Direct Data-based Decision Making under Uncertainty

In a typical one-period decision making model under uncertainty, unknown consequences are modeled as random variables. However, accurately estimating probability distributions of the involved random variables from historical data is rarely possible. As a result, decisions made may be suboptimal or even unacceptable in the future. Also, an agent may not view data occurred at different time moments, e


Introduction
A typical process of decision making under uncertainty is as follows data → uncertainty modeling → → risk preference modeling → choice/decision (1) Let X be a set of available (feasible) actions.Scheme (1) can be formally stated as: (i) modeling unknown consequences of every action X ∈ X as a random variable (r.v.) R(X), (ii) establishing a numerical representation U : R → R for agent's preference relation, defined on a space R of all r.v.'s and (iii) finding best action by maximizing U with respect to X ∈ X : max X∈X U (R(X)). ( What an agent has readily available is only historical/experimental data and his/her preferences towards risk and reward.The rest is statistical inference from the data about corresponding uncertain outcomes based on various assumptions, which largely depend on the nature of data.For example, measurements of the length of some object can be reliably assumed to be realizations of independent and identically distributed (i.i.d.) r.v.'s-timing of those measurements can be safely ignored.By the central limit theorem (CLT), the average of a large number (a) The distributions of rates of return of financial assets are typically non-symmetric with left tails being much heavier than right tails [50].
(b) Increments of actual price processes are not stationary, and consequently, Lévy processes cannot be calibrated with real data [36]. 1c) "Periods of lower returns are systematically followed by compensating periods of higher returns" [51] ("mean reversion" phenomenon)-evidence that price increments are not independent.
In fact, the above issues with stochastic processes can be "fixed" by time-series models.For example, autoregressive models AR(p) assume that asset's rate of return depends on p previous ones, moving-average models MA(q) involve last q values of a stochastic error, autoregressive-moving-average models ARMA(p, q) generalize AR(p) and MA(q), whereas ARIMA models generalize ARMA(p, q), suitable to describe a wide range of non-stationary processes [8].However, any time-series model is merely another inference from the historical data and its parameters are subject to estimation errors.
The discrepancy between a real-life phenomenon and its model is called model error-in contrast to approximation error, which can be resolved by simply increasing the sample size, the model error implies that an increase of observations of asset rates does not directly translate into the accuracy/precision in estimation of the probability distributions of the rates.There are various existing approaches that address model uncertainty.For example, bootstrapping [6] generates different scenarios for the variable of interest from a given time series, robust optimization [4] assumes that probabilities in question belong to certain intervals, whereas dual characterization of risk and deviation measures [2,44] relies on risk envelopes, which can be viewed as sets of distortions of an underlying probability measure, see [32].Notably, Pflug et al. [42] showed that the naive 1/n investment strategy could be optimal in portfolio selection when model uncertainty is high.Savage [49] suggested to study decisions as functions from some state space Ω to a set of outcomes Y ⊂ R, which are now known as Savage acts.This approach involves no probability measure on Ω-a critical feature that gave rise to various Savage-act versions of the expected utility theory (EUT) [11,29].For example, Gilboa and Schmeidler [21] proposed to study preference relations over acts, i.e., "functions from states of nature into finite-support distributions over a set of deterministic outcomes."In this case, the agent ends up with the same optimization problem (2), where R is a functional from X to the set A of all acts, and U : A → R is a numerical representation of Gilboa & Schmeidler's preference relation.Of course, the list of existing approaches goes far beyond these examples, see e.g.[3,10,13,55] for alternative approaches and [38,20] for recent surveys.
However, accurately modeling of outcomes of real-life actions in the context of any of these theories is difficult.For example, modeling of financial portfolio returns in terms of Gilboa-Schmeidler acts [21] includes forecasting of a set of finite-support distributions, and therefore, could, in fact, be harder than that in terms of r.v.'s.The main problem with uncertainty modeling is that, contemplating a choice among several alternatives, an agent ponders what alternative he/she would be most benefited from in the future, while the only available information is often the data representing their historical performances in the past.
