Entropic risk measures and their comparative statics in portfolio selection: Coherence vs. convexity

We conduct a decision-theoretic analysis of optimal portfolio choices and, in particular, their comparative statics under two types of entropic risk measures, the coherent entropic risk measure (CERM) and the convex entropic risk measure (ERM). Starting with the portfolio selection between a risky and a risk free asset (framework of Arrow (1965) and Pratt (1964)), we find a restrictive all-or-nothing investment decision under the CERM, while the ERM yields diversification. We then address a portfolio problem with two risky assets, and provide comparative statics with respect to the investor’s risk aversion (framework of Ross (1981)). Here, both the CERM and the ERM exhibit closely interrelated inconsistencies with respect to the interpretation of their risk parameters as a measure of risk aversion: for any two investors with different risk parameters, it may happen that the investor with the higher risk parameter invests more in the riskier one of the two assets. Finally, we analyze the portfolio problem “risky vs. risk free” in the presence of an independent background risk, and analyze the effect of changes in this background risk (framework of Gollier and Pratt (1996)). Again, we find questionable predictions: under the CERM, the optimal risky investment is always increasing instead of decreasing when a background risk is introduced, while under the ERM it remains unaffected.     

Risk taking with background risk under recursive rank-dependent utility

This paper examines how background risk affects risk taking under rank-dependent utility. I assume that a decision-maker facing a risk taking decision in the presence of background risk views these risks as composing a compound lottery, and recursively evaluates this compound lottery using rank-dependent utility. I show that adding background risk increases risk aversion whenever the utility-for-wealth function is risk vulnerable (Gollier and Pratt, 1996) in this model.     

Optimal prevention and prudence in a two-period model

This work shows that, in a two-period framework, prudence has a positive effect on optimal prevention. This conclusion is the opposite to that obtained in a one-period framework [Eeckhoudt L., Gollier C., 2005. The impact of prudence on optimal prevention. Economic Theory 26, 989–994]. This is due to the opposite effect of prevention on wealth in the period where the risk occurs.     

Risk sharing and information revelation mechanism of a one-manufacturer and one-retailer supply chain facing an integrated competitor

This paper develops an information revelation mechanism model of a one-manufacturer and one-retailer supply chain facing an outside integrated-competitor under demand uncertainty. We investigate how the manufacturer designs a wholesale price-order quantity contract to induce the retailer to report his risk sensitivity information truthfully. We try to explore the effects of the outside competitor and the risk-sharing rule on the optimal price-service level decisions of the retailer and the optimal wholesale prices of the manufacturer. We find that the strategic interaction plays an important role in the effect of risk sensitivity on the order quantity for the retailer. When the fraction of the risk cost shared by the manufacturer is sufficiently large (small), the optimal wholesale price for the high risk-averse retailer is higher (lower) than that for the low risk-averse retailer.     

Do investors like to diversify? A study of Markowitz preferences

We study rankings of completely and partially diversified portfolios and also of specialized assets when investors follow so-called Markowitz preferences. It turns out that diversification strategies for Markowitz investors are more complex than in the case of risk-averse and risk-inclined investors, whose investment strategies have been extensively investigated in the literature. In particular, we observe that for Markowitz investors, preferences toward risk vary depending on their sensitivities toward gains and losses. For example, it turns out that, unlike in the case of risk-averse and risk-inclined investors, Markowitz investors might prefer investing their entire wealth in just one asset. This finding helps us to better understand some financial anomalies and puzzles, such as the well known diversification puzzle, which notes that some investors tend to concentrate on investing in only a few assets instead of choosing the seemingly more attractive complete diversification.     

A multi-product risk-averse newsvendor with exponential utility function

We consider a multi-product newsvendor using an exponential utility function. We first establish a few basic properties for the newsvendor regarding the convexity of the model and monotonicity of the impact of risk aversion on the solution.When the product demands are independent and the ratio of the degree of risk aversion to the number of products is sufficiently small, we obtain closed-form approximations of the optimal order quantities. The approximations are as easy to compute as the risk-neutral solution. We prove that when this ratio approaches zero, the risk-averse solution converges to the corresponding risk-neutral solution. When the product demands are positively (negatively) correlated, we show that risk aversion leads to lower (higher) optimal order quantities than the solution with independent demands.Using a numerical study, we examine convergence rates of the approximations and thoroughly study the interplay of demand correlation and risk aversion. The numerical study confirms our analytical results and further shows that an increased risk aversion does not always lead to lower order quantities, when demands are strongly negatively correlated.     

