Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion

In this paper, a continuous time mean-variance portfolio optimization problem is considered within a game theoretic framework, where the risk aversion function is assumed to depend on the current wealth level and the discounted (preset) investment target. We derive the explicit time consistent investment policy, and find that if the current wealth level is less (larger) than the discounted investment target, the future wealth level along the time consistent investment policy is always less (larger) than the discounted investment target.     

Delegated portfolio management under ambiguity aversion

We examine the problem of setting optimal incentives for a portfolio manager hired by an investor who wants to induce ambiguity–robust portfolio choices with respect to estimation errors in expected returns. Adopting a worst-case max–min approach we obtain the optimal compensation in various cases where the investor and the manager, adopt or relinquish an ambiguity averse attitude. We also provide examples of applications to real market data.     

Optimal portfolio choice for an insurer with loss aversion

The problem of optimal investment for an insurance company attracts more attention in recent years. In general, the investment decision maker of the insurance company is assumed to be rational and risk averse. This is inconsistent with non fully rational decision-making way in the real world. In this paper we investigate an optimal portfolio selection problem for the insurer. The investment decision maker is assumed to be loss averse. The surplus process of the insurer is modeled by a Lévy process. The insurer aims to maximize the expected utility when terminal wealth exceeds his aspiration level. With the help of martingale method, we translate the dynamic maximization problem into an equivalent static optimization problem. By solving the static optimization problem, we derive explicit expressions of the optimal portfolio and the optimal wealth process.     

Possibilistic risk aversion and coinsurance problem

The coinsurance problem is an important topic in insurance decisions. A risk-averse agent should choose a coinsurance rate maximizing the expected final wealth. In this paper, we propose a possibilistic model of coinsurance problem. A decision problem whose solution is the optimal coinsurance is formulated. Some of its properties, the calculation modality and its behavior towards the changes of risk aversion are studied.     

Increasing risk aversion and life-cycle investing

Mathematics and Financial Economics - We derive the optimal portfolio for an investor with increasing relative risk aversion in a complete continuous-time securities market. The IRRA assumption...     

The emergence of risk aversion

Economic decision making under uncertainty is universally characterized by aversion to risk. One of the most basic concepts in economic theory, risk aversion is usually explained by the concavity of the utility function, which, in turn, is based on a person's satiability for wealth. I use genetic algorithms to show that risk aversion, and some related consequences, emerge naturally as a result of evolutionary pressures. In analogy to the well-known hillclimbing metaphor, it is helpful in this context to characterize optimizing under uncertainty as “surfing in a fitness seascape.” © 1997 John Wiley & Sons, Inc.     

Comparative higher-degree Ross risk aversion

We provide generalized comparative global conditions for higher-degree Ross risk aversion, which are similar to those studied by Ross for risk aversion. This generalization corresponds to the special cases of comparative risk aversion as developed by Ross (1981) and of comparative downside risk aversion as developed by Modica and Scarsini (2005).     

Multivariate risk aversion with applications

Two measures of multivariate risk aversion, closely related t ocertain eigenvalue problems are proposed here. These measures are shown t obe ralated to multivariate distance concepts in statistics and its applications include risk analysis in portfolio theory, vector efficiently and compparison of minimax strategies in applied modelling.     

Comparative higher-degree Ross risk aversion

We provide generalized comparative global conditions for higher-degree Ross risk aversion, which are similar to those studied by Ross for risk aversion. This generalization corresponds to the special cases of comparative risk aversion as developed by Ross (1981) and of comparative downside risk aversion as developed by Modica and Scarsini (2005).     

Univariate and multivariate measures of risk aversion and risk premiums

This paper develops univariate and multivariate measures of risk aversion for correlated risks. We derive Rubinstein's measures of risk aversion from the risk premiums with correlated random initial wealth and risk. It is shown that these measures are not only consistent with those for uncorrelated or independent risks, but also have the corresponding local properties of the Arrow-Pratt measures of risk aversion. Thus Rubinstein's measures of risk aversion are the appropriate extension of the Arrow-Pratt measures of risk aversion in the univariate case. We also derive a risk aversion matrix from the risk premiums with correlated initial wealth and risk vectors. This matrix measure is the multivariate version of Rubinstein's measures and is also the generalization of Duncan's results for non-random initial wealth. The univariate and multivariate measures of risk aversion developed in this paper are applied to portfolio theory in Li and Ziemba [15].This research was partially supported by the National Research Council of Canada.     

Duopolistic competition under risk aversion and uncertainty

A monopolist typically defers entry into an industry as both price uncertainty and the level of risk aversion increase. By contrast, the presence of a rival typically hastens entry under risk neutrality. Here, we examine these two opposing effects in a duopoly setting. We demonstrate that the value of a firm and its entry decision behave differently with risk aversion and uncertainty depending on the type of competition. Interestingly, if the leader’s role is defined endogenously, then higher uncertainty makes her relatively better off, whereas with the roles exogenously defined, the impact of uncertainty is ambiguous.     

