Expected Shortfall Constraint Research Articles

Modelization and Calibration of the Power-Law Distribution in Stock Market by Maximization of Varma Entropy

Proper description of the return distribution is crucial for investment practitioners. The underestimation of the tail risk may lead to severe consequences, even for assets with moderate fluctuations. However, many empirical studies found that the distribution tails of many financial assets drop off more slowly than the Gaussian distributions. Therefore, we intend to model and calibrate the heavy tails observed in financial fluctuations in this study. By maximizing the Varma entropy with value-at-risk and expected shortfall constraints, we obtain the probability distribution of stock return and observe that the tail of stock return distribution is a power law. Since the variance of the real stock portfolio may be a random variable, using the mean-VaR-ES constraints to maximize the Varma entropy effectively avoids the problem of assuming that the variance is a constant value under the traditional mean-variance constraint. Therefore, the deduced theoretical model would be more consistent with the real market. Using high-frequency data from China’s stock markets, we calibrate our theoretical model and give the concrete form of probability density distribution p(x) for different time intervals. The calibration results show that the tail of the stock return distribution is a power law with most of the power-law orders between −2 and −7. We prove the robustness of our results by calibrating the Varma entropy for S&P 500 of the USA stock market and different stock market indices in China’s A-share market. Our research’s findings not only offer a theoretical perspective for researchers but also give investing professionals a theoretical foundation on which to base their decisions.

Coherent risk measures alone are ineffective in constraining portfolio losses

We show that coherent risk measures alone are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term ρ-arbitrage for a risk measure ρ. We show how to determine analytically whether such ρ-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether Expected-Shortfall arbitrage exists in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk.

Good-Deal Bounds for Option Prices under Value-at-Risk and Expected Shortfall Constraints

In this paper, we deal with the pricing of European options in an incomplete market. We use the common risk measures Value-at-Risk and Expected Shortfall to define good-deals on a financial market with log-normally distributed rate of returns. We show that the pricing bounds obtained from the Value-at-Risk admit a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller’s bound for a call option is smaller than the buyer’s bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between good-deal bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of risk measures based on multiple eligible assets in our setting.

Linking Risk Management Under Expected Shortfall to Loss-Averse Behavior

We introduce and solve an optimal asset allocation problem under a weighted expected shortfall (WES) constraint, which contains the risk management problem under an expected shortfall constraint of Basak and Shapiro (2001) as a special case. Furthermore, we link our risk management problem under the WES constraint with an optimal asset allocation with a multiple-reference-based preference (MRBP) and find that the optimal wealth with MRBP owns the same form as the optimal solution under the WES constraint. For the degenerate case with a fixed reference level, we are able to determine the critical maximal allowed expected shortfall constraint as a function of the loss aversion parameters to achieve equivalence. It is interesting to observe that, while no equivalence can be in general obtained between the WES and the MRBP solution, the optimal terminal wealth of the WES can be made to coincide with the MRBP terminal wealth in the most favorable and in the worst market states. In addition, we carry out a general equilibrium analysis in the presence of a WES/MRBP risk manager.

Short-horizon regulation for long-term investors

We study the effects of imposing repeated short-horizon regulatory constraints on long-term investors. We show that Value-at-Risk and Expected Shortfall constraints, when imposed dynamically, lead to similar optimal portfolios and wealth distributions. We also show that, in utility terms, the costs of imposing these constraints can be sizeable. For a 96% funded pension plan, both an annual Value-at-Risk constraint and an annual Expected Shortfall constraint can lead to an economic cost of about 2.5–3.8% of initial wealth over a 15-year horizon.

