Strong Increases in Downside Risk Aversion

A ranking of risk preferences is of economic interest insofar as it leads to unambiguous comparative statics predictions, and for this to be the case, the ranking must be a strict partial ordering. The ranking by greater risk aversion meets this demand at the second order, and yields a variety of well-known predictions concerning the effect of greater risk aversion on demands for insurance and risky assets, among many other applications. There has been less success at the third order, where ranking preferences by aversion to downside risk has not produced a strict partial ordering. The problem is that account has not been taken of the fact that an increase in downside risk aversion must induce changes in risk aversion as well. We propose a definition of stronger downside risk aversion that does yield a strict partial ordering by requiring a nested increase in both second- and third-order risk aversion, so that v is more strongly downside risk averse than u if v is more risk averse and more downside risk averse than u. We demonstrate that v being more strongly downside risk averse than u is characterized by v never liking any change in the probability distribution for y that induces a third-order stochastic dominance deterioration in the distribution for u(y). We apply the definition to obtain intuitive comparative statics predictions in the precautionary saving problem, and relate the definition to alternatives proposed in the literature.     

Risk Aversion,Downside Risk Aversion and Paying for Stochastic Improvements

This paper considers the relationship between risk preferences and the willingness to pay for stochastic improvements. We show that if the stochastic improvement satisfies a double-crossing condition, then a decision maker with utility v is willing to pay more than a decision maker with utility u, if v is both more risk averse and less downside risk averse than u. As the condition always holds in the case of self-protection, the result implies novel characterizations of individuals’ willingness to pay to reduce the probability of loss. By establishing a general result on the correspondence between an individual's willingness to pay, and his optimal purchase of stochastic improvement when there is a given relationship between stochastic improvements and the amount paid for them, we further show that all results on the willingness to pay can be applied directly to characterize the conditions under which a more risk averse individual will optimally choose to buy more stochastic improvement. Generalizations of existing results on optimal choice of self-protection can be obtained as corollaries.     

Does Inefficient Risk Sharing Increase Public Self-Protection?

This paper studies how the risk of having an unequal distribution of income across the population affects the investment in a public self-protection policy, such as financial regulation or climate change mitigation. Two economies are compared. In the first economy, there is perfect risk sharing, i.e., individuals can credibly commit on a set of transfers that will remove ex-post inequalities in consumption. In the second economy, no risk sharing takes place. By referring to the literature on background risks, I determine some conditions in terms of change in risk aversion and prudence, which guarantee an increase in self-protection under inefficient risk sharing. Generally speaking, if self-protection reduces the risk of inequality, the investment tends to rise when either the probability of a catastrophic event and/or the risk of inequality are sufficiently low. If self-protection increases the risk of inequality, the investment tends to rise when both the probabilities of aggregate loss and the increase in the risk of inequality are sufficiently small.     

The price of shelter - Downside risk reduction with precious metals

Investor aversion to extreme losses may motivate them to seek out investments perceived to function as a safe haven during times of crisis. In this study, we consider the potential for precious metals to mitigate downside risk when combined with equities, and evaluate the impact on portfolio risk-adjusted returns. Each of gold, silver and platinum are found to contribute to downside risk reduction at short horizons, but diversification into silver and platinum may result in increased long horizon portfolio risk. The price of sheltering an equity portfolio from downside risk is a relative reduction in portfolio risk-adjusted returns. Variance and kurtosis properties of precious metals are identified as marginal contributors to downside risk reduction. Futures markets on precious metals are also shown to present an interesting and viable diversification alternative to physical metals.     

Decision making with Expected Shortfall and spectral risk measures: The problem of comparative risk aversion

We analyze spectral risk measures with respect to comparative risk aversion following Arrow (1965) and Pratt (1964) for deterministic wealth, and Ross (1981) for stochastic wealth. We argue that the Arrow–Pratt-concept per se well matches with economic intuition in standard financial decision problems, such as willingness to pay for insurance and simple portfolio problems. Different from the literature, we find that the widely-applied spectral Arrow–Pratt-measure is not a consistent measure of Arrow–Pratt-risk aversion. Instead, the difference between the antiderivatives of the corresponding risk spectra is valid. Within the framework of Ross, we show that the popular subclasses of Expected Shortfall, and exponential and power spectral risk measures cannot be completely ordered with respect to Ross-risk aversion. Thus, for all these subclasses, the concept of Ross-risk aversion is not generally compatible with Arrow–Pratt-risk aversion, but induces counter-intuitive comparative statics of its own. Compatibility can be achieved if asset returns are jointly normally distributed. The general lesson is that these restrictions have to be considered before spectral risk measures can be applied for the purpose of optimal decision making and regulatory issues.     