In view of failure of common statistical assumptions in application to a stock market [50,36,51] and in view of sensitivity of optimal decisions (portfolios) to errors in estimation of probability distributions of financial assets [28,30], this work aims to identify intertemporal principles for comparing historical time series of asset rates of return and to develop an axiomatic framework for a rational decision making in portfolio theory on the space of historical time series.For example, an agent may postulate that if A always outperformed B in the past, then A B, even though better past performance does not guarantee better future performance.
The idea of making decisions based directly on historical data is not new, 2 but it has received relatively little attention in economic and financial literature.Gilboa and Schmeidler [22,23] introduced a case-based decision theory, which makes decisions based on past experience in similar situations. 3In a financial market setting, this theory would identify the moment in the past when the market behavior was most similar to the current one and would prescribe to invest all money into the financial asset which had the highest rate of return in that "similar" situation.However, it is not clear what "similarity measure" to use, and the resulting investment strategy may con-tradict the diversification principle.There are other objections for the use of direct data-based decision making in portfolio selection: (i) Information such as recent market trends and news about particular companies may provide valuable insights for selecting a financial portfolio.(ii) The future may have little in common with the past, for instance, due to unique events such as BREXIT.(iii) New financial assets lack historical data, but it is unlikely that agents would veiw, say, a new bank and a startup IT company similarly.
However, incorporating news and other non-quantitative information, e.g. a recent hire of a highly regarded CEO, into a mathematical model requires human participation and is, therefore, expensive and slow.In contrast, calibrating stochastic models based only on historical data can be fully automated and performed in milliseconds, which is particularly valuable for high-frequency trading.Thus, if the choice of optimal portfolio is based on some uncertainty modeling, which in turn uses historical data only, then the uncertainty modeling stage could be omitted, and decisions could be made based on data directly.
The contribution and organization of this work are as follows.Section 2 introduces the notion of time profile and discusses numerical representation of time series.Section 3 introduces intertemporal principles of rational choice.Section 4 reinterprets the mean-variance and maxmin utility analyses in the context of direct data-based decision making.Section 5 concludes the work.Appendix A contains proofs of key results in Section 3 and Appendix B provides an axiomatic foundation for a data-based analogue of the EUT.

Time Profiles and Numerical Representation of Time Series
Let T = {s 1 , . . ., s T } be a finite set of discrete time moments s 1 < • • • < s T in the past, and let x 1 , . . ., x T be corresponding rates of return of some financial asset.Since x 1 , . . ., x T encode a time structure and are not realizations of i.i.d.r.v.'s, the agent would unlikely view x 1 , . . ., x T as equally valuable data and may assign them corresponding weights q 1 , . . ., q T of historical data "depreciation" to be collectively referred to as time profile Q.For example, the agent may postulate that fraction q t /q t+1 is a constant q ∈ (0, 1] independent of t, which implies that q t = q T q T−t , t = 1, . . ., T. Alternatively, q 1 , . . ., q T can be chosen to be proportional to the (normalized) autocorrelation profile of the asset-if for some time periods (usually far in the past), the autocorrelation vanishes, then those past values play little role in predicting asset's behavior.Suppose, for example, the FTSE 100 index is such an asset.Figure 1 depicts the sample autocorrelation function (ACF) of daily prices of the index from 1-April-2015 to 1-April-2016 with the lag up to 80, taken from [1].For a lag longer than 80 days, the autocorrelation is negligible, so that T = 80, and weights q 1 , . . ., q 80 are then proportional to the ACF in Figure 1 and satisfy ∑ T t=1 q t = 1.Behavioral evidence supporting the notion of time profile includes, but is not limited to the following (a) The effect of fading memory and emotions [18,Part 6]: an individual is much more likely to rely on and act upon recent experience rather than that occurred far in the past.(b) A reversion of the behavioral time discounting principle stating that "money available at the present time is worth more than the same amount in the future" [18,Part 3].(c) Only 21% of agents agree that historical data should be equally weighted [15].
