Shiller P/E and Macroeconomic Conditions

**Robert D. Arnott Denis B. Chaves ^{1} Tzee-man Chow **

Research Affiliates, LLC The Vanguard Group Research Affiliates, LLC

Abstract

Abstract It is well documented that the Shiller P/E (or Cyclically-Adjusted PE, or CAPE) is a powerful predictor for long-horizon capital market returns, all over the world, as a consequence of long-horizon mean-reversion. It is also well-known that the Shiller PE, and other valuation metrics, are poor predictors of short-term returns. We find that this is because the normal level of the Shiller P/E ratio varies with changing economic conditions. Leibowitz and Bova (2007) show that periods of moderate real interest rates empirically allow higher market valuations, and that P/Es tend to fall in times of abnormally high or low real interest rates. Our work builds on Leibowitz and Bova, showing a similar linkage between P/Es and inflation. Moderate, rather than rock-bottom, levels of inflation and real interest rates are associated with the highest valuation multiples, creating a valuation “mountain.” We also extend these findings to other developed markets. We then examine whether these findings can improve the efficacy of the Shiller P/E as a predictor of short-term returns. If we assume that P/Es mean-revert toward levels that are suggested by macroeconomic conditions, rather than toward long-term averages, the P/E becomes a statistically significant and economically meaningful predictor of shorter-term returns. Valuation does indeed predict short-term returns, as long as we recognize that the “normal” level of P/E ratios is non-stationary and will vary with changing economic conditions.

^{1} The views and opinions expressed herein are those of the author and do not necessarily reflect the views of The Vanguard Group, its affiliates or employees.

One of the most common misattributions in finance is Ben Graham’s supposed-and almost certainly apocryphal-remark that “in the short run the market is a voting machine and in the long run it is a weighing machine.”^{1} The history of financial analysis is, in part, a quest for metrics that can help us better predict the voting and more accurately gauge the weight (the fair value) of an investment. Robert Shiller and John Campbell introduced one of the most powerful measures of value when they developed the Cyclically-Adjusted Price/Earnings ratio, or CAPE.^{2}Quite simply, CAPE divides the current price of a broad market index, like the S&P 500, by a ten-year average of its earnings. Earnings are adjusted for inflation, so that past earnings will not be understated as a consequence of inflation. Peak earnings no longer create an illusion of low P/E ratios; trough earnings no longer create artificially elevated P/E ratios.CAPE is a powerful predictor for long-horizon capital market returns all over the world. This result is expected, as long as markets exhibit long-horizon mean-reversion towards the historical average for the CAPE ratio.

The Shiller P/E, like other valuation metrics, is a much less powerful predictor of short-term returns. Can we fix this? Building on the work of Leibowitz and Bova (2007), we find that the average CAPE ratio (hereafter referred to simply as the P/E) varies with both real interest rates and inflation. Moderate levels of inflation and real interest rates coincide with the highest average valuation multiples. Unusually high or unusually low real yields-or inflation rates-tend to coincide with much lower average valuation multiples, creating a valuation “mountain.”We find that this relationship spans the developed world. We will speculate on the direction of causality shortly.

Suppose we measure the abnormal P/E by comparing the current P/E with a conditional normal P/E adjusted to reflect current inflation and real yields. If P/Es mean-revert toward levels that are suggested by macroeconomic conditions, rather than toward long-term averages, the conditional abnormal P/E may be a better predictor of short-term market returns. We find this to be the case to an extent that is statistically significant and economically meaningful. P/E is an excellent predictor of long-term returns, over spans of three to ten years (and even better over longer spans, though this goes beyond the scope of this study). P/E, relative to a conditional normal P/E, adjusted to reflect current inflation and real yields, proves to be better than simple P/E as a predictor of shorter-term returns of one month to one year.

