Yoram Kroll

Experimental tests of the mean-variance model for portfolio selection

Abstract Statistically knowledgeable male and female undergraduate students participated in 40 portfolio selection problems with monetary payoff contingent on performance. The portfolio selection task included two independent risky assets with normally distributed returns. It provided access to information about previous returns, allowed borrowing and lending at a fixed interest rate, and forced on each decision period a choice between the two risky assets. The findings show a high percentage of inefficient mean-variance portfolios which does not decrease with practice, a high rate of requests for useless information, a large frequency of switches between the two risky assets, and sequential dependencies. Theoretical and practical implications of the findings are briefly discussed.

Sampling Errors and Portfolio Efficient Analysis

Studies which deal with portfolio efficiency analysis can be divided into two main categories: (a) those concerned with the development of normative decision rules; and (b) those that discuss the application of the normative rules to empirical data. Most of the research on portfolio efficiency analysis uses some set of empirical data, without considering the possible errors which may arise when a sample rather than the entire population is examined. The prevailing neglect of the sampling errors is a clear reflection of the complexity of the issue.

VAR Risk Measures Versus Traditional Risk Measures: An Analysis and Survey

The article presents an analysis and survey regarding the validity of VaR risk measures in comparison to traditional risk measures. Individuals are assumed to either maximize their expected utility or possess a lexicographic utility function. The analysis is carried out for generally distributed functions and for the normal and lognormal distributions. The main conclusion is that although VaR is an inadequate measure within the expected utility framework, it is at least as good as other traditional risk measures. Moreover, it can be improved by modified versions such as the Accumulated-VaR (Mean-Shortfall) Assuming a lexicographic expected utility strengthens the argument for using AVaR as a legitimate risk measure especially in the case of a regulated firm.

Stochastic Dominance with Riskless Assets

Investment decision making under conditions of uncertainty, and in particular portfolio selection, is carried out mainly in the Mean-Variance framework which has been developed by Markowitz [29], [30] and Tobin [42]. By assuming the lending and borrowing of money at a given riskless interest rate, Sharpe [39], [40], Lintner [27], [28], Mossin [34], and others derived and extended the Capital Asset Pricing Model, under which an equilibrium price of each risky asset is determined. However, though the mean variance rule is quite convenient to apply, its limitations are well known, i.e., one must assume either normal probability distributions with risk aversion or quadratic utility functions.

Voluntary Insurance Coverage, Compulsory Insurance, and Risky-Riskless Portfolio Opportunities

A decision involving a portfolio of an uninsurable risk, two other risks (one of which can be insured voluntarily, while the second is covered by compulsory insurance), and a risk-free investment is examined. The focus is on the effects of compulsory insurance coverage on the demand for voluntary insurance, and extends the results of Schulenburg (1986) by elaborating upon the impact of the risk-free opportunity on the tradeoff between voluntary and compulsory coverages.

The Limited Relevance of the Multiple IRRs

We find that the incidence of multiple internal rate of return (IRR) pitfalls are rare. This might partially explain the widespread use of IRRs by practitioners. When a projects cash flow has only two sign variations, the IRR rule is simple. For analyzing the case of three or more sign variations in the cash flow, we assume a non-negative sum of the future anticipated cash flows. Given this assumption, we present necessary and sufficient conditions for an investments cash flows with three or more sign variations to have multiple IRR solutions. Based on these conditions we show that the confusing multiple IRR solutions are only possible under unrealistic large fluctuations of the cash flows.

Stochastic Dominance With a Riskless Asset: An Imperfect Market

The assumption that investors can borrow and lend at a riskless interest rate reduces the Mean-Variance (M-V) efficient set to only one optimal unlevered portfolio. However, once we realize that the market is generally imperfect and that the borrowing rate is higher than the lending rate, we can no longer use the mean-variance Separation Theorem. Instead, a number of unlevered portfolios must be included in the efficient set, while the optimal unlevered portfolio is selected on the basis of the investors preference. The size of the efficient set of unlevered portfolios is a function of the type of empirical data used and of the disparity between the borrowing and lending interest rates.

