We examine the models of 2013 Nobel laureate Robert Shiller and see what predictive role they have in the performance of the S&P 500.
The joys of receiving the 2013 Nobel Prize for economics – shared between Eugene Fama, Robert Shiller and Lars Peter Hansen – may well have been tempered. The ideas of Shiller and Fama on how financial markets behave are about as far apart as they get. Fama made his reputation exposing the economic logic, the efficiency of the market. Shiller made his attempting to prove the opposite.
We consider below just how useful price/earnings (PE) ratios are for market timing decisions. We ask whether the Shiller approach has merit – it continues to receive attention from market timers – or whether Shiller in his call that the S&P 500 market was greatly overvalued in 2000 (as it subsequently proved to be), was just lucky enough to be in the right place and time to draw favourable attention to his work.
Bubbles are good for some
Shiller’s analysis of an overbought stock market in 2000 was based on apparently unsustainably high price-earnings ratios for the S&P 500. He used a 120 month (10 year) moving average of real (CPI deflated) earnings as the denominator and real share prices, represented by the S&P 500 for which he derived data going back to 1871, as the initial starting point for the analysis. The Shiller PE is also described as the Cyclically Adjusted Price Earnings ratio (CAPE).
In the figure below, we compare the Shiller price/smoothed earnings multiple to the conventional measure using reported or what are often called trailing earnings. The two series differ most dramatically during the Global Financial Crisis (GFC) of 2008 when prices held up, even though reported earnings had collapsed. Shiller earnings, being a 120 month moving average, moderated this fall in earnings, so leading to a much lower multiple in 2008-09 and a much higher one therafter, given the recovery in trailing earnings. As may be seen in the chart below, the current Shiller PE multiple is now significantly more demanding than a multiple based on reported earnings.
The problem with a moving average when earnings collapse
The application of a 120 month moving average to estimate earnings means that the low level of reported bottom line S&P earnings in 2009-10 will continue to drag down Shiller earnings, relative to the much higher subsequent reported earnings, for 10 years thereafter. The current Shiller PE ratio is 24.42, while the conventional trailing PE is 19.43 (see Figure 2). The long term average (1871- 2013) monthly Shiller PE ratio is 16.5 and the conventional PE has a long run average of 15.83.
Incidentally, in contradiction to the Shiller notion that share prices are more variable than reported earnings – indicating some degree of irrationality in valuations – the opposite has been true since the late 19th century using the data suppled by Shiller. The average move in the real S&P has been 3.56% a year, calculated over 1578 months (over 131 years), while the average annual growth in real reported S&P earnings has been 6.86% over the same period.
Since 1960, the average real price move has remained at 3.6% a year while the average growth in earnings has been higher, at 11.6% a year. The standard deviation (SD) of real price moves, the conventional measure of volatility, has been 16.09% a year since 1960, while that of real earnings growth has been much greater, a staggering SD of 78.5% (influenced by the post GFC collapse in earnings and changes in accounting conventions that now discourage any protection of the bottom line that might have helped smooth earnings in the past).
The Shiller theory of value and the evidence – problems with a price earnings model that does not revert to some consistent long term average
The Shiller theory is that the Shiller price to smoothed earnings multiple will provide a superior method for recognising an over- or undervalued share market than the conventional measure of a price to earnings ratio. Or, in other words, comparing the Shiller PE to its long run average can assist market timing decisions. The problem for those using either measure is that neither of them can be regarded as reverting over time to their long term averages. The ability to revert (called mean reversion by econometricians) can be calculated – and it can be very clearly shown that these ratios do not converge to long term averages. These PE multiples can remain well above or well below their long term averages for long periods of time. There can thus be little confidence, based on the statistical evidence, that the PE multiple will consistently gravitate to some long term average within some operationally useful period of time for investors to time entry to or exit from the share market
Using either measure of the PE multiple, conventional or Shiller, the market appeared to be as significantly over valued long after Fed chairman at the time, Alan Greenspan, spoke memorably in 1996 of irrational exuberance. The S&P and the price to earnings multiples moved significantly higher for another four years.
It is expected earnings that drive value
It should be appreciated that investors use past performance, as represented by realised earnings, as a proxy or starting point for expected earnings when undertaking any valuation exercise. Future earnings will determine future valuations and the path of future earnings may well be expected to diverge from past earnings for any number of reasons – for example underlying economic growth can realistically be expected to gain or lose momentum for an extended period of time, adding to or subtracting from expected profits.