Empirical Evidence for Advantageous Selection in the Commercial Fire Insurance Market

De Meza and Webb (2001) indicated that individuals with a higher degree of risk aversion would demand more insurance and invest in self-protection to reduce risk probability when both the preference type and investment in self-protection are hidden from insurers. They referred to the negative correlation between market insurance and risk type as advantageous selection. However, the relationship between risk type and the degree of risk aversion is debatable in both theoretical and empirical research. This paper therefore proposes that advantageous selection could be supported from another angle by directly examining the relationships that exist among market insurance, self-protection, and risk probability. By focusing on the commercial fire insurance market, information on the purchase of market insurance, investment in self-protection, and fire accident records is hand-collected by means of a unique survey. It is found that firms purchasing market insurance have a greater tendency to channel efforts into self-protection. It is also found that firms expending effort on self-protection are less likely to suffer a fire accident. Furthermore, it is found that firms with commercial fire insurance have less chance of suffering a fire accident than those without such insurance. Each of the above three findings jointly supports the view that advantageous selection could play a critical role in the commercial fire insurance market.     

A Note on Risk Aversion,Prudence and Portfolio Insurance

This paper examines some properties of portfolio insurance that are linked to the risk aversion and the prudence of the investor. We provide explicit conditions to measure portfolio sensitivity to downside risk. We also characterize the degree of portfolio insurance by means of the ratio of absolute prudence to absolute risk aversion.     

Skewness Preference,Risk Taking and Expected Utility Maximisation

Available empirical evidence suggests that skewness preference plays an important role in understanding asset pricing and gambling. This paper establishes a skewness-comparability condition on probability distributions that is necessary and sufficient for any decision-maker's preferences over the distributions to depend on their means, variances, and third moments only. Under the condition, an Expected Utility maximizer's preferences for a larger mean, a smaller variance, and a larger third moment are shown to parallel, respectively, his preferences for a first-degree stochastic dominant improvement, a mean-preserving contraction, and a downside risk decrease and are characterized in terms of the von Neumann-Morgenstern utility function in exactly the same way. By showing that all Bernoulli distributions are mutually skewness comparable, we further show that in the wide range of economic models where these distributions are used individuals’ decisions under risk can be understood as trade-offs between mean, variance, and skewness. Our results on skewness-inducing transformations of random variables can also be applied to analyze the effects of progressive tax reforms on the incentive to make risky investments.     

Market or limit orders?

In this paper we investigate the problem of optimal order placement of an asset listed on an exchange using both market and limit orders in a simple model of market dynamics. We seek to understand under which settings it is optimal to place limit or market orders. Limit orders typically lower transaction costs but increase the risk of incomplete order execution, whereas market orders typically have higher transaction costs but are guaranteed to be executed. Rather than considering order book dynamics to determine if a limit order is executed we rely on price dynamics for this. We look at implementation shortfall in this setup with market impact of trading and propose a dynamic program to find the optimal placement of both market and limit orders for risk-neutral and risk-averse traders. With this we find a bound on the expected cost of trading and show that a trader who behaves optimally should always expect to pay less to trade less. We then solve the dynamic program numerically and examine optimal order placement strategies. We find that the decision between market and limit orders is sensitive to price volatility, risk aversion, and trading costs.     

Investment option under CIR interest rates

We analyze extensively the characteristics of the solution to an irreversible investment decision when the only source of uncertainty comes from interest rates. They are assumed to be driven by the popular Cox–Ingersoll–Ross (CIR) stochastic process. Particular attention is paid to the impact that both CIR parameters and risk aversion have on the threshold rate.     

