Title: UNDERSTANDING DISTRIBUTIONAL AMBIGUITY VIA NON-ROBUST CHANCE CONSTRAINT

Abstract: We propose a non-robust interpretation of the distributionally robust optimization (DRO) problem by relating the impact of uncertainties around the distribution on the impact of constraining the objective through tail probabilities. Our interpretation allows utility maximizers to figure out the size of the ambiguity set through parameters that are directly linked to the chance parameters. We first show that for general φ-divergences, a DRO problem is asymptotically equivalent to a class of mean-deviation problems, where the ambiguity radius controls investor's risk preference. Based on this non-robust reformulation, we then show that when a boundedness constraint is added to the investment strategy. The DRO problem can be cast as a chance-constrained optimization (CCO) problem without distributional uncertainties. Without the boundedness constraint, the CCO problem is shown to perform uniformly better than the DRO problem, irrespective of the radius of the ambiguity set, the choice of the divergence measure, or the tail heaviness of the center distribution. Besides the widely-used Kullback-Leibler (KL) divergence which requires the distribution of the objective function to be exponentially bounded, our results apply to divergence measures that accommodate well heavy tail distribution such as the student t-distribution and the lognormal distribution. Comprehensive testings on synthetic data and real data are provided.

1. INTRODUCTION:

Optimization models with an expectation as the objective function are always used when studying decision making problems. These problems require the knowledge of the "true" distribution, which is uncertain, so its estimation is required. Many studies have been focused on different estimation methods, see for example Merton (1980) , Bai et al. (2007) , El Karoui et al. (2008) , and Qiu et al. (2015) . However, it is often not accurate since the number of observations is inadequate. For example, as stated in Chen & Yuan (2016) , around 3, 000 months of historical data are needed to give accurate approximations for a portfolio of 25 assets. In reality, it is impossible to have such long time series data sets. An inaccurate estimation of the distribution can cause erroneous judgments for a practitioner based on corresponding solutions. It is necessary to seek an approach that can incorporate uncertainty under a scarcity of data.

Formulating a distributionally robust optimization (DRO) model is recognized as a solution to deal with the issue. In a DRO model, we add an extra layer of inner optimization over an ambiguity set which contains distribution as uncertainty. Generally, there are 3 ways to describe the ambiguity set. The first way is the geometric approach, in which the ambiguity set is described as desired geometric shapes such as box, ellipsoid and polyhedral, etc. (e.g., Kim et al. (2014 ), Zhu et al. (2009 ), and Zhu & Fukushima (2009) ). The second way is the moment-based approach, and was studied in Delage & Ye (2010) , Scarf (1957) , Chen et al. (2011), and Zymler et al. (2013) for instance. The ambiguity set is usually defined so that all distribution in it have the same n-th moments. The last way is the statistical distance approach so that the ambiguity set can be thought as a ball with ambiguity radius ρ such that it contains all the distribution of distance/divergence measures from a center distribution less than ρ.

We adopt the statistical distance approach to define the ambiguity set. This is because under the geometric approach and moment-based approach, it is not assured that the "true" distribution lies in the ambiguity set. As a result, optimal solutions become meaningless although it is easier than using the statistical distance approach. Only a small portion of the DRO problems under the statistical distance approach are known to be solved analytically. For example, Hu & Hong (2013) solves the DRO problem if the ambiguity set is constructed using KL divergence. Instead of seeking an analytical solution, we can apply an expansion approach (see e.g., Lam (2016) and Gotoh et al. (2018) ) to solve the DRO problem. The advantage of using the expansion approach is that we can consider solving DRO under general divergence measures.

In this paper, we would like to understand the ambiguity set in the financial context and further find a non-robust interpretation of the ambiguity set, which is characterized by the radius parameter ρ. This is because the value has a great impact on the optimal solution. Small values of ρ exclude the true distribution being examined, while large values of ρ make the DRO problem conservative. Both are not very useful in practice. Pardo (2005) gives a statistical meaning of ρ asymptotically and Ben-Tal et al. (2013) applies it to interpret the ambiguity set. The interpretation of the ambiguity radius is based on the assumption that the true distribution is within the same parameterized distribution family as the center distribution. Such assumption limits the universal understanding of the ambiguity radius. In this paper, we relate the DRO problem to a CCO problem. We find that a DRO problem can be reformulated as a class of optimization problems with mean-deviation as the objective function after using the expansion method to remove randomness. We then reformulate a chance-constrained optimization CCO problem and derive equivalence between the DRO problem and the CCO problem. This equivalence opens a door for financial practitioners to interpret the radius parameter through parameters that are directly linked to investment performance.

The rest of this paper is organized as following. In Section 2, we provide background information and a motivation on the optimization problems being considered, including notations and φdivergences in defining the ambiguity set. In Section 3, we reformulate the DRO problem and CCO problem, respectively, and solve the corresponding optimal decisions and optimal values. Section 4 gives numerical experiments, while Section 5 concludes our findings. All detailed proofs are postponed to the Appendix.

2. PROBLEM SETUP:

In this section, we introduce the notations used in this paper and give a brief introduction to the φ-divergence for defining the ambiguity set. Furthermore, we present the motivation to explain why we focus on the ambiguity radius in the DRO problem.

Notations. Let r ∈ R n , an n-dimensional real-valued random vector, be the vector of asset returns and suppose that the joint probability distribution of r is P . Let P 0 be the nominal probability distribution of r. Let x ∈ R n be the asset allocation strategy and e ∈ R n be a vector with all entries equal to 1, respectively. We assume that x lies in a convex set X and P lies in an ambiguity set U. φ-divergence. φ-divergence is a commonly used statistical distance to describe the ambiguity set U. It quantifies how one probability distribution diverges from another. It's defined by a convex function φ(t) which satisfies φ(1) = 0, 0φ 0 0 := 0, and φ a 0 := a lim t→∞ φ(t) t when a > 0. Given the function φ(t), the φ-divergence D(Q||P ) between distribution Q and distribution P is:

D(Q||P ) := Ω φ dQ dP P (dt) = E P φ dQ dP := E P [φ (L)] ,

The quantity L is called the Radon Nikodym derivative (or likelihood ratio) such that L ≥ 0 almost surely and E P [L] = 1. Associated with the function φ(t) is its conjugate function φ * (s), defined as

EQUATION

Table 1 lists the two divergences used in this paper. The KL divergence is commonly used because of the appealing properties its conjugate function has. In order to use the KL divergence, the distribution of the objective function needs to be exponentially bounded, which excludes important heavy tail distribution used ubiquitously for financial asset returns, especially the lognormal distribution and the student t-distribution. The Cressie-Read divergence, also called the α-divergence in Glasserman & Xu (2014) , overcomes this limitation and can well accommodate heavy tail distribution. However, our interpretation of the ambiguity radius ρ as a chance constraint applies to all 

Kullback-Leibler t log(t) -t + 1 e s -1 Cressie-Read 1-θ+θt-t θ θ(1-θ) , θ = 0, 1 (1-s(1-θ)) θ θ-1 θ -1 θ , s < 1 1-θ

the φ-divergences, including Burg entropy, J-divergence, χ 2 -distance, modified χ 2 -distance, and Hellinger distance.

Motivation. The goal is to maximize the expectation of investors' utility function f (x, r) over a set of admissible allocation strategies X, namely,

max x∈X E r∼P [f (x, r)].

Utility function is ubiquitous in economics and it measures one's preference for alternatives. In general, it is either in the form of exponential utility function f (x, r) = 1-e -ax T r a (a = 0) or in the form of power utility function (Pratt (1978) ). Whatever the concrete utility function is, the stochastic optimization problem itself depends strongly on assumptions and properties of the asset return r, the true distribution of which is usually unknown. To address the distribution uncertainty, a common approach is to formulate a robust counterpart of the original problem, called as a distributionally robust optimization problem. A DRO formulation addresses the uncertainty by constructing an ambiguity set of distributions centered at the nominal distribution P 0 and then optimizing the worst-case utility among all distributions within the ambiguity set. In this paper, we focus on an ambiguity set U defined by the φ-divergence and controlled by a radius parameter ρ > 0, i.e., U := {P : D(P ||P 0 ) ≤ ρ}. Thus, the distributionally robust counterpart of the original stochastic optimization problem is: max

(a.k.a. CRRA utility function) f (x, r) = (x T r) 1-η -1 1-η when η = 1 & η > 0 and f (x, r) = ln(x T r) when η = 1

EQUATION

The choice of ambiguity radius ρ is critical. One cannot set it too large since the maximal utility decreases in ρ. However, if it is too small, one loses the robust protection. There is a trade-off in choosing its magnitude. This situation requires an alternative interpretation of the ambiguity radius ρ. Pardo (2005) proves that, assuming that the true distribution P and the nominal distribution P 0 belong to the same parameterized distribution family with parameter dimension d, then the normalized estimated φ-divergence 2N φ (2) (1) D(P ||P 0 ) asymptotically follows a χ 2 d -distribution. It thus relates the ambiguity radius ρ to a confidence level at which the true distribution P falls within this ambiguity set. However, in financial practice with real data, the assumption that the true distribution is in the same parameterized family with the center distribution often fails to hold. A wrong guess of the nominal distribution might lead to a meaningless confidence level interpretation of the ambiguity radius ρ. Focusing on the impact of the ambiguity radius ρ on the maximal utility, we realize that, through a CCO problem, the ambiguity radius ρ can be explained by the parameters of a chance-constrained problem, using only information of the nominal distribution P 0 .