Technical arguments in favor of time profiles include (a) In time-series analysis, (4) is known as the weighted moving average, in which more weight is often given to the most recent data.In particular, (3) are weights in exponential smoothing [9].(b) The ACF decreases with time and almost vanishes after 80 days (see Figure 1).(c) In mean-variance portfolio selection, the optimal portfolios with time profiles based on geometric progression (3) with various q outperform optimal portfolio in which asset rates of return are modeled by an ARIMA time-series model -to be discussed in Example 9 (Figure 5).
For a time series X = (x 1 , . . ., x T ) and time profile Q = (q 1 , . . ., q T ), the weighted average and the mean-square deviation of x 1 , . . ., x T are defined by respectively.Here, E Q [X] and σ Q (X) are not assumed to be estimates of the future expected value and standard deviation, respectively, they are just weighted average and standard deviation of the time series X.For two time series X = (x 1 , . . ., x T ) and Y = (y 1 , . . ., y T ) and time profile Q = (q 1 , . . ., q T ), the covariance is defined by In fact, the agent may contemplate a whole set Q of various time profiles.First, the agent may postulate that q t are nonnegative, impose normalization ∑ T t=1 q t = 1 and define the set of time profiles to be Q = Q max = (q 1 , . . ., q T ) ∈ R T q 1 0, . . . ,q T 0, ∑ T t=1 q t = 1 .
(7) Next, the agent may assume that q 1 . . .q T for every Q ∈ Q-more recent data is more valuable.A maximal "time averse" subset Q ⊂ Q max is given by Q = (q 1 , . . ., q T ) ∈ R T 0 q 1 . . .q T , ∑ T t=1 q t = 1 .
With a chosen time profile set, Q, the agent may define the utility of the time series X = (x 1 , . . ., x T ) by and then can use (11) for comparing different time series.In fact, ( 11) is a data-based analogue of Gilboa & Schmeidler's maxmin model [21].
Example 2 (data-based version of drawdown measure) Let X = (x 1 , . . ., x T ) be a historical time series of the rate of return of some financial asset, and let x t = ∑ t j=1 x j be uncompounded cumulative rate of return over period [1, t].The drawdown of X can be defined by ξ t = max 1 k t x k − x t [14,56].Then the maximum drawdown max 1 t T ξ t can be represented in the form of (11) with Also, the average of the k largest drawdowns can be defined as where (ξ 1 , . . ., ξ T ) is a permutation of (ξ 1 , . . ., ξ T ) such that ξ 1 . . .ξ T [14, 56].The functional U(X) = −DD α (X) is a particular case of (11).Remarkably, the time-series definition (12) of drawdown, which is a dynamic measure, is as simple as time-series analogue of a one-period risk measure, e.g.CVaR.
Thus, while representation (11) is known, the time profile Q here has the meaning of historical data "depreciation."See Appendix B for a nonlinear generalization of (11).
Let Q * = (q * 1 , . . ., q * T ) ∈ Q max with at least three of q * 1 , . . ., q * T being non-zero.The utility of the time series X = (x 1 , . . ., x T ) can also be measured by the mean-standard deviation functional with a continuous function V strictly increasing in the first argument and strictly decreasing in the second one.

Example 3 (data-based version of the mean-standard deviation utility)
The mean-standard deviation utility is defined by V(m, σ) = m − λσ with a specified "level of risk aversion" λ > 0 [25, Example 6].With this V, (13) takes the form Note that ( 14) is a particular case of (11) with [46, Example 1] Example 4 (data-based version of the mean-variance analysis) With a specified threshold µ on E Q * [X], a data-based analogue of the mean-variance analysis corresponds to the utility functional In contrast to the existing decision theories, the proposed direct-data based approach does not try to make any statistical inference from the historical data, but rather incorporates agent's perception of the historical data into a decision process through the time profiles, e.g. as in (11) and ( 13)- (15), and the goal of this work is to identify intertemporal principles of rational choice for constructing time profile sets.An axiomatic framework for intertemporal principles is laid out in §3, and then the direct data-based decision making approach is demonstrated in portfolio optimization with (11) ( §4.1) and in mean-variance portfolio selection ( §4.2).

