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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.