This article focuses on the price earnings ratio (PE) for the S&P 500 index. It is about more than just a number. It is about the valuation of the 500 largest companies in the US, whose aggregate value is currently in excess of $31 trillion. All publicly traded stocks are valued at over $36 trillion. It is also about the paradox of market efficiency and deviations from long-term norms.

At this time of year, many analysts are providing prognostications for the stock market based on PE analysis. Bulls, bears, billionaires, pundits and Nobel Prize winners are at it with the same gusto most of us have dipped into the punch bowl this holiday season. Most prognosticators declare that the PE is too high – we are in a bubble – and we are headed into a period of subpar returns for the next ten years. The trembling and fear are palpable.

Perhaps people are looking at the wrong PE ratio.

The market keeps moving up, largely because most credible forecasters are painting a rosy picture of the economy and in particular the growth of earnings of the publicly traded companies. That is what is important to focus on. My purpose is not to provide advice but rather give another perspective on different approaches to earnings valuation.

If you accept the efficient market hypothesis, you presume that the market is always fairly priced in the sense that all publicly available information is incorporated in setting current prices.

Can the market be in a “bubble” and be fairly priced at the same time? Might the current price equilibrium be fragile? If disequilibrium (breakdown of equilibrium) were to occur would it be more likely to the upside or the downside? Are there not times when errors of assumptions can be immaterial to market prices and at other times lead to large price adjustments?

Those paradoxical types of questions are routine to the world of investment research. New information (news) is by definition unknown and random and errors of judgement around that news abound. Fifty years ago, Burton Malkiel argued in his book *A Random Walk Down Wall Street* that these realities mean you can’t beat the market.

That is not to say the market doesn’t go up on balance, just as the US economy goes up on balance, but the period-to-period moves, both in the aggregate and on a stock by stock basis, are unpredictable. So, stock selection and market timing strategies that appear to result from skill are in fact a result of chance. Therefore these strategies do not work in the long run.

The process by which the market incorporates (or discounts) new information into prices is complicated because there are so many forces at work. Some are micro, some macro; some psychological and some empirical in nature. The stock market never remains in a state of sustained equilibrium as is found in the case for consumer goods and service. Prices are set by the invisible intersection of supply and demand curves. But those curves are shifting up and down every day – a constant tug of war of ideas that change unpredictably and constantly.

### How Important are Financial Markets to the Economy?

While we reduce the stock market to a single number each day – “the Dow is up 45 points today” – that simplification obscures the enormous importance of financial markets to the US economy. The stock market is at the heart of the American financial capitalist system, which encompasses vast debt and equity markets. Our financial markets are where capital is raised from investors to finance entrepreneurs. The results of these investments not only add to the balance sheet of the public in the form of pensions and investment funds but also increase the net worth of professional investors who make the decisions of whom to provide the capital to fund entrepreneurship.

The financial industry comprises investors (both public and private), brokers, investment banks, banks, insurance companies, real estate financiers, leasing companies, venture capitalists and private equity firms. This list is not exhaustive. Collectively, they comprise 21% of the value of GDP – that is the biggest component of our economy relative to any other industry sector.

Equity markets, that is the stock market, are the most visible of the capital markets. Cable TV programs and a plethora of publications provide a seemingly endless stream of comment on the stock market’s daily behavior and outlook. This report focuses on the market’s valuation of earnings.

Pundits, market technicians, and financial economists, including several Nobel laureates, have contributed to the vast literature surrounding the issue of how to assess the stock market. The different approaches to valuation range from technical analysis, to supply/demand analysis, normative valuation and relative valuation. Each has validity up to a point. But it must be obvious to the observer that none of these approaches are deterministic. Further, these methods to forecast market levels are not relevant to all time horizons. Some are appropriate to the short term, others to the longer term. Finally, all these approaches suffer from shortage of overt information – facts. Much has to be inferred because much is not directly observable.

### Price Level vs Value

The price level of a stock is a judgement of its value. The price level cannot be understood in the absolute. Value is relative and largely subjective. Furthermore, the valuation varies across all the participants in the markets.

Colloquially speaking we say that one price is high or that another is low. What we really mean is that it is *relatively* high – high relative to an alternative assessment of its underlying value or high relative to the value of another security. A stock price of $5,000 per share of one security is not high relative to another at a per share price of $50 if the value of the former is a hundred times greater than the latter.