Intertemporal Principles of Rational Choice
This section discusses intertemporal principles of rational choice for constructing time profile sets introduced in §2.For any asset A, let be its historical excess rate of return over the risk-free rate.
Let X be a portfolio consisting of risky assets A 1 , . . ., A n with portfolio weights (α 1 , . . ., α n ) ∈ R n (short selling is allowed) and of a risk-free asset A 0 with the weight where x i (t) is defined by ( 16) for asset A i and where I and J are sets of indices i such that x i (t) = k and x i (t) = k, respectively.
Thus, the portfolio X corresponds to a function x : T → F, where F = {a + bk, | a ∈ R, b 0} is a real vector space 5 with addition and multiplication by a constant defined by The set X of all possible portfolios is identified with set F T of all vectors (x 1 , . . ., x T ), where x t = x(s t ), t = 1, . . ., T, or, equivalently, X ⊂ R 2T is the set of vectors (a 1 , b 1 , a 2 , b 2 , . . ., a T , b T ) with b t 0, t = 1, . . ., T, where x t = a t + b t k, t = 1, . . ., T.
Let be a preference relation on X : X Y if X is strictly preferred to Y, X ∼ Y if an agent is indifferent between X and Y, and X Y if either X Y or X ∼ Y.One of the fundamental principles of rational choice is that forms a complete weak order on X .
Axiom 1 (complete weak order) is complete and transitive: (i) X Y or Y X for all X, Y ∈ X (completeness).
(ii) X Y and Y Z imply that X Z for all X, Y, Z ∈ X (transitivity).
Axiom 1(i) asserts that decision making is based solely on historical data and no other information is available to the agent (important and non-trivial assumption).Under a mild technical assumption, 6 axiom 1 implies that admits a numerical representation U : X → R such that X Y ⇐⇒ U(X) U(Y), see Theorem 2.6 in [19].
Set X = F T is a metric space with distance ρ(X, Y): where X = (x 1 , . . ., x T ), Y = (y 1 , . . ., y T ), and We can then define open and closed subsets of X with respect to topology induced by this metric.
Axiom 2 states that does not change when X and Y are slightly perturbed.With axiom 2, Theorem 2.15 in [19] implies that a numerical representation U for can be chosen as a continuous function on X .
It is well known that rational agents diversify their portfolios rather than "keep all eggs in one basket." A numerical representation U of that follows this principle is a quasi-concave function: U(αX + (1 − α)Y) min{U(X), U(Y)}.See [12] for a recent study of quasiconcave utility functions.A trivial sufficient condition for axiom 3 to hold is the existence of concave numerical representation U of : i.e., U(αX Often investment decisions are made in two steps: (i) decide on portion α of the capital to be invested into risky assets (and keep the remaining money in a savings account), and (ii) select a risky portfolio for the portion α.Next axiom states that the choice of the risky portfolio in step (ii) does not depend on α.
Proposition 1 Let X + = {X | X X 0 }, where X 0 denotes investment into the risk-free asset only.satisfies axioms 1-4 on X + if and only if it has a continuous numerical representation U on X + satisfying (i) (ii) Positive homogeneity: U(αX) = αU(X), for every X ∈ X + and α 0.
Proof See Appendix A. 2 For the agent who considers only portfolios strictly preferable to the risk-free investment X 0 , restriction to the set X + in Proposition 1 is inessential.Suppose U satisfies (i) and (ii) on the whole set X .Then, it can be represented by where X = (x 1 , . . ., is a set of vectors Q = (q 1 , q 1 , q 2 , q 2 , . . ., q T , q T ), which can be chosen convex, closed, and bounded, and where 17) is attained will be called identifiers of X.
If Q = {Q} is a singleton, (17) simplifies to U(X) = Q, X .In general, q t (or q t ) is interpreted as the weight/ importance that the agent assigns to the historical data (or absence of data) at time t ∈ T .The agent may consider a set Q of possible weights and may select Q ∈ Q such that Q, X is the worst possible: this is an interpretation of (17).Portfolio optimization with (17) takes the form where X ⊂ X is the set of all feasible portfolios, which motivates calling (17) maxmin utility theory (MMUT).