The CCO problem considered in this paper is:

EQUATION

It shares the same objective function as that of problem (2) . The expectation is taken under the nominal distribution P 0 , not subject to distributional robustness. The new component is the probability constraint with the parameters δ and characterizing downside risk, which incorporates the tail of r into consideration. Similar CCO formulations appear often, from the basic Kelly problem to the newsvendor problem (see for example, Busseti et al. (2016) and Özler et al. (2009) ). Our work aims to interpret U incorporating the tail information of r. To be specific, we try to find the relation between ρ in (2) and ( , δ) in (3). In the coming section, we first describe the expansion approach in solving DRO problem (2) . For CCO problem (3) with general chance constraints, there is no consensus on explicit solutions. So as a special case, we focus on the utility function f (x, r) := x T r in the CCO problem. By transforming constraints in (3) to VaR, we obtain analytically the relation between ( 2) and (3) under an unbounded feasibility set of allocation strategies. For bounded feasibility set, we look into the equivalence on ( 2) and (3) through numerical experiments. Although our methodology is investigated in the financial context, it can also be applied to other contexts.

3. INTERPRET ρ AS A CHANCE CONSTRAINT:

In this section, we discuss how to reformulate ( 2) and (3) so that we can compare the optimal values of the two problems.

3.1 REFORMULATION OF (2) The Lagrangian dual to the inner minimization problem in ( 2) is:

EQUATION

)

The last equality is from the definition of conjugate function φ * (s) in Eq. ( 1). Difficulty in solving the dual problem lies in the term

E r∼P0 [φ * (-η 2 (x T r + η 1 ))].

For general φ-divergence, φ * (s) may not have explicitly formula (see Ben-Tal et al. (2013) ). We hereby follow the idea in Gotoh et al. (2018) 1), respectively. Suppose that under mild conditions, the strong duality holds. Defining

D η2,φ,P0 (f (x, r)|E[f (x, r)]) := min η1 η 1 + 1 η 2 E [φ * (η 2 (E[f (x, r)] -f (x, r) -η 1 ))] .

Then, ( 2) is equivalent to:

EQUATION

Through Theorem 3.1, we can eliminate the distribution uncertainty in (2) and reformulate the DRO problem to a deterministic optimization problem as in (5) . The only unknown term

D η2,φ,P0 (f (x, r)|E[f (x, r)]

) is indeed a function of φ * . Therefore, we apply the expansion method to the quantity D η2,φ,P0 (f (x, r)|E[f (x, r)]) and expand it as a series of terms, the coefficients of which can be computed under the nominal distribution P 0 . In such a way, we can reformulate ( 2) as a single-layer maximization problem. Lemma 3.2. Suppose that n is an even number, φ(t) ∈ C n+1 is a convex function which satisfies φ(1) = φ (1) (1) = 0 and φ (2) (1

EQUATION

where

b k = (-1) k+1 z (k) (0) (k+1)!

, and η * 1 is the optimal solution to min

η1 n-1 k=1 b k E (X + η 1 ) k+1 η k 2 . Specifically, z(•) is a function satisfying z(0) = 1, z (1) (•) = 1 φ (2) (z(•)) , and z (k) (•) can be obtained recursively for k ≥ 2.

Note that most of the φ-divergences (KL divergence, Cressie-Read divergence, Burg entropy, Jdivergence, χ 2 -distance, modified χ 2 -distance, and Hellinger distance) satisfy the smoothness conditions. Taking KL divergence and Cressie-Read divergence as example, for n = 4, we can explicitly solve the terms in Eq. ( 6), as shown in the following corollary.

EQUATION

with η * 1 being the real root to the following cubic equation

3 k=1 (k + 1)b k η k 2 • η k 1 + 12b 3 η 3 2 E[X 2 ] • η 1 + (4b 3 η 3 2 E[X 3 ] + 3b 2 η 2 2 E X 2 ) = 0. (8)

For KL divergence, the coefficients are b et al. (2018) . The main difference between our expansion in Eq. ( 7) and the expansion given by Gotoh et al. lies in the calculation of η * 1 . In Eq. ( 7), η * 1 is directly solved through the polynomial equation, while in Gotoh et al. (2018) , η * 1 is approximated as a function of η 2 . In the sequel, we consider the 2 nd order expansion of

1 = 1/2, b 2 = -1/6, b 3 = 1/24; for Cressie-Read divergence with θ > 2, the coefficients are b 1 = 1/2, b 2 = (θ -2)/6, b 3 = (θ -2)(2θ -3)/24. Gotoh et al. gives a similar expansion of D η2,φ,P0 (f (x, r)|E[f (x, r)]) (see Proposition 3.5 of Gotoh

D η2,φ,P0 (f (x, r)|E[f (x, r)]), which is η2E[X 2 ] 2φ (2) (1)

. Substituting this back to ( 5), we achieve the 2 nd order reformulation of (2) in Theorem 3.4. Theorem 3.4. Suppose that φ(t) is convex, twice continuously differentiable, and that φ(1) = φ (1) (1) = 0 and φ (2) (1) > 0. 2) is asymptotically equivalent to a mean-deviation problem:

Define X := f (x, r) -E[f (x, r)]. (

EQUATION

)

Theorem 3.4 tells that the ambiguity radius ρ controls the investor's risk preference. Notice that, in the 2 nd order reformulation of ( 2), the optimal Lagrangian multiplier η * 2 = 2ρφ (2) (1)

E[X 2 ]

, which increases simultaneously with ρ. This suggests, when ρ is small, the optimal Lagrangian multiplier η * 2 is also small and the expansion in equation ( 6) is accurate. Later theoretical analysis over the DRO problem would be based on the 2 nd order reformulation in Eq. ( 9). As stated before, we consider 9), where we denote E[r] = µ and Cov(r) = Σ.

f (x, r) = x T r in the sequel such that E[f (x, r)] is replaced by x T µ and E[X 2 ] is replaced by x T Σx in Eq. (

3.2. REFORMULATION OF (3):

Note that when f (x, r) = x T r, the chance constraint in (3) is of the same form as the definition of VaR, which focuses on the probability of losses. This motivates us to reorganize the tail chance constraint in (3) with VaR. The VaR is defined as the minimal level γ such that the probability that the portfolio loss -x T r exceeds γ is below , i.e., V (x) := inf{γ ∈ R :

P 0 {-x T r ≥ γ} ≤ }.

With the definition of V (x), we have the equivalent form of chance constraint P 0 {-x T r ≥ δ} ≤ as V (x) ≤ δ. Hence, (3) can be reformulated as

max x∈X x T µ s.t. V (x) ≤ δ. If P 0 is a normal distribution, then V (x) = κ( ) √ x T Σx -x T µ, where κ( ) = -Φ -1 ( ).

For general elliptical distribution, Lesniewski et al. (2016) gives an asymptotic expansion of V (x), which takes the form κ( )

√

x T Σx -x T µ asymptotically when → 0. For example, if P 0 is student t distribution with degree of freedom parameter ν, then κ(

) = D -1 ν , where D = c d π d-1 2 Γ( ν+1 2 ) νΓ( ν+d 2 ) 1 ν and c d = Γ( ν+d 2 ) Γ( ν 2 ) ν ν 2 π -d 2 .

