Erik Kole

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.

Portfolio Implications of Systemic Crises

Systemic crises can have grave consequences for investors in international equity markets, because it causes the risk-return trade-off to deteriorate severely for a longer period. In this paper we propose a novel approach to include the possibility of systemic crises in asset allocation decisions. By combining regime switching models with Merton (1969)-style portfolio construction, our approach captures persistence of crises much better than existing models. Our analysis shows that incorporating systemic crises has a large impact on asset allocation decisions, while the costs of ignoring such crises are substantial. For an expected utility maximizing US investor, who can invest globally these costs range from 1.13% per year of his initial wealth when he has no prior information on the likelihood of a crisis, to over 3% per month if a crisis occurs with almost certainty. If a crisis is taken into account, the investor allocates less to risky assets, and particularly less to emerging markets, being most prone to a crisis. An investor facing short selling constraints withdraws completely from equity markets if the likelihood of a crisis increases.

Interpreting financial market crashes as earthquakes: A new Early Warning System for medium term crashes

We propose a modeling framework which allows for creating probability predictions on a future market crash in the medium term, like sometime in the next five days. Our framework draws upon noticeable similarities between stock returns around a financial market crash and seismic activity around earthquakes. Our model is incorporated in an Early Warning System for future crash days. Testing our EWS on S&P 500 data during the recent financial crisis, we find positive Hanssen-Kuiper Skill Scores. Furthermore our modeling framework is capable of exploiting information in the returns series not captured by well known and commonly used volatility models. EWS based on our models outperform EWS based on the volatility models forecasting extreme price movements, while forecasting is much less time-consuming.

How to Identify and Forecast Bull and Bear Markets

The state of the equity market, often referred to as a bull or a bear market, is of key importance for financial decisions and economic analyses. Its latent nature has led to several methods to identify past and current states of the market and forecast future states. These methods encompass semi-parametric rule-based methods and parametric regime-switching models. We compare these methods by new statistical and economic measures that take into account the latent nature of the market state. The statistical measure is based directly on the predictions, while the economic mea- sure is based on the utility that results when a risk-averse agent uses the predictions in an investment decision. Our application of this framework to the S&P500 shows that rule-based methods are preferable for (in-sample) identification of the market state, but regime-switching models for (out-of-sample) forecasting. In-sample only the direction of the market matters, but for forecasting both means and volatilities of returns are important. Both the statistical and the economic measures indicate that these differences are significant.

Time Variation in Asset Return Dependence: Strength or Structure?

The dependence between asset returns varies. Its strength can become stronger or weaker. Also, its structure can change, for example, when asymmetries related to bull and bear markets become more or less pronounced. To analyze these different types of variations, we develop a model that separately accommodates these changes. It combines a mixture of structurally different copulas with time variation. Our model shows both types of changes in the dependence between several equity market returns. Ignoring them leads to biases in risk measures. An underestimation of Value-at-Risk by maximum 15% occurs exactly when most harmful, during crisis periods.

Forecasting Value-at-Risk Under Temporal and Portfolio Aggregation

textabstractWe examine the impact of temporal and portfolio aggregation on the quality of Value-at-Risk (VaR) forecasts over a horizon of ten trading days for a well-diversified portfolio of stocks, bonds and alternative investments. The VaR forecasts are constructed based on daily, weekly or biweekly returns of all constituent assets separately, gathered into portfolios based on asset class, or into a single portfolio. We compare the impact of aggregation to that of choosing a model for the conditional volatilities and correlations, the distribution for the innovations and the method of forecast construction. We find that the degree of temporal aggregation is most important. Daily returns form the best basis for VaR forecasts. Modelling the portfolio at the asset or asset class level works better than complete portfolio aggregation, but differences are smaller. The differences from the model, distribution and forecast choices are also smaller compared to temporal aggregation

Specification Testing in Hawkes Models

We propose various specification tests for Hawkes models based on the Lagrange Multiplier (LM) principle. Hawkes models can be used to model the occurrence of extreme events in financial markets. Our specific testing focus is on extending a univariate model to a multivariate model, that is, we examine whether there is a conditional dependence between extreme events in markets. Simulations show that the test has good size and power, in particular for sample sizes that are typically encountered in practice. Applying the specification test for dependence to US stocks, bonds and exchange rate data, we find strong evidence for cross-excitation within segments as well as between segments. Therefore, we recommend that univariate Hawkes models be extended to account for the cross-triggering phenomenon.

Cyclicality in Losses on Bank Loans

Based on unique data we show that macro variables, the default rate and loss given default of bank loans share common cyclical components. The innovation in our model is the distinction between loans with either severe or mild losses. The variation in the proportion of these two types drives the cyclic behavior of the loss given default, and constitutes the links with the default rate and macro variables. These links vary according to loan and borrower characteristics. During downturns, the proportion of defaults with severe losses increases, but the distribution of losses conditional on their being mild or severe does not change. Though loans are monitored more closely than bonds and are more senior, the cyclical variation in their losses resembles those for bonds, albeit around a lower average level. This variation leads to an increase in the capital reserves required for loan portfolios.

Exploiting Spillovers to Forecast Crashes

We develop Hawkes models in which events are triggered through self as well as cross-excitation. We examine whether incorporating cross-excitation improves the forecasts of extremes in asset returns compared to only self-excitation. The models are applied to US stocks, bonds and dollar exchange rates. In-sample, a Lagrange Multiplier test indicates the existence of cross-excitation for these series. Out-of-sample, we find that the models that include spillover effects forecast crashes and the Value-at-Risk significantly more accurately than the models without.