CAPM and APT-like models with risk measures

Balbás, Alejandro; Balbás, Beatriz; Balbás, Raquel

doi:10.1016/j.jbankfin.2009.11.013

Cited by 22 publications

(12 citation statements)

References 36 publications

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“…3 In particular, ρ (y 0 ) = 0 for every G-deviation ρ and every y 0 ∈ Lp (μ G , E). Notice that it is sufficient to impose ρ to be (Λ, G)-translation invariant and positively homogeneous.…”

Section: Definition 32mentioning

confidence: 99%

Vector Risk Functions

Balbás

Jiménez-Guerra³

2011

Mediterr. J. Math.

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Abstract. The paper introduces a new notion of vector-valued risk function, a crucial notion in Actuarial and Financial Mathematics. Both deviations and expectation bounded or coherent risk measures are defined and analyzed. The relationships with both scalar and vector risk functions of previous literature are discussed, and it is pointed out that this new approach seems to appropriately integrate several preceding points of view. The framework of the study is the general setting of Banach lattices and Bochner integrable vector-valued random variables. Sub-gradient linked representation theorems and practical examples are provided.Mathematics Subject Classification (2010). 91B30, 91G80.

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Section: Definition 32mentioning

confidence: 99%

Vector Risk Functions

Balbás

Jiménez-Guerra³

2011

Mediterr. J. Math.

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“…Balbás et al (2010) proposed a general optimal investment problem with coherent and expectation bounded risk measures and perfect pricing models. The natural extension for a market with friction is Good deals in markets with friction 5 minimizes the risk of a pay-off y whose global ask price is not higher than one dollar and whose expected value is at least R. Thus, it is a standard risk/return approach with ρ as the risk measure.…”

Section: Primal and Dual Optimal Investment Problemsmentioning

confidence: 99%

“…Balbás et al (2010) 40 have shown the existence of 'pathological results' when combining some risk measures (VaR, CVaR, dual power transform (DPT), etc.) and very popular pricing models (Black and Scholes, Heston, etc.).…”

Section: Introductionmentioning

confidence: 99%

Good deals in markets with friction

Balbás

2013

Quantitative Finance

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This paper studies an optimization problem involving pay-offs of (perhaps dynamic) investment strategies. The pay-off is the decision variable, the expected pay-off is maximized and its risk is minimized. The pricing rule may incorporate transaction costs and the risk measure is continuous, coherent and expectation bounded. We will prove the necessity of dealing with pricing rules such that there exists an essentially bounded stochastic discount factor that must also be bounded from below by a strictly positive value. Otherwise, good deals will be available to traders, i.e. depending on the selected risk measure, investors can choose pay-offs whose (risk, r etur n) will be as close as desired to (−∞, ∞) or (−1, ∞). This pathological property still holds for vector risk measures (i.e. if we minimize a vector-valued function whose components are risk measures). It is worth pointing out that, essentially, bounded stochastic discount factors are not usual in the financial literature. In particular, the most famous frictionless, complete and arbitrage-free pricing models imply the existence of good deals for every continuous, coherent and expectation bounded (scalar or vector) measure of risk, and the incorporation of transaction costs will not guarantee the solution of this caveat.

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“…In a recent study, Rockafellar, Uryasev, and Zabarankin (2007) use the so-called diversion measures (an example is Conditional Value at Risk, CVaR) to investigate equilibrium in capital markets. Balbás, Balbás, and Balbás (2010) use coherent risk measures, expectation bounded risk measures and general deviations in optimal portfolio problems, and study CAPM-like relations. Grechuk and Zabarankin (2012) consider an optimal risk sharing problem among agents with utility functionals depending only on the expected value and a deviation measure of an uncertain payoff.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium in an ambiguity-averse mean–variance investors market

Pınar

2014

European Journal of Operational Research

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a b s t r a c tIn a financial market composed of n risky assets and a riskless asset, where short sales are allowed and mean-variance investors can be ambiguity averse, i.e., diffident about mean return estimates where confidence is represented using ellipsoidal uncertainty sets, we derive a closed form portfolio rule based on a worst case max-min criterion. Then, in a market where all investors are ambiguity-averse mean-variance investors with access to given mean return and variance-covariance estimates, we investigate conditions regarding the existence of an equilibrium price system and give an explicit formula for the equilibrium prices. In addition to the usual equilibrium properties that continue to hold in our case, we show that the diffidence of investors in a homogeneously diffident (with bounded diffidence) mean-variance investors' market has a deflationary effect on equilibrium prices with respect to a pure mean-variance investors' market in equilibrium. Deflationary pressure on prices may also occur if one of the investors (in an ambiguity-neutral market) with no initial short position decides to adopt an ambiguity-averse attitude. We also establish a CAPM-like property that reduces to the classical CAPM in case all investors are ambiguity-neutral.

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CAPM and APT-like models with risk measures

Cited by 22 publications

References 36 publications

Vector Risk Functions

Vector Risk Functions

Good deals in markets with friction

Equilibrium in an ambiguity-averse mean–variance investors market

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