For distributions other than elliptical distributions, Ghaoui et al. (2003) suggests an approximation of κ( ): κ( ) = 1-. This suggests that (3) can be reformulated as a second-order cone optimization problem:

EQUATION

)

3.3. SOLUTIONS:

It has been shown that ( 2) and ( 3) can be reformulated to a deterministic mean-deviation problem in (9) and a linear optimization with a second-order cone constraint in (10), respectively. In this section, we look into the optimal solution and the optimal value to optimizations ( 9) and (10) to interpret the impact of the ambiguity radius ρ. We would denote the optimal solution and the optimal value to optimization (9) by x * and v * , respectively. Correspondingly, the optimal solution and the optimal value to optimization (10) are denoted by x * and ṽ * . We begin with the unbounded feasibility set X := x ∈ R n | x T e = 1 . By examining the convexity of the function a √ x T Σx -x T µ, we know that problem ( 9) is a convex optimization and problem ( 10) is a convex optimization only when κ( ) > 0. Recall that in a convex optimization, any local optimum is certainly a global optimum. This motivates us to study the optimal solution to problem (9), x * , and the optimal solution to problem (10), x * , through KKT conditions. The results for (x * , v * ) and ( x * , ṽ * ) are summarized in Theorem 3.5 and Theorem 3.6, respectively. Theorem 3.5. Suppose φ (2) (1) > 0. Define A := e T Σ -1 e, B := µ T Σ -1 e, and C := µ T Σ -1 µ. Then for optimization (9) with the feasibility set X := x ∈ R n | x T e = 1 , we have:

• If ρ > φ (2) (1)(C -B 2

A )/2, then the optimal solution and optimal value for problem (9) are

x * = Σ -1 (µ -λ * e) µ T Σ -1 e -λ * e T Σ -1 e , v * = λ * = B/A -B 2 -A C -2ρ/φ (2) (1) /A. • If ρ ≤ φ (2) (1)(C -B 2 A )/2

, then no local optimal solution for problem (9) and the optimal value v * = +∞.

Notice that, for ( 2) with an unbounded feasibility set, the optimal value is +∞ when there is no distributional uncertainty (equivalently, ρ = 0). Thus, Theorem 3.5 establishes a threshold for the ambiguity radius ρ, beyond which the distributional uncertainty is indeed effective. Theorem 3.6. Given A, B, and C defined in Theorem 3.5. Suppose κ( ) > 0, and then for optimization (10) with the feasibility set X := x ∈ R n | x T e = 1 , the following conclusions hold:

• If ( , δ) satisfies C -B 2 /A < (κ( )) 2 < δ 2 A + 2δB + C and B + δA > 0,

then the optimal solution and optimal value are:

x * = Σ -1 [(1 + λ)µ -θe] e T Σ -1 [(1 + λ)µ -θe]

, ṽ * = λδ + θ,

where λ = (AC -B 2 )/(Aκ( ) 2 -AC + B 2 ) + (κ( )(B + Aδ) √ AC -B 2 )/((Aκ( ) 2 - AC + B 2 ) Aδ 2 + 2Bδ + C -κ( ) 2

) and θ = ((C + δB)( λ + 1) -λκ( ) 2 )/(B + δA). Furthermore, when the optimal value v * for problem (9) is finite, ṽ * ≥ v * .

• If B > 0 and if ( , δ) satisfies C -B 2 /A < (κ( )) 2 < δ 2 A + 2δB + C and B + δA < 0,

then the feasibility set for problem (10) is ∅.

• If ( , δ) satisfies (κ( )) 2 ≤ C -B 2 /A, then there exists no local optimal solution for problem (10) and the optimal value ṽ * = +∞.

In Theorem 3.6, we first provide the sufficient and necessary conditions of ( , δ) for the optimization problem (10) to have a finite optimal value. We further compare the finite optimal values to problem (10) and to problem (9), ṽ * and v * , and find that the CCO reformulation performs uniformly better than the DRO reformulation. In addition, we identify one sufficient condition under which optimization (10) is infeasible. We also identify one sufficient condition for the chance constraint to be redundant in optimization (10), and the optimal value tends to be positive infinity.

4. EXPERIMENTS:

In Section 3, we proved that the CCO reformulation performs better than the DRO reformulation when the optimal values are both finite for an unbounded strategy set. For a bounded strategy set, SOCP problem with boundedness is generally explicitly unsolvable. So we resort to numerical analysis. In Section 4.1, we numerically test the reformulation accuracy of optimization ( 9) to (2) . In Section 4.2, we conduct experiments to understand the ambiguity radius ρ via the chance constraints and find that the tail heaviness indeed affects the interpretation of ρ. In Section 4.3, we test the interpretation of the ambiguity radius based on empirical data.

4.1 REFORMULATION ACCURACY OF (2) In this section, we numerically test the reformulation accuracies of the 2 nd order and 4 th order reformulations with respect to (2) . The φ-divergence we take includes both KL divergence and Cressie-Read divergence. It is known that under KL divergence, ( 2) can be exactly solved. Under Cressie-Read divergence, we take the optimal value solved by the Robust Monte Carlo method introduced in Glasserman & Xu (2014) as the benchmark to compare the results.

Table 2 records under KL divergence, the relative errors (in the 3 rd & 4 th rows) w.r.t. the exact optimal value (the 2 nd row). It is demonstrated that the higher order improvement is particularly notable when data exhibits a heavier tail. In the case of Cressie-Read divergence, which we do not record in the table due to the page limit, we observe a 50 times improvement: when ρ is set to 0.78, relative error for the 4 th order reformulation is 1.53%, while it is 56.54% for the 2 nd order reformulation given that the optimal value is -0.2787. Here, we assume that the ambiguity set under the KL divergence centers at a six-dimensional multivariate exponential distribution with mean=0.2, std=0.2, skewness= 2, and kurtosis= 6. We set the dimensions to be i.i.d to see a clear impact from the heavy tail. And the center distribution P 0 under Cressie-Read divergence is multivariate t. We see that the larger the size of the ambiguity set, i.e., larger ρ, the better the improvement of the 4 th order reformulation. In fact, the error reduction is about 10 folds in this example. However, using the 2 nd order equivalent formulation is good enough to study relations between ( 2) and (3) when ρ is small. This experiment shows that the tail heaviness of the nominal distribution P 0 indeed affects the interpretation of the ambiguity radius ρ. We focus on three distribution of 5 assets: multivariate normal/lognormal distribution and student t 3 -distribution. The set of allocation strategies is bounded below by -1, and the ambiguity radius ρ is fixed at 0.27. We say that the ambiguity radius ρ can be explained by a chance constraint with parameters ( , δ) if the optimal value of problem ( 9) is equal to that of problem (10) under the same distribution P 0 . We plot the results of equivalent ( , δ) in Figure 1 . It shows that, first, the ambiguity radius ρ can be explained by a set of pairs ( , δ) in terms of the impact on the optimal value. Second, tail heaviness affects the interpretation of ρ and distribution with heavier tail result in a larger loss threshold for a given loss probability .

ǫ(%) 

4.3. EMPIRICAL STUDIES:

To see more clearly the financial interpretation of the ambiguity radius ρ, we undergo experiments based on empirical data. We extract past 40 years' daily simple returns of four major asset classes: Equity indexes (DAX, FTSE, HSI, NASDAQ, NIKKEI250, SP500) , US Treasuries (2year, 10year, 30year), Currencies (AUD/USD, CHF/USD, EUR/USD, GBP/USD, JPY/USD), and Commodities (Crude oil, Silver, Gold). For the DRO problem, we use the Cressie-Read divergence instead of KL divergence since all data exhibits quite heavy tail. For the CCO problem, we choose the negative daily return threshold -δ to be the 1%, 3% and 5% empirical quantile of the daily simply return series for each asset class so that they can differ across assets. We choose the chance level ranging from 1% to 20%. The results are given in the section A.1 such that Table 4 , Table 5 , Table 6 and Table 7 corresponds to asset class equity indexes, US Treasuries, currencies and commodities accordingly.

In order to study the relation between ρ in (2) as well as δ and in (3) on the four assets classes at the same time, we report the equivalent ambiguity radius ρ of the DRO problem, together with the corresponding optimal portfolio return (annualized), at a given pair of the CCO parameters ( , δ) for the four asset classes simultaneously in Table 3 . In the table, we extract the results when δ corresponds to 3% and to be 2% and 5%, which mincing (rounded) event frequencies at quarterly (4 out of 252) and monthly (12 out of 252) so that investors can relate to the degree of event rareness. The portfolio weights are limited to be bounded below by -1. Both multivariate tand normal distribution are tested as the center P 0 of the ambiguity set U when fitting data. Also, we test the 4 th order and 2 nd order reformations of the DRO problem.

We read from Table 3 that, by relating the size of ambiguity set ρ in the DRO problem to the CCO chance parameters, it then becomes tangible, without which even the appropriate order is hard to guess. In our tests, its magnitude can range from 10 -3 to 10 -14 depending on asset classes and the investor's tolerance level. What's more, the heavy tail nature of financial data demands the use of divergence measures such as the Cressie-Read divergence, which allows heavy tail distributions if one takes the robust approach for portfolio optimization. Ambiguity sets constructed by the KL divergence, however, require the objective function to be exponentially bounded, which exclude important heavy tail distribution used ubiquitously for financial asset returns, e.g., the student tdistribution. Among the 16 tests in Table 3 , the larger returns in bold show 12 favor fitting data by assuming that P 0 is multivariate t-distributed.