In economics we summarize the collective behavior of buyers and sellers with supply and demand curves that intersect at a quantity and a price. The shape of the curves, particularly the demand curve, reflects the tradeoffs each consumer makes, subject to his budget constraints, between more of one thing at the expense of less of another thing. Loosely speaking, all other things being equal, the desire for more bonds implies less demand for stocks.

In financial markets, the supply curve, that is the supply of new securities, is highly inelastic in the short term. That is, the number of shares outstanding do not change much in the short term. The importance of that observation of fact is that it means changes in market prices are mostly due to changes in the demand curve.

### Is PE a Correct Metric?

Common ways to value stocks include measuring a stock price in relation to some underlying fundamental measure, such as the firm’s book value, dividend, cash flow, earnings, etc., on a per share basis. Again, the PE of the US stock market, as measured by the S&P 500, is our core concern,

Depending on the analyst, the price earnings ratio (PE) is usually calculated and judged relative to:

- trailing 12-month earnings, PE[-12],
- current earnings PE, or
- prospective earnings for next year, PE[+12].

In arriving at PE norms, analysts are divided over whether they should look backward, look forward, or look at current earnings. Historical earnings are known whereas forward earning involves a forecast. One would think that looking forward makes more sense. After all, is not the stock market forecasting coming prosperity or hardship? Is not the stock price index considered a leading indicator of the economy? The counter argument is that, though a forward looking PE[+12] may make sense, the market tends to return to historical norms PE[-12], because market returns are mean-reverting.

When asked if the market is too high, reference is made to one of these measures. Currently you might hear statements such as: “Oh, a PE of 35 is much too high relative to the long-term average of 15.”

Is it really that simple? NO!

### Alternative Measures of PE

Classic valuation theory holds that the price of a stock reflects the present value of the expected stream of income a company will generate over the long term. This *valuation formula* simplifies to the following equation:

P = E/(k-g)

where k is the cost of equity capital and g is the expected long-term growth of earnings.

The cost of equity capital, “k,” is also referred to as the total return investors expect a company to generate over time. It is a cost to the firm because it is the rate of return the firm must provide in order to attract investors to place their savings at risk in stocks rather than in other assets. The value of “k” is made up of two components: an interest rate that could be earned from a relatively riskless US government security (*i*), and an additional return component to compensate for the risks inherent in that company, or in all stocks in the aggregate (r). Therefore, “k” boils down to:

k = *i* + r

When looked at this way, the real PE or P/E (which we will hereafter refer to as PE*) is something a bit more complex than the three measures suggested earlier. Combining the two equations above, and solving for P/E, we get:

P/E = 1/(*i* + r -g) = PE*

This measure, PE*, looks through recent and one-year out earnings. The current calculated value of PE takes into account a long-term projection of earnings. The value of PE is justified by the future, not the present.

Let’s look at some graphs to get perspective. First, 135 years of the S&P 500 PE calculated conventionally, i.e. price divided by current earnings are shown in Chart 1a. In Chart 1b the underlying components are shown; that is, the price and earnings per share of the S&P 500. The later chart uses log scales to reveal trends.

It is clear from Chart 1a that conventional PE is very volatile. The long-term average PE is 16.3 with a plus or minus one-standard deviation (+/-8.8) band around the values encompassing a range of PEs between 7.5 and 25.1. In the hypothetical world of normally distributed numbers the plus-minus range encompasses 2/3rds of all observations. But as can be seen there have been many extreme moves in PE, such as during the Dotcom boom of 2000 and the real estate crisis of 2008 when conventional PE was over 100.

The price and earnings data in Chart 1b, while volatile at times (Great Depression and the Great Recession), show fairly steady growth.

Looking at Chart 1a, it is curious and notable that during the last 20 years, (2000-2020) the PE tended to be at a higher level than during the preceding 115 years (1885-2000). Furthermore, it has been far more volatile (it varied within a far greater range) in the last two decades than in the prior 115-year period.

This is important to understand. What has changed and why? What may have explained the higher level and the higher volatility of PE in recent decades?

We conclude that both the higher and more volatile PE is due to lower interest rates. Chart 2 plots the path of PE relative to the interest rate on a 10-year US Government bond.

The chart clearly shows a generally negative relationship between interest rates and the PE. But of course, this is what one would expect based on the equation for the PE* (above). The interest rate is in the denominator, so lower rates lead to a higher PE*.

The chart also reveals that at lower interest rates, the dispersion of PE increases; in other words, PE has historically been more variable during periods of low interest rates.

