Rational Pessimism and Tobin’s q

I found out about Rational Pessimism: Predicting Equity Returns using Tobin's q and Price/Earnings Ratios only in the last few days. I am impressed. It is available from Duke University. It was written by Matthew Harney and Edward Tower. It was written for “The Journal of Investing, Fall 2002.”

Rational Pessimism by Harney and Tower

My single complaint is with Salomon Smith Barney and Citigroup. They allowed the article to be published with a standard disclaimer. They should have bragged about it.

In this review, I show how this fits into my own research.

Tobin’s q

Tobin’s q is a market measure similar to price-to-book-value. It is the ratio of the value of the stock market and corporate net worth, where the net worth is based on the replacement cost of physical assets plus financial assets minus liabilities.

Their Findings

A. The writers investigated S&P500 (real, annualized, total) returns as a function of Tobin’s equity q (which differs slightly from Tobin’s Q) and P/Ex, using x = 1, 10, 20 and 30 years. They examined timeframes of 1, 5, 10 and 20 years.

The best fits were, in order, q, P/E30, P/E20 and P/E10.

P/E1 had almost no predictive power.

The investment returns covered the end of 1900 through the end of 2001. Earlier years were used in the calculation of P/Ex.

B. The writers introduced a simple feedback mechanism, which they refer to as momentum. They added an adjustment to their predictions based on recent history.

The greatest merit from this part of their investigation is that they measured the timeframe and impact of such memory. Although the details of the model influence results somewhat, their key findings are worth knowing.

When projecting 5 to 10 years into the future, any excess performance from the previous decade carries over, almost one-to-one, into the next decade. As the timeframe increases, this effect diminishes. It disappears when predicting 20 years into the future.

[Refer to footnote 5 in the article for additional details.]

C. The writers calculated scaling equations.

Since q is similar to a price-to-book ratio, an equation of the dividend-to-book ratio would have the form (dividend/book) = (dividend/price)*(price/book). Assuming that the dividend-to-book ratio is stable (similar to a payout ratio), (dividend/book) = constant (approximately) = (dividend yield) * q.

This allows us to calculate an equivalent for the dividend yield in the Gordon Equation. They determined the calculated value to use in the equation: the (real) Investment Return = dividend yield plus the long-term growth rate of earnings = (a calculated value)/q + 1.5% per year.

Similarly, they calculated the price of the S&P500 equivalent to this value of q. They determined the second calculated value to use in the equation: the (real) Investment Return = (the second calculated value)/P + 1.5% per year where P is the current value of the S&P500 index.

My Research

My research differs in two important respects:

1) I do not use P/E10 directly. I use the percentage earnings yield, 100/[P/E10].

2) Almost all of my calculations include withdrawals or deposits.

Long ago, I discovered that the percentage earnings yield works better than the price-to-earnings ratio. Not only that, I was able to identify why: earnings yield serves as a proxy for dividends in the Gordon Equation. An additional benefit of using earnings yield is that it captures the effect of prices in the Speculative Return. Using earnings yield captures the main variables in both the Investment Return and the Speculative Return. The remaining term, the growth rate, has been relatively stable.

In addition, I have discovered that the relationship between stocks and the percentage earnings yield 100/[P/E10] in the earliest years contained in the database differs from what happens later. Gummy (Professor Peter Ponzo) was able to narrow this further to the first decade in the twentieth century.

New Data

I collected new data. Although it leaves much to be desired, I am able to make some early observations.

My initial data source was:

Gold-Eagle Web Site

My preferred source will be the data from Andrew Smithers and Stephen Wright of Smithers and Co. Ltd. Although I have downloaded data, I am not sure about which values to use.

Smithers and Co. Ltd q and FAQs

New Safe Withdrawal Rate Investigations

I calculated 30-Year regression equations for HSWR50 and HSWR80 using 1945-1975 values of the percentage earnings yield 100E10/P, Tobin’s q and 1/[Tobin’s q].

Using the percentage earnings yield for x, the calculated withdrawal rates were:

HSWR50 = 0.5414x + 1.926 with R-squared = 0.8627.
HSWR80 = 1.0066x – 0.3161 with R-squared = 0.8212.

Using Tobin’s q for x, the calculated withdrawal rates were:

HSWR50 = -0.045x + 8.2677 with R-squared = 0.752.
HSWR80 = -0.0834x +11.464 with R-squared = 0.7125.