Table 3 : Equivalent ambiguity radius ρ and optimal annualized return R at a given pair of ( , δ).

ρ t , ρ n R t , R n

Equity: -δ = -3.35% Bond: -δ = -6.58% FX: -δ = -1.40% Commodity: -δ = -4.4%

4 th order 2 nd order 4 th order 2 nd order 4 th order 2 nd order 4 th order 2 nd order = 2%

3.5e-4, 1.2e-4 6.1e-4, 1.2e-4 2e-6, 2.8e-14 9.5e-6, 2.8e-14 2.6e-4, 6.1e-5 3.1e-4, 6.1e-5 9.6e-5, 3.7e-9 1.5e-4, 3.7e-9 30.7%, 15.3% 30.7%, 15.3% -1.1%, -2.6% -1.1%, -2.6% 2.3%, 3.6% 2.3%, 3.6% 17.3%, 4.6% 17.3%, 4.6% = 5%

3.4e-4, 1.2e-4 6.1e-4, 1.2e-4 2e-6, 2.8e-14 4.8e-6, 2.8e-14 1.5e-4, 3.1e-5 3.1e-4, 3.1e-5 6.5e-5, 1.9e-9 7.6e-4, 1.9e-9 39.2%, 19.8% 39.2%, 19.8% 0.7%, -2.6% 0.7%, -2.6% 4.4%, 5.0% 4.4%, 5.0% 22.7%, 4.6% 22.6%, 4.6%

5. CONCLUSIONS:

We delved into the ambiguity radius for DRO problems with a distribution ambiguity set regulated by φ-divergence. We showed that for general φ-divergences, a DRO portfolio optimization problem is asymptotically equivalent to a mean-deviation problem, where the ambiguity radius controls an investor's risk preference parameter. Theoretical analysis over the mean-deviation problem sets a threshold for the ambiguity radius, across which the optimal value suffers from a drastic phase transition. It is only beyond that radius threshold can the distributional uncertainty take effect. We also showed both numerically and theoretically that, when the investment strategy is bounded, the ambiguity radius can be cast as a chance constraint in a deterministic optimization with the same objective. Otherwise, within the set of unbounded investment strategies, a chance-constrained deterministic optimization consistently performs better than the DRO problem. 

A APPENDIX:

A.1 NUMERICAL RESULTS

In this section, we give numerical results on ρ and annualized return R under different δ and using 4 th and 2 nd order of formulation of the DRO problem for the four major asset classes. Two types of distribution, multivariate distribution t and multivariate normal distribution are considered.

Table 4 : (Equity Indexes) Equivalent ambiguity radius ρ and optimal annualized return R at a given pair of ( , δ). '-' means that there is no equivalent ρ for the given pair of ( , δ). Bond: -δ = -5.18%

ρ t , ρ n R t ,

Bond: -δ = -6.58% Bond: -δ = -9.98% 4 th order 2 nd order 4 th order 2 nd order 4 th order 2 nd order =1% 1.0e-10 ,1.5e-08 Inf ,1.5e-08 3.0e-06 ,5.8e-11 1.9e-05 ,5.8e-11 2.0e-06 ,5.7e-14 9.5e-06 ,5.7e-14 Inf ,-2.7%

Inf ,-2.7% -2.09% ,-2.58% -2.09% ,-2.58% -1.09% ,-2.57% -1.09% ,-2.57% =2% 3.0e-06 ,1.9e-09 1.9e-05 ,1.9e-09 2.0e-06 ,2.8e-14 9.5e-06 ,2.8e-14 2.0e-06 ,2.8e-14 4.8e-06 ,2.8e-14 -1.65% ,-2.63% -1.65% ,-2.63% -1.07% ,-2.57% -1.07% ,-2.57% 0.143% ,-2.57% 0.143% ,-2.57% =3% 2.0e-06 ,2.9e-11 9.5e-06 ,2.9e-11 2.0e-06 ,2.8e-14 9.5e-06 ,2.8e-14 1.0e-08 ,2.8e-14 2.4e-06 ,2.8e-14 -1.05% ,-2.58% -1.05% ,-2.58% -0.393% ,-2.57% -0.393% ,-2.57% 1.1% ,-2.57% 1.1% ,-2.57% =4% 2.0e-06 ,2.8e-14 9.5e-06 ,2.8e-14 2.0e-06 ,2.8e-14 4.8e-06 ,2.8e-14 1.0e-08 ,2.8e-14 1.2e-06 ,2.8e-14 -0.58% ,-2.57% -0.58% ,-2.57% 0.174% ,-2.57% 0.174% ,-2.57% 1.93% ,-2.57% 1.93% ,-2.57% =5% 2.0e-06 ,2.8e-14 4.8e-06 ,2.8e-14 2.0e-06 ,2.8e-14 4.8e-06 ,2.8e-14 1.0e-08 ,2.3e-10 1.5e-07 ,2.3e-10 -0.17% ,-2.57% -0.17% ,-2.57% 0.677% ,-2.57% 0.677% ,-2.57% 2.68% ,-2.59% 2.68% ,-2.59% =6% 2.0e-06 ,5.7e-14 4.8e-06 ,5.7e-14 1.0e-08 ,2.8e-14 2.4e-06 ,2.8e-14 1.0e-08 ,4.7e-10 3.6e-14 ,4.7e-10 0.201% ,-2.57% 0.201% ,-2.57% 1.14% ,-2.57% 1.14% ,-2.57% 3.17% ,-2.6% 3.17% ,-2.6% =7% 2.0e-06 ,2.8e-14 4.8e-06 ,2.8e-14 1.0e-08 ,2.8e-14 1.2e-06 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 0.545% ,-2.57% 0.545% ,-2.57% 1.57% ,-2.57% 1.57% ,-2.57% 3.17% ,-2.6% 3.17% ,-2.6% =8% 1.0e-08 ,2.8e-14 2.4e-06 ,2.8e-14 1.0e-08 ,2.8e-14 6.0e-07 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 0.868% ,-2.57% 0.868% ,-2.57% 1.98% ,-2.57% 1.98% ,-2.57% 3.17% ,-2.61% 3.17% ,-2.61% =9% 1.0e-08 ,1.9e-09 2.4e-06 ,1.9e-09 1.0e-08 ,2.8e-14 3.0e-07 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.17% ,-2.62% 1.17% ,-2.62% 2.36% ,-2.57% 2.36% ,-2.57% 3.17% ,-2.61% 3.17% ,-2.61% =10% 1.0e-08 ,4.7e-10 2.4e-06 ,4.7e-10 1.0e-08 ,1.9e-09 1.5e-07 ,1.9e-09 1.0e-08 ,1.9e-09 3.6e-14 ,1.9e-09 1.47% ,-2.6% 1.47% ,-2.6% 2.74% ,-2.62% 2.74% ,-2.62% 3.17% ,-2.62% 3.17% ,-2.62% =11% 1.0e-08 ,2.3e-10 1.2e-06 ,2.3e-10 1.0e-08 ,9.3e-10 2.3e-09 ,9.3e-10 1.0e-08 ,1.9e-09 3.6e-14 ,1.9e-09 1.75% ,-2.59% 1.75% ,-2.59% 3.09% ,-2.61% 3.09% ,-2.61% 3.17% ,-2.63% 3.17% ,-2.63% =12% 1.0e-08 ,2.8e-14 6.0e-07 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 2.02% ,-2.57% 2.02% ,-2.57% 3.17% ,-2.61% 3.17% ,-2.61% 3.17% ,-2.63% 3.17% ,-2.63% =13% 1.0e-08 ,2.8e-14 6.0e-07 ,2.8e-14 1.0e-08 ,9.3e-10 7.1e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 2.28% ,-2.57% 2.28% ,-2.57% 3.17% ,-2.6% 3.17% ,-2.6% 3.17% ,-2.63% 3.17% ,-2.63% =14% 1.0e-08 ,2.8e-14 3.0e-07 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 2.54% ,-2.57% 2.54% ,-2.57% 3.17% ,-2.6% 3.17% ,-2.6% 3.17% ,-2.64% 3.17% ,-2.64% =15% 1.0e-08 ,2.8e-14 7.5e-08 ,2.8e-14 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 2.78% ,-2.57% 2.78% ,-2.57% 3.17% ,-2.61% 3.17% ,-2.61% 3.17% ,-2.64% 3.17% ,-2.64% =16% 1.0e-08 ,2.3e-10 9.3e-09 ,2.3e-10 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 3.03% ,-2.59% 3.03% ,-2.59% 3.17% ,-2.61% 3.17% ,-2.61% 3.17% ,-2.64% 3.17% ,-2.64% =17% 1.0e-08 ,2.3e-10 3.6e-14 ,2.3e-10 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 3.17% ,-2.59% 3.17% ,-2.59% 3.17% ,-2.61% 3.17% ,-2.61% 3.17% ,-2.64% 3.17% ,-2.64% =18% 1.0e-08 ,4.7e-10 3.6e-14 ,4.7e-10 1.0e-08 ,1.9e-09 3.6e-14 ,1.9e-09 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 3.17% ,-2.6% 3.17% ,-2.6% 3.17% ,-2.62% 3.17% ,-2.62% 3.17% ,-2.64% 3.17% ,-2.64% =19% 1.0e-08 ,4.7e-10 7.1e-14 ,4.7e-10 1.0e-08 ,1.9e-09 3.6e-14 ,1.9e-09 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 3.17% ,-2.6% 3.17% ,-2.6% 3.17% ,-2.62% 3.17% ,-2.62% 3.17% ,-2.64% 3.17% ,-2.64% =20% 1.0e-08 ,9.3e-10 3.6e-14 ,9.3e-10 1.0e-08 ,1.9e-09 3.6e-14 ,1.9e-09 1.0e-08 ,3.7e-09 3.6e-14 ,3.7e-09 3.17% ,-2.61% 3.17% ,-2.61% 3.17% ,-2.63% 3.17% ,-2.63% 3.17% ,-2.64% 3.17% ,-2.64%