Some agents prefer investing into portfolios which performed well in the past, while avoiding assets with poor or unknown performance, e.g.those which just appear on the market.This intuition is formalized in the following axiom.
then X Y.In other words, if the proportion of money invested into assets with no historical data in X is less than that in Y and if collectively the assets with known data in X always outperformed those in Y, then X Y.
If no data is missing and the agent uses direct historical simulation for forecasting, axiom 5 is equivalent to monotonicity axiom for r.v.'s.
However, financial interpretations of axiom 5 and ( 18) are completely different.With simple historical simulation, the agent interprets inequalities x 1 y 1 , . . ., x T y T as evidence that a portfolio associated with X = (x 1 , . . ., x T ) will outperform the one associated with Y = (y 1 , . . ., y T ) in the future with probability 1 and, thus, strongly prefers X to Y.This again shows that historical data can be easily misinterpreted (mis-modeled), which could result in poor decisions.In contrast, axiom 5 does not imply that the portfolio associated with X will outperform the one associated with Y in the future-it merely states historical facts.
In fact, when forecasting uses methods other than direct historical simulation, axiom 5 differs from (18) as the following example demonstrates.
Example 5 In a "Gaussian world," the future excess rate of return of a portfolio with past excess rates of return X = (x 1 , . . ., x T ) can be modeled by a normally distributed r.v.R(X) with mean µ X and variance σ 2 X estimated from X by Then (18) simplifies to which neither implies axiom 5 nor follows from it.
Axiom 5 assumes that X outperformed Y at every single time moment in the past.Some agents, however, may consider cumulative past performance.

Axiom 6 (time aversion)
then X Y.
Proposition 2 Let (17) be a numerical representation of on X .The following statements are equivalent.
Proof See Appendix A. 2 Many models of financial market assume that observations of asset's rates of return at different times are independent.Black & Scholes [5] priced options based on the assumption that stock prices follow a Brownian motion, whereas Sato [48] replaced the Brownian motion by a Levy process, which also assumes price increments (and consequently returns) to be independent.In such models, discrete historical data x 1 , . . ., x T can be considered as realizations of independent and identically distributed r.v.'s.In this case, the order of data is not important, which corresponds to the following no-time-structure principle X ∼ Y whenever (y 1 , . . ., y T ) is a permutation of (x 1 , . . ., x T ), (22) which implies that X = (a 1 + b 1 k, . . ., a T + b T k) with a 1 . . .a T (positive trend) and Y = (a T + b T k, . . ., a 1 + b 1 k) with the reverse order of data (negative trend), are equally preferable-this does not seem to be a rational behavior.
An axiomatic framework for data-based expected utility theory (36) is presented in Appendix B.
When there is no missing data or when the agent deliberately excludes assets with missing/incomplete data, every portfolio X can be identified with its historical time series (x 1 , . . ., x T ) ∈ R T .In this case, axioms on a preference relation are introduced and studied on X = R T .
• Axiom 1 states that defines a complete and transitive weak order on R T .• Axiom 2, formulated in usual topology of R T , states that the sets {Y ∈ X |Y X} and {Y ∈ X |X Y} are closed in R T and implies the existence of continuous function U : R T → R such that X Y ⇔ U(X) U(Y).• Axiom 3 states that U is a quasi-concave function on R T .
• Axioms 3 and 4 imply that U is concave and positive homogeneous, hence admit the form (11). • Axiom 5 states that if X = (x 1 , . . ., x T ) and Y = (y 1 , . . ., y T ) are such that x 1 y 1 , . . ., x T y T , then X Y.It implies that q t in ( 11) are non-negative.• Axiom 6 states that X = (x 1 , . . ., x T ) Y = (y 1 , . . ., y T ) provided that ∑ T t=τ x t ∑ T t=τ y t , τ = 1, . . ., T (portfolios with better average recent performance are preferable).It implies that q 1 . . .q T for every Q ∈ Q in (11).Now, with what introduced axioms is the mean-standard deviation functional (13) consistent?Proposition 3 (13) is a numerical representation U of if and only if satisfies axioms 1 and 2, and the following two additional axioms (a) X + C X for all C > 0, and Next propositions address consistency of the mean-standard deviation utility (14) with axiom 5 (monotonicity).
