proposition 2 is proved. Example 3 (Loss averse investor with a random benchmark). There is evidence that investors are distinctively more sensitive to losses than to gains (Kahneman & Tversky 1979). This phenomenon is called loss aversion and introduces non-differentiability (kink) in the utility function at the reference level used to distinguish losses from gains; see also Segal & Spivak (1990). Berkelaar et al. (2004) studied the optimal portfolio choice in this setting, and we refer to this problem as the optimal portfolio problem for the loss averse investor. Specifically, these authors derive the optimal portfolio for a concave utility function defined over losses and gains relative to a fixed reference level R, i.e., they consider ( C γ1 XTγ + (B − C) γ1 Rγ for XT 6 R, U (R, XT ) = (37) 1 γ B γ XT for XT > R, in which 0 < γ < 1 and C > B is required for modeling loss aversion. It appears more natural to consider a stochastic reference point. In what follows, we replace R by a stochastic benchmark AT . This benchmark could be the terminal wealth from a bank account yielding a stochastic interest rate, the portfolio of a competitor, or a sector index representing the sector in which the agent is investing. Replacing R by AT > 0, we first get ( C γ1 XTγ + (B − C) γ1 AγT U (AT , XT ) = B γ1 XTγ

for XT 6 AT , for XT > AT .

(38)

Also observe that 0

U (AT , XT ) =



CXTγ−1 BXTγ−1

11

for XT 6 AT for XT > AT ,

(39)

From Proposition 2, the optimal strategy depending on AT is    1  CAγ−1  λξT γ−1  for ξT > λT ,  C  BAγ−1 T YT∗ = [U 0 (AT , ·)]−1 (λξT ) = A for < ξT < T λ  1    γ−1  γ−1 BA  T  λξT for ξ 6 , T B λ

CAγ−1 T , λ

(40)

where λ is determined by the budget constraint E[ξT YT∗ ] = ω0 . For AT = R, we recover the result of Berkelaar et al. (2004) (see their Proposition 1 and formula (8)). We illustrate our findings using AT = St with t = 1/2, and choosing the parameters as in Tables 1 and 2. In Figure 4, we represent the optimal terminal wealth YT∗ as a function of ST and of the benchmark value AT . Here λ = 0.814 to match the budget. Table 2: Parameters in (38)

B 1

C 2.25

γ 0.88

The parameters B, C and γ are needed to compute U (R, XT ) in (38).

500

400

Y*T

300

200

100

0 250 250

200 200

150 150 100

S

100 50

50

T

ST/2

Figure 4: Three dimensional graph representing the optimal wealth YT∗ as a function of the benchmark AT = St for t = 1/2 and of ST .

5

Inferring the state-dependent utility From the joint distribution with the benchmark

Optimal portfolios for standard expected utility maximizers have a certain distribution of terminal wealth. It is intuitive and also easy to show that they are the cheapest possible in attaining this distribution of terminal wealth (see for instance Lemma 1 in Bernard et al. 2015a). Such payoffs are called cost-efficient. However, cost-efficient payoffs typically do not offer protection in times of economic hardship. To see this, observe that the optimal payoffs derived for standard

12

expected utility maximizers in the examples of Section 3 are all decreasing in ξT (see expressions (15) and (23)) and thus increasing in ST (when µ > r). Optimal portfolios of state-dependent utility maximizers are typically not decreasing in ξT and are thus not cost-efficient. However, they are conditionally cost-efficient as we show in the first paragraph. This property is key to construct a state-dependent utility function that may explain the demand for a strategy in a context when the performance is not solely driven by the probability distribution of the portfolio but by its interaction with some benchmark index in the market. Consider an optimal portfolio for a state-dependent utility optimizer. It has a joint distribution with the benchmark (appearing in the state-dependent utility) and it turns out it is cheapest in obtaining this joint distribution, i.e., they are a solution to Problem 3 and are thus conditionally cost-efficient. Proposition 3 (Optimal payoffs of state-dependent utility maximizers are conditionally cost– efficient). Let AT be a n-dimensional random vector. There exists a (n + 1)-multivariate joint distribution G such that the optimal solution of Problem 1, when it exists, also solves Problem 3 min (AT ,XT )∼G

E[ξT XT ].