Is the inf-convolution of law-invariant preferences law-invariant?

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In other words, we try to understand whether the representative economic agent (after risk redistribution) only cares about the distribution of the total risk, assuming all individual agents do so. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer becomes positive. These conditions do not require any specific forms or convexity of the risk functionals, and they are the richness of the underlying probability space, and monotonicity or continuity of one of the risk functionals. We provide several examples and counter-examples to discuss the subtlety of the question on law-invariance.     

Optimal prevention and prudence in a two-period model

This work shows that, in a two-period framework, prudence has a positive effect on optimal prevention. This conclusion is the opposite to that obtained in a one-period framework [Eeckhoudt L., Gollier C., 2005. The impact of prudence on optimal prevention. Economic Theory 26, 989–994]. This is due to the opposite effect of prevention on wealth in the period where the risk occurs.     

考虑保守卖空与财务困境的背景风险投资组合模型

如何合理地考虑投资者所面临的背景风险及现实市场限制来进行有效地投资决策是人们所广泛关注的重要实际管理决策问题。本文研究投资者同时面临加性和乘性两类背景风险的前提下具有保守卖空与财务困境的投资组合选择问题。假定投资者寻求使得投资收益最大、投资风险最小及证券主体财务困境最小的最优投资组合策略,进而提出考虑保守卖空与财务困境的背景风险投资组合模型。然后,利用具有精英策略的非支配排序遗传算法对模型进行求解。最后,通过实例来阐述模型的实用性。研究结果表明：考虑保守卖空能为投资者提供更大的收益;两类背景风险的变化均导致有效前沿面的变化。     

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

We consider the problem of optimizing a portfolio of n assets, whose returns are described by a joint discrete distribution. We formulate the mean–risk model, using as risk functionals the semideviation, deviation from quantile, and spectral risk measures. Using the modern theory of measures of risk, we derive an equivalent representation of the portfolio problem as a zero-sum matrix game, and we provide ways to solve it by convex optimization techniques. In this way, we reconstruct new probability measures which constitute part of the saddle point of the game. These risk-adjusted measures always exist, irrespective of the completeness of the market. We provide an illustrative example, in which we derive these measures in a universe of 200 assets and we use them to evaluate the market portfolio and optimal risk-averse portfolios.     

Risk-averse stochastic optimal control: An efficiently computable statistical upper bound

In this paper, we discuss an application of the Stochastic Dual Dynamic Programming (SDDP) type algorithm to nested risk-averse formulations of Stochastic Optimal Control (SOC) problems. We propose a construction of a statistical upper bound for the optimal value of risk-averse SOC problems. This outlines an approach to a solution of a long standing problem in that area of research. The bound holds for a large class of convex and monotone conditional risk mappings. Finally, we show the validity of the statistical upper bound to solve a real-life stochastic hydro-thermal planning problem.     

The newsvendor problem under multiplicative background risk

This note studies the single-period newsvendor problem when the newsvendor faces a multiplicative neutral independent background risk in an expected utility framework. It is shown that multiplicative risk vulnerability is a sufficient condition to guarantee a decrease in the optimal order. A weaker sufficient condition which has more interpretability is also provided and discussed. This result sheds light on situations where exchange, tax or inflation rates risks, which apply multiplicatively to the final wealth, are at work.     

