Portfolio insurance under a risk-measure constraint

Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization

Insider Information, Arbitrage and Optimal Portfolio and Consumption Policies

Convex risk measures based on generalized lower deviation and their applications

Portfolio management using lower partial risk (downside risk) measures is attracting more attention of practitioners in recent years. The purpose of this paper is to review important characteristics of these riskmeasures and conduct simulation using four alternative measures, lower semi-variance, lower semi-absolute deviation, first order below targetrisk and conditional value-at-risk.We will show that these risk measures are useful to control downside risk whenthe distribution of assets is non-symmetric. Further, we will propose a computational scheme to resolve the difficultyassociated with solving a large dense linear programming problems resulting from these models. We will demonstrate that this method can in fact solve problems consisting of104 assets and 105 scenarios within a practical amount of CPU time.

Entropy-Based Analysis of Spectral Risk Measures for Practical Implementation

Risk is one of the important parameters in portfolio optimization problem. Since the introduction of the mean-variance model, variance has become the most common risk measure used by practitioners and researchers in portfolio optimization. However, the mean-variance model relies strictly on the assumptions that assets returns are multivariate normally distributed or investors have a quadratic utility function. Many studies have proposed different risk measures to overcome the drawbacks of variance. The purpose of this paper is to discuss and compare the portfolio compositions and performances of four different portfolio optimization models employing different risk measures, specifically the variance, absolute deviation, minimax and semi-variance. Results of this study show that the minimax model outperforms the other models. The minimax model is appropriate for investors who have a strong downside risk aversion.

Risk-adjusted probability measures in portfolio optimization with coherent measures of risk

Even with unbiased judgments and project calculations, an insidious bias will still be present in an approved projects portfolio. The screening and ranking process used to select projects causes an optimizer's curse effect (Smith and Winkler, 2006). A portfolio's true outcome value will tend to fall below the forecast distribution mean. Stated another way, we should expect to be disappointed in the value of the eventual outcomes. This optimizer's curse effect can be huge: the true portfolio value might be half its forecast value when forecast in the usual way. This paper describes a correction process based upon Bayesian inversion obtained with Monte Carlo simulation. Estimate/Actual distributions characterize component evaluation errors. The mean and standard deviation of the E/A distribution measures the quality of judgments and estimates. Examples demonstrate the value of better information in project or asset evaluation, portfolio optimization, and competitive bidding. The premise of our approach to correct for optimizer's curse is straightforward enough, though implementing this method requires judging the population of evaluated projects—or at least assessing the shape of this population's distribution. We offer a process and guidelines for correcting for the optimizer's curse in both project and portfolio value calculations. Correcting estimates for systematic biases restores luster to the recommended expected value-maximizing decision policy. We also tie-back to the winner's curse (Capen, Clapp and Campbell, 1971) experienced in competitive bidding and how it affects bid optimization.

Mean-risk models have been widely used in solving portfolio selection problems in the last years. Since the mean-variance theory of Markowitz, an enormous amount of papers have been published extending or modifying the basic model in several directions like the simplification of the type and amount of input data, the introduction of alternative measures of risk, the incorporation of additional criteria or constraints. Recently, risk measures concerned with the left tails of distributions that evaluate the extremely unfavourable outcomes are used. The most used risk measure for such purposes is Value-at-Risk (VaR). In this paper we concentrate on the second direction of incorporating of a new risk measure in portfolio modeling. We define a new risk measure, which take into consideration the values exceeding a certain threshold in the extreme tail of the loss distribution, called Limited Value-at-Risk (LVaR). We study the properties of this risk measure. We build a new model for portfolio selection, named mean-LVaR model, in which risk is evaluated using LVaR risk measure. We study the properties of the new mean-risk model and compare it with the classical mean-VaR model. We derive the analytical form of LVaR risk measure in the case of normal distribution. We provide computational results and analyze the implications of using the mean-LVaR risk model in portfolio optimization problem.