(41)

Proof. Let X ∗ be an optimal solution of Problem 1 then let G be the joint cumulative distribution function of (AT , X ∗ ) then X ∗ also solves (41). Let Y be a solution of (41) if X ∗ does not solve (41) then E[ξT Y ] < E[ξT X ∗ ]. (42) Let Y ∗ := Y + (E[ξT X ∗ ] − E[ξT Y ])e−rT . Then the initial cost of Y ∗ is equal to the one of X ∗ . Then E[U (AT , X ∗ )] = E[U (AT , Y )] because (AT , X ∗ ) and (AT , Y ) have the same joint distribution. As U is increasing in the second variable, E[U (AT , Y )] < E[U (AT , Y ∗ )],

(43)

which violates the optimality of Y and thus X ∗ also solves (41). Bernard et al. (2015a) seek to infer preferences of consumers who are investing in the financial market. They show that if a portfolio is cost-efficient (cheapest way to achieve the distribution of the portfolio), then it can be rationalized by standard expected utility theory with a deterministic non-decreasing and concave function. Specifically, given the distribution of portfolio returns, they give the utility function such that this portfolio is optimal when optimizing the expected utility of the portfolio. To do so, they construct a concave utility function U (x) such that the solution to max E[U (XT )] (44) E [ξT XT ] = ω0 has a given cumulative distribution function F . The proof of their main results builds on Dybvig’s work on cost-efficiency (Dybvig 1988a,b, Bernard et al. 2014) and on the explicit form of the optimal portfolio of an expected utility maximizer (Merton 1971). In this section, we extend this result building on the conditional cost-efficiency property of optimal portfolios for state-dependent expected utility maximizers (Proposition 3) and on earlier results in the literature that solve conditional cost-efficiency problems explicitly. Let us recall the following proposition without its proof as it was proved in Theorem 4.4 of Bernard et al. (2015b). See also Theorem 4.4 of Bernard & Tang (2016) for the case when AT is multidimensional. 13

Proposition 4 (Characterization of conditional cost-efficient payoffs). Assume that (AT , ξT ) has joint density with respect to the Lebesgue measure. Let G be a bivariate cumulative distribution function. The optimal state-dependent strategy determined by min E[ξT XT ], (AT , XT ) ∼ G

(45)

has an almost surely unique solution XT∗ , which is almost surely decreasing in ξT , conditionally on AT , and given by XT∗ := FX−1T |AT (1 − FξT |AT (ξT )).

(46)

Following Proposition 3, a solution XT∗ to the optimization Problem 1 is completely characterized by its conditional distribution FXT |AT . We then combine the explicit expression of conditionally cost-efficient strategies in Proposition 4 and the explicit solutions to the state-dependent utility problems (Propositions 1 and 2) to find an explicit expression for a state-dependent utility function U (AT , XT ), which can rationalize the demand for some conditionally cost-efficient wealth XT . This will be made explicit in the next proposition. Proposition 5 (Inferring utility). Assume that (AT , ξT ) has joint density with respect to the (0) = 0. Let XT be a conditionally cost(1) = +∞ and Fξ−1 Lebesgue measure and that Fξ−1 T |AT T |AT efficient payoff (i.e. it solves Problem 3) for a benchmark asset AT . Assume that its cost (see also (45)) is finite and denote it by ω0 . Let G be the bivariate cumulative distribution function G and (XT , AT ) ∼ G. We assume that the conditional distribution of XT given AT , FXT |AT , is strictly increasing and continuous on (a, b) ⊆ R. Then XT is also the optimal solution of the state-dependent utility maximization Problem 1 with the following explicit expression Z x (1 − FXT |AT (y))dy (47) U (AT , x) = Fξ−1 T |AT c

for some c such that FXT |AT (c) > 0. Other state-dependent utility functions, which lead to an optimal solution Y with (Y, AT ) ∼ G write as x 7→ U (AT , x) + h(AT ) for some measurable function h. Proof. To prove Proposition 5, we use both Propositions 1 and 4. Since ω0 is the cost of XT∗ and its conditional distribution, FXT |AT , is defined on (a, b), we find that ω0 ∈ (E[ξT a], E[ξT b]). To find the expression of the state-dependent utility that can explain the demand for the distribution G, we start from the following two observations. First, we observe that U (AT , ·) satisfies the conditions of Proposition 1, thus that the optimal solution to Problem 1 can be written as XT∗ := [U 0 (AT , ·)]−1 (λ∗ ξT ),