What levels of inflation and real interest rates are more favorable to stock markets? Many investors, commentators, and policy makers seem to believe that rock-bottom levels of inflation and real interest rates provide the best economic conditions for stock prices to soar. Their logic is straightforward. Consider the textbook valuation formula, relating the current price of a stock or portfolio, P_{t}, to the sum of discounted expected future dividends, D_{t}:

Given that the discount rate, r_{tâ†’t+τ}, is the sum of expected inflation, real interest rates, and an equity risk premium, it seems easy to conclude that a reduction in any of those three variables should always reduce the discount rate and consequently raise stock prices. In other words, this logic implies that the relationship between stock prices and either inflation or real interest rates is monotonically inverse.

This straightforward logic-that stock prices always go up when inflation rates or real interest rates go down-is not supported by the data. In the messy real world, market participants appear to value stocks based on the Goldilocks principle: the levels of inflation and of real interest rates have to be “just right” to sustain high valuation levels. When inflation or real interest rates deviate from their “sweet spot”-in either direction-valuations tend to fall. The relationship between stock prices and inflation, and between stock prices and real interest rates, can be simplistically described as a mountain. It has a peak at medium levels of inflation and real interest rates, and it slopes downward from that point in any direction.

How can we explain this mountain-shaped relationship? Where does the logic behind the monotonic relationship described above break down? To answer these questions, notice that for this monotonic relationship to hold, all other variables must remain constant when the inflation rate or real interest rate changes. Why should they not be interconnected? It turns out that cash flows, measured here as earnings, and risk premiums exhibit nonlinear and non-monotonic interrelationships with inflation and real interest rates.

According to the logic presented earlier, very low inflation rates would drive the discount rate down and stock prices up. However, this regression shows that the growth rate in profits falls even faster, given the linear term in excess of one and the highly negative quadratic term, which would cause stock prices to go down and not up. At times of very low-even negative-inflation rates, market participants should evidently worry about the economy and reduce their expectations of real earnings growth. Notice that the negative quadratic term creates a mountain-like relationship between earnings growth-the numerator in Equation (1)-and inflation rates, influencing stock prices to display a similar relationship with inflation rates.

Similar arguments can be made about real interest rates. A prolonged period of low real interest rates may suggest a market expectation of slow macroeconomic growth, or increased fear, which might require a higher equity risk premium, hence lower valuations.^{5} For instance,

using the same data starting in 1871, consider a forecasting regression of future three-year

According to the logic presented earlier, very low real interest rates would drive the discount rate down and stock prices up. However, the strongly positive quadratic term shows that abnormally low (or high) real interest rates tend to indicate higher uncertainty in prices. It seems reasonable if market participants actually increased the discount rate-reduce the valuation multiples-as inflation volatility rises, which would cause stock prices to go down and not up. Notice that the positive quadratic term creates a U-shaped relationship between inflation volatility and real interest rates, again influencing stock prices to display a mountain-like relationship with real interest rates.

In this study, we formalize the mountain-shaped relationship between the stock market valuation, and inflation rates and real yields. Under the right conditions, i.e., moderate levels of inflation and real yields, the market P/E empirically tends to reside well above the unconditional long-term historical average of 16.6. In contrast, when either the inflation rate or the real yield is at an

extreme, in either direction, we observe a markedly lower valuation as measured by P/E.

If we use histogram bins to identify the mountain, then each bin will have enough samples that we will find a relationship between inflation, real interest rates, and average P/E that has considerable “noise” in estimating the n

ormal P/E for each histogram bin. So, we define a continuous non-linear Gaussian model to estimate the normal P/E ratio, hence the short-term P/E mean-reversion target, given the inflation rate and the real yield. The formula for a simple bell-shaped Gaussian curve has far fewer degrees of freedom, leading to far higher confidence in the accuracy of our estimates of the conditional average P/E ratios, than a more conventional histogram bin approach.^{6}

It is widely accepted that the deviation of the current stock market valuation from its long-term unconditional average is a good predictor of long-horizon (10-year) return. Our model of a short-term mean-reversion target significantly enhances the predictability of stock market returns at short horizons (1 year or less ). Our observations are robust within the U.S. market as well as across the global developed markets.