Portfolio optimization is crucial for investors seeking to maximize returns while minimizing risk. This study determines the optimal portfolio composition of IDX High Dividend 20 stocks using two distinct approaches: Single Index Model (SIM) and Multi Index Model (MIM). SIM utilizes only the Jakarta Composite Index (IHSG) as a market factor, while MIM incorporates three factors: IHSG, USD/IDR exchange rate, and Bank Indonesia (BI) Rate. The study analyzes daily closing price data from 20 stocks in the IDX High Dividend 20 index during the period of June 1 to November 28, 2025, obtained from investing.com.The optimal portfolio is constructed by comparing the Excess Return to Beta (ERB) with the cut-off rate (C*) for each model. Risk measurement and volatility testing include stationarity tests, normality tests, heteroskedasticity tests, and volatility estimation using historical volatility and EWMA methods. The optimal portfolio is constructed by comparing the Excess Return to Beta (ERB) with the cut-off rate (C*) for each model. Risk measurement and volatility testing include stationarity tests (Augmented Dickey-Fuller and KPSS), normality tests (Kolmogorov-Smirnov), heteroskedasticity tests (Breusch-Pagan), and volatility estimation using historical volatility and EWMA methods. The results show that SIM outperforms MIM significantly, with the SIM optimal portfolio consisting of 7 stocks (UNTR 32.93%, ASII 30.84%, UNVR 12.80%, GGRM 9.03%, TLKM 5.92%, INTP 4.91%, HMSP 3.57%) yielding an expected return of 60.63% per annum with a portfolio risk of 17.01% and Sharpe Ratio of 0.201. In contrast, MIM produces only 5 stocks with negative expected return (-57.55% p.a.), higher risk (25.14%), and negative Sharpe Ratio (-0.160). This study demonstrates that for the IDX High Dividend 20 index during the observed period, the simpler SIM approach provides superior risk-adjusted returns compared to the more complex MIM approach.

This paper discusses and analyzes risk measure properties in order to understand how a risk measure has to be used to optimize the investor's portfolio choices. In particular, we distinguish between two admissible classes of risk measures proposed in the portfolio literature: safety-risk measures and dispersion measures. We study and describe how the risk could depend on other distributional parameters. Then, we examine and discuss the differences between statistical parametric models and linear fund separation ones. Finally, we propose an empirical comparison among three different portfolio choice models which depend on the mean, on a risk measure, and on a skewness parameter. Thus, we assess and value the impact on the investor's preferences of three different risk measures even considering some derivative assets among the possible choices.

Portfolio Choice with State-Dependent Adjustments: Analyzing Leveraged Positions Without Parametric Assumptions

Worst-case risk measures provide a means of calculating the largest value of risk when only partial information of the underlying distribution is available. For popular risk measures such as value-at-risk (VaR) and conditional value-at-risk (CVaR) it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known. We show in this paper that closed-form solutions exist for a general class of law invariant coherent risk measures, which consist of spectral risk measures (and thus CVaR also) as special cases. Moreover, we provide worst-case distributions characterized in terms of risk spectrums, which can take any form of distribution bounded from below. As applications of the closed-form results, new formulas are derived for calculating the worst-case values of higher order risk measures and higher order semideviation, and new robust portfolio optimization models are provided. The online appendix is available at https://doi.org/10.1287/opre.2018.1736 .

In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

In this paper the concept of quantile-based optimal portfolio selection is introduced and a specific portfolio connected to it, the conditional value-of-return (CVoR) portfolio, is proposed. The CVoR is defined as the mean excess return or the conditional value-at-risk (CVaR) of the return distribution. The portfolio selection consists solely of quantile-based risk and return measures. Financial institutions that work in the context of Basel 4 use CVaR as a risk measure. In this regulatory framework sufficient and necessary conditions for optimality of the CVoR portfolio are provided under a general distributional assumption. Moreover, it is shown that the CVoR portfolio is mean-variance efficient when the returns are assumed to follow an elliptically contoured distribution. Under this assumption the closed-form expression for the weights and characteristics of the CVoR portfolio are obtained. Finally, the introduced methods are illustrated in an empirical study based on monthly data of returns on stocks included in the S&P index. It is shown that the new portfolio selection strategy outperforms several alternatives in terms of the final investor wealth.