Good deals and benchmarks in robust portfolio selection

Balbás, Alejandro; Balbás, Beatriz; Balbás, Raquel

doi:10.1016/j.ejor.2015.09.023

Cited by 20 publications

(17 citation statements)

References 30 publications

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“…Accordingly, the decision-maker may prefer to deal with an ambiguous framework in which probabilities are not perfectly known. Although there are several ways to introduce ambiguity, we will follow the approach of Balbás et al, 22,26 since it seems to be quite intuitive and general. Consider a couple of random variables u, U ∈ L ∞ such that 0 ≤ u ≤ 1 ≤ U, the closed interval [u, U] = { ∈ L ∞ ; u ≤ ≤ U}, and the ( L ∞ , L 1 ) −closed and norm-bounded (and therefore (…”

Section: Ambiguous Frameworkmentioning

confidence: 99%

“…‖ Solving (26) under (27) for several values of and k, one has that this optimization problem is unbounded in a neighborhood of 1 − = 90% and k = 8740. In particular, for this level of confidence and this strike the investor purpose is feasible, in the sense that one can buy one future contract, sell one European call and invest the call price (1456.65, # Simultaneous negative prices and risks are consistent with the absence of arbitrage, 26 and ambiguity does not alter this consistency. ‖ Further details are available at the Spanish market website: www.meff.es according to the BS formula) in the riskless asset.…”

Section: Financial Examplementioning

confidence: 99%

“…**IP 1 is the probability measure on the Borel −algebra of (0, ∞) generated by the solution of (27), and (26) has been studied with p = 2. † † Bearing in mind the put/call parity, the proposed strategy may be replicated by the sale of the European put and the investment of 26.65 (put price) in the riskless asset.…”

Section: Example Of Marketingmentioning

confidence: 99%

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V@R representation theorems in ambiguous frameworks

Balbás

Charron

2019

Appl Stoch Models Bus & Ind

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The value at risk (V@R) is a very important risk measure with significant applications in finance (risk management, pricing, hedging, portfolio theory, etc), insurance (premium principles, optimal reinsurance, etc), production, marketing (newsvendor problem), etc. It also plays a critical role in regulation about risk (Basel, Solvency, etc), it is very appreciated by practitioners due to its intuitive interpretation, and it is the unique popular risk measure remaining finite for heavy tailed risks with unbounded expectation. Besides, ambiguous frameworks are becoming more and more usual in applications of risk analysis. Lack of data or committed errors may provoke discrepancies between real probabilities and estimated ones. This paper combines both V@R and ambiguous settings, and a new representation theorem for V@R is given. Consequently, inspired by previous studies dealing with coherent risk measures and their representation, we will give new methods to compute and optimize V@R under ambiguity. This seems to be a relevant finding because the analytical properties of V@R are very weak if one compares with a coherent risk measure. Indeed, V@R is neither continuous nor convex, which makes it very complicated to deal with it in mathematical approaches. Nevertheless, the results of this paper will allow us to transform computation and optimization problems involving V@R into continuous and differentiable problems. KEYWORDSambiguity, heavy tail, representation theorem, value at risk INTRODUCTIONRisk measures beyond the standard deviation are becoming more and more important in many applications of risk analysis. Indeed, most of them can be interpreted in terms of potential capital losses, while the standard deviation does not satisfy this property in many situations. Besides, the standard deviation is not consistent with the second-order stochastic dominance and the standard utility functions in the presence of asymmetries, 1 and this drawback is overcome by other risk measures.Among many other examples, the coherent risk measures 2 and the expectation bounded risk measures 3 are very popular for practitioners and researchers since they have good analytical properties such as subadditivity, convexity, and continuity, making it feasible to deal with them in complex mathematical approaches. Important examples of such measures are the conditional value at risk (CV@R) and the weighted conditional value at risk. 3 With respect to the coherent and expectation bounded risk measures above, the value at risk (V@R) presents several shortcomings. Indeed, it is neither continuous nor convex, making it complex many analytical studies. Nevertheless, V@R 414

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Section: Ambiguous Frameworkmentioning

confidence: 99%

Section: Financial Examplementioning

confidence: 99%

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V@R representation theorems in ambiguous frameworks

Balbás

Charron

2019

Appl Stoch Models Bus & Ind

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“…or the

V a R

risk measure (despite the fact that

V a R

does not satisfy Assumption ). Furthermore, this caveat may also arise if one incorporates ambiguity in the pricing model (i.e.,