(48)

where λ∗ > 0 is chosen such that E[ξT XT∗ ] = ω0 . Second, from Proposition 4, the optimal state-dependent strategy with (XT , AT ) ∼ G has the following expression YT∗ := FX−1T |AT (1 − FξT |AT (ξT )),

(49)

where E[YT∗ ξT ] = ω0 . Now we need to verify that our candidate utility function (47) satisfies the properties listed in Definition 1 and equates (48) and (49). Let c be an arbitrary value such that FXT |AT (c) > 0 and define U (AT , x) as in (47). Using the properties of continuity and increasingness of FξT |AT , we can show that U (AT , x) satisfies 14

the properties listed in Definition 1 for a state-dependent utility functions and thus satisfies all conditions of Proposition 1: it is continuously differentiable on (a, b) for given AT and U 0 (AT , x) := Fξ−1 (1 − FXT |AT (x)) for x ∈ (a, b) and given AT . Then U (AT , x) is strictly T |AT increasing and U 0 (AT , x) is strictly decreasing on (a, b) for given AT . Note also that U (AT , c) = 0, limx&a U 0 (AT , x) = +∞ and limx%b U 0 (AT , x) = 0 because Fξ−1 (1) = +∞ and Fξ−1 (0) = 0. T |AT T |AT Furthermore, U (AT , x) is such that U 0 (AT , x) = Fξ−1 (1 − FXT |AT (x)) T |AT

(50)

and thus [U 0 (AT , ·)]−1 (y) = FX−1T |AT (1 − FξT |AT (y)). Therefore the solution XT∗ to Problem 1 writes as XT∗ = [U 0 (AT , x)]−1 (λ∗ ξT ) = FX−1T |AT (1 − FξT |AT (λ∗ ξT )),

(51)

where λ∗ > 0 is chosen such that E[ξT XT∗ ] = ω0 . By assumption, ω0 is the cost of the payoff solving (45). Hence, λ∗ = 1 and the utility function (47) equates (48) and (49). If there is another utility U2 (AT , x) such that XT∗ is an optimal solution to Problem 1, then, by the same reasoning as above, we find that ξT = U20 (AT , XT∗ ) = U 0 (AT , XT∗ ). Since XT∗ |AT has a strictly increasing conditional distribution on (a, b), U20 (AT , x) = U 0 (AT , x) for all x ∈ (a, b) and U2 (AT , x) = U (AT , x) + h(AT ) in which h is a measurable function. If an expected state-dependent utility maximizer chooses a particular investment with a bivariate cumulative distribution function G, then a state-dependent utility function that rationalizes her choice is given by (47). Its expression involves properties of the financial market at the horizon time T (through the cumulative distribution function FξT |AT of the pricing kernel ξT |AT ). Remark 1. It is possible to generalize Proposition 5 to include more general distributions (discrete and mixed distributions), but this will be no further pursued in this paper. Obviously, any distribution G can always be approximated by a sequence of continuous strictly increasing distributions, Gn . Then, for each Gn , Proposition 5 allows us to obtain the corresponding state-dependent utility function Un (AT , x) so that the optimal investment for an expected statedependent utility maximizer with state-dependent utility function Un (AT , x) is distributed with the cumulative distribution function Gn . Thus, Proposition 5 already explains approximately the demand for all distributions. Example 4. Recall that in Example 1, we derived the optimal strategy for a CRRA investor who aimed at beating the benchmark AT = Stc , 0 < t < T . The optimal strategy we derived was given as c− ηc

YT∗ = St

− η1

(λ2 (c)ξT )

.

(52)

It is joint log-normally distributed with St with correlation coefficient (as a parameter of the Gaussian dependence) given as √ (cη − c + β) t

, ρ= p (η − 1)2 t + 2β(η − 1)t + β 2 T 15

(53)

and with conditional distribution    ln λ2 (c)αStc−cη−β S0β xη − β(µ − √ FYT∗ |St (x) = Φ  θ T −t

σ2 2 )(T

− t)

 .