The relationship between stock prices and nominal or real interest rates is amply studied and well understood. Leibowitz and Bova (2007), for instance, introduce the “intriguing conjecture that P/E s in the U.S. market may decline in times of both significantly lower, as well as significantl y higher, real interest rates.” We extend the ir study in two different directions before turning to the core goal of improving the efficacy of P/E as a predictor of shorter-term market returns.

First, we show that the same non-monotonic, mountain-shaped relationship between P/E s and real interest rates also holds between P/Es and inflation. Second, we show that this relationship is found not only in the U.S. market, but also exists in a sample including a large number of developed markets across the globe. We then turn attention to creating a bell-shaped Gaussian “mountain” that describes this relationship, and finally test the markets’ tendency to mean-revert towards this new variable normal P/E, conditioned on current inflation and real interest rates, as a predictor of future market returns.

This paper contrasts with the literature on what is commonly referred to as the inflation or money illusion. Modigliani and Cohn (1979), Ritter and Warr (2002), Asness (2003), and Campbell and Vuolteenaho (2004), among others, argue that investors extrapolate past trends in nominal cash flow growth when forming their expectations about future nominal growth, failing to adjust them for changes in inflation. As a consequence, in times of low inflation their cash flow growth assumptions are too high, resulting in inflated P/Es, and in times of high inflation their cash flow growth assumptions are too low, resulting in depressed P/E s. This ” inflation illusion” or ” money illusion ” generates long periods of time during which one observes a negative relationship between P/Es and inflation. If cash flow growth expectations were correctly adjusted for inflation, P/E s should not move as much with inflation.

A clear example of the inflation illusion took place during the late 1990s and early 2000s, when a rule of thumb dubbed the “Fed model” became the preferred argument used by market pundits to justify extremely high stock prices (debunked by Asness 2003). According to this simple rule, one need only compare the stock market earnings yield, or E/P , with nominal interest rates to know whether stocks are fairly priced: Buy when E/P is above nominal interest rates and sell when it is below them. Since nominal interest rates were relatively low during that time, and since historically inflation expectations have been the major driver of nominal interest rates, supporters of the Fed model failed to lower their nominal cash flow growth expectations, but discounted those prospective cash flows using low nominal interest rates, resulting in high P/Es.

We are not proponents of this simple trading rule. Yields, growth rates and inflation are interconnected, with non-linear connections, that make the “Fed model” an exercise in folly. We propose a more sophisticated model to a ss ess the fair valuation of the stock market and to forecast return.

A growing literature in finance shows that forecasting stock market returns can be significantly improved by deviating from static mean-reversion targets. The most common approach, exemplified by Lettau and Van Nieuwerburgh (2007) and Pettenuzzo and Timmermann (2011), is to assume that markets suffer structural breaks and thus mean-reversion targets are dependent on some unobservable state of the economy that needs to be inferred using advanced econometric techniques. We add to the literature with a more direct and practical approach. P/E declines when either the inflation rate or the real yield deviates from moderate levels, in either direction. Conditioning P/E on inflation and real yield forms a sensible short- term mean-reversion target. We define a 3-dimensional parametrized continuous bell curve to model the relationship between the expected P/E and both the inflation and the real yield. The difference between the observed and this modeled P/E is more powerfully – and significantly – related to near-term future stock market returns than a simple P/E relationship.

It is our hope that this work stimulates further research into the linkage between macroeconomic conditions and equilibrium valuation levels, and that our parametrized Gaussian model can be useful to help others extract more statistical significance from these linkages.

Leibowitz and Bova (2007) framed their analysis within the practical considerations of pension funding ratios. When real interest rates rise, the value of pension fund liabilities decreases in the same proportion as the value of pension fund assets, composed of stock and bonds, decreases. When real interest rates fall, however, the value of pension fund liabilities and of fixed-income assets increases, but the value of equity assets decreases. This perverse relationship between stock prices and interest rates can create a stark and economically important mismatch between the value of assets and liabilities.

In the wake of the financial crisis of 2008 and the ensuing Great Recession, it becomes more important to gain a deeper understanding of the relationship between interest rates and stock prices. Slow to no macroeconomic growth, accompanied by high unemployment, has driven central banks in many developed countries – including the globally dominant economies of the United States, Japan, the Eurozone, and the United Kingdom – to cut interest rates to extremely low levels, and to engage in varied forms of unconventional monetary policy . In particular, different versions of quantitative easing – buying long-term government or asset- backed bonds with the goal of lowering long-term interest rates – became a common remedy in the medicine cabinet of many central banks. In cases where these policies have led to negative real interest rates, we should want to understand the implications for market valuation.

In the words of former Fed chairman Ben Bernanke (2010): “This approach [quantitative easing] eased financial conditions in the past and, so far, looks to be effective again. Stock prices rose and long-term interest rates fell when investors began to anticipate the most recent action… And higher stock prices will boost consumer wealth and help increase confidence, which can also spur spending.” ^{8} (Emphasis added.)

As the evidence we present suggests, reducing inflation or real interest rates can be helpful to market valuations, to a point. But, beyond a certain threshold, it may cause the opposite of the intended effect on stock markets

: in the long run, valuation multiples could actually fall. It also means that investors may wish to be wary of the efficacy of the ” Bernanke (now Yellen, Draghi, and Kuroda) put. ” The notion that a new round of quantitative easing will surely follow any meaningful drop in share prices, rescuing the market, may not be as reliable as many observers hope.^{9}

At the time of this writing, the P/E of the U.S. stock market has stayed above 24 for almost 2 years. Our model indicates the U.S. market is expensive: priced to offer anemic real returns, or worse if P/Es revert towards historical norms. That said, the Fed has done an admirable job in keeping the inflation rate (and, to a lesser extent, real interest rates) at levels that are favorable to stock prices. From these “Goldilocks” conditions, any positive shock to inflation and further reduction in real yield can be the catalyst for a serious reversal in the US stock market.

All data are from Global Financial Data and are in U.S. dollars. Table 1 shows a few summary statistics for the individual countries. While the U.S. sample starts in 1880, the international sample starts in 1972, the first year the requisite data are available for at least three countries : Canada, Japan, and the United Kingdom. Other countries included in the international sample, with later start dates, are Australia, Austria, Belgium, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Italy, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, and Switzerland.

We use the yield on 10-year sovereign bonds as our measure of nominal interest rates. We use one-year inflation from the consumer price index (CPI) as our measure of inflation, but we calculate real yields by subtracting trailing 3-year inflation. ^{10} For purposes of our study, we are interested in using a valuation ratio that is not subject to the short-term volatility in

earnings. Therefore, we rely on the Shiller P/E, which is calculated as the current real price divided by a 10-year moving average of real earnings. ^{11}

Median P/E s are mostly in the 15 — 20 range, with a few exceptions. Most notably, Japan had a median P/E of 34, while the Netherlands’ was close to 14. The United Kingdom had the highest median inflation rate at 4.2 percent, while Singapore’s was just 1.1 percent. The median real yield was lowest in Ireland at 1.2 percent and Japan at 1.8 percent, and highest in Denmark at 3.8 percent and Belgium at 3.7 percent.

We start our analysis by examining the univariate relationship between P/Es and either real yields or inflation. Leibowitz and Bova (2007), using data from 1978 to 2004, find that stock market prices, as a multiple of earnings, tend to peak when real yields are situated between two and three percent. Using a far longer sample, spanning 134 years instead of 26 years, Figure 1 shows very similar results. ^{12} The tent-shaped chart indicates that U.S. stock market valuation multiples tend to be at their highest when real yields range from 3 to 4 percent. Outside this narrow interval – in either direction

– median P/E s fall rather quickly from a peak of 19.6 to 10.7 when real yields are