Despite the claims by pundits that the world is riskier, the reason behind increasing variability in PE*over the past two decades is in fact nothing more than a simple mathematical reality. All other things held constant, as *i* decreases PE* rises, but this relationship is not linear. As the level of “*i”* declines, “*i”* will produce larger and larger changes in in the level of PE*. This also means that small perturbations in “*i”* will result in larger perturbations in PE* when interest rates are low than when they are at high levels.

In econometric analysis, this is known as *h*** eteroscedasticity**, which occurs when a dependent variable is not constant across the range of values of an independent variable. In this case it is by a natural process that PE becomes far more variable as interest rates decline. This is more than just a pedantic point of interest. Many forecasts we come across fail to take account of this phenomenon and are misled, as a result, in their conclusions that conventional PE is too high and too variable.

Low interest rates are therefore responsible for increased volatility in PE, regardless of *why *rates are low. Pundits often cite Fed policy, for example, when explaining PEs and market bubbles. But the Fed can affect PE only indirectly, for two reasons. First, it only controls one, very short-term, interest rate, which may or may not affect longer-term rates, and then only by affecting inflationary expectations. Second, even if longer term rates are affected, the impact on PE is likely to be muted except during periods when low interest rates prevail.

Interest rate changes are driven by myriad factors that are not the focus of this paper. But it is clear that the inherent relationship between prevailing interest rates and PE must be considered when judging whether PE is “too high” or “too low.”

### Historical Returns to the S&P 500

How much has the investor in the S&P 500 made over time? By raising this question, we shift the focus from an earnings-based calculation to actual realized returns. We do so because it provides an alternate way to estimate “k.” The cost of equity capital can be derived from historical earnings data and the expected growth in earnings (as described in the *valuation formula above*), or it can be extracted from the actual total return performance of the market.

Chart 3 shows the 135-year record of total returns on a year-over-year basis; that is, the dividend plus the price appreciation as a percent of the price level 12 months earlier. Two things are revealed by the underlying data that may not be readily apparent in the chart: (1) other than during the Great Depression, the *volatility* of total return appears high but fairly constant, (2) *returns* are slightly lower during the 50 years prior to WW2 and slightly higher for the 80 years following the war.

Next we will consider the connection between the *valuation Formula* (discussed earlier) and the total return model as a means to estimating the value of “k.”

### Two Methods for Estimating “k,” the Cost of Equity Capital

What is the value of “k,” that is, the return required by the market?

By one method the answer is obtained using the *valuation formula* for P and PE*, which is dependent on measures of earnings growth (g) and the average earnings to price ratio (E/P).

By another method, “k” can be estimated using a *total return model*, that is, calculating the average annual return the market has generated for investors.

Both methods are valid but use different data. As a result, they will produce different results. These differences are understandable and will be discussed later.

*Valuation Formula*

First, rearranging the terms of the *valuation formula* and solving for “k” we get:

k = E/P + g

where E/P is the historical earnings yield, and g is the long-term expected growth in earnings.

From the data we calculate that the long-term (1885-Dec 2020) average earnings yield (E/P) is 6.15% and the long-term historical growth in earnings (g) is 4.37%. Using historical growth assumes that what has transpired in the past will continue into the future. Therefore, “k” is simply the sum of the two values, or 10.51%.

*Total Return Model*

In Chart 4 we show the cumulative compound monthly returns since 1885 (the red line). That is, what the pattern of cumulative growth would be if $1 were invested in the market in 1885 and every dollar of dividend received were also reinvested in the market thereafter.

Chart 5 is simply a “zooming in” on the post-war years (1945-Dec 2020) portion of Chart 4.

Both charts use logarithmic scales in order to fit the page and better show growth patterns. In logarithmic charts a constant growth rate appears as a straight line.

Though we could directly calculate averages for different periods, we chose to use regression analysis instead. Regression estimates provide more reliable and useful statistical measures. The slope of a fitted curve, using log values, is a derived estimate of the average annual total return.

Using regression analysis, over the 135-year period (1885-2020) investors have realized an average annual total return of 9.73%.

However, looking more carefully at Chart 4 we discern a noticeable change in the rate of return when comparing the pre-1945 period and post-1945 period. Fitting a curve to the two subsets of data, we find that prior to the end of WW2 the annual total return averaged 7.23%. After WW2, annual total return stepped up to 10.82%.

In both charts the dotted lines are constant total returns, or the regression lines.