4 Data-based Portfolio Optimization

Maxmin Portfolio Optimization
In the direct data-based decision approach, historical excess rates or return of the risk-free asset and n risky assets during last T time periods are given by X 0 = (0, . . ., 0), X 1 = (x 11 , . . ., x 1T ), . . ., X n = (x n1 , . . ., x nT ), respectively.With no short sales and with U being the utility (11), a portfolio optimization problem is formulated by Example 6 (single time profile) For a singleton Q = {Q * }, problem (24) simplifies to In this case, u * = max i u i , and an optimal strategy is v i * = 1, v i = 0, i = i * , where i * = arg max i u i , i.e. investing the whole capital in the "best" asset-no diversification.
Example 7 ("ultimate" risk aversion) is equivalent to the problem of finding mixed-strategy Nash equilibrium in a two-player zero-sum game [53] with (n + 1) × T payoff matrix X having elements x it .In this case, u * is equal to the value of the game, and the optimal investment strategy v * ∈ V is a solution to the linear program If Q is an arbitrary closed convex subset of Q max , von Neumann minimax theorem [54] implies that u * = max v∈V min Q∈Q v X Q = min Q∈Q max v∈V v X Q , and u * can be found as Example 8 For Q being the maximal "time averse" set (8), or for Q belonging to the one-parameter family (9), problem (25) is a linear program.
The set Q in (11) can also be found by the inverse portfolio approach introduced in [26,27] in terms of r.v.'s.The idea is that the agent recovers Q from the time series X * = (x * 1 , . . ., x * T ) of the rate of return of a portfolio that he/she is relatively satisfied with-such a portfolio should solve (24) with U given by (11).Proposition 7 in [26] implies that the maximal possible (most robust) such Q is given by provided that the following no-perfect-history assumption holds: there is no time series (x 1 , . . ., x T ) of the rate of return of a feasible portfolio such that x 1 0, . . ., x T 0 with at least one inequality being strict.For example, for T n, there is unlikely to be a portfolio that outperforms the risk-free asset for every time period in the past.Thus, T can be chosen sufficiently large to guarantee that the no-perfect-history assumption holds, otherwise the historical time series can be perceived to be too short for making a reliable decision.

Mean-Variance Portfolio Selection
Suppose there is a risk-free asset with constant rate of return r 0 , and there are n risky assets.In the typical approach (1), rates of return of the risky assets are modeled as r.v.'s r 1 , . . ., r n on some probability space.With v i being the fraction of the capital invested into asset i, the rate of return of a portfolio is then ∑ n i=0 v i r i .The mean-variance portfolio selection problem [39] with desired premium ∆ > 0 over r 0 , continues to be a cornerstone of the modern portfolio theory: one-fund theorem, twofund theorem, capital asset pricing model (CAPM), Sharpe ratio and asset beta all stem from (26), see [34],-this is one of the reasons to consider its data-based analogue: where X i = (x i1 , . . ., x iT ) ∈ R T is the time series of the rate of return for risky asset i ∈ {1, . . ., n}, and 4), ( 5) and ( 6), respectively, with a time profile Q = (q 1 , . . ., q T ) ∈ Q max .Note that ( 27) is equivalent to max v 0 ,v 1 ,...,v n U (∑ n i=0 v i X i ) subject to ∑ n i=0 v i = 1 with U defined by (15).
