Kees Koedijk

The tail index of exchange rate returns

In the literature on the empirical distribution of foreign exchange rates there is now consensus that exchange rate yields are fat-tailed. Three problems, however, persist: (1) Which class of distribution functions is most appropriate? (2) Are the parameters of the distribution invariant over subperiods? (3) What are the effects of aggregation over time on the distribution? In this paper we employ extreme value theory to shed new light on these questions. We apply the theoretical results to EMS data.

Capital structure policies in Europe: Survey evidence

textabstractIn this paper we present the results of an international survey among 313 CFOs on capitalnstructure choice. We document several interesting insights on how theoretical concepts arenbeing applied by professionals in the U.K., the Netherlands, Germany, and France and wendirectly compare our results with previous findings from the U.S. Our results emphasize thenpresence of pecking-order behavior. At the same time this behavior is not driven by asymmetricninformation considerations. The static trade-off theory is confirmed by the importance of a targetndebt ratio in general, but also specifically by tax effects and bankruptcy costs. Overall, we findnremarkably low disparities across countries, despite the presence of significant institutionalndifferences. We find that private firms differ in many respects from publicly listed firms, e.g. listednfirms use their stock price for the timing of new issues. Finally, we do not find substantialnevidence that agency problems are important in capital structure choice.

Tail-Index Estimates in Small Samples

Financial returns are known to be nonnormal and tend to have fat-tailed distributions. This article presents a simple methodology that accurately estimates the degree of tail fatness, characterized by the tail index, in small samples. Our method is a weighted average of Hill estimators for different threshold values that corrects for the small-sample bias apparent in the latter. Using this estimator we produce tail-index estimates for returns on exchange rates that are close to nonbiased estimates obtained from extremely large datasets. The results indicate that many documented conclusions concerning the tail behavior of financial series are likely to have overestimated the tail fatness in small samples.

Optimal portfolio selection in a Value-at-Risk framework

Abstract In this paper, we develop a portfolio selection model which allocates financial assets by maximising expected return subject to the constraint that the expected maximum loss should meet the Value-at-Risk limits set by the risk manager. Similar to the mean–variance approach a performance index like the Sharpe index is constructed. Furthermore when expected returns are assumed to be normally distributed we show that the model provides almost identical results to the mean–variance approach. We provide an empirical analysis using two risky assets: US stocks and bonds. The results highlight the influence of both non-normal characteristics of the expected return distribution and the length of investment time horizon on the optimal portfolio selection.

Selecting Copulas for Risk Management

Copulas offer financial risk managers a powerful tool to model the dependence between the different elements of a portfolio and are preferable to the traditional, correlation-based approach. In this paper we show the importance of selecting an accurate copula for risk management. We extend standard goodness-of-fit tests to copulas. Contrary to existing, indirect tests, these tests can be applied to any copula of any dimension and are based on a direct comparison of a given copula with observed data. For a portfolio consisting of stocks, bonds and real estate, these tests provide clear evidence in favour of the Students t copula, and reject both the correlation-based Gaussian copula and the extreme value-based Gumbel copula. In comparison with the Students t copula, we find that the Gaussian copula underestimates the probability of joint extreme downward movements, while the Gumbel copula overestimates this risk. Similarly we establish that the Gaussian copula is too optimistic on diversification benefits, while the Gumbel copula is too pessimistic. Moreover, these differences are significant.

