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S&P 500 Daily and Annual Return Statistics

Daily S&P 500 returns have highly leptokurtic distribution but annual returns are close to normal. What does this mean for traders and investors?

Below is the distribution of daily returns of S&P 500 from 08/1940 to 04/09/2020.

This is a highly leptokurtic distribution with kurtosis of 23.3866. Mean, standard deviation and results of Lilliefors normality test are shown below:

Mean daily return is 0.0327% and standard deviation is 0.975%. The p-value of the test is practically 0 and normality is rejected .

We skip weekly, monthly and quarterly returns because the results for normality are the same, i.e., normality is rejected, and we go to annual returns next.

Below is the distribution of annual returns of S&P 500 index from 08/1940 to 04/09/2020.

Mean, standard deviation and results of Lilliefors normality test are shown below:

Mean annual return is 8.528% and standard deviation is 16.409%. But p-value in this case is 0.3482 and normality cannot be rejected.

It may also be seen from the yearly chart that all returns are within three standard deviations.

What do the above results mean?

1. Daily returns distribution is fat-tailed and this is basically what matters for traders and investors because profit and losses are market-to-market daily.  This means that those not careful with risk management can hit uncle point whether they are day traders or longer-term investors. The risk is there, no matter what the underline frequency of return is.

2. There is a fundamental difference here for strategies operating in daily versus annual timeframe: T-statistics for annual strategies are valid under normality but not valid for any other timeframes. This means that longer-term investing (usually buy and hold or variants with some timing) can use metrics to measure significance such as the T-statistic in annual timeframe.

Caveat emptor: The data sample did not include Great Depression. The assumption is that such period cannot occur again due to proactive role of central banks and fiat money credit. This is only an assumption because depressions can take many forms. Therefore, it may be the case that the annual returns sample is not representative and in reality the distribution of returns is not normal but fat-tailed. In that case, longer-term investors face the same problems in measuring statistical significance of their strategies as day traders.

However, after repeating the analysis with DJIA annual data since 1915, we find again that normality cannot be rejected with p-value of 0.3337 although there are two large returns, a positive and a negative, above and below the three-standard deviation bands.  Regardless, this appears to be more of a problem of lack of sufficient sample.

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Charting and backtesting program: Amibroker

Data provider: Norgate Data

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