If equity returns are normally distributed, why are average equity returns not zero [closed]

So I am getting confused between assumption of equity returns normality and why then equity markets in the long term on average go up i.e equity risk premium.

Does this not already poke wholes in the assumption of normality, if most returns fall in the mean region, then why do equity markets go up more than down, implying the mean is positive.

Well there are two misconceptions in your assessment of how returns behave.

1) Returns can be normally distributed or not;

2) Even if they are normally distributed it does not mean that returns have a mean of zero. In fact the mean as you say is slightly positive.

So what can we do? Well we can test the data. I took the SPX returns between 1980 and 2012 at daily, weekly and monthly frequency.

Some summary statistics below:

Just by looking to the table above can you tell me if returns are normally distributed? Well at the daily level, they could have a normal distribution with mean of 0.02% and standard deviation of 1.16%. But look the skewness and kurtosis! Skewness of a normal distribution should be zero and kurtosis should be 3! So at the daily level they are not normally distributed. At the monthly level however, they look more "normally distributed".

Can we test this formally? Yes. We can test using a Jarque-Bera test for normality:

$$JB = \frac{n-k+1}{6} \bigg ( S^2 + \frac{1}{4}(K-3)^2 \bigg )$$

where $S$ is the sample skewness and $K$ is the sample kurtosis. What happens if we run the test? Well we get:

So indeed returns are not normally distributed!

• (+1) An issue though with formal normality tests is that they in a sense ask the wrong question, "can we statistically reject normality?" rather than "is normality close enough to 'work'?" With large data sets, you can almost always reject normality using high powered tests. An interesting discussion is here. – Matthew Gunn Jun 2 '18 at 13:23
• I agree with you. In any case for returns we (economists) are pretty sure they are not normally distributed. – phdstudent Jun 2 '18 at 13:24
• I agree.(The most obvious failing is that returns are fat tailed.) I think sometimes these deviations from a distribution are more easily seen in a qqplot? – Matthew Gunn Jun 2 '18 at 13:28

It's news to me that in today's world anybody really believes that equity returns are normally distributed. For instance in US Senate testimony by a Goldman Sachs CFO, under assumptions of Gaussian normality market returns of, e.g. the drops in the DJIA that presaged the 2008 Downturn were 25 std dev (1 in 3.6 x 10e88) events -- several days in a row. This statement is patently absurd since there isn't enough time since the beginning of the universe for a single 25 sd event much less several in a row. Regardless Gaussian assumptions of normality are baked into and retained by many, many financial models, one of the most important being Black-Scholes. In a sense then these assumptions can be described as legacies of earlier eras in financial thought when they were just beginning to diffuse through financial minds and literature. Today while they may be folk, commonsense knowledge the problem is that there is no industry-wide agreement on what to replace Gaussian assumptions with -- until such agreements are reached, outdated Gaussian frameworks remain in place.

Gaussian normality makes several assumptions but first one needs to distinguish between theoretical and empirical (observed, actual, real world) information. In theory, normally distributed information is perfectly symmetric, ranges between +/- infinity, has central tendencies (mean, median, mode) equal to zero and is expressed with two parameters: the mean and sd. In reality these assumptions can only be described as sloppy since they are so rarely met, e.g., the Jacques-Bera test for normality almost always rejects assumptions of normality.

Given that it should come as no surprise that observed equity returns do not conform to theoretical assumptions of normality.

A distribution may be normal and have a mean different from zero. For example, IQs, weights, heights and so forth. All normal distributions assume a mean and a standard deviation. These two parameters completely describe the distribution. The standard normal is the special case where the mean is zero and the standard deviation is 1.0. So stocks can be normally distributed and still have a mean different from zero. But no one thinks stock returns follow a standard normal distribution.

Many folks don't think of stocks as precisely normally distributed but approximately normally distributed. That is, for some purposes the approximation of normality is close enough. In some models, such as the Black-Scholes option-pricing model, it's not stock returns that are normally distributed. Rather, it's the natural log of (1+return) (e.g. 1.08) where (1+return) is produced by S*/S which is the ending price relative to the initial price. This is the assumption that stock returns are log-normally distributed.

Another real world point - the mean and standard deviation parameters of a normal distribution are fixed. In my opinion, the mean and standard deviation of stock returns can vary through time (in statistical terminology they are non-stationary). That makes the application of a single normal distribution difficult.

Normality does not mean that mean return has be zero. The assumption you are talking about is of standard normal distribution which has mean and SD (0,1) respectively. As your question indicates that why positive returns are higher than negative returns. First of all let us understand the mathematics behind normal distribution which says that distribution should be symmetric on both sides. Being a symmetric distribution does not imply that mean has to be zero. It could be 4/5/6 anything. half of the distribution values should be on one side and rest on other half. If you talk about single asset/equity, return for the same could follow normal/non normal distribution. If it is a well diversified index suppose S&P 500 then probability of index return would follow normal distribution,are very high. Obviously people are working to create value in the firms so if firms value go up, index consisting of various frims would also go up.