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Your Financial Models Are Wrong!

Investment Managers Use Incorrect Return Assumptions in Their Models

Measuring and modeling financial returns are important for investors to understand their risks and expectations. Understanding how frequently worst-case events can happen has become increasingly important as we're in the 9th year of a bull cycle and valuations are above their long term averages. Money managers use financial models to measure the risk and return profiles of their investment returns, the problem is that they're doing it wrong.

Most of the widely used financial time series models assume financial returns or the logarithm of financial returns are normally distributed. However, many empirical studies have shown that the distributions of financial returns are non-normal and heavy-tailed, indicating that extreme values, especially negative shocks, are more likely to occur than the investment managers model for (Müller et al., 1998; Cont et al., 2001; Ibragimov et al., 2013). Due to the increasing market volatility over the past decades and the challenges for traditional financial models to capture the heavy tail risks, it is crucial to examine the true distribution of financial returns and the risks inherited in using stylized financial models.

1. Assuming the market behaves normally is wrong

Widely used financial models, including the famous Nobel winner models, Black Scholes Merton and Generalized Autoregressive Conditional Heteroskedasticity (GARCH), assume that returns or the noise of returns are normally distributed. The main reason these models assume return normality is simplicity. The symmetric bell-shaped normal distribution with most of the densities concentrated around the mean is easy to model. The problem - financial returns are actually not normally distributed! The significant divergence between the true return distribution and normal distribution could result in inaccurate estimates and forecasts of future returns or risks.

2. The market has significantly more negative returns than accounted for

In this section, several techniques are used to examine the behavior of financial returns. We take daily S&P 500 returns from 4/19/2005 – 4/18/2018, 3273 observations.We plot the S&P 500 returns as well a normal distribution with the same mean and standard deviation as the S&P 500 returns.

Figure 1: Density plots of daily S&P500 returns and normal distribution: the black normal density plot is symmetric and the left and right limit of the normal density on the x-axis are set to match the minimum and maximum values of the S&P500 return observations.

Figure 1 shows that daily S&P500 return density significantly diverges that of a normal distribution. First, we implemented Jarque-Bera normality test and Quantile-to-Quantile Plot to further confirm that the S&P 500 returns are statistically significantly non-normal. Jarque-Bera normality test is a common tool for normality testing. The null hypothesis is a joint hypothesis of the skewness being zero and the excess kurtosis being zero, which implies a normal distribution. From Table 2 we could see that the JB Test Statistic is very high and the p-Value is very small, indicating that the S&P 500 returns are significantly non-normal.

Table 1: Jarque-Bera Test for S&P 500 Returns

Figure 2 shows the Quantile-to-Quantile Plot for the S&P 500 returns. We could see that the empirical distribution quantiles diverge from normal distribution quantiles as well as the 95% confidence intervals for the normal quantiles. Therefore, we could conclude that the S&P 500 returns are significantly different from a normal distribution with the same mean and standard deviation.

Figure 2: Quantile-to-Quantile Plot for S&P 500 Returns. The red solid line is the quantiles of the normal distribution. The red dashed line is the 95% confidence interval for the quantiles of the normal distribution. The black dots are the quantiles for the S&P 500 returns empirical distribution.

From Figure 1, we could also see that the left tail of the S&P500 returns are slightly heavier than the normal benchmark. To further confirm the heavy-tailedness of returns, we compare the number of observations (exceedances) that are beyond 2 standard deviations from the mean for both the S&P500 returns and the empirical distribution.

Note: the number of exceedances for S&P500 returns are taken directly from the empirical return density. The number of exceedances for the normal distribution is estimated through the normal density function, so the decimals are expected here. The lower tail cut-off point is -2.36% and the upper tail cut-off point is 2.43%.

Table 2 shows that the number of exceedances of the lower tail is higher than that of the normal distribution and the number of exceedances of the upper tail is slightly lower than that of the normal distribution. This result confirms our initial guess that financial returns have heavier lower tails, which infers that negative shocks are more likely to happen than the widely used normal assumption.

In addition, to illustrate the heavy-tailedness of returns over year. We also plotted the number of lower tail exceedances over year from 2005-2018 on Figure 3. It is shown that the highest number of exceedances occurs on 2008 and the number of exceedance drops sharply after 2011. Moreover, 3 exceedances have already been observed in the first 4 months on 2018 while there are only 11 exceedances in total between 6 years from 2012 to 2017. The 3 largest negative shocks on 2018 occur on 2/5/2018, 2/8/2018, and 3/22/2018. Our observations indicate that financial returns on 2018 could be more volatile than previous years and thereby investors should be more cautious about the heavy-tailedness in the markets.

Figure 3: The number of exceedances on the lower tail from 2015-2018: the number of exceedances is obtained from S&P 500 returns that are lower than -2.36%, which is 2 standard deviation from the return mean from 2015-2018.

