Valuation Measure of the Stock Market using Stochastic Volatility and Stock Earnings

TL;DR

<t>The paper develops a stochastic-volatility framework to reframe stock-market valuation beyond the traditional CAPE by introducing a mean-reverting valuation measure $H(t)=\ln W(t)-\ln \overline{E}(t)-ct$, where $W$ is wealth and $\overline{E}$ is trailing earnings. It jointly models three asset classes (US stocks $Q$, international stocks $I$, and US corporate bonds $B$) with four factors: annual volatility $V$, BAA rate $R$, term spread $S$, and earnings growth $G$, using $V$ to normalize returns into IID residuals and constructing a comprehensive, regression-based simulator. The authors prove stationarity for the extended model under specified conditions, provide detailed data preprocessing, and implement an online simulator to study retirement scenarios and rule-of-thumb withdrawals, illustrating practical implications for asset allocation and risk of ruin. The work offers a tractable, testable framework that connects valuation, macro factors, and wealth dynamics, with public code and data to support further research and practical retirement planning.}

Abstract

We create a time series model for annual returns of three asset classes: the USA Standard & Poor (S&P) stock index, the international stock index, and the USA Bank of America investment-grade corporate bond index. Using this, we made an online financial app simulating wealth process. This includes options for regular withdrawals and contributions. Four factors are: S&P volatility and earnings, corporate BAA rate, and long-short Treasury bond spread. Our valuation measure is an improvement of Shiller's cyclically adjusted price-earnings ratio. We use classic linear regression models, and make residuals white noise by dividing by annual volatility. We use multivariate kernel density estimation for residuals. We state and prove long-term stability results.

Figures (7)

• Figure 1: On the left: Shiller cyclically-adjusted price-earnings (CAPE) ratio on the log scale versus the new valuation measure proposed in Valuation. They track each other before 2000, but after that they diverge. On the right: annual realized volatility for the Standard & Poor.
• Figure 2: The long-short (10 year minus 3 month) Treasury spread, and the Moody's BAA rate: Average daily December data.
• Figure 3: Top: The quantile-quantile plot for $Q$, and the ACF for $Q$ and for $|Q|$, where $Q$ is S&P returns. This shows $Q$ are IID but not Gaussian. Bottom: The quantile-quantile plot for $Q/V$, and the ACF for $Q/V$ and for $|Q/V|$, where $Q$ is total S&P returns. This shows $Q/V$ are IID Gaussian.
• Figure 4: The new valuation measure, computed in this article, using end-of-year S&P data instead of January average S&P data, and 10-year instead of 5-year averaging window. It closely tracks but is slightly different from the new valuation measure on Figure\ref{['fig:compare']}. However, it shows the three main peaks in 1920s, 1960s, and 1990s. As of 2024, the stock market is not overvalued.
• Figure 5: Top: The quantile-quantile plot for $G$, and the ACF for $G$ and for $|G|$, where $G$ is S&P returns. This shows $G$ are not IID and not Gaussian. Bottom: The quantile-quantile plot for $G/V$, and the ACF for $G/V$ and for $|G/V|$, where $G$ is total S&P returns. This shows $G/V$ are IID Gaussian.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof