My Finance

My undergraduate student Angel Piotrowski replicated my previous unpublished work A New Stock Market Valuation Measure with Applications to Retirement Planning (available at arXiv:1905.04603) with annual volatility which she computed previously. Here I try to explain my research. In a further post, I will explain Angel’s contributions.

In this manuscript, I considered total Standard & Poor 500 returns (and their predecessors before 1957) taken from Robert Shiller’s data library (published on my web site) and compared them with annual earnings growth. Often, price-earnings or price-dividend ratio is used to analyze the stock market: If such ratios are high the market is overvalued, like at the top of the dotcom bubble. The average price-earnings historical ratio is around 15-20. So if this ratio is much more than 15-20, the market is overvalued. This is classic research by John Campbell and Robert Shiller, which won Nobel Prize in Economics. They used 10-year trailing averaged earnings to reduce noise and adjust for recessions and expansions (which take on average no more than 10 years), and their ratio is called cyclically adjusted

But comparing only prices with earnings might not be enough: In the XXI century, earnings recently are used for buybacks which raise prices, not dividend payouts. This artificially increases prices but does not mean overvaluation. Use of price-earnings or price-dividend ratios will incorrectly show the market is overvalued. The true comparison must be between stock returns (total, including dividends) and earnings growth. Historically, earnings growth is ~2% per year, and total market returns are ~6-7%. Here we consider only real (inflation-adjusted) returns and growth terms.

Thus if we take the difference and it is much higher than 4-5% per year for a few years, then the stock market starts to be overvalued. We formally can write this as an autoregression of order 1 for the cumulative sum after subtracting the trend where . Our bubble measure is and it is high when the market is overvalued. Let us write linear regression:

After fitting this as multiple linear regression, we get: We can see that the estimate for is within this 4-5% range. The Student test for gives so we can reject the random walk hypothesis: The model is stationary after detrending. Moreover, we can apply the Student test, since the residuals (innovations) are well modeled by independent identically distributed Gaussian with This is shown by the autocorrelation function plot for and the autocorrelation plot for as well as the quantile-quantile plot for versus the Gaussian distribution. See Figure 5 from the article.

Here we used 5-year averaged earnings, similarly to Campbell-Shiller’s 10-year averaged earnings. But we also do a separate analysis for simple annual earnings, simple annual dividends, and 5-year averaged dividends. The results fit the best for 5-year averaged earnings. The best means the residuals are closest to independent identically distributed normal.

Previously, Campbell and Shiller did not do analysis for residuals: whether they are truly distributed as independent identically distributed normal. Unfortunately, this is common in economics research. But I did this here.

According to the analysis, the classic cyclically adjusted price-earnings ratio in 2024 is very high compared to the historical average. Thus the classic analysis shows that the Standard \& Poor 500 is overvalued, almost like the dotcom bubble. But our new stock market valuation measure is not historically high. So our own measure does not show the market is overvalued. The current situation is not like the top of the dotcom bubble. See Figure 7 in the article.