Is stock market skewed more towards negative returns? I am not talking about the steep downturn in the last few months due to special reasons such as COVID, Inflation worries etc. Looking at the data for S&P 500 and NASDAQ over a period of 2005 - 2019, the monthly return distributions are skewed towards negative returns (Figure 1). The risk measure (Tsallis Relative Entropy in our case) is quite different when calculated from negative and positive changes. Hence methods based on theories such as put forth by Black-Scholes which implicitely assumes a symmetric distribution are questionable both for long term investments such as portfolio management and short term investments like options.

Figure 1. Monthly return distributions (a) S&P 500 and (b) Nasdaq. The blue curve is the symmetric bell curve which the standard Capital Asset Portfolio Management assumes.

Taking into account the asymmetry, we estimated the risk measure seperately for

1) Symmetric model stock return distribution (S)

2) Asymmetric model stock return distribution (SA)

3) Risk measure from negative returns only (S-)

Figure 2 shows the managed Portfolio earnings kept for 10 yrs (see Portfolio Management), as a function of the starting time of the portfolio for the 3 differrent risks shown above,

Figure 2. Comparison of ten year SPY and portfolio earnings as a function of the start time of portfolio. The x-axis is the maturity date. Time digitization is 1 month. The risks used are the 90th percentile values. The first 10 year portfolio maturity date is 2 December 2010 and the last 27 October 2021 (includes COVID period).

The average earnings and the corresponding standard deviations are shown in Table below. The mean/median portfolio earnings in all three risk cases are much higher than the SPY earnings, with that from S- being the highest. Surprisingly the portfolio with symmetric risk is a close second!

The above studies show that the chaotiicity is much more pronounced when we look at the negative changes in the stock dynamics. Hence portfolios constructed with our novel risk measure ( from entropy suitable for a chaotic system) estimated from the negative changes in the market, seem to yield earnings well above the market earnings.

For details see publication 1 in Publications