(54)

Let us derive the utility function that rationalizes this given distribution of terminal wealth, conditionally on the benchmark (or equivalently, that rationalizes the corresponding joint distribution of terminal wealth and benchmark). To this end, we first compute Fξ−1 (y) T |St



√

−1

= exp θ T − tΦ

    −β σ2 St (y) + ln α − β µ − (T − t) . 2 S0

(55)

Next, from Proposition 5 and choosing d > 0, the corresponding state-dependent utility function U (St , x) is given as Z x (1 − FXT |St (y))dy Fξ−1 U (St , x) = T |St d  1−η  1−η ! Z x (56) 1 x d −1 cη−c −η = λ2 (c) St y dy = − . λ2 (c)(1 − η) Stc Stc d Hence, up to a linear multiplier 1/λ2 (c) and up to a measurable function h(St ) = (d/Stc )1−η (which does not affect the optimization), the state-dependent utility function is given as U (St , x) = (x/Stc )1−η /(1 − η) and we obtain correspondence (and thus consistency) with the CRRA utility function used in Example 1.

Example 5. We assume that the investor aims for a payoff XT that conditionally on AT is ˆ (AT ) and σ log-normally distributed with parameters M ˆ 2 (AT ) that depend on the outcomes of 2 ˆ ˆ (AT ) might be decreasing in AT so AT , i.e., XT |AT ∼ LN (M (AT ), σ ˆ (AT )). For instance, M that protection is obtained when AT is low. In addition, we require that ξT |AT is log-normally distributed, i.e., ξT |AT ∼ LN (MξT (AT ), σξ2T (AT )). Applying Proposition 5 we obtain the statedependent utility function Z x U (AT , x) = Fξ−1 (1 − FXT |AT (y))dy T |AT c   ˆ (AT ) σξT (AT )M (57) exp + MξT (AT ) σξ (AT ) σ ˆ (AT ) 1− σˆT(A ) T = x . σ (AT ) 1 − σξˆT(AT ) In other words, we obtain that XT maximizes the expected utility for a powerutility U (AT , x) =  ˆ (AT ) + Mξ (AT ) . g(AT )x1−γ(AT ) /(1−γ(AT )) where γ(AT ) = σξT (AT )/ˆ σ (AT ) and g(AT ) = exp γ(AT )M T As a special case of the above results let us consider a constant-mix strategy with payoff XT at T in which the proportion invested in the risky asset is π and choose AT = St . We aim at rationalizing the investment choice, i.e., we infer the utility function that leads to the distribution

16

of (St , XT ). In this regard notep that St and XT are Log-Normal coupled by a Gaussian copula with correlation parameter ρ = t/T so that the above results can be applied. We obtain that    ˆ (AT ) = M ˆ (St ) = π ln St + (1 − π)rT + (πµ − 0.5π 2 σ 2 )T − π(µ − 0.5σ 2 )t,  M   S0      −β  2 − β(µ − σ2 )(T − t), MξT (AT ) = MξT (St ) = ln α SS0t (58)   2 2 2 2  σ ˆ (AT ) = σ ˆ (St ) = π σ (T − t),    2 σξT (AT ) = σξ2T (St ) = β 2 σ 2 (T − t). From the general result (57) we thus get that  exp β( π1 − 1)rT + β(µ − 0.5πσ 2 )T − (r + U (St , x) = 1 − βπ

θ2 2 )T

 β

x1− π .

(59)

Hence, up to a linear multiplier, the utility function is given as U (St , x) = x1−β/π /(1 − β/π). Note that this utility does no longer depend on the benchmark and is given as a CRRA utility function (see also Example 1) in which the risk aversion coefficient is driven by the market parameter β and proportion invested in the risky asset π. The feature of a utility function that does not depend on the benchmark is to be expected (and confirms the results) since optimal mix strategies are known to be optimal for law-invariant investors employing a CRRA utility.

6

Conclusions

The standard expected utility theory allows to rationalize investment choices of investors who only care about the distribution of terminal wealth and who prefer more to less (law-invariant increasing preferences), and note that most decision theories comply with this paradigm. For any probability distribution there exists a unique concave utility function (up to a linear transformation) such that the optimal portfolio for the standard expected utility theory maximizer has this particular distribution (Bernard et al. 2015a). The utility function is known explicitly, and there is a bijection between the set of probability distributions and the set of concave utility functions. However this correspondence only allows to rationalize decisions that are solely driven by distributions of terminal wealth (i.e., when investors maximize a law-invariant increasing objective function). In practice, many investors make decisions by considering additional features. For instance, investment decisions can be also driven by the level of volatility (e.g., an index volatility in the market), by the level of interest rates or by a portfolio reflecting the performance of a sector. In these cases, decisions are driven by “state-dependent” considerations in that the optimal portfolio choice is not solely driven by some target distribution but also based on (desired) interaction with some benchmark. We are able to solve explicitly optimal portfolio for expected utility maximizers who optimize a state-dependent utility function. We then show that this setting is very general in that it can rationalize any decision that is consistent with achieving a given joint distribution with a benchmark. Our results assume that the state-price density is agreed upon by all decision makers. By doing so we include the case of complete markets in which case the optimal strategies that we derive are attainable. By contrast, in incomplete market settings the optimal payoffs are not necessarily attainable. Using results of R¨ uschendorf & Vanduffel (2018) it is possible to extend 17

