Risk Factor Distributions: Before and After the Lehman Bankruptcy

By Chris Porter, CFA
christopher.n.porter@jpmorgan.com

On September 15, 2008, Lehman Bros. filed for Chapter 11 bankruptcy. This event sent a shudder throughout world markets which has culminated in a drawn out period of economic uncertainty and market volatility. The below histogram shows the plot of the S&P 500 index returns for the three years before and three years after the Lehman Bankruptcy.

Histogram of S&P 500 Index returns 3 years before and 3 years after the Lehman Bankruptcy

Source: J.P. Morgan's Investment Analytics & Consulting group.

At a glance, this depiction of the asset return distribution illustrates the difference in the pattern of returns between the two periods.

When using financial models to assess risk management and portfolio allocation decisions, in order for the model assumptions to be understood and rationalized, it is important to appreciate the potential shape and behavior of asset return distributions. The purpose of this study is to analyze the shape and characteristics of the asset return distributions of a number of financial variables three years prior to and three years after the Lehman Bankruptcy. The motivation for this study is to understand if asset return distributions have changed in behavior and analyze if the simple methods used for calculating the risk of portfolios are appropriate for assessing a portfolio's risk profile and making asset allocation decisions.

Firstly, the structure of the analysis will be to review the raw empirical asset returns looking at the third and fourth moments of the distributions, namely, Skew and Kurtosis¹. Next, we will test for violations of two assumptions used in many risk models of independent and normally distributed return time series.² After review of the raw empirical asset returns, the analysis will then move on to adjust the data to take account of violations of these assumptions by using an Exponentially Weighted Moving Average (EWMA) model.³

Why are Independence and Normality important? Independence and Normality are important from a risk modeling perspective as these assumptions are convenient for forecasting volatility and determining the risk profile of a portfolio. To put it briefly, independence means that prior returns have no influence on future returns (i.e., are essentially random variables). This leads to the assumption of normality, which draws on the central limit theorem that independent random variables are approximately normally distributed. If these two assumptions hold then future asset returns can be forecast using a random process drawn from a normal distribution.

Why are Skew and Kurtosis important? Skew and Kurtosis can be used to describe a distribution compared to the normal distribution. If a distribution is significantly more skewed or exhibits more kurtosis than the normal distribution, a model that has made these assumptions may lack precision. For example, a given model may forecast risk assuming future returns are independently and normally distributed, whereas the real distribution is negatively skewed with more kurtosis than a normal distribution. In this instance, the model will understate risk because a distribution with these characteristics will have more large negative returns than expected by normality.

Distributions

J.P. Morgan's Investment Analytics & Consulting team analyzed various return distributions to cover multiple asset classes and geographic locations. The
selected distributions are listed in Figure 1.

Figure 1: Asset return distributions

Analysis

To illustrate how the analysis was conducted, the EuroStoxx 50 data is shown in detail and then a summary of the full results is displayed for discussion. Figure 2 shows a histogram of the EuroStoxx 50 index returns before and after the Lehman bankruptcy.

Figure 2: Histogram of asset returns before and after bankruptcy

Source: J.P. Morgan's Investment Analytics & Consulting group.

The red line in Figure 2 shows the plot of the Normal Distribution. It can be seen from the diagram that a number of returns are present in the tails of the distributions before and after the bankruptcy. These long tails are reflected by the high excess kurtosis. The distribution is negatively skewed before the bankruptcy, while returns appear less skewed after the bankruptcy.

This visual study gives us reason to be concerned about the assumptions we might make about independence and normality in the asset return distribution. Before moving on to a more rigorous statistical analysis, Figure 3 shows the dates of the top five and bottom five returns of the EuroStoxx 50 index before and after the bankruptcy.

Figure 3: Top and bottom 5 returns of the Eurostoxx 50 before and after the bankruptcy

Figure 3 shows that the largest returns appear to be clustered around certain periods. For example, the top three worst returns before and after the crisis occurred within a one month period. This is an example of volatility clustering; if the day-to-day returns were independent and not influenced by each other, they would be expected to be more evenly spaced over the period. The data is now adjusted using a EWMA model which corrects for such an effect. Figure 4 shows a histogram of the returns adjusted with the EWMA model.

Figure 4: Histogram of asset returns before and after bankruptcy

Source: J.P. Morgan's Investment Analytics & Consulting group.

It can be seen that the distributions now appear to better resemble a normal distribution. The EWMA process has also reduced the kurtosis of the distribution and not as many observations appear in the tail. This is because a EWMA process adjusts volatility quicker when a period of large returns occurs. Figure 5 shows the top and bottom returns after the EWMA adjustment.

Figure 5: Top and bottom 5 returns of the Eurostoxx 50 before and after the bankruptcy (EWMA adjusted)

Figure 5 shows that the bottom five and top five returns are drawn more evenly from the three years before and after the bankruptcy. Therefore, because a EWMA process adjusts for a lack of independence, the assumption of normality in this return distribution becomes more plausible.

Moving on to the formal statistical tests, Figure 6 displays the results of a Jarque-Bera Test for Normality and a Ljung-Box Test for Independence (or autocorrelation in time series).⁴

Figure 6: Tests for independence and normality

These statistical tests show that non-independence and non-normality existed in the raw data before the bankruptcy. After the bankruptcy, the hypothesis of independence was not rejected; however, the hypothesis of normality was rejected. After adjusting the raw data with a EWMA process, the prebankruptcy hypothesis of independence is not rejected. The test for normality is no longer rejected at the 99% level, but it is still rejected at the 95% level. After the crisis, the EWMA adjustment led to the hypothesis of normality not being rejected.

