The Mystery of m

The dividend futures-based model we invented to project the potential future trajectories the S&P 500 (Index: SPX) starts from a very simple observation:

Ap = m * Ad

In this relationship, A_p represents the change in the rate of growth of stock prices and A_d is the change in the rate of growth of dividends per share. The value m is an amplification factor that varies over long periods of time but can be nearly constant for short-to-intermediate periods of time.

Since we first formulated this relationship in April 2009, we’ve found that short-to-intermediate periods of time can be as long as decades. But eventually, the value of m does change and whenever it does it’s a big deal because it means the market regime in which stock prices are set has changed.

The following chart tracks how the value of m has changed from January 2014 through the end of November 2025, which covers the period after we first developed the alternative futures chart we use to visualize the dividend-based model’s projections. Our initial observations that set the value of m = 5.0 however go back to March 2010, when dividend futures as we know them today became a reality and made that estimation possible. We should also note that m was almost certainly at that same level for years before that point in time.

So what is m really?

A potential solution to that mystery was advanced by Xavier Gabaix and Ralph S.J. Koijen in their June 2021 working paper In Search of the Origins of Financial Fluctuations: The Inelastic Market Hypothesis. For us, this paper immediate leapt to the front of the pack for its potential explanatory power of what m represents because of a simple example they developed to explore one of their propositions. Here is a screenshot of the proposition:

Here is their example:

To think through the economics of Proposition 3, we found the following simple, undergraduate-level example useful. Suppose that there are just two funds: the pure bond fund and the representative mixed fund, which always holds 80% in equities (the magnitude suggested by Figure 1). Then, theta = .08, kappa = 0, so that zeta = 1 – zeta = 0.2 and and 1/zeta = 5. Then an extra 1% inflow into the stock market increases the total market valuation by 5%.

Or to put it more simply, a multiplier of 5 for this simple example, which puts it in the right ballpark for our observations.

It certainly is an intriguing possibility, especially if it can explain for how the value of m has changed in the period since 19 February 2020, during which the value of m has held at various constant levels for much shorter periods of time.

References

Xavier Gabaix and Ralph S.J. Koijen. In Search of the Origins of Financial Fluctuations: The Inelastic Markets Hypothesis. National Bureau of Economic Research Working Paper 28967. [PDF Document]. June 2021.