Using 1/[Tobin’s q] for x, the calculated withdrawal rates were:

HSWR50 = 1.4471x + 2.8631 with R-squared = 0.7679.
HSWR80 = 2.3898x + 1.8525 with R-squared = 0.6872.

Because there is only one set of Historical Surviving Withdrawal Rates, we can make direct comparisons using R-squared values. In all cases, the percentage earnings yield had the highest value of R-squared.

The best choice when calculating Safe Withdrawal Rates is the percentage earnings yield, 100/[P/E10].

As an excursion, I calculated withdrawal rates using P/E10 for x. To my surprise, the fit was excellent. P/E10 varied from 8.9 through 24.1. The curvature of the data (as opposed to the straight line regression equation) is readily apparent.

HSWR50 = -0.2394x + 9.3958 with R-squared = 0.8622.
HSWR80 = -0.4455x + 13.579 with R-squared = 0.8221.

Single Decade Projections

I calculated the 10-Year real, annualized, total return of the S&P500 for 1945-1980. [There was no issue involving completed sequences.] I determined regression equations using the percentage earnings yield 100E10/P, Tobin’s q and 1/[Tobin’s q].

Using the percentage earnings yield for x, the calculated return was:

Return = 1.8704x – 6.804 with R-squared = 0.4506.

Using Tobin’s q for x, the calculated return was:

Return = -0.211x + 18.686 with R-squared = 0.5615.

Using 1/[Tobin’s q] for x, the calculated return was:

Return = 5.7118x – 4.7893 with R-squared = 0.4485.

Because there is only one set of stock market returns, we can make direct comparisons using R-squared values. Tobin’s q was best. The percentage earnings yield and 1/[Tobin’s q] were almost identical.

Visual inspection of the Tobin’s q plot shows a data dropout problem not apparent in the percentage earnings yield plot. There is a clustering in the 1/[Tobin’s q] plot, with data either above or below the regression line, indicative of an unexplained relationship as opposed to randomness.

Twenty-Year Projections

I calculated the 20-Year real, annualized, total return of the S&P500 for 1945-1980. [There was no issue involving completed sequences.] I determined regression equations using the percentage earnings yield 100E10/P, Tobin’s q and 1/[Tobin’s q].

Using the percentage earnings yield for x, the calculated return was:

Return = 1.566x – 4.9989 with R-squared = 0.8009.

Using Tobin’s q for x, the calculated return was:

Return = -0.147x + 14.635 with R-squared = 0.6908.

Using 1/[Tobin’s q] for x, the calculated return was:

Return = 4.5775x – 2.8958 with R-squared = 0.7303.

Because there is only one set of returns, we can make direct comparisons using R-squared values. The percentage earnings yield 100/[P/E10], was best. Next was 1/[Tobin’s q]. Tobin’s q came in third.

Data Analysis

My initial investigations show that Tobin’s q has considerable promise for predicting stock market returns ten years into the future. Somewhat bothersome, the best estimates at year 10 use q in the numerator. Theory tells us that it should be in the denominator.

There are other artifacts that tell us to proceed with caution. Regardless, Tobin’s q is promising. It deserves further consideration.

My initial investigations tell us to stick with the percentage earnings yield 100/[P/E10] when calculating Safe Withdrawal Rates and when calculating 20-Year returns.

My excursion using P/E10 directly (instead of 100/[P/E10]) is bothersome. The curvature of the data is readily apparent. But it is not captured by R-squared.

Additional Remarks

The report is excellent. It needs to include confidence limits along with its projections as do all academic studies. Adding something meaningful would be straightforward.

The section involving the return from buy-and-hold could have been improved. It uses the Investment Return portion of the Gordon Model. It needs an adjustment for the Speculative Return. In spite of this, the writers successfully determined scaling equations for updated values of q and for updated values of the S&P500 index P.

The feedback measurement is quite valuable. It supplies an adjustment that narrows down the range of year 10 projections. Look up the most recent 10-year (real, annualized, total) return of stocks and compare it with the historical average. Add (or subtract) the surplus (or deficit). Guess what? Today’s adjustment is very close to zero.

Be sure that you understand the nature of the feedback model. In essence, it matches historical returns with the best fixed period oscillator. Do not depend on the model for fine grain details. But as a first order adjustment, it is outstanding.

Have fun.

John Walter Russell
August 4, 2006