Unforeseen economic policy events, taxes, regulation or even natural events can alter the present value calculations that investors make. They may view the economic outlook with more or less confidence. The less their confidence, the more risk they will attempt to allow for when they buy or sell assets and so the higher or lower the discount rate or equity risk premium they would use to calculate a present value for a stream of benefits. The real cost of capital may well change as global and domestic propensities to save increase or decline, given degrees of capital mobility.
The future may well be different and investors can rationally hope to benefit from the difference. Most assets, including the real plant and equipment that corporations invest in, have a natural limited economic life. Earnings that might add value to an asset if it survives beyond a twenty year horizon cannot add much present value.
It can be demonstrated that small adjustments to two key influences on the price earnings multiple can drive the PE ratio dramatically lower or higher. These are in the required rate of return (or the cost of capital used to discount expected earnings) or in the expected growth rates in earnings themselves. Even small changes in the discount rate or the expected growth in earnings or dividends can have an explosive impact on the PE ratio.
Time will tell whether investors were too optimistic or too pessimistic about these forces that drive valuations. Market prices set in advance may be proved wrong by subsequent market moves – but this would not prove the valuation process as irrational. The unexpected might well prove the norm to which valuations adjust. This is a point Eugene Fama would no doubt make in response to accusations of market inefficiency – because markets may appear to change their collective minds and forecast poorly. It does suggest however that forecasting share prices is difficult to do successfully and also that using a simple metric like the price earnings multiple – compared to its long run average – is not the holy grail of share market valuations.
The growth in earnings and share prices
As an alternative to the PE ratio, we examine below the relationship between the annual growth in earnings and the annual growth in share prices. It seems reasonable to expect the relationship between the growth in earnings and prices, or between price moves and earnings moves, to cancel out over a period of time. Either prices catch up with faster earnings growth or, vice versa, earnings growth can catch up with higher share prices, so closing the gap between growth in prices and growth in earnings. If earnings fail to grow as expected, prices will then tend to retreat, thus closing the gap between price and earnings movements.
We examined the difference between annual price movements in the S&P Index and the simultaneous growth in earnings, both conventional and Shiller. The results of the exercise are pictured in Figure 4 below.
These differences in the annual growth in share prices and the growth in earnings do appear to revert to zero over time, though not necessarily in any regular way, either within the same year or even over the next few years. This suggests that it may well take a long time for the adjustment process of share prices to corporate performance or the other way round, to work out. The annual differences between the growth in share prices and earnings have an almost random, rather than persistent, character and so these differences in growth rates provide little notice of whether the current gap between the recent growth in prices and earnings will subsequently widen or narrow.
When we compare the growth in the S&P over 29 consecutive five year periods, with the growth in earnings over the same five years, we do get a somewhat better statistical fit. However the results are not convincing: even if we could successfully forecast earnings and earnings growth over the next five years we could not be confident that we would derive accurate predictions of the stock market . In the figure below, we show the results of such an exercise for the five year changes in the real S&P and real S&P reported earnings since 1871.
Connecting the level of the market to the level of earnings
While the PE ratio compared to its long run average provides little help in predicting the direction of the market , we may be able to gain some insight by comparing the level of the market with the level of earnings using regression analysis, especially if we add the influence of interest rates to the description of the level of the market relative to reported earnings. The difference between this approach and those of Shiller or conventional PE ratios to indicate an over- or undervalued market, is that it allows for a discount rate to be added to earnings or dividends as an additional explanation of value. The opportunity cost of holding equities in the form of interest income foregone surely influences the prices investors are willing to pay for a share in a listed company.
Adding long term US interest rates to both of these equations improves the fit of these models. The interest rate variable included in these models is the yield on a 10 year US Treasury Bond, for which data is also available back to 1871. The interest rate betas have the predicted negative association with share prices and are statistically significant.
The equation drawn, sometimes described as the Fed Model, with the natural log of the real S&P explained by the log of real S&P dividends and long term interest rates since 1880 and compared to actual values, is shown below. Judged by this model, the S&P 500 is currently about 9% undervalued.
By any conceivable measure of past performance, the S&P 500 was greatly overvalued in 2000
Utilising any of the variations of the Fed model, the US equity market was very demandingly valued in 2000. Using the Shiller definition of earnings as the explanatory variable without interest rates, a regression equation run over the period 1880-2000 indicates that the real value of the S&P was 2.7 times its value predicted by the Fed model with Shiller earnings in early 2000. Substituting reported earnings for Shiller earnings in this PE model indicates a market 2.2 times above that predicted by real earnings.
