scholarly journals Non-Cash Risk Measure on Nonconvex Sets

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 186 ◽  
Author(s):  
Chang Cong ◽  
Peibiao Zhao
Keyword(s):  
Optimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Financial Position ◽  
Nonconvex Sets ◽  
The Relationship

Monetary risk measures are interpreted as the smallest amount of external cash that must be added to a financial position to make the position acceptable. In this paper, A new concept: non-cash risk measure is proposed and this measure provides an approach to transform the unacceptable positions into the acceptable positions in a nonconvex set. Non-cash risk measure uses not only cash but also other kinds of assets to adjust the position. This risk measure is nonconvex due to the use of optimization problem in L 1 norm. A convex extension of the nonconvex risk measure is derived and the relationship between the convex extension and the non-cash risk measure is detailed.

AN ERGODIC BSDE RISK REPRESENTATION IN A JUMP-DIFFUSION FRAMEWORK

2021 ◽  
pp. 2150015
Author(s):  
CALISTO GUAMBE ◽  
LESEDI MABITSELA ◽  
RODWELL KUFAKUNESU
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Risk Measures ◽  
Risk Measure ◽  
Jump Diffusion ◽  
Backward Stochastic Differential Equations ◽  
Financial Position ◽  
Entropic Risk Measure ◽  
Risk Representation

We consider the representation of forward entropic risk measures using the theory of ergodic backward stochastic differential equations in a jump-diffusion framework. Our paper can be viewed as an extension of the work considered by Chong et al. (2019) in the diffusion case. We also study the behavior of a forward entropic risk measure under jumps when a financial position is held for a longer maturity.

Weighted Scoring Rules and Convex Risk Measures

2022 ◽  
Author(s):  
Zachary J. Smith ◽  
J. Eric Bickel
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Mathematical Finance ◽  
Scoring Rules ◽  
Convex Risk Measures ◽  
University Of Texas ◽  
Probabilistic Forecasts ◽  
Expected Utility Maximization ◽  
The University ◽  
The Relationship

In Weighted Scoring Rules and Convex Risk Measures, Dr. Zachary J. Smith and Prof. J. Eric Bickel (both at the University of Texas at Austin) present a general connection between weighted proper scoring rules and investment decisions involving the minimization of a convex risk measure. Weighted scoring rules are quantitative tools for evaluating the accuracy of probabilistic forecasts relative to a baseline distribution. In their paper, the authors demonstrate that the relationship between convex risk measures and weighted scoring rules relates closely with previous economic characterizations of weighted scores based on expected utility maximization. As illustrative examples, the authors study two families of weighted scoring rules based on phi-divergences (generalizations of the Weighted Power and Weighted Pseudospherical Scoring rules) along with their corresponding risk measures. The paper will be of particular interest to the decision analysis and mathematical finance communities as well as those interested in the elicitation and evaluation of subjective probabilistic forecasts.

scholarly journals Optimal Design of Dynamic Default Risk Measures

2012 ◽  
Vol 49 (4) ◽  
pp. 967-977 ◽  
Author(s):  
Leo Shen ◽  
Robert Elliott
Keyword(s):  
Optimal Design ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Default Risk ◽  
Optimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Jump Process ◽  
Entropic Risk Measure

We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.

Optimal Design of Dynamic Default Risk Measures

2012 ◽  
Vol 49 (04) ◽  
pp. 967-977
Author(s):  
Leo Shen ◽  
Robert Elliott
Keyword(s):  
Optimal Design ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Time ◽  
Default Risk ◽  
Optimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Jump Process ◽  
Entropic Risk Measure

We consider the question of an optimal transaction between two investors to minimize their risks. We define a dynamic entropic risk measure using backward stochastic differential equations related to a continuous-time single jump process. The inf-convolution of dynamic entropic risk measures is a key transformation in solving the optimization problem.

scholarly journals Scenario aggregation method for portfolio expectile optimization

2016 ◽  
Vol 33 (1-2) ◽  
Author(s):  
Edgars Jakobsons
Keyword(s):  
Portfolio Optimization ◽  
Value At Risk ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Risk Measures ◽  
Risk Measure ◽  
Conditional Value At Risk ◽  
Aggregation Method ◽  
Portfolio Optimization Problem ◽  
Scenario Aggregation

AbstractThe statistical functional expectile has recently attracted the attention of researchers in the area of risk management, because it is the only risk measure that is both coherent and elicitable. In this article, we consider the portfolio optimization problem with an expectile objective. Portfolio optimization problems corresponding to other risk measures are often solved by formulating a linear program (LP) that is based on a sample of asset returns. We derive three different LP formulations for the portfolio expectile optimization problem, which can be considered as counterparts to the LP formulations for the Conditional Value-at-Risk (CVaR) objective in the works of Rockafellar and Uryasev [

DESIRABLE PROPERTIES OF AN IDEAL RISK MEASURE IN PORTFOLIO THEORY

2008 ◽  
Vol 11 (01) ◽  
pp. 19-54 ◽  
Author(s):  
SVETLOZAR RACHEV ◽  
SERGIO ORTOBELLI ◽  
STOYAN STOYANOV ◽  
FRANK J. FABOZZI ◽  
ALMIRA BIGLOVA
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Approximation Problem ◽  
Performance Ratio ◽  
Probability Metrics ◽  
Time Aggregation ◽  
Computational Advantage ◽  
The Impact ◽  
The Relationship ◽  
Multidimensional Concept

