Estimating market index valuation from macroeconomic trends

We discuss USA stock market data from 1789 until 2020, focusing our attention on the S&P 500 index (1957-2020). We find that the data can be split into two periods, (1789-1948) and (1948-2020), displaying roughly 2% and 7% growth rates, respectively. The index variations from each trend appear similar, suggesting some degree of stationarity in market fluctuations. We then correlate market behavior to macroeconomic data, such as world (and USA) population growth and gross domestic product (GDP), on different time horizons. The analysis signals that the S&P 500 might be overvalued, possibly undergoing a series of bubbles, since the 1990s. To understand this behavior, we introduce a model for bubbles, showing that they can be caused by a lack of correlations between stock prices and a virtual market index, the latter calculated self-consistently from the stock prices. We argue that variations, Delta gamma, in the "bubble parameter" (or decoupling factor gamma), are anticorrelated to variations of the Federal Funds Rate (FFR), which may trigger a bubble phenomenon (gamma -> 1) when persistent rate cuts become too pronounced. The FFR are confronted with the consumer price index (CPI) in the period (1955-2020) as an attempt to complete the picture. Our analyses suggest that the strong departure of the S&P 500 from historical fundamental trends within (1990-2020) may reflect the development of financial anomalies, in part related to monetary policies, which should be carefully addressed in the near future.