考虑风险规避的供应链融资与信用保险组合模型研究

针对供应链的资金约束问题,考虑了由一个资金约束的零售商、资金充足的供应商、风险规避型银行以及保险公司组成的供应链融资系统,分别构建了风险规避型银行融资模型以及银行融资与信用保险的组合模型,给出了两种模型中供应链融资系统成员的最优决策。研究表明:与传统风险规避型的银行融资模型相比,银行融资与信用保险的组合模型能够有效地增加零售商的融资规模,并降低银行的损失风险,从而信用保险能够给整个供应链融资系统带来严格帕累托改进。最后,通过数值算例说明了研究结论的有效性。     

Risky asset allocation and consumption rule in the presence of background risk and insurance markets

This study examines joint decisions regarding risky asset allocation and consumption rate for a representative agent in the presence of background risk and insurance markets. Contrary to the conclusion of the “mutual fund separation theorem”, we show that the optimal risky asset mix will reflect an agent’s risk attitude as long as background risk is not independent of investment risk. This result can, however, be used to solve the “riskyasset allocation puzzle”. We also unveil that optimal insurance to shift background risk is determined through establishing a hedging portfolio against investment risk and is an arrangement maintaining the balance between growth and volatility of expected consumption. Because the optimal insurance we obtain generally leads to a smoother consumption path, it may plausibly explain the “equity premium puzzle” in the financial literature.     

从管理风险的角度看金融风险度量

本文得出了连续时间下均值-VaR模型的最优投资策略。在这个最优解的基础上,我们比较说明了概率和分位数作为风险度量方法在管理风险中发挥的作用。我们的分析结果表明:从管理风险的角度出发控制损失发生的概率要比控制损失的水平更为有意义;并且选择的VaR置信度水平越高,监管的效果会越好。     

All-pay auctions with risk-averse players

We study independent private-value all-pay auctions with risk-averse players. We show that: (1) Players with low values bid lower and players with high values bid higher than they would bid in the risk neutral case. (2) Players with low values bid lower and players with high values bid higher than they would bid in a first-price auction. (3) Players’ expected utilities in an all-pay auction are lower than in a first-price auction. We also use perturbation analysis to calculate explicit approximations of the equilibrium strategies of risk-averse players and the seller’s expected revenue. In particular, we show that in all-pay auctions the seller’s expected payoff in the risk-averse case may be either higher or lower than in the risk neutral case.     

基于CVaR风险测度标准的价格补贴策略下的协调研究

基于条件风险值模型(CVaR),探讨了在一个风险中性制造商和一个风险规避零售商组成的制造商领导的斯塔克伯格博弈供应链中,制造商如何与风险规避零售商订立批发价契约以最大化其期望利润的问题。设计了价格补贴的契约协调机制,给出了该机制下风险规避程度对零售商和制造商最优决策的影响。证明了在一定的实施条件下,制造商通过设立价格补贴机制,可改善供应链双方利润与供应链效率。最后,用算例验证了模型和理论分析的可行性。     

Would a risk-averse newsvendor order less at a higher selling price?

We model a risk-averse newsvendor’s decision-making behavior with some commonly used classes of utility functions within the expected utility theory (EUT) framework. Under fairly general conditions of EUT, we show that a risk-averse newsvendor will order less than an arbitrarily small quantity as selling price gets larger if price is higher than a threshold value, i.e., the optimal order quantity decreases as the selling price increases.     

Efficient option risk measurement with reduced model risk

Options require risk measurement that is also computationally efficient as it is important to derivatives risk management. There are currently few methods that are specifically adapted for efficient option risk measurement. Moreover, current methods rely on series approximations and incur significant model risks, which inhibit their applicability for risk management.In this paper we propose a new approach to computationally efficient option risk measurement, using the idea of a replicating portfolio and coherent risk measurement. We find our approach to option risk measurement provides fast computation by practically eliminating nonlinear computational operations. We reduce model risk by eliminating calibration and implementation risks by using mostly observable data, we remove internal model risk for complex option portfolios by not admitting arbitrage opportunities, we are also able to incorporate liquidity or model misspecification risks. Additionally, our method enables tractable and convex optimisation of portfolios containing multiple options. We conduct numerical experiments to test our new approach and they validate it over a range of option pricing parameters.     

Consistent modeling of risk averse behavior with spectral risk measures

This paper clarifies the relation between decisions of a risk-averse decision maker, based on expected utility theory on the one hand, and spectral risk measures on the other.