This paper examines the properties that a risk measure should satisfy in order to characterize an investor's preferences. In particular, we propose some intuitive and realistic examples that describe several desirable features of an ideal risk measure. This analysis is the first step in understanding how to classify an investor's risk. Risk is an asymmetric, relative, heteroskedastic, multidimensional concept that has to take into account asymptotic behavior of returns, inter-temporal dependence, risk-time aggregation, and the impact of several economic phenomena that could influence an investor's preferences. In order to consider the financial impact of the several aspects of risk, we propose and analyze the relationship between distributional modeling and risk measures. Similar to the notion of ideal probability metric to a given approximation problem, we are in the search for an ideal risk measure or ideal performance ratio for a portfolio selection problem. We then emphasize the parallels between risk measures and probability metrics, underlying the computational advantage and disadvantage of different approaches.

scholarly journals Optimization Problem of Insurance Investment Based on Spectral Risk Measure and RAROC Criterion

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Xia Zhao ◽  
Hongyan Ji ◽  
Yu Shi
Keyword(s):  
Risk Aversion ◽  
Optimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Risk Attitude ◽  
Investment Strategy ◽  
Marginal Effect ◽  
Investment Ratio ◽  
Spectral Risk Measure ◽  
The Impact

This paper introduces spectral risk measure (SRM) into optimization problem of insurance investment. Spectral risk measure could describe the degree of risk aversion, so the underlying strategy might take the investor's risk attitude into account. We establish an optimization model aiming at maximizing risk-adjusted return of capital (RAROC) involved with spectral risk measure. The theoretical result is derived and empirical study is displayed under different risk measures and different confidence levels comparatively. The result shows that risk attitude has a significant impact on investment strategy. With the increase of risk aversion factor, the investment ratio of risk asset correspondingly reduces. When the aversive level increases to a certain extent, the impact on investment strategies disappears because of the marginal effect of risk aversion. In the case of VaR and CVaR without regard for risk aversion, the investment ratio of risk asset is increasing significantly.

scholarly journals Market and Accounting Measures of Risk: The Case of the Frankfurt Stock Exchange

Risks ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 14
Author(s):  
Anna Rutkowska-Ziarko
Keyword(s):  
Stock Exchange ◽  
Risk Measures ◽  
Risk Measure ◽  
Statistical Significance ◽  
Correlation Coefficients ◽  
Downside Risk ◽  
Average Value ◽  
Market Rate ◽  
Measures Of Risk ◽  
The Relationship

The main purpose of this study was to explore the relationship between market and accounting measures of risk and the profitability of companies listed on the Frankfurt Stock Exchange. An important aspect of the study was to employ accounting beta coefficients as a systematic risk measure. The research considered classical and downside risk measures. The profitability of a company was expressed as ROA and ROE. When determining the downside risk, two approaches were employed: the approach by Bawa and Lindenberg and the approach by Harlow and Rao. In all the analyzed companies, there is a positive and statistically significant correlation between the average value of profitability ratios and the market rate of return on investment in their stocks. Additionally, correlation coefficients are higher for the companies included in the DAX index compared with those from the MDAX or SDAX indices. A positive and in each case a statistically significant correlation was observed for all DAX-indexed companies between all types of market betas and corresponding accounting betas. Likewise, for the MDAX-indexed companies, these correlations were positive but statistical significance emerged only for accounting betas calculated on ROA. As regards the DAX index, not every correlation was positive and significant.

scholarly journals Modified Mean-Variance Risk Measures for Long-Term Portfolios

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 111
Author(s):  
Hyungbin Park
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Investment Strategy ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Volatility Model ◽  
Variance Risk ◽  
Investment Portfolios ◽  
Mean Variance

This paper proposes modified mean-variance risk measures for long-term investment portfolios. Two types of portfolios are considered: constant proportion portfolios and increasing amount portfolios. They are widely used in finance for investing assets and developing derivative securities. We compare the long-term behavior of a conventional mean-variance risk measure and a modified one of the two types of portfolios, and we discuss the benefits of the modified measure. Subsequently, an optimal long-term investment strategy is derived. We show that the modified risk measure reflects the investor’s risk aversion on the optimal long-term investment strategy; however, the conventional one does not. Several factor models are discussed as concrete examples: the Black–Scholes model, Kim–Omberg model, Heston model, and 3/2 stochastic volatility model.

scholarly journals Minimizing spectral risk measures applied to Markov decision processes

2021 ◽  
Author(s):  
Nicole Bäuerle ◽  
Alexander Glauner
Keyword(s):  
Minimization Problem ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Planning Horizon ◽  
Insurance Company ◽  
Existence Of A Solution ◽  
Discounted Cost ◽  
Spectral Risk Measures ◽  
Markov Decision

AbstractWe study the minimization of a spectral risk measure of the total discounted cost generated by a Markov Decision Process (MDP) over a finite or infinite planning horizon. The MDP is assumed to have Borel state and action spaces and the cost function may be unbounded above. The optimization problem is split into two minimization problems using an infimum representation for spectral risk measures. We show that the inner minimization problem can be solved as an ordinary MDP on an extended state space and give sufficient conditions under which an optimal policy exists. Regarding the infinite dimensional outer minimization problem, we prove the existence of a solution and derive an algorithm for its numerical approximation. Our results include the findings in Bäuerle and Ott (Math Methods Oper Res 74(3):361–379, 2011) in the special case that the risk measure is Expected Shortfall. As an application, we present a dynamic extension of the classical static optimal reinsurance problem, where an insurance company minimizes its cost of capital.

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