### Comparing the Two Estimates of “k”

The value of “k” calculated by the *Valuation Formula*, i.e., directly from the data using the average for E/P and earnings growth “g,” are somewhat higher estimates overall and for the two subperiods used in the *Total Return Model*. The comparisons between the models and for the different periods are shown below:

Time Period | Valuation Formula | Total Return Model |

All 135 years | 10.51 | 9.77 |

1885-1944 | 8.63 | 7.25 |

1945-2020 | 11.64 | 10.87 |

It is not surprising the *Valuation Model* estimates for “k” are higher than the *Total* *Return Model* estimates. Prices and dividends are known with certainty, whereas reported earnings are accounting calculations that are managed and subject to revision.

For example, share buybacks have steadily reduced the number of shares outstanding since the mid-1980s. Prior to that, routine buybacks were illegal. In recent years buybacks have averaged nearly 3% of the shares outstanding. The effect of buybacks over the past 30 years has been to increase the per share calculation each year, hence boosting the growth rate and reducing the PE. Oftentimes buybacks were financed by new debt issuance and didn’t come out of earnings. In addition, earnings have not always been reported on a consistent basis using today’s FASB accounting standards. Furthermore, the more economically relevant concept of operating earnings is not the same as reported earnings. Reported earnings are generally higher than operating earnings.

So, while the estimates for “k” for the pre- and post-war periods for the two calculation methods are not the same, the pattern is similar. We are partial to the estimates derived from the *Total* *Return Model* both for the earnings accounting reasons explained above, but also because realized total returns are unambiguous – they are real. We take comfort from the fact that both calculations are of similar magnitudes.

Looking at Charts 4 and 5 again, what we see is that areas above the dotted lines are periods of relative overvaluation and the areas below the dotted lines are periods of relative undervaluation relative to the theoretical fair value, i.e. relative to the historical norm, derived from the *Total Return Model*.

In this context it is noteworthy that since the real-estate/mortgage-backed-securities crisis of 2008, the market has been undervalued relative to historical norms. Furthermore, as of December 31, 2020 the market is approximately back to theoretical fair value.

### Conclusion

Market pundits who focus on PE, PE[-12], PE[+12] have come to the conclusion that the US stock market, as measured by the S&P 500, is overvalued and due for correction. Some believe we are in a bubble and there will be a sharp correction. Others opine that from current levels we are headed for a low rate of market performance for the next 10 years. These views may turn out to be correct.

However, a more formal analysis of classic valuation theory that relates the current price level to the present value of the future stream of expected income suggests otherwise; that is, the market is more fairly valued and in keeping with its historical norms. From a statistical point of view, there is uncertainty around such a conclusion. Market returns routinely fluctuate. Two thirds of the time total returns realized in the market fluctuate by plus or minus 4.4% . Nonetheless, there is analytical support for the conclusion that from both a market efficiency point of view and from the perspective of long-term valuation theory, the market is roughly aligned with long-term norms.

At the risk of seeming repetitive, it should be stressed that the market trades within a range around its long-term fair value norms. For sure prices have trended for long periods below and above the historical norm. But it cannot be dismissed lightly that those historical norms are real and give every appearance of remaining intact for the foreseeable future.

The executives who manage US corporations are capable of generating strong steady growth. It is truly remarkable that despite all the challenges of international competition, American entrepreneurs and labor have shifted from industries with declining prospects to industries with brighter prospects; from raw material-intensive industries such as mining and heavy industry to more intellectual property-dependent industries in the computer, telecommunications, internet, medical and services industries; and from dependency on domestic sales to global sales.

Though US industry has been buffeted by domestic and international economic storms, history shows with high certainty that entrepreneurs steady the ship and get back on track and generate returns consistent with the historical norms. From time to time the unexpected occurs which would appear to change the paradigm of what is possible. Old industries die off; rust belts emerge. And then came the commercialization of the internet, which has created seemingly limitless new opportunities that have changed the growth path upward on a sustained basis.

At other times government intervention in the marketplace has both disrupted and created doubt about the ability of executives to ever get back on track and catch up. That is particularly the case when government intervention impedes mobility of capital, the efficient allocation of resources, the ability to exploit opportunity, or changed incentives in a way that leads to malinvestment. When those impediments are removed, or circumvented, optimism returns, and the economy gets back on track.

Despite the severe disruptions and ‘creative destruction’ that occur from time to time, the returns to capital that fuel innovation and prosperity remain strong.

So, to recapitulate, the market is always efficient and fairly priced relative to what is known, but the market can also be above or below its long-term enduring norms for good and rational reasons. In the end the historical norms prevail.