Wealth Effects on Self-Insurance and Self-Protection against Monetary and Nonmonetary Losses

This paper considers the wealth effects on self-insurance and self-protection activities against possible losses of monetary wealth such as properties and nonmonetary wealth such as health. Increased initial income or monetary wealth decreases the demand for self-insurance against monetary wealth loss under the decreasing absolute risk aversion assumption, and has an ambiguous effect on self-protection. However, increased initial monetary wealth increases both self-insurance and self-protection against health loss, explaining empirical trends, if wealth and health are complements. When multiple self-insurance activities against both types of losses are considered, the effect of an increase in initial monetary wealth on self-insurance against health loss remains the same, but the effect on self-insurance against wealth loss depends on the preferences. JEL Classification No: D81, G22     

Optimal order placement in limit order markets

To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by characteristics of the order flows and queue sizes in each limit order book, as well as the structure of transaction fees and rebates across exchanges. We propose a quantitative framework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows the study of how the optimal order placement decision depends on the interplay between the state of order books, the fee structure, order flow properties and the aversion to execution risk. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general case of order placement across multiple exchanges, we propose a stochastic algorithm that computes the optimal routing policy and study the sensitivity of the solution to various parameters. Our algorithm does not require an explicit statistical model of order flow but exploits data on recent order fills across exchanges in the numerical implementation of the algorithm to acquire this information through a supervised learning procedure.     

Hybrid Cat Bonds

Natural catastrophes attract regularly the media attention and have become a source of public concern. From a financial viewpoint, they represent idiosyncratic risks, diversifiable at the world level. But for various reasons, reinsurance markets are unable to cope with this risk completely. Insurance-linked securities, such as catastrophe (cat) bonds, have been issued to complete the international risk transfer process, but their development is disappointing so far. This article argues that downside risk aversion and ambiguity aversion explain their limited success. Hybrid cat bonds, combining the transfer of cat risk with protection against a stock market crash, are proposed to complete the market. The article shows that replacing simple cat bonds with hybrid cat bonds would lead to an increase in market volume.     

Locally Greater Vulnerability to Background Risk

Willingness to take on risk is influenced by the presence of fair and unfair background risks for decision makers who are risk vulnerable as defined by Gollier and Pratt [1996], for these decision makers are more risk averse when they possess such an uninsurable background risk. We present an alternative derivation of the index of local vulnerability based on Diamond and Stiglitz [1974] compensated increases in risk, such that risk aversion increases with the introduction of any small fair background risk if and only if the index of local vulnerability is positive. We establish that the increase in risk aversion is greater for those who are more vulnerable as measured by the index of local vulnerability.     

Self-Protection, Information and the Precautionary Principle

In this paper we examine the effect of information on investment in self-protection. We show that the relationship between more information and investment in self-protection is ambiguous in general. If absolute risk aversion is constant, then investment in self-protection always decreases with a better information structure. We show that if we interpret the Precautionary Principle as requiring more self-protection today, it is difficult to accept it on the grounds of efficiency, except for a particular subset of information structures.     

Recovering risk aversion from option prices and realized returns

A relationship exists between aggregate risk-neutral and subjectiveprobability distributions and risk aversion functions. We empiricallyderive risk aversion functions implied by options prices andrealized returns on the S&P500 index simultaneously. Theserisk aversion functions dramatically change shapes around the1987 crash: Precrash, they are positive and decreasing in wealthand largely consistent with standard assumptions made in economictheory. Postcrash, they are partially negative and partiallyincreasing and irreconcilable with those assumptions. Mispricingin the option market is the most likely cause. Simulated tradingstrategies exploiting this mispricing show excess returns, evenafter accounting for the possibility of further crashes, transactioncosts, and hedges against the downside risk.     

Comparative Ambiguity Aversion in Intertemporal Decisions

This article examines the effects of ambiguity aversion on intertemporal decisions when there is ambiguity about a future state. Compared to the existing literature, we allow for a three-way separation between intertemporal substitution, risk aversion, and ambiguity aversion. Holding risk preferences, beliefs, and time preferences fixed, we explore how a change in ambiguity aversion increases the strength of the current willingness to pay. We apply our results to saving, self-protection, and self-insurance problems.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.     

Dynamic Portfolio Choice with Stochastic Wage and Life Insurance

We study optimal insurance, consumption, and portfolio choice in a framework where a family purchases life insurance to protect the loss of the wage earner's human capital. Explicit solutions are obtained by employing constant absolute risk aversion utility functions. We show that the optimal life insurance purchase is not a monotonic function of the correlation between the wage and the financial market. Meanwhile, the life insurance decision is explicitly affected by the family's risk preferences in general. The model also predicts that a family uses life insurance and investment portfolio choice to hedge stochastic wage risk.     

A Generalisation of the Mean-Variance Analysis

In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above this reference point. We also suppose that the decision maker generally distorts the objective probabilities. First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This 'mean-partial moments' utility generalises not only mean-variance utility of Tobin and Markowitz, but also mean-semivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present a solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure which generalises the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker's skewness preferences have first-order impact on risk measurement even when the risk is small.