Optimal portfolio weights v * 0 and where e = (1, . . ., 1) is the n-dimensional unit vector and where X and Λ Q are matrices with entries x it , i = 1, . . ., n, t = 1, . . ., T, and cov Q (X i , X j ), i, j = 1, . . ., n, respectively.If e v * = 1, which implies that v * 0 = 0 (no investment into the risk-free asset), then the optimal portfolio is called a master fund of positive type (market portfolio) [45] with the weights [57, (8.2.4)] and the time series of the rate of return X M = v M X.The optimality conditions for the master fund can be restated as the capital asset pricing model (CAPM) [46]: Problem (26) requires knowing the expected values and variance-covariance matrix of asset rates of return.If historical rates of return of each asset are assumed to be realizations of i.i.d.r.v.'s, then ( 26) is a particular case of ( 27) with is what is solved in practice.However, (26) does not distinguish portfolios with different trends but having the same histogram of historical rates over the same period of time.Also, it is well-known that ( 26) is inconsistent with ordinary monotonicity axiom 9 (18).However, in the light of the historical data "depreciation," this could be a result of mis-interpretation of historical data as a "forecast" for future rates of return.In fact, what is violated is axiom 5 (monotonicity).Indeed, the agent may well believe that assets with good historical performance are now overpriced and may prefer a portfolio with acceptable past performance and least historical volatility.
Example 10 Figure 6 depicts the mean-variance efficient frontier for the Markowitz's portfolio problem (27) with the time 9 The mean-variance approach may violate "ordinary" monotonicity axiom: if R X is the rate of return of a solution of (26), then there may exist another feasible portfolio with the rate of return R Y , such that P[R Y R X ] = 1 and P[R Y > R X ] > 0, see [25,Example 5].The best monotone approximation of a mean-variance functional is obtained in [35].Alternatively, one may obtain a monotone preference relation if the standard deviation in (26) is replaced by a general deviation measure [44,45,46]. 10Namely, Apple Inc. (AAPL), Amazon.comInc (AMZN), Bank of America Corp (BAC), Twenty-First Century Fox Inc Class B (FOX), IBM Common Stock (IBM), The Coca-Cola Co (KO), McDonald's Corporation (MCD), Microsoft Corporation (MSFT), Nike Inc (NKE), and Visa Inc (V). 11Negative portfolio weights correspond to short selling.weights chosen according to the ACF of the FTSE 100 index in Figure 1.We select n = 70 most actively traded 12 assets from the FTSE 100 index and use assets' daily rates of return from 1-April-2015 to 1-April-2016.It is assumed that r 0 = 0.01%.
Remark 1 The weights Q = (q 1 , . . ., q T ) can be viewed as parameters, and sensitivity of the optimal value V(Q) = σ Q (X * (Q)) in (27) with respect to changes in q 1 , . . ., q T can 12 The comparison is made based on average trading volume.

Conclusions
In a typical one-period decision making under uncertainty, outcomes of feasible actions are modeled as r.v.'s.As a result, optimal decisions depend on the accuracy of estimation of the time (1 May 2016 -1 May 2017) out-of-sample price evolution of $1 portfolio q = 1 q = 0.99 q = 0.95 q = 0.9 ARIMA corresponding probability distributions.Agents who believe that probability distributions of asset rates cannot be reliably estimated would unlikely use any random variable-based decision theory.They will also find it hard to apply Gilboa & Schmeidler's case-based decision theory [22,23], which requires a similarity measure for market behavior over different time periods.As an alternative, this work has formulated "intertemporal" principles/axioms for a preference relation on the space of historical time series to facilitate making a rational choice in portfolio selection.It does not suggest to dismiss existing decision theories which include uncertainty modeling.Instead, it shows how to adapt them to deal with historical time series.This adaptation, however, is not "mechanical": some of the proposed axioms, e.g."time aversion," have no direct analogue in the existing theories.Example 5 demonstrates that the same axiom (in this case, monotonicity) may lead to completely different decisions when applied to r.v.'s and to time series.Thus, instead of making statistical inference from the historical data, an agent may incorporate his/her perception of the data through time profiles and make a decision based on the data directly.Figure 5 (Example 9) shows that in mean-variance portfolio selection, the optimal portfolios with the exponential time profiles with various q outperform optimal portfolio in which asset rates of return are modeled by the ARIMA model.However, no matter what advantage on a particular dataset, the direct databased decision making approach demonstrates over approaches with uncertainty modeling, those agents who believe that asset prices (rates of return) can be reliably predicted by merely statistical means could hardly be discouraged-they will continue either relying on some "trusted" statistical model or searching for an ideal one.This work aims to provide an alternative decision making approach for those who do not have such a belief.