Determination of Optimal Contribution Rate and Optimal Investment Portfolio of Defined Benefit Pension Plan Under the Expected Shortfall Constraint

We establish stochastic models considering stochastic growth rate of salary, stochastic return rate of investment portfolio and stochastic mortality to evaluate pension surplus. We also establish the stochastic objective function of minimizing the contribution rate satisfying the constraint of expected shortfall less than a constant. We consider the time varying of means and variance of return rate of investment and deduct the calculation formula of time varying covariance of it. We apply the Monte Carlo simulation, and stochastic optimization techniques with the help of Matlab to find the optimal solutions of contribution rate and investment portfolio. Finally, we discuss the effect of the changes of the age of participating pension plan, the retirement age and mortality improvement on the optimal contribution rate and the effect of changing retirement age on the replacement rate.

Fixed-Income Portfolio Allocation Including Hedge Fund Strategies: A Copula Opinion Pooling Approach

This paper adapts Meucci9s [2006a, 2006b] copula opinion pooling (COP) framework to examine whether fixed income hedge fund strategies enhance the risk-return spectrum of traditional bond portfolios. In contrast to the Black-Litterman setup, the COP approach does not rely on linear dependencies, and avoids the problems associated with the assumption of normally distributed asset returns. We analyze three scenarios that represent investor expectations about the performance of fixed income portfolios, and we add fixed income hedge fund strategies such as fixed income arbitrage, convertible bond arbitrage, and distressed securities, given expected shortfall constraints. Our results suggest that investor market expectations and attitudes toward potential losses are both important in determining the relative weight of hedge funds in the optimal portfolio. Depending on the model parameters, the allocation to hedge funds can vary greatly, from 0% to 85%.

The Marginal Price of Risk With a Cvar Constraint

Applying the approach used by Eisenberg (2007) to derive the marginal price of risk for an expected value maximizing manager who has a Var constraint, I derive the marginal price of risk given a Cvar (Acerbi and Tasche, 2001), also known as an expected shortfall constraint. Despite the criticism that Var is not as good a measure of risk as Cvar, under typical conditions the marginal price of risk with a Cvar constraint is approximately the same as the price with a Var constraint.

Portfolio selection with transaction costs under expected shortfall constraints

An investor subject to proportional transaction costs allocates funds to multiple stocks and a bank account, to maximise the expected growth rate of the portfolio value under Expected Shortfall (ES) constraints. In a numerical example with ten time steps and one stock important innovations are caused by the introduction of the Expected Shortfall constraint: First, expected returns are reduced by less than one-tenth when the ES constraint is introduced. In comparison, economic capital as measured by ES, is reduced to amounts between one-half and three-quarters, when the ES constraint is introduced. Second, the dependence of expected return and ES on the initial portfolio, in particular when transaction costs are high, is largely removed by the introduction of the ES constraint.

Dividing gains between a client and her agent

A client(she) contracts with an agent(him), who has limited liability, as follows: she lends him one dollar at time 0 and he uses the money to trade in a security market. As return, he promises to give her a fixed amount \(e^{r_0T}\) at the final time T; in addition, if the real return rate of the strategy is larger than \(r_0\), she can also get a fixed proportion \((1-\alpha)\) of the “excess profit” and he will take the rest. Assume that the market is complete and the agent aims to maximize the risk-neutral value of his profit subject to some expected shortfall constraint. The reasonable benchmark return rate \(r_0\) and the proportion \(\alpha\) are explicitly worked out.

Trading Strategy Performance when Using Value at Risk or Expected Shortfall as a Risk Constraint

This paper studies the performance of three trading strategies: the sample Sharpe ratio, the momentum and the contrarian strategies subjected to the value at risk and expected shortfall constraint using 30 or 90 stocks with an equal weight or mean-variance optimization allocation. The results show that imposing the risk constraint deteriorates the performance, except for the case of using Sharpe ratio as a stock selection criteria. However, both unconstrained and constrained strategies outperform the market, especially for the contrarian strategies. However under the risk constrained strategies, the value at risk constraint strategy performs better than the expected shortfall constraint strategies. Moreover, the performance is improved when we use mean-variance optimization allocation and allow for leveraging.