the results to some incomplete market settings of interest. We illustrate the results of our paper with various examples. These examples are presented in the context of a multidimensional Black Scholes model and may appear of limited practical interest. They show however how to apply the theoretical results and help the reader in understanding the implications of the theoretical results. We believe that the methodology presented in this paper has many practical applications that we leave for future research. For example, it is possible to infer the statedependent utility function of hedge fund managers and to infer their risk aversion. We could also use our tool to show evidence of state-dependent risk aversion and to quantify it. In this regard, it is commonly believed that investors may be increasingly risk seeking when facing losses. As far as we know inferring state-dependent utility functions and quantifying statedependent risk aversion has not been done before in the literature but in this paper we provide the tools to do so. Another application is to improve existing investment strategies. If one knows very clearly the benchmark that is used as reference for performance assessment, one may be able to improve a given investment strategy by making use of conditionally cost-efficient strategies. Finally, the characterization result that we provide for cheapest portfolios having a certain dependence with a benchmark could be used to develop a tool for detecting fraud of some investment managers (extending results in Bernard & Vanduffel 2014b).

References Allais, M. (1953). Le comportment de 1homme rational devant 1e risque, critique des postulates et axiomes de 1eco1e americaine. Econometrica, 21 , 803–815. Berkelaar, A. B., Kouwenberg, R., & Post, T. (2004). Optimal portfolio choice under loss aversion. The Review of Economics and Statistics, 86 (4), 973–987. Bernard, C., Boyle, P. P., & Vanduffel, S. (2014). Explicit representation of cost-efficient strategies. Finance, 35 (2), 5–55. Bernard, C., Chen, J. S., & Vanduffel, S. (2015a). Rationalizing investors choices. Journal of Mathematical Economics, 59 , 10–23. Bernard, C., Moraux, F., R¨ uschendorf, L., & Vanduffel, S. (2015b). Optimal payoffs under statedependent preferences. Quantitative Finance, 15 (7), 1157–1173. Bernard, C., & Tang, J. (2016). Simplified hedge for path-dependent derivatives. International Journal of Theoretical and Applied Finance, (p. 1650045). Bernard, C., & Vanduffel, S. (2014a). Financial bounds for insurance claims. Journal of Risk and Insurance, 81 (1), 27–56. Bernard, C., & Vanduffel, S. (2014b). Mean–variance optimal portfolios in the presence of a benchmark with applications to fraud detection. European Journal of Operational Research, 234 (2), 469–480. Dybvig, P. H. (1988a). Distributional analysis of portfolio choice. The Journal of Business, 61 (3), 369–393. Dybvig, P. H. (1988b). Inefficient dynamic portfolio strategies or how to throw away a million dollars in the stock market. The Review of Financial Studies, 1 (1), 67–88. Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of The Econometric Society, (pp. 263–291). Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3 (4), 373–413. Quiggin, J. (1992). Generalized Expected Utility Theory: The Rank Dependent Model . Springer Science & Business Media. Rozeff, M. S. (1994). Lump-sum investing versus dollar-averaging. The Journal of Portfolio Management, 20 (2), 45–50.

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R¨ uschendorf, L., & Vanduffel, S. (2018). On the construction of optimal payoffs. Working paper, Available on SSRN . Segal, U., & Spivak, A. (1990). First order versus second order risk aversion. Journal of Economic Theory, 51 (1), 111–125. Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. Journal of Economic Literature, 38 (2), 332–382. Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5 (4), 297–323. Vanduffel, S., Ahcan, A., Henrard, L., & Maj, M. (2012). An explicit option-based strategy that outperforms dollar cost averaging. International Journal of Theoretical and Applied Finance, 15 (02), 1250013. Von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior . Princeton university press. Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica: Journal of the Econometric Society, (pp. 95–115).

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