While the focus of this work is on direct data-based analogues of the Gilboa & Schmeidler's maxmin model, the meanvariance approach, and the EUT, other existing decision theories can be reinterpreted and analyzed in a similar fashion.is strictly monotone.With (31), f cannot be decreasing, hence it is strictly increasing.Thus, f has a strictly increasing inverse function f −1 , and U(X) := f −1 (U (X)) is another numerical representation of .Then U(αX * ) = f −1 (U (αX * )) = α for all α > 0. Then X ∼ U(X)X * for all X ∈ X + , and, by axiom 4, αX ∼ αU(X)X * .Hence, U(αX) = U(αU(X)X * ) = αU(X), and (ii) follows.

Proof of Proposition 2
For (iii) ⇒ (i) part, select any X, Y satisfying (19) and choose any Q ∈ Q.Let δ 1 = q 1 0 and δ t = q t − q t−1 0, t = 2, . . ., T. Then T t=1 q t c t , where the inequality follows from (19).By similar argument, ∑ T t=1 q t b t ∑ T t=1 q t d t , so that U(X) U(Y), and, consequently, X Y.
(i) ⇒ (ii) is straightforward.For (ii) ⇒ (iii), let Q Y = (q Y 1 , q Y 1 , . . ., q Y T , q Y T ) ∈ Q be an identifier of any Y = (a 1 + b 1 k, . . . ,a T + b T k) ∈ X and let q Y i > q Y j for some i < j.Then for X defined by x(t) = y(t) − δI T 1 ,T 2 (t) with T 1 = {i}, T 2 = {j}.This contradicts (20), and, consequently, 0 q Y 1 . . .q Y T for every Y ∈ X .Similarly, (21) yields 0 q Y 1 . . .q Y T .Let Q ⊆ Q be the closure of the convex hull of all Q ∈ Q, which are identifiers of some Y ∈ X , and let U be given by (17) with Q .Then U(Y) = Q Y , Y U (Y) for every Y ∈ X , so that Q ⊆ Q , which yields Q = Q.Consequently, Q is the closure of the convex hull of some vectors satisfying (ii), and thus, this condition holds for every Q ∈ Q.
(c) → (a): Fix j such that q * j = z, and let X x , x ∈ R, be a one-parameter family in R T such that x j = x, and x ] = q * j x 2 j + ∑ t =j q * t x 2 t = x 2 z + z 2 1−z , and Consequently, Since U(X x ) is non-decreasing in x by (c), ( 23) follows.

B Appendix:
Data-Based Expected Utility Theory (EUT) A data-based analogue of the independence axiom for r.v.'s is stated as follows.
Axiom 7 (independence) Let A and B be any disjoint sets such that A ∪ B = T .For any X, Y ∈ X denote X A ⊕ Y B a function z : T → F such that z(t) = x(t) for t ∈ A and z(t) = y(t) for t ∈ B. Then X A ⊕ Z B Y A ⊕ Z B implies that X A ⊕ W B Y A ⊕ W B for any X, Y, Z, W ∈ X . is a numerical equivalent of the appeal for the historical rate of return x of a given portfolio at time t.If u t is differentiable, u t (x) measures sensitivity of investor's utility to changes in data at time t, and (37) represents the principle "recent observations are more important than past ones."If u t (x) = q t x, t = 1, . . ., T, (36) and (37) simplify to (11) with a singleton Q = {Q} with q 1 . . .q T .

Figure 5 :Figure 6 :
Figure5: In-sample and out-of-sample price evolution of 1$ invested at the beginning of the corresponding periods into the master fund in(27) with q = 1, 0.99, 0.97, and 0.95, and in(26) with the ARIMA model, respectively.