The problem for the investor utilising the Fed model is that in mid 1996 the S&P was already 1.9 times its value with Shiller earnings and 1.6 times its value, as predicted by reported earnings. Utilising either the Shiller earnings approach or reported earnings would have told essentially the same story. Yet the demanding valuations persisted for many years.
The case for dividends rather than earnings as the measure of past performance
It can be argued, especially taking into account the recent earnings turbulence, that real S&P dividends per share are a superior measure of past performance to reported S&P earnings. Definitions of bottom line earnings may change over the years as accountants revise their conventions. Dividends are much less complicated: they are simply cash in money of the day paid to shareholders. Buying back shares, subject to changes in financial fashion, serves the same purpose but does not show up immediately as dividends per share.
Reported dividends per share held up much better than trailing earnings per share through the GFC. They therefore provide a much more realistic estimate of realised corporate performance and perhaps also the performance expected by management, than bottom line earnings that were so much affected by the write offs of financial institutions (see Figure 8 below).
Regression models of the real S&P 500 over the period 1960-2013, using either real trailing earnings, real Shiller earnings or real dividends as the explanatory variable, together with the 10 year Treasury yield, indicates that the S&P 500 is currently under rather than overvalued: by 13% using real Shiller earnings and long term interest rates; fairly valued using trailing real earnings and interest rates; and as much as 32% undervalued using real dividends per share, when combined with interest rates.
Statistical issues with these regression equations
The statistical issues with these linear regression models, using levels of earnings or dividends with interest rates to explain the market, is that, as with the Shiller or conventional PE, the under- or overvaluation identified by the models are persistent. One of the essential conditions for unbiased regression estimates is that the error term, that part of the dependent variable not explained by the model, should have an expected value of zero and be normally distributed about zero (that is the regression estimate should have the same probability of being above or below the estimated value).
This is not the case with these models, which reveal what statisticians would describe as serial correlation and therefore do not provide unbiased estimates of the relationships identified. When this persistence of errors occurs, the results of any linear regression analysis and the validity of the estimated coefficients are compromised. As we indicated earlier, this persistence of Shiller’s PE away from its long run average is subject to the same bias. Lars-Peter Hansen was awarded his share of the Nobel Prize for providing methods to overcome the serial correlation and many other problems caused when the data is compromised.
How to apply these models, with their statistical weaknesses recognised
It is best to regard these models not so much as helping time entry or exit from the market (trading models), but rather as a way to interrogate and perhaps understand the level of the market and the earnings and interest rate expectations implied by current market valuations. If the model suggests the market is significantly overvalued by its own standards, a degree of caution may be called for. And vice versa, if the market is deeply undervalued a degree of confidence in its future recovery may be encouraged by undemanding earnings expectations. Again, there should be no confidence that these identified valuation gaps will be closed rapidly. Earnings can catch up with prices or prices can catch up with earnings as the truth about underlying economic performance is revealed and this may take time.
Conclusion: The market proposes – economic reality disposes
The lesson to be drawn from this analysis is that in the long run, valuations will catch up with economic performance or performance will catch up with valuations. However predicting the direction of the market over the next month, year or five years (even knowing where earnings and dividends are going) is not easy to do.
Trading theories that rely on price/earnings ratios or the relationship between the level of earnings and prices may indicate degrees of investor exuberance or despondency by the standards of the past. But there can be no certainty that temporary exuberance or despondency will prove excessive or short lived.
There will always be more than earnings or dividends (or even expected earnings or dividends) at work in driving the market in one direction or another. The growth in expected earnings as well as the discount rate attached to them may be in a constant flux, making accurate predictions of market returns extremely difficult to make. Not only would it be necessary to predict earnings with accuracy, but also the direction of interest rates and the degree of risk aversion or tolerance that may emerge.
The direction of causation may well be from prices to earnings rather than from earnings or expected earnings to share prices. For example, the higher the share price, the more willing and able the company will be to expand its operations and increase its bottom line earnings. Market volatility furthermore provides no proof of market irrationality. It indicates only the difficulty in forecasting the behaviour of complex systems with feedback loops – in the case of the share market from performance to prices and also from share prices to the economic performance of a company.