From the perspective of re-engineering the CAPM, Sections 1 and 2 are best seen as preparatory. Section 1 determined the best possible earnings to use to calculate earnings yields (EY). Section 2 laid the groundwork for modeling the various risks equities and Treasuries face. With this done, we are now ready to re-engineer the CAPM.
The equity risk premium (ERP) is at the core of the CAPM because it quantifies the extra return that investors demand to carry the risks inherent in equity ownership. Its calculation is straightforward for us12 because of the way we have calculated the earnings yield. To recap, the dispersion adjustment raises the earnings to compensate for both dispersion and real growth. Therefore, the perceived EY corresponds to what the EY of the S&P 500 would be if it were zero real growth and had no dispersion. Said differently, we have converted the 500 stocks of the S&P 500 into a single equivalent perpetual bond generating a real yield equal to the earnings yield of the day.
In light of this, the ERP is simply the difference between this real yield and the real truly riskfree rate (RTRR). In contrast to traditional ERP calculations, no growth parameter is required here because growth has already been accounted for in the dispersion calculation.
The objectives of this section are threefold. First, to extract the RTRR out of the TIPS by removing the risks embedded in the TIPS to get to a truly risk-free rate. Second, to explain the ERP by quantifying the risk factors that drive it. Third, to do so in a way that (i) minimizes the overdetermination issues we discussed in Section 2.3; (ii) reflects the divergence of the asset classes as time goes by; and (iii) reflects the multi-frequency nature of the risks being priced in as they are increasingly recognized over time.
Exhibit 38 shows the position at the eve of this process, at which the ERP is calculated off the pre-tax TIPS rather than the post-tax RTRR, and while there is not yet an explanation for its variations.
The approach section that follows is very detailed, as we lay bare the details of the reengineering process. But from 30,000 feet, once we have done the great deal of preparation required to get to the correct EY and RTRR, the ERP model that links the two is surprisingly simple. A good analogy for the model is a locomotive pulling risk premia railcars. The locomotive has been pulling three railcars since 1871. 14 As time goes by, the train gets longer. A fourth railcar is added during the interwar period. Two more join in the early postWorld War II period, and the seventh, and the last to date, in the 1970s. Furthermore, the railcars come in three classes. We have three high-frequency railcars, three mediumfrequency railcars, and one low-frequency railcar.
Approach
Foreword
Large quantitative models can often appear like black boxes to outside observers, which fuels concerns of look-ahead bias and overfitting. To alleviate those concerns, we compile Appendix E, which sheds more light on what we do than the main text below, and shows why our equity model does not suffer from these issues.
Overview
Any successful re-engineering of the CAPM starts with the removal of high-frequency equity risks. There are two reasons for this:
- Such risks are confounding factors that drive those rapid variations in risk aversion that move equity markets in the short term. They hide what we seek to uncover in our focus on secular forces.
- High-frequency risks suffer much less from overdetermination than their lowerfrequency counterparts, so removing them early on simplifies the modeling challenge.
But there is no way to directly reconstruct what risk aversion was in every single of the 1,66215 months under observation, so we are forced to reverse-engineer that monthly risk aversion rather than observe it directly. We do this primarily by studying the behavior of other risk assets, which provide clear pointers of the market mood in a particular month. More specifically, we look not only at bond spreads and the VIX, but also at Treasuries, which, as we established earlier, are not risk free.
Turning to overdetermination, high-frequency risks are much less sensitive to it because they are usually (but not always) highly ergodic (please see Section 2.3 and Footnote 100). In layman’s terms, this mathematical concept means that a time series comes close to its extremes quite frequently. In turn, this makes it easier to determine whether a correlation is causal or not. We show below that bond spreads, the VIX, the high-frequency part of Treasuries, and consumer confidence all follow this general rule, which somewhat simplifies our task. Having removed high-frequency risks from equities, we follow a similar process to remove them from TIPS.
Turning to medium frequency risks, we start by removing them from equities. Thereafter we remove them from TIPS, which is the core step in establishing the RTRR. And then, finally, we review the low frequency risk factor in the equity risk premium.
Outcome of the CAPM Re-engineering Process
At the eve of entering the process of re-engineering the CAPM, we assessed the do-ability of the endeavor. Exhibits 58 and 59 show how poorly the traditional CAPM explains historical reality, particularly with respect to the period between 1920 and 2010.
The degree of empirical underperformance is such that the possibility of the CAPM being irredeemably flawed had to be considered. Could it be that the CAPM is an academic construct of no practical value? Fortunately, our findings show that the CAPM is conceptually sound and its failure is more apparent than real. This apparent failure is driven by a long list of execution shortfalls that can be corrected. To summarize: (a) the earnings need to be adjusted to be fully consistent over a century and a half, based on the Warren Buffett accounting standard rather than GAAP, and put on a perceived earnings footing;18 (b) the nominal Treasury rate needs to be replaced by the RTRR; and (c) investor personal taxes need to be accounted for.
Furthermore, the correctly calculated True North ERP is well explained by a four-factor model. The first factor focuses on three high frequency risks;19 the second and third factors focus on the two medium frequency risks of the Ulcer Index and distance to Goldilocks respectively; lastly, the fourth factor focuses on the 85% rather than 100% passthrough of the RTRR to EYs. The model currently lacks its fifth factor, a quantified bubble indicator, which is why it underexplains periods such as 1929 and the tech boom. This factor will be added in due course.
Exhibits 60 and 61 compare the results based on the Fed Model with those based on our reengineered CAPM. The latter reduces the standard deviation of the error by two thirds overall. 20 This will get better when we introduce the fifth factor, because we currently massively under-explain equity prices during both the late 1920s and the tech boom.
Expressed in another way, the annual average price per share in 2021 is 38 times of what it was in 1871. We explain this variation with four drivers, going from the most ‘obvious’ to the most ‘subtle’:
- Secular sales per share: The 2021 average is 24 times of the 1871 average.
- Perceived margins: They are the other component of perceived earnings besides secular sales, and in the post-WWII period, they have been as low as 4.0% and as high as 9.5%.
- The post-tax RTRR: It has been at the all-time low of -0.5% in 2021 and reached as high as 4.1% in 1921.
- The Equity Risk Premium core of the model23: See Sections 3.2.3, 3.2.4, and 3.2.6 above for details of the four factors of the model.
Exhibits 62 and 65 below highlight how our explanation of price improves when we add back the RTRR, perceived margins, and secular sales per share to the core ERP model. The core ERP model has an R^2 of 80%; adding the impact of the RTRR increases the R^2 to 82%; adding perceived margins further increases the R^2 to 96%24 . And finally, adding back the 24x increase in secular sales per share increases the R^2 to 99% in the logarithmic space, via the mechanical effect of introducing the long-term growth trend.