3. Normality in financial models

In the previous section, we concluded that S&P 500 returns are significantly non-normally distributed. In this section, we examine the widely applied financial models that assume normality and why these models may generate inaccurate results during periods of high volatility.

a. Value at Risk (VaR)

VaR is the most commonly used method to quantify the future risks of an asset or a portfolio. An example of VaR is represented as “portfolio i has 95% probability of losing no more than 1 million dollars in the next 10 days". "Variance and Covariance" method, or "Analytical" method is used very often to estimate VaR. This method typically assumes the asset return or the logarithm of return if normally distributed. The VaR estimation starts with estimating the mean and variance of the portfolio using time decaying methods, such as Exponentially Weighted Average (EWMA), and then derive the VaR estimator by taking the quantile value on the normal distribution density with previously estimated mean and variance. It is easy to observe the VaR could be highly biased when the asset returns are not normally distributed and have heavy tails because the VaR will simply underestimate the unexpected risks that assets will lose more than the value drawn from a normal density. More advanced methods, such as Expected Shortfall (ES), has been proposed to better quantify the tail risks, but the VaR still relies on the assumption of normally distributed returns. Obviously, VaR is very likely to fail to give descriptions of unexpected risks, especially the "black swan" events in the past few decades.

b. GARCH volatility models

The GARCH volatility models have been widely applied in return volatility forecasting due to its power to explain the inherited relationship between lagged returns and volatilities.

The simplest GARCH (1,1) model could be represented as:

Under GARCH (1,1) model, Maximum Likelihood Estimator (MLE) is applied to estimate parameters and . Since the noise term that is directly linked to asset return is assumed to have the normal distribution, MLE with normal likelihood function is commonly applied in the estimation. The normal likelihood function measures the probability (likelihood) of the real market observation (i.e. stock returns) given a set of parameters and using the normal distribution density function. Then, the set of parameters and that has the highest probability is chosen as the MLE estimator.

The normality assumption for GARCH(1,1) greatly reduces the complexity of choosing the best combination of and by taking the logarithm of the normal likelihood function. However, many empirical studies show that the observations of is not normally distributed and have heavy tails (Diebold et al., 1988, Fan et al., 2014). Because is the only random part contributing to the return , these studies support our findings that return also have heavy-tailed distributions.

Based on the findings above, the asset return and the return noise could be more heavy-tailed and volatile than what the normality assumption describes. Thus, the normality assumption in GARCH(1,1), as well as the MLE method to estimate parameters, may fail to capture the abnormal changes of return volatilities. An overestimate of an underestimate of market volatilities could result in further inaccuracies in different fields, such as trading strategy design, derivative pricing, and risk control.

c. Multi-factor models

Since the invention of the famous CAPM and Fama-French 3 factor models, multi-factor models have been studied and refined extensively to discover mispriced financial assets and forecast future stock returns. Typically, Multi-factor models are constructed using the formula:

In the Multi-factor model, to estimate the parameters bm and bj in this model, the normality of the input data is not required given an input data size large enough. However, the only part that explains the randomness of the asset returns is the noise term ei . In most of the applications,ei is required to have normal distribution with mean 0 and variance 1. Under circumstances when the market is highly volatile, the Multi-factor model may fail to capture the negative shocks of individual assets that are assumed to be unlikely to happen, especially in prediction tasks.

There are far more popular financial models widely used in the industry that assumes normality, such as the Black-Scholes-Merton. Though the normality assumption largely reduces the complexity of the problem and increase the efficiencies of parameter optimization, more research is needed to provide better assumptions to these models to account for the dynamic heavy-tailed patterns in returns distributions.

4. Conclusion

This report first explains the normality assumption in modeling financial returns and uses several techniques to prove the non-normality and heavy-tailedness in financial returns. Then the paper demonstrates why using popular time series models, such as GARCH, could be biased in modeling heavy tail risks. Based on this conclusion of this report, investors and researchers should be cautious in modeling financial returns using commonly used models, especially during periods with rising return volatility.

Reference:

Cont, Rama. "Empirical properties of asset returns: stylized facts and statistical issues." (2001): 223-236.

Diebold, Francis X. Empirical modeling of exchange rate dynamics. Vol. 303. Springer Science & Business Media, 2012.

Fan, Jianqing, Lei Qi, and Dacheng Xiu. "Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods." Journal of Business & Economic Statistics 32.2 (2014): 178-191.

Ibragimov, Marat, Rustam Ibragimov, and Paul Kattuman. "Emerging markets and heavy tails." Journal of Banking & Finance 37.7 (2013): 2546-2559.

Müller, Ulrich A., Michel M. Dacorogna, and Oliver V. Pictet. "Heavy tails in high-frequency financial data." A practical guide to heavy tails: Statistical techniques and applications (1998): 55-78.

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