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Cross-country and intertemporal indexes of risk aversion Authors: Mark Kritzman, Kenneth Lowry, Anne-Sophie Van Royen Source: Journal of Asset Management, volume 3, number 1 Date: July 2002 |
Abstract
It is widely recognized that investors are averse to risk. This notion of risk aversion, although it has been under consideration for a long time, needs to be studied with more precision, particularly to understand its link with the risk premiums offered by equities. This article takes a step forward on the subject, by deriving a formula for risk aversion and by showing how it helps to predict bond and stock returns.
A characterisation of risk aversion was first proposed by the mathematician Daniel Bernoulli in 1738. His definition relied on the fact that people confronted with a choice between two equal expected values of gain, but one certain and the other uncertain, will choose the certain one. The reason is that the utility associated with the certain gain is higher than the utility associated with the uncertain gain.
To derive a formulation for risk aversion, the authors consider Markowitz’s problem, which consists of maximising the investor's expected utility. The problem is solved using Lagrange multipliers and the optimal solution is calculated. This solution consists of the weight of each asset class in the portfolio. To go further and calculate the risk aversion, the authors observe that at the optimum, the partial derivatives of the expected utility with respect to the weights are all equal. The risk aversion value can then be determined by taking any two of the equations and solving them. The two asset classes considered can be stocks and bonds. Risk aversion appears then to depend on the expected return of stocks and bonds, the standard deviation of stocks and bonds, the correlation between stocks and bonds and the respective weights of stocks and bonds in the portfolio. So risk aversion evaluation requires an estimation of all of those parameters. The formula obtained can be applied to each country or can be used to evaluate the global risk aversion.
Calculation was performed for several countries for a period beginning in December 1989 and ending in December 1999. A global risk-aversion coefficient was computed for the same period. The result shows an evolution in risk aversion around economic events: the Mexican and Asian crises, for example, caused an increase in global risk aversion. Another observation is that there is a high correlation between the risk aversion of different countries, but there are also important country-specific effects.
The authors also performed a regression to study the impact of risk aversion on the prediction of returns. Their results indicate that risk aversion predicts bond and stock returns better than it predicts risk premiums. Another regression was performed using the change in risk aversion, instead of the risk aversion. The result shows a tendency for a change in risk aversion to predict returns. But the authors conclude that it is difficult to determine whether the nature of the relationship between risk aversion and returns is a delayed reaction or an immediate reaction to this modification.
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