ON THE ATTITUDE TOWARDS INEQUALITY

This paper presents an experimental framework for separating the attitude toward inequality from the attitude toward risk. This exploratory experimental study examines the attitude toward inequality while keeping risk constant. The results support the hypothesis of inequality aversion only among middle-income subjects. More interestingly we found that higher equality motivates individuals to take more risk and challenge. This result is a counterpoint to the standard line that inequality is needed to encourage effort.

Increasing risk, decreasing absolute risk aversion and diversification

Abstract This paper defines conditions for ‘Increasing Risk’ when the utility functions of risk averse investors are characterized by decreasing absolute risk aversion (DARA). Rothschild and Stiglitz ( Journal of Economic Theory 1970, 2, 225–243, and 1971, 3, 66–84) define cases when a random variable Y is ‘more risky’ (or ‘more variable’) than another variable X for the utility functions of risk averse investors. They conclude that Y is riskier than X if G , the cumulative distribution of Y , can be formed from F , the cumulative distribution of X , by adding a series of mean preserving spread (MPS) steps to F . This paper suggests considering a sequence of steps which are denoted by ‘mean preserving spread and antispread’ (MPSA). We define the condition under which a random variable (r.v.) Y is ‘more risky’ (or ‘more variable’) than another variable X for DARA utility functions. We prove that for DARA utility functions, Y is riskier than X if and only if G , the cumulative distribution function of Y , can be formed from F , the cumulative distribution function of X by adding a series of MPSA steps to F , under the restrictions stated in the paper. The economic intuition and impact of MPS and MPSA steps on the optimum diversification strategy are demonstrated.

Efficiency analysis of deductible insurance policies

The Expected Utility Theory has been used extensively in the analysis of optimal insurance strategies. See, for example, Borch (1960), Mossin (1968), Smith (1968), Gould (1969), Arrow (1970, 1973), and Raviv (1979). The optimal amount and type of protection against casualty or liability which is purchased by an individual is a function of his risk preferences, as well as the distribution function of his assets under the various available insurance strategies. However, a major impediment to the practical determination of the optimal strategy is the limited information on the specific utility function of the individual. In this study, it is assumed that there is only partial and very general information on the utility function. Therefore, we do not attempt to calculate he optimal insurance strategy (for which we need full information on the utility function). Instead, we establish upper and lower bounds of admissible insurance strategies. The bounds on the prices of a given insurance policy are based on the dominance argument which is based on the arbitrage possibilities between riskfree assets and risky assets. The essence of this dominance argument is as follows: An Insurance policy on an asset is overpriced, if all individuals can have a better risk reduction simply by increasing the proportion of a riskfree asset, rather than purchasing the insurance policy. In such a case, we say that the insurance poficy is a dominated one. Similarly, we claim that the insurance policy is underpriced, when all investors would obtain a better risk reduction by purchasing the insurance policy rather than increasing the proportion of the riskfree asset. If the price of the insurance policy is within the bounds, one cannot establish dominance for all the potential investors. In this paper we use the Stochastic Dominance with the riskless asset (SDR) framework ’ to establish a relatively narrow range for the premium of a given excess insurance policy. The narrow range which is obtained under the assumption of risk aversion is due to the arbitrage process in which we implicitly assume that the insurance policy should compete in the market with other means for reducing risk. These means include the opportunity to vary the amount of the riskless asset in the overall portfolio. In the next section we present the assumptions, notations and the Stochastic Dominance rule with borrowing and lending of a riskfree asset which is established by Levy & Kroll(l978). In the third section we apply this dominance rule to establish lower and upper bounds of the premiums of a given deductible. These bounds are constructed under relatively weak and general assumptions on the utility function. In the third section, we present a numerical example by which we demonstrate the features of our bounds. Long, and relatively complicated proofs are relegated to the appendixes.