Table 6 : (Currencies) Equivalent ambiguity radius ρ and optimal annualized return R at a given pair of ( , δ).

ρ t , ρ n R t , R n FX: -δ = -1.16% FX: -δ = -1.40% FX: -δ = -1.89%

4 th order 2 nd order 4 th order 2 nd order 4 th order 2 nd order =1%

Inf ,1.2e-04 Inf ,1.2e-04 3.5e-04 ,6.1e-05 6.1e-04 ,6.1e-05 2.1e-04 ,3.1e-05 3.1e-04 ,3.1e-05 -Inf ,1.76% -Inf ,1.76% 0.17% ,2.89% 0.17% ,2.89% 3.21% ,4.68% 3.21% ,4.68% =2%

3.5e-04 ,6.1e-05 6.1e-04 ,1.2e-04 2.6e-04 ,6.1e-05 3.1e-04 ,6.1e-05 1.4e-04 ,3.1e-05 1.5e-04 ,3.1e-05 0.21% ,2.53% 0.21% ,2.53% 2.28% ,3.61% 2.28% ,3.61% 4.85% ,5.47% 4.85% ,5.47% =3% 3.0e-04 ,6.1e-05 6.1e-04 ,6.1e-05 2.1e-04 ,6.1e-05 3.1e-04 ,6.1e-05 1.0e-04 ,3.1e-05 1.5e-04 ,3.1e-05 1.59% ,3.04% 1.59% ,3.04% 3.23% ,4.14% 3.23% ,4.14% 5.87% ,6.04% Applying changing of measure, it is equivalent to solving

min L E [f (x, r)L] s.t. E [φ(L)] ≤ ρ and E [L] = 1. (A.1)

Here, L is the likelihood ratio. The Lagrangian function is

l(L, η 1 , η 2 ) = E [f (x, r)L] + η 2 (E [φ(L)] -ρ) + η 1 (E [L] -1).

Under mild conditions such that strong duality holds, Eq. (A.1) is equivalent to solving its dual max η1∈R,η2≥0 min L l(L, η 1 , η 2 ), which is

max η1∈R,η2≥0 min L {E [f (x, r)L] + η 2 (E [φ(L)] -ρ) + η 1 (E [L] -1)} . (A.2)

Eq. (A.2) allows us to eliminate L. Now,

max η1∈R,η2≥0 min L {E [f (x, r)L] + η 2 (E [φ(L)] -ρ) + η 1 (E [L] -1)} = max η1∈R,η2≥0 -η 2 ρ -η 1 -η 2 max L E - (f (x, r) + η 1 ) η 2 L -E [φ(L)] .

Suppose that the maximization sign 'max' and the expectation sign E can be interchanged. Making use of the definition of conjugate φ * (•), we have

max η1∈R,η2≥0 -η 2 ρ -η 1 -η 2 E max L - (f (x, r) + η 1 ) η 2 L -φ(L) = max η1∈R,η2≥0 -η 2 ρ -η 1 -η 2 E φ * - (f (x, r) + η 1 ) η 2 = -min η2≥0 ρ η 2 + min η1∈R η 1 + 1 η 2 E [φ * (-η 2 (f (x, r) + η 1 ))] = -min η2≥0 ρ η 2 + D η2,φ,P0 (f (x, r)|E [f (x, r)]) -E [f (x, r)] =E [f (x, r)] -min η2≥0 ρ η 2 + D η2,φ,P0 (f (x, r)|E [f (x, r)]) .

The second last equality holds, since if we let η1 = η 1 -E [f (x, r)], we have

D η2,φ,P0 (f (x, r)|E[f (x, r)]) := min η1 η 1 + 1 η 2 E [φ * (η 2 (E[f (x, r)] -f (x, r) -η 1 ))] = min η1 η1 + 1 η 2 E [φ * (η 2 (-f (x, r) -η1 ))] + E [f (x, r)] .

The second last equality holds by letting η1 = η 1 -E [f (x, r)]. Lastly, we apply Result A.2 which guarantees that the maximization sign 'max' and the expectation sign E can be interchanged (see for example, Ben-Tal & Teboulle (2007) and Hu et al. (2013) ). The proof is completed. Followed by Theorem 3.1, we prove Lemma 3.2. What we are going to do is to express the quantity D η2,φ,P (f (x, r)|E [f (x, r)]) as a series.

Proof of Lemma 3.2. Denoting X = f (x, r) -E [f (x, r)], we recall that

D η2,φ,P0 (f (x, r)|E [f (x, r)]) = min η1 η 1 + 1 η 2 E [φ * (η 2 (E [f (x, r)] -f (x, r) -η 1 ))] = min η1 η 1 + 1 η 2 E [φ * (η 2 (-X -η 1 ))] .

Applying Taylor expansion on φ * (•) around 0 under the assumption that E X k exists for all k ≤ n, the expansion of φ * (η 2 (-X -η 1 )) up to order n is:

n k=2 η k-1 2 (-1) k (φ * ) (k) (0) k! E (X + η 1 ) k = n-1 k=1 η k 2 (-1) k+1 (φ * ) (k+1) (0) (k + 1)! E (X + η 1 ) k+1 .

It remains to compute (φ * ) (k+1) (ξ). We start from the definition of φ * (ξ) = sup z≥0 {zξ -φ(z)}.

Let z(ξ) be the optimizer, then we obtain φ * (ξ) = z(ξ)ξ -φ(z(ξ)). Differentiating it gives

(φ * ) (1) (ξ) = z (1) (ξ)ξ + z(ξ) -φ (1) (z(ξ))z (1) (ξ) = z(ξ).

The last equality follows from the first order optimality condition, which gives that ξ -φ (1) (z(ξ)) = 0. As a result, (φ * ) (k+1) (ξ) = z (k) (ξ) and it suffices to find z (k) (ξ).

The term z (k) (ξ) can be found from the first order optimality condition. If we differentiate it on both sides w.r.t. ξ, we obtain φ (2) (z(ξ))z (1) (ξ) = 1. This means that z (1) (ξ) = 1 φ (2) (z(ξ))

and hence z (k) 

(ξ) = 1 φ (2) (z(ξ)) (k-1)

for k ≥ 2. Denoting η * 1 be the optimal solution to min

η1 n-1 k=1 b k E (X + η 1 ) k+1 η k 2 .

We then show that z(0) = 1. From the definition of φ * (ξ), we know that φ * (0) = 0 so that φ(z(0)) = 0. If z(0) := a = 1, then φ(x) = 0 for any x lying between 1 and a according to the definition of φ. This means that φ (n) (x) = 0 for any n ∈ N and x is a point lying between 1 and a.

Note that φ (2) (1) = 0 and φ is an ∞-differentiable function. Ee can find a point x lying in between 1 and a (indeed, close to 1) such that φ (2) (x) = 0, so an contradiction arises. The proof of Lemma is completed.

We move to the proof of Corollary 3.3:

Proof of Corollary 3.3. We prove for the case when φ-divergence is chosen as Cressie Read divergence with θ > 2. The case when φ-divergence is chosen as KL divergence is similar and is omitted. First, we compute the first four derivatives of φ-divergence when it is a Cressie Read divergence, which gives

φ (1) (t) = 1 -t θ-1 1 -θ , φ (2) (t) = t θ-2 , φ (3) (t) = (θ -2)t θ-3 , φ (4) (t) = (θ -2)(θ -3)t θ-4 .

In the proof of Lemma 3.2, we know that (φ * ) (k) (•) can be obtained from z (1) (•), z (2) (•) and z (3) (•).

We omit detailed deviations, and we can obtain z

(1) (0) = 1, z (2) (0) = -(θ -2) and z (3) (0) = (θ -2)(2θ -3). Now, assuming X = f (x, r) -E [f (x, r)], this means that D η2,φ,P0 (f (x, r)|E [f (x, r)]) = min η1    η 2 E (X + η 1 ) 2 2! + η 2 2 (θ -2)E (X + η 1 ) 3 3! + η 3 2 (θ -2)(2θ -3)E (X + η 1 ) 4 4!    . (A.3)

We need to show that the local optimizer η 1 of Eq. (A.3) is a root of a cubic equation. Indeed, the objective function is a polynomial of η 1 with degree 4, which can be rewritten as

a 0 + a 1 η 1 + a 2 η 2 1 + a 3 η 3 1 + a 4 η 4 1

, where

a 0 = η 2 E X 2 2 + (θ -2)η 2 2 E X 3 6 + (θ -2)(2θ -3)η 3 2 E X 4 24 , (A.4) a 1 = (θ -2)η 2 2 E 3X 2 6 + (θ -2)(2θ -3)η 3 2 E 4X 3 24 , (A.5) a 2 = η 2 2 + (θ -2)(2θ -3)η 3 2 E 6X 2 24 , (A.6) a 3 = (θ -2)η 2 2 6 , (A.7) a 4 = (θ -2)(2θ -3)η 3 2 24 . (A.8) Since θ > 2, we have (θ -2)(2θ -3) > 0 and so (θ-2)(2θ-3)η 3 2 4! > 0.

We can therefore find a local minimum (global minimum). The local optimal points should satisfy a 1 +2a 2 η 1 +3a 3 η 2 1 +4a 4 η 3 1 = 0. It is not difficult to obtain Eq. (8) from Eqs. ( 6), (A.5), (A.6), (A.7) and (A.8).

It remains to show that the local optimal obtained is indeed a local minimum. This can be done by examining if 2a 2 + 6a 3 η 1 + 12a 4 η 2 1 ≥ 0 for local optimal η 1 . Indeed,

2a 2 + 6a 3 η 1 + 12a 4 η 2 1 =2η 2 (θ -2)(2θ -3) 4 η 2 1 η 2 2 + θ -2 2 η 1 η 2 + 1 2 + (θ -2)(2θ -3)η 2 2 E X 2 4 = 2η 2 4 (θ -2)(2θ -3) η 1 η 2 + 1 2θ -3 2 + 3θ -4 2θ -3 + (θ -2)(2θ -3)η 2 2 E X 2 ≥ 0.

Therefore, we can further simplify D η2,φ,P0 (f (x, r)|E r∼P0 [f (x, r)]) if η 1 exists for a real solution of a 1 + 2a 2 η 1 + 3a 3 η 2 1 + 4a 4 η 3 1 = 0 and can be expressed explicitly. The remaining in the proof shows how to find the explicit formula of the local optimizer η 1 . We examine for a triple polynomial equation. When > 0, we know that the local optimal η 1 can be expressed as u -p 3u -a3 4a4 such that

p = a 2 2a 4 - 3a 2 3 16a 2 4 , q = a 3 3 32a 3 4 - a 2 a 3 8a 2 4 + a 1 4a 4 , = q 2 + 4p 3 27 , u = 3 -q + √ 2 ,

which follows from Cardano formula for a cubic equation. This follows from the fact that p ≥ 0, since

p = a 2 2a 4 - 3a 2 3 16a 2 4 = 1 16a 2 4 8 η 2 2 + (θ -2)(2θ -3)η 3 2 E r∼P 6X 2 24 (θ -2)(2θ -3)η 3 2 24 -3 (θ -2)η 2 2 6 2 = η 4 2 192a 2 4 (θ -2)(3θ -4) + (θ -2) 2 (2θ -3) 2 η 2 2 E X 2 ≥ 0.

This completes the proof.

Before presenting proofs of Theorem 3.5 and 3.6, it is necessary to study any properties of the function f (x) = a √ x T Σx -x T b with a > 0 and a vector b. Indeed, it is a convex function. The proof is tedious and can be completed using Cauchy Schwartz inequality. Now, we are ready to present the proofs of Theorem 3.5 and Theorem 3.6. The proofs are presented sequentially.

Proof of Theorem 3.5. According to the symbols defined and let b = 2ρ φ (2) (1) for notational simplification, Eq. ( 9) is equivalent to

-min x∈X -x T µ + b √ x T Σx ⇔ -min x -x T µ + b √ x T Σx s.t. x T e = 1

.

Clearly, it is a convex optimization problem. Applying Lagrangian multiplier method, let us examine

-µ + bΣx √

x T Σx + λe = 0, (A.9)

x T e = 1. (µ -λe) T Σ -1 (µ -λe) (µ T Σ -1 e -λe T Σ -1 e) 2 µ T Σ -1 e -λe T Σ -1 e + λe = 0, iff √ (µ T Σ -1 e-λe T Σ -1 e) 2 µ T Σ -1 e-λe T Σ -1 e = 1. Note that µ T Σ -1 e -λ + e T Σ -1 e < 0 < µ T Σ -1 e -λ -e T Σ -1 e, so the point µ T Σ -1 e -λ + e T Σ -1 e is rejected. Therefore, there is one local optimal solution and it is at x(λ -) with optimal values µ T Σ -1 (µ-λ-e) µ T Σ -1 e-λ-e T Σ -1 e -b µ T Σ -1 µ-2λ-µ T Σ -1 e+λ 2 -e T Σ -1 e (µ T Σ -1 e-λ-e T Σ -1 e) 2 = λ -, where the last equality follows from Eq. (A.9).

If (µ T Σ -1 e) 2 -(e T Σ -1 e)(µ T Σ -1 µ -b 2 ) < 0, no real roots for λ + and λ -exist. This means that no local optimal solutions exist.

We can draw the same conclusion as in Theorem 3.5 using the same notations in Theorem 3.5.

Finally, we give the proof of Theorem 3.6.

Proof of Theorem 3.6. According to the symbols defined, problem (10) in the main paper is equivalent to 

:



Figures and Tables:

Figure 1: Given ρ = 0.27, tail heaviness affects the equivalent loss threshold δ.

Let Ω be a σ-finite measure space, and let Y := L p (Ω, F, P ), p ∈ [1, +∞]. Let g : R × Ω → (-∞, +∞] be a normal integrand, and define on Y the integral functional I g (y) := Ω g(y(ω), ω)P (dω). Then, inf x∈Y Ω g(y(ω), ω)P (dω) = Ω inf s g(s, ω)P (dω), provided that the left-hand side is finite. Moreover, ȳ ∈ arg min y∈Y I g (y) = ȳ(ω) ∈ arg min s∈R g(s, ω), a.e. on ω ∈ Ω. Proof of Result A.2. The proof can be found in Theorem 14.60 of "Variational analysis (Rockafellar & Wets, 2009)".

(A.10) Let x(λ) = Σ -1 (µ-λe) µ T Σ -1 e-λe T Σ -1 e . From the assumption of λ, λ satisfies e T Σ -1 e • λ 2 -2λµ T Σ -1 e + (µ T Σ -1 µ -b 2 ) = 0for two real roots, where we denote them asλ + = µ T Σ -1 e + (µ T Σ -1 e) 2 -(e T Σ -1 e)(µ T Σ -1 µ -b 2 ) e T Σ -1 e , λ -= µ T Σ -1 e -(µ T Σ -1 e) 2 -(e T Σ -1 e)(µ T Σ -1 µ -b 2 ) e T Σ -1 e . If (µ T Σ -1 e) 2 -(e T Σ -1 e)(µ T Σ -1 µ -b 2 ) > 0, then we have -µ + bΣx(λ)x(λ) T Σx(λ) + λe = -µ + b(µ -λe)