Credit portfolio risk and asset price cycles

It is a stylized fact that credit risk is high at the same time when asset values are depressed. However, most of the standard credit risk models ignore this kind of correlation, leading to underestimation of risk measures of portfolio credit risk such as Value at Risk and Expected Shortfall. In our paper we make an attempt to quantify the underestimation of these risk measures when the dependence between credit risk and asset values is ignored and show that credit risk is underestimated by a significant margin.      

Robust portfolio asset allocation and risk measures

Many financial optimization problems involve future values of security prices, interest rates and exchange rates which are not known in advance, but can only be forecast or estimated. Several methodologies have therefore been proposed to handle the uncertainty in financial optimization problems. One such methodology is Robust Statistics, which addresses the problem of making estimates of the uncertain parameters that are insensitive to small variations. A different way to achieve robustness is provided by Robust Optimization, which looks for solutions that will achieve good objective function values for the realization of the uncertain parameters in given uncertainty sets. Robust Optimization thus offers a vehicle to incorporate an estimation of uncertain parameters into the decision making process. This is true, for example, in portfolio asset allocation. Starting with the robust counterparts of the classical mean-variance and minimum-variance portfolio optimization problems, in this paper we review several mathematical models, and related algorithmic approaches, that have recently been proposed to address uncertainty in portfolio asset allocation, focusing on Robust Optimization methodology. We also give an overview of some of the computational results that have been obtained with the described approaches. In addition we analyze the relationship between the concepts of robustness and convex risk measures.     

Total risk aversion and the pricing of options

The objective of this paper is to derive the price of a call option in a stable market for amarket maker who is totally risk averse in the sense that he has a utility function with infinite index of absolute risk aversion. We prove that in the case when the borrowing rateR is the same as the lending rater the option price must be the Black—Scholes price with rateR = r. In the more interesting caseR > r, we prove that two option pricing functions are necessary and sufficient for making a market in the option. These two functions are the Black—choles prices with rater orR. Which price to use at each time will depend on the circumstances (buying or selling) and the capitalization of the market maker. We determine which price to use at each time and the optimal riskless portfolio.Partially supported by a grant, No. AFOSR-86-0202, from the Air Force Office for Scientific Research and a grant from Loyola University of Chicago.Partially supported by Grant No. AFOSR-86-0202 from the Air Force Office for Scientific Research, a grant from the National Science Foundation, and a grant from Loyola University of Chicago.     

风险度量模型的研究

分析了方差法和半方差法作为风险测度的不足之处,考虑到资产离散程度和实际投资者的风险偏好,引入风险偏好系数,建立加权的半方差证券组合决策模型,作出了理论分析并给出了相应的数值比较结果。     

Estimation risk and stability of optimal portfolio composition

The mean-variance portfolio models indicate that for optimal investment decisions, the ‘true’ ex-ante values of the model parameters should be used. Instead, in practice, ex-post parameter estimates are used. If in the estimation process, the probability distribution of estimators is not known, there is a problem of estimation risk. This paper investigates the impact of estimation risk on the composition of optimal portfolios. As the multivariance distribution of the vector of optimal portfolio weights allocated to risky assets is analytically intractable, a use of the Monte Carlo simulation experimental is made. This study suggests that the composition of optimal portfolio is relatively more stable when the estimates of model parameters are obtained from longer series of historical observations or the expected portfolio return is low.     

Distribution assumptions and risk constraints in portfolio optimization

Empirical distributions are often claimed to be superior to parametric distributions, yet to also increase the computational complexity and are therefore hard to apply in portfolio optimization. In this paper, we approach the portfolio optimization problem under constraints on the portfolios Value at Risk and Expected Tail Loss, respectively, under empirical distributions for the Standard and Poors 100 stocks. We apply a heuristic optimization method which has been found to overcome the restrictions of traditional optimization techniques. Our results indicate that empirical distributions might turn into a Pandoras Box: Though highly reliable for predicting the assets risks, employing these distributions in the optimization process might result in severe mis-estimations of the resulting portfolios actual risk. It is found that even a simple mean-variance approach can be superior despite its known specification errors.AMS Classification: G11, C61Dietmar G. Maringer: Im grateful to two anonymous referees, Peter Winker, Manfred Gilli, Berç Rustem, Erricos Kontoghiorghes, Alfred Lehar, Josef Zechner, Suresh Sundaresan, and conference participants at Aix-en-Provence, Limassol, and Sydney for valuable discussions and comments on earlier versions of this paper.