I P

is not perfectly known) and deals with robust risk measures (Balbás et al., ). Henceforth, strategies producing the pathology above will be called GD s in this paper.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Good deal indices in asset pricing: actuarial and financial implications

Balbás

Garrido

Okhrati

2017

Int Trans Operational Res

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We integrate into a single optimization problem a risk measure, beyond the variance, and either arbitrage-free real market quotations or financial pricing rules generated by an arbitrage-free stochastic pricing model. A sequence of investment strategies such that the couple (expected return, risk) diverges to (+∞, −∞) will be called a good deal (GD). The existence of such a sequence is equivalent to the existence of an alternative sequence of strategies such that the couple (risk, price) diverges to (−∞, −∞). Moreover, by appropriately adding the riskless asset, every GD may generate a new one only composed of strategies priced at one. We will see that GDs often exist in practice, and the main objective of this paper will be to measure the GD size. The provided GD indices will equal an optimal ratio between both risk and price, and there will exist alternative interpretations of these indices. They also provide the minimum relative (per dollar) price modification that prevents the existence of GDs. Moreover, they will be a crucial instrument to detect those securities or marketed claims that are over-or underpriced. Many classical actuarial and financial optimization problems may generate wrong solutions if the used market quotations or stochastic pricing models do not prevent the existence of GDs. This fact is illustrated in the paper, and we point out how the provided GD indices may be useful to overcome this caveat. Numerical experiments are also included. Keywords: risk measure; compatibility between prices and risks; good deal size measurement; actuarial and financial implications A. Balbás et al. / Intl. Trans. in Op. Res. 26 (2019) are combined in a single problem, one often faces the existence of sequences of investment strategies (good deals or GDs) whose pairs (expected return, risk) diverge to (+∞, −∞). The existence of GDs is equivalent to the existence of alternative sequences of investment strategies whose pairs (risk, price) diverge to (−∞, −∞). This pathological finding has been analyzed in Balbás and Balbás (2009) and Balbás et al. (2016a), where explicit examples of the sequences above have been constructed and their performance empirically tested. The main conclusion was that the divergence of (expected return, risk) to (+∞, −∞) is more theoretical than real, but the performance of the constructed GD was good enough. The GD were collections of options providing much better realized Sharpe ratios than their underlying assets.In this paper, we will deal with a couple (ρ, ) composed of the risk measure ρ and the pricing rule . The pair (ρ, ) will be called noncompatible if it implies the existence of a GD, and the main objective of this paper will be the measurement of the GD size by means of a new index denoted byÑ orÑ(ρ, ). An important precedent in financial theory is the notion of arbitrage. Although the absence of arbitrage always holds in theoretical approaches, real market quotations sometimes reflect the existence of arbitrage. For this reason, some years ago many authors defined severa...

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“…Wang and Cheng [28] considered the robust portfolio selection problem which has a data uncertainty described by the (p, w)-norm in the objective function. Balbás A., Balbás B. and Balbás R. [1] handled portfolio selection problems under risk and ambiguity. Yu X.…”

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confidence: 99%

Portfolio Optimization Under Parameter Uncertainty Using the Risk Aversion Formula

İNAN¹

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

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Abstract. The Markowitz portfolio optimization model has certain di¢ culties in practise since real data are rarely certain. The robust optimization is a recently developed method that is used to overcome the uncertainty situation. The technique has been recently suggested in the portfolio selection problems. In this study, two kinds of portfolio optimization problems are presented: (i) the risk aversion portfolio optimization problem based on the classical Markowitz framework, and (ii) the max-min counterpart problem based on the robust optimization framework. In the application, the two models are performed on a real-world data set obtained from BIST (Borsa Istanbul). Numerical results show that the objective function values of the classical solution and the robust solution are similar to each other. It can be said that the robust model, which works as well as the classical model in the uncertainty situations, can be used instead of the classical model and also that the optimal solution obtained in the uncertainty situation is robust to parameter perturbation.

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Good deals and benchmarks in robust portfolio selection

Cited by 20 publications

References 30 publications

V@R representation theorems in ambiguous frameworks

V@R representation theorems in ambiguous frameworks

Good deal indices in asset pricing: actuarial and financial implications

Portfolio Optimization Under Parameter Uncertainty Using the Risk Aversion Formula

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