Increased correlation in bear markets

A number of studies have provided evidence of increased correlations in global financial market returns during bear markets. Other studies, however, have shown that some of this evidence may be biased. We derive an alternative to previous estimators for implied correlation that is based on measures of portfolio downside risk and that does not suffer from bias. The unbiased quantile correlation estimates are directly applicable to portfolio optimization and to risk management techniques in general. This simple and practical method captures the increasing correlation in extreme market conditions while providing a pragmatic approach to understanding correlation structure in multivariate return distributions. Based on data for international equity markets, we found evidence of significant increased correlation in international equity returns in bear markets. This finding proves the importance of providing a tail-adjusted mean–variance covariance matrix. A generally accepted concept today is that, over time, returns when the markets are experiencing large negative movements are more highly correlated than returns during more normal times. If true, this phenomenon has serious implications for portfolio and risk management because it means that the benefits of diversification are curtailed precisely when investors most need them. The correlation, however, depends on how the returns are conditioned on the size of the returns. Previous studies have provided alternative correlation structures with which to compare conditional empirical correlations, but these estimates have upward or downward biases that need to be corrected. In this article, we provide a quantile correlation approach that is not biased by the size of the return distribution. The result is a simple and pragmatic approach to estimating correlations conditional on the size of the returns. Based on empirical data, we show how the correlation estimates can be used directly in portfolio and risk management. We derive a conditional correlation structure based on the quantile of the joint return distribution; that is, correlation is conditioned on the size of the return distribution. In a bivariate framework, the correlation is estimated by using those observations that fall below the portfolio return of the two assets. The approach is thus in line with current correlation measures used in Markowitz-style portfolio analysis and in current risk management techniques. The quantile correlation structure is determined by the weights of the assets in the portfolio and the quantile estimates of the distribution of returns on the two assets and of the portfolio return. When the distribution is normal, the conditional correlation structure is constant; hence, the conditional quantile correlation will equal the unconditional correlation. Therefore, because the correlation structure is constant over the distribution for normality, one can easily compare empirical estimates of conditional correlation with their theoretical values under conditions of normal distribution. We examine a variety of daily returns from international stock market indexes for the period May 1990 through December 1999 to establish, first, their unconditional correlations. For example, this procedure produced a correlation between the U.S. market (S&P 500 Index) and the U.K. market (the FTSE 100 Index) of 0.349. Assuming bivariate normality for the whole distribution, we would expect the quantile conditional correlation also to be 0.349. For quantiles up to the 95 percent level, we found that the assumption of normality cannot be rejected at the 95 percent confidence level for all the series. For higher quantiles, however—that is, large negative returns in the bivariate return distribution—the conditional correlation structure increased the correlations; in the case of the U.S. and U.K. markets, the correlation increased to 0.457. The effect on mean–variance portfolio optimization is a reduction in the recommended weight of the risky assets held in the portfolio. These results imply that the gains from diversification are not reaped in periods when diversification benefits are most crucial from a mean–variance perspective—in bear markets. Practitioners, therefore, need to know what sort of model is generating the correlations they are relying on. If the underlying model assumed normality, then the correlation estimates used in the model need to be adjusted to incorporate the bear markets higher-than-normal correlation structure.

Continental factors in international real estate returns

This paper examines the extent to which real estate returns are driven by continental factors. This subject is relevant for determining the country allocation of international real estate portfolios. If returns are driven by a continental factor, investors should look for diversification opportunities outside their own continent. This paper finds strong continental factors in North America and especially in the United States. For the Asia–Pacific region, real estate returns are not driven by a continental factor. The results suggest that, for European, North American and Asia—Pacific real estate portfolio managers, the Asia—Pacific region provides attractive international diversification opportunities.

Extreme support for uncovered interest parity

Abstract Concerning UIP, the common conclusion is that it may be valid but undetectable for many reasons. In this paper we take a complementary route in that we base our methodology on a random time effects panel model that controls for various biasing factors and which is invariant to the choice of the numeraire currency. We show that the rejection of UIP is not as severe as is commonly found and that it almost perfectly holds in periods where the forward premiums are large.

Portfolio selection with limited downside risk

A safety-first investor maximizes expected return subject to a downside risk constraint. Arzac and Bawa [Arzac, E.R., Bawa, V.S., 1977. Portfolio choice and equilibrium in capital markets with safety-first investors. Journal of Financial Economics 4, 277–288.] use the Value at Risk as the downside risk measure. The paper by Gourieroux, Laurent and Scaillet estimates the optimal safety-first portfolio by a kernel-based method, we exploit the fact that returns are fat-tailed, and propose a semi-parametric method for modeling tail events. We also analyze a portfolio containing the two stocks used by Gourieroux et al. and discuss the merits of the safety-first approach.

Capturing downside risk in financial markets: the case of the Asian Crisis

Using data on Asian equity markets, we observe that during periods of financial turmoil, deviations from the mean-variance framework become more severe, resulting in periods with additional downside risk to investors. Current risk management techniques failing to take this additional downside risk into account will underestimate the true Value-at-Risk with greater severity during periods of financial turnoil. We provide a conditional approach to the Value-at-Risk methodology, known as conditional VaR-x, which to capture the time variation of non-normalities allows for additional tail fatness in the distribution of expected returns. These conditional VaR-x estimates are then compared to those based on the RiskMetricsTM methodology from J.P. Morgan, where we find that the model provides improved forecasts of the Value-at-Risk. We are therefore able to show that our conditional VaR-x estimates are better able to capture the nature of downside risk, particularly crucial in times of financial crises.