Results

The analysis for each asset class is now considered, unless stated otherwise the comments refer to the raw data. In the Appendix, we summarize the Skew, Kurtosis, and p-values of the Ljung-Box test for independence and Jarque-Bera Test for normality before and after the bankruptcy.⁵

First, we analyze the results for equities. Considering the raw data, all distributions exhibit negative skew before the crisis. This skew increases in all cases apart from the EuroStoxx 50 after the bankruptcy. Introducing the EWMA process leads to a broadly consistent skew before and after the crisis. Kurtosis increases considerably for the Topix and S&P 500, particularly the Topix, whose distribution was affected by the earthquake and subsequent tsunami in March 2011. The EWMA process reduces the effect on Kurtosis after the crisis and brings the Kurtosis to a similar level before and after the bankruptcy. Non-independence existed in three of the distributions before the crisis, but only one after the crisis. After EWMA adjustment, independence exists after the bankruptcy according to this test. Normality is rejected in all four distributions before and after the bankruptcy. The EWMA process leads to normality not being rejected for the Eurostoxx 50 at the 95% confidence level after the crisis, but normality is still rejected for all other distributions before and after the crisis.

Next, we consider the results for yields. Skew remained consistent before and after the crisis across yields, except for Italian Sovereign 10-year yields which became positively skewed after the bankruptcy. Italian sovereign yields experienced a large increase in kurtosis after the bankruptcy. Independence is found in all the 10-year yields before the crisis (except U.S. Treasury), and in all of the time series after EWMA adjustment. After the bankruptcy, nonindependence exists in the 10-year Italian yields. Normality is rejected in all four distributions after the crisis and only in the Italian yields after the EWMA adjustment. Before the crisis, all distributions rejected normality or were close to rejecting normality at the 95% significance level with or without the EWMA process.

Next, we review foreign exchange results. Skew became very negative for the CHF-USD exchange rate after the bankruptcy. The EWMA process removes this large change in skew. Kurtosis increased considerably for the CHF-USD exchange rate which was again removed by the EWMA process. Non-independence does not exist in exchange rates before and after the crisis, except for the JPY-USD rate before the crisis. Normality is rejected in all distributions before the crisis. The EWMA process does not reject normality for the EUR-USD exchange rate. After the crisis, normality is rejected in three out of four distributions and two out of four with EWMA adjustment.

Finally, we consider the results for commodities. Skew became negative for the West Texas and Brent Oil after the crisis and this was less pronounced after the EWMA adjustment. Kurtosis increased in all four distributions after the crisis, the EWMA process reduced kurtosis considerably post bankruptcy. Nonindependence does not exist in the commodities before and after the crisis. After EWMA adjustment, normality is not rejected in any of the distributions except for Gold where non-normality is present in all tests.

Conclusion

Figure 7 summarizes the independence and normality tests across the 16 asset return distributions.

Figure 7: Summary of independence and normality tests

Figure 7 shows that on average Independence of returns exist in the majority of the distributions, both before and after the bankruptcy. The EWMA adjustment reduces the impact of non-independence in time series on the data, so that it exists in only one distribution before and after the bankruptcy. However, non-normality exists in the majority of distributions before and in all but one distribution after the bankruptcy. The EWMA process removes non-normality from a number of distributions, leaving nine and seven distributions with non-normality before and after the bankruptcy, respectively.

The EWMA model modifies the data by making return distributions closer to being independently and normally distributed. This can be seen in distributions that incur a shock, such as the Japanese stock market after the catastrophic earthquake in March 2011 and the Italian Sovereign yields in the wake of the Eurozone crisis. In these cases, skew and kurtosis are reduced, moving closer to normality.

In summary, the results show that skew, kurtosis, dependence and non-normality exist in many of the raw empirical distributions before and after the Lehman bankruptcy. Implementing a EWMA model improves the asset return distribution modeling by removing some of the effects of nonindependence and non-normality. The EWMA adjustment also leads to more consistent estimates of skew and kurtosis before and after the bankruptcy. However, even with this adjustment non-normality is still present in approximately half of the distributions. Therefore, a further extension to this analysis could be to test how a GARCH process could improve on the EWMA model, and if any improvement is justified by the increased complexity. Additionally, the analysis could be extended to analyze more risk factors or other risk types such as credit spreads and implied volatilities.

Appendix

Bibliography:
Alexander, C., & Sheedy, E. (2010). Advanced Value at Risk Models. PRM Handbook: Volume III.
Brooks, C. (2008). Introductory Econometrics for Finance. Cambridge University Press.
Jorion, P. (2001). Value at Risk, 2nd Edition. McGraw-Hill.
Sheikh, A., & Qiao, H. (2009). Non-normality of Market Returns. J. P. Morgan Asset Management.

¹A skewed distribution has returns that are not evenly distributed around the mean. For example, a characteristic of a negatively skewed distribution is that it has the majority of returns above the mean and a number of larger negative returns below the mean. Kurtosis measures the shape of a distribution compared to a normal distribution. The Kurtosis used in this article is excess kurtosis. A normal distribution has an excess kurtosis of zero. A distribution with wider tails than a normal distributions has an excess kurtosis greater than zero.
² A Normal (or Gaussian) distribution is a probability distribution. It is widely recognized by the bell shape of its probability density function.
³ The decay factor used will be 0.94.
⁴ The hypothesis in these tests are as follows:
     Ljung-Box Test for independence.
           Null: Asset returns are independent
           Alternative: Asset returns are not independent
     Jarque-Bera Test for Normality.
          Null: Asset returns are normally distributed
          Alternative: Asset returns are not normally distributed
⁵ The p-value is the confidence level at which the null hypothesis is rejected. The larger the p-value indicates that there is more evidence to reject the null hypothesis.