My Finance

Model description: Continuing research from the previous post on earnings growth and bond spreads, consider the annual December data 1927-2024. We have Standard & Poor returns in four versions:

• total returns (including dividends) vs price returns (excluding dividends);
• nominal returns (not inflation-adjusted) vs real returns (inflation-adjusted).

We have the following three factors, known at end of year

• Earnings Yield: annual earnings / end-of-year price
• Bond spread BAA-AAA (both are rates by Moody’s)
• Bond spread BAA-10YTR (Moody’s BAA rate minus 10-year Treasury rate)

Also, we add annual realized volatility which we add both additively to the linear regression, as a factor, and multiplicatively, to the innovations (residuals). We have the following model:

Summary of results: Instead of giving large tables, we simply provide a short summary and refer an interested reader to the Python code.

• Regression residuals pass Shapiro-Wilk and Jarque-Bera normality test with flying colors. This matches quantile-quantile plots versus the Gaussian distribution.
• The autocorrelation function plots for and for show that white noise conjecture is reasonable. Although, similarly to this post, we have strange large value at lag 4.
• Applying L1 norm for the autocorrelation function and comparing with this simulation, we get values which lie between 0.4 and 0.5 for and between 0.6 and 0.7 for , which are within the 99% interval. Thus we fail to reject the white noise hypothesis.
• Regression factors have large p-values according to the Student T-test. Thus we fail to reject the hypothesis that or or We did not test the joint hypothesis when all these three parameters are zero. But we reject because
• Point estimates in all four cases are

A modification: Instead of earnings yield (which is always positive) take its logarithm: Results are almost the same as described above. Although for nominal total returns the L1 norm for is 0.706.

See the code and data at https://github.com/asarantsev/growth-spread