min x∈X {-x T µ} s.t. κ( )√x T Σx -x T µ ≤ δ ⇔ min x {-x T µ} s.t.x T e = 1 κ( )√ x T Σx -x T µ ≤ δ. Σx -x T µ -δ} = 0, (A.13) λ ≥ 0.(A.14)From Eq. (A.11), we know that x is linearly dependent of (1 + λ)µθe.Let x = x( λ, θ) = Σ -1 [(1+ λ)µ-θe] e T Σ -1 [(1+ λ)µ-θe]. What is left is to find λ and θ. From Eq. (A.13), λ = 0, otherwise we obtain θe = µ from Eq. (A.11), which is a contradiction in general. Hence, we only need to consider Σx -x T µ -δ = 0. (A.16)Now, the terms x T µ, Σx, and√x T Σx are:x T µ = µ T Σ -1 [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] , Σx = [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] , √ x T Σx = [(1 + λ)µθe] T Σ -1 [(1 + λ)µθe] |e T Σ -1 [(1 + λ)µθe]| .This suggests that Eq. (A.15)-(1 + λ)µ + λκ( )[(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] |e T Σ -1 [(1 + λ)µθe]| [(1 + λ)µθe] T Σ -1 [(1 + λ)µθe]+ θe = 0, (A.17) and Eq. (A.16)κ( ) [(1 + λ)µθe] T Σ -1 [(1 + λ)µθe] |e T Σ -1 [(1 + λ)µθe]| -µ T Σ -1 [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] -δ = 0 ⇒ κ( ) [(1 + λ)µθe] T Σ -1 [(1 + λ)µθe] |e T Σ -1 [(1 + λ)µθe]| = µ T Σ -1 [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] + δ. (A.18)We obtain the first relation between λ and θ by substituting Eq. (A.18) into Eq. (A.17):λ(κ( )) 2 = µ T Σ -1 [(1 + λ)µθe] + δe T Σ -1 [(1 + λ)µθe]. (A.19)Now, from Eq. (A.17), together with the constraint that x T e = 1, we have the second relation(1 + λ) µ T Σ -1 [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] -θ(e T Σ -1 [(1 + λ)µθe]) 2 = + λκ( ) [(1 + λ)µθe] T Σ -1 [(1 + λ)µθe]|e T Σ -1 [(1 + λ)µθe]|.Squaring the above relation on both sides, we have the reformulations of the left hand side and the right hand side: L.H.S (1 + λ) µ T Σ -1 [(1 + λ)µθe] e T Σ -1 [(1 + λ)µθe] -θ(e T Σ -1 [(1 + λ)µθe]) 2 2 = e T Σ -1 [(1 + λ)µθe] 2 (1 + λ) µ T Σ -1 [(1 + λ)µθe] -θ(e T Σ -1 [(1 + λ)µθe]) 2

The two φ-divergences used in this paper. The KL divergence applies to light-tail distribution, while the Cressie-Read divergence is compatible with heavy tail distribution.

Table content: <table><tr><td>Divergence</td><td>φ(t), t ≥ 0</td><td>φ</td></tr></table>

Relative errors of the 4 th order reformulation and 2 nd order reformulation w.r.t. the optimal value of (2). Ambiguity sets are defined by KL divergence centered at 6-d exponential distribution.

Table content: <table><tr><td>ρ</td><td>0.01</td><td>0.02</td><td>0.03</td><td>0.04</td><td>0.05</td><td>0.06</td><td>0.07</td><td>0.08</td><td>0.09</td></tr><tr><td>vopt1</td><td>0.1887</td><td>0.1841</td><td>0.1807</td><td>0.1778</td><td>0.1753</td><td>0.1730</td><td>0.1710</td><td>0.1691</td><td>0.1673</td></tr><tr><td>4 th order</td><td>0.0002%</td><td>0.0038%</td><td>0.0128 %</td><td>0.0274%</td><td>0.0479%</td><td>0.0748%</td><td>0.1082%</td><td>0.1483%</td><td>0.1951%</td></tr><tr><td>2 nd order</td><td>0.1172%</td><td>0.2397%</td><td>0.3659%</td><td>0.4951%</td><td>0.6270%</td><td>0.7613%</td><td>0.8979%</td><td>1.037%</td><td>1.778%</td></tr><tr><td colspan="3">4.2 INTERPRETATION</td><td/><td/><td/><td/><td/><td/><td/></tr></table>

R Tyrrell Rockafellar and Stan Uryasev. The fundamental risk quadrangle in risk management, optimization and statistical estimation. Surveys in Operations Research and Management Science,

Table content: <table><tr><td>18(1-2):33-53, 2013.</td></tr><tr><td>R Tyrrell Rockafellar and Roger J-B Wets. Variational analysis, volume 317. Springer Science &amp;</td></tr><tr><td>Business Media, 2009.</td></tr><tr><td>Herbert E Scarf. A min-max solution of an inventory problem. Technical report, RAND CORP</td></tr><tr><td>SANTA MONICA CALIF, 1957.</td></tr><tr><td>Shushang Zhu and Masao Fukushima. Worst-case conditional value-at-risk with application to ro-</td></tr><tr><td>bust portfolio management. Operations research, 57(5):1155-1168, 2009.</td></tr><tr><td>Shushang Zhu, Duan Li, and Shouyang Wang. Robust portfolio selection under downside risk</td></tr><tr><td>measures. Quantitative Finance, 9(7):869-885, 2009.</td></tr><tr><td>Steve Zymler, Daniel Kuhn, and Berc ¸Rustem. Distributionally robust joint chance constraints with</td></tr><tr><td>second-order moment information. Mathematical Programming, 137(1-2):167-198, 2013.</td></tr></table>

R n

Table content: <table><tr><td/><td colspan="2">Equity: -δ = -2.68%</td><td colspan="2">Equity: -δ = -3.35%</td><td colspan="2">Equity: -δ = -4.64%</td></tr><tr><td/><td>4 th order</td><td>2 nd order</td><td>4 th order</td><td>2 nd order</td><td>4 th order</td><td>2 nd order</td></tr><tr><td>=1%</td><td colspan="6">3.5e-04 ,2.4e-04 1.2e-03 ,2.4e-04 3.5e-04 ,1.2e-04 1.2e-03 ,1.2e-04 3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04</td></tr><tr><td/><td colspan="5">18.8% ,9.76% 18.8% ,9.76% 25.2% ,13.2% 25.2% ,13.2% 35% ,19.3%</td><td>35% ,19.3%</td></tr><tr><td>=2%</td><td colspan="6">3.5e-04 ,1.2e-04 1.2e-03 ,1.2e-04 3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 3.0e-04 ,6.1e-05 6.1e-04 ,6.1e-05</td></tr><tr><td/><td colspan="5">24.4% ,11.7% 24.4% ,11.7% 30.7% ,15.3% 30.7% ,15.3% 42% ,22.1%</td><td>42% ,22.1%</td></tr><tr><td>=3%</td><td colspan="6">3.5e-04 ,1.2e-04 1.2e-03 ,1.2e-04 3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 2.3e-04 ,6.1e-05 3.1e-04 ,6.1e-05</td></tr><tr><td/><td colspan="3">27.5% ,13.1% 27.5% ,13.1% 34.2% ,17%</td><td colspan="3">34.2% ,17% 46.9% ,24.4% 46.9% ,24.4%</td></tr><tr><td>=4%</td><td colspan="4">3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04</td><td>1.9e-04 ,-</td><td>3.1e-04 ,6.1e-05</td></tr><tr><td/><td colspan="3">29.8% ,14.3% 29.8% ,14.3% 37% ,18.5%</td><td colspan="3">37% ,18.5% 50.6% ,26.4% 50.6% ,26.4%</td></tr><tr><td>=5%</td><td colspan="4">3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 3.4e-04 ,1.2e-04 6.1e-04 ,1.2e-04</td><td>1.6e-04 ,-</td><td>3.1e-04 ,3.1e-05</td></tr><tr><td/><td colspan="6">31.7% ,15.4% 31.7% ,15.4% 39.2% ,19.8% 39.2% ,19.8% 53.7% ,28.3% 53.7% ,28.3%</td></tr><tr><td>=6%</td><td colspan="4">3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 3.1e-04 ,6.1e-05 6.1e-04 ,6.1e-05</td><td>1.3e-04 ,-</td><td>1.5e-04 ,3.1e-05</td></tr><tr><td/><td colspan="6">33.3% ,16.5% 33.3% ,16.5% 41.2% ,21.1% 41.2% ,21.1% 56.4% ,30.1% 56.4% ,30.1%</td></tr><tr><td>=7%</td><td colspan="4">3.5e-04 ,1.2e-04 6.1e-04 ,1.2e-04 2.9e-04 ,6.1e-05 3.1e-04 ,6.1e-05</td><td>1.1e-04 ,-</td><td>1.5e-04 ,3.1e-05</td></tr><tr><td/><td colspan="3">34.7% ,17.5% 34.7% ,17.5% 43% ,22.4%</td><td colspan="3">43% ,22.4% 58.7% ,31.8% 58.7% ,31.8%</td></tr><tr><td>=8%</td><td/><td/><td/><td/><td/></tr></table>

(US Treasuries) Equivalent ambiguity radius ρ and optimal annualized return R at a given pair of ( , δ).

Table content: <table><tr><td>ρ t , ρ n</td></tr><tr><td>R t , R n</td></tr></table>

Proof of Theorem 3.1. We need to check that the quantity D η2,φ,P0 (f (x, r)|E[f (x, r)]) defined in Theorem 3.1 satisfies the conditions stated in definition A.1. It is easy and is thus omitted.Write minP ∈U E r∼P [f (x, r)] as min P E r∼P [f (x, r)] s.t. D(P ||P 0 ) ≤ ρ.

Table content: <table><tr><td/><td/><td>5.87% ,6.04%</td></tr><tr><td>=4%</td><td colspan="2">2.6e-04 ,6.1e-05 3.1e-04 ,6.1e-05 1.8e-04 ,3.1e-05 3.1e-04 ,6.1e-05 8.3e-05 ,1.5e-05 1.5e-04 ,1.5e-05</td></tr><tr><td/><td>2.31% ,3.46% 2.31% ,3.46% 3.9% ,4.59%</td><td>3.9% ,4.59% 6.66% ,6.52% 6.66% ,6.52%</td></tr></table>

Peer review (iclr2020):
This paper studies the relationship between Distributionally Robust Optimisation (DRO) and Chance-Constrained Optimisation (CCO). In particular, the paper finds that with a particular cost function and a series of approximations applied to both DRO and CCO, the two optimisation problems start to resemble each other. This enables the authors to relate the hyperparameters of CCO and DRO. The paper then concludes by a series of toy and real-world experiments demonstrating this relationship. I am currently leaning towards rejection. The main reasons are: (1) The paper does not seem to be a good fit with the aims of the conference. Instead of using standard statistical/ML terminology, it seems to be written for an audience with quantitative finance background. While some of the employed techniques could be of interest to the broader ML community, I am afraid most readers of the ICLR proceedings will not appreciate these insights due to the unusual terminology and framing of this paper which obfuscates the underlying theory. Relatedly, the authors claim that the main benefit of their discovered relationship between DRO and CCO is that it “opens a door for financial practitioners to interpret the radius parameter [of DRO] through parameters [of CCO] that are directly linked to investment performance.” While this statement may be appreciated by readers of a quantitative finance journal, it only left me wondering why not use CCO directly and not bother with the DRO relationship, if the “investment performance” interpretation is already embedded within CCO?! (2) I am not sure whether the amount of the contributions relevant to the ICLR community goes far enough beyond those made in the cited Gotoh et al. (2018) paper. (3) The paper is difficult to read with a significant number of grammatical errors (especially within the first two pages) and in contrast to its relative preciseness within the proofs section in the appendix, contains a noticeably high number of confusing or imprecise statements within the main body. Major comments: - I would have liked to see more discussion of the o(eta_2^(n - 1)) from Lemma 3.2 and of the kappa(epsilon) approximations from sect.3.2. From your experiments in sect.4, it seems like the error introduced by the approximations can be quite substantial. If this is true, I am not sure how to interpret the results in sect.3.3. For example, one of the claims that you emphasise is that the approximation of CCO achieves better results than the approximation of DRO when both have finite optimal value (1st bullet point in thm.3.6). However the original formulations of both the DRO and CCO problems are intended to provide an additional set of guarantees about the obtained solution beyond how good is the value they achieve (robustness and low probability of negative return respectively) which AFAICT are not guaranteed to be satisfied by their approximate versions derived in this paper. Can you please clarify whether the approximate versions of DRO and CCO satisfy the original constraints and/or whether you can quantify by how much these can be violated? - On a related note, can you please clarify what do you mean by “optimal value” in sect.3.3? In particular, is it x^T mu for both (9) and (10), or do you take the value of the whole expression inside the curly braces in (9)? - Could you please provide some citations or discuss why you have chosen the particular set-up of experiments in sections 4.1 and 4.2? Minor comments: - In the abstract, you say “Based on this non-robust reformulation, we then show that when a boundedness constraint is added to the investment strategy. The DRO problem can be cast as a chance-constrained optimization (CCO) problem without distributional uncertainties.” Wasn’t this supposed to be a single sentence with a comma in the middle? - Can you please provide citation for the first sentence in the sect.1? Also, in the second sentence, by “true” distribution, do you mean population distribution? - In the 4th sentence in sect.1, I’d recommend replacing “it”; e.g., you can say “However, the estimates are often inadequate due to an insufficient number of available data points.” - Last sentence of par.1 in sect.1, “under a scarcity of data” -> “when data is scarce”. - Can you please provide a citation for the first sentence in par.2 of sect.1? Also can you please reword the very next sentence (particularly the phrase “contains distribution as uncertainty” the meaning of which was unclear to me until sect.3.1)? - In par.2 of sect.1, within the statement starting with “The second way is the moment-based approach ...”, I would suggest moving the words “for instance” before the citations (i.e., “was studied for instance in …”). - There are many more places which are not entirely grammatical. To shorten my review, I will not include further comments related to grammar, and would instead strongly recommend that you proof-read the paper again please, particularly the abstract and sect.1. - In the 2nd sentence of par.3 in sect.1, you say “As a result, optimal solutions become meaningless ...“ while speaking about the moment-based and geometric approaches. Can you provide citation please? While I generally agree with the sentiment that we would like to ensure that the true distribution is within the ambiguity set with high probability, this is AFAIK not guaranteed even by the statistical approaches without further assumptions on the underlying population distribution (which allow to derive an appropriate setting of the diameter rho). - By the end of the introduction, the reader still did not learn what you mean by the “non-robust interpretation”. Please consider either explaining this within sect.1 or removing all mentions within sect.1. - In the “Notations” paragraph within sect.2, does “nominal probability distribution” mean the same as what is commonly called the “empirical distribution”, or rather just any distribution (not necessarily the empirical one) that you will use as the center of the ambiguity set? - In the “phi-divergence” paragraph within sect.2, please consider restating the expressions which include division by zero in terms of limits. Also, for the benefit of the reader, I would add that for the Radon-Nikodym derivative to exist, Q must be absolutely continuous w.r.t. P, and that if it exists then Q(A) = int_A dQ/dP dP for any measurable set A. - Please reconsider use of contractions throughout the paper (e.g., “It’s defined by a convex …” -> “It is defined by a convex …” in the “phi-divergence” paragraph within sect.2). - In eq.1, it is somewhat strange to writhe P(dt) given the variable “t” does not appear anywhere in the integral. Please consider either writing int phi((dP / dQ)(t)) P(dt) or simply int phi(dP / dQ) dP. - On the penultimate line of p.2, you cite Glasserman and Xu (2014) for the name “alpha-divergence”. I find this somewhat strange as alpha-divergences (or Renyi alpha-divergences) date back much further---see [1, 2]. On a related note, some of the other quoted examples like Hellinger distance and xi-squared distance are (up to a scaling factor) members of the alpha-divergence family. - In the “Motivation” section, you say that the CRRA utility function is ln(x^T r) when eta = 1; however, you did not constrain the vectors x and r to always have positive inner product (only constraint is x, r in R^n). Can you clarify please? - At multiple places, you use max and min instead of sup and inf but it is not entirely clear to me the optimum exists in all those cases. Some examples are the definition of the objective max_x E_r [f(x, r)] in sect.2 (with f an arbitrary measurable function), or of max_x min_P instead of sup_x inf_P in eq.2 (which AFAICT allows substitution of any phi-divergence?!). Can you please either switch to sup and inf or provide a justification for using the min and max operators where the existence of an optimum is not obvious? - Below eq.2 in the sentence starting with citation of Pardo (2005), is N supposed to be n? Also, I am assuming phi^{(2)} stands for the second derivative of phi (?), which should be probably better to state explicitly. Also, AFAICT you have only assumed “phi” is convex which implies differentiable almost everywhere, but does not imply phi is twice differentiable. Is this discussion restricted to the two choices for the phi function from tab.1? Please clarify. - Also in the paragraph below eq.2, you use the term “center distribution” likely to what you were referring to before as the “nominal distribution”. Perhaps it would make it easier for the reader if you chose to stick with single name?! - Please consider explicitly stating delta > 0, epsilon > 0 (or similar according to the intended use) somewhere around eq.3. - In the paragraph under eq.3, I cannot see where the abbreviation “VaR” was introduced?! - In the proof of thm.3.1, it would have been nicer if you explicitly commented on incorporating the L >= 0 constraint (which you eventually introduce with the convex conjugate of phi), and why dividing by eta_2 is ok in the second unnumbered equation after A.2. A subtler point is that by introducing L, you are implicitly assuming P is absolutely continuous w.r.t. P_0 which is in principle implied by the fact you are taking infinum over a phi-divergence ball of size rho (which only contains distributions that satisfy this condition), but never discussed explicitly. References: [1] A. Rényi, “On measures of entropy and information,” Fourth Berkeley symposium on mathematical statistics and probability, vol. 1, 1961. [2] S. Amari, Differential-Geometrical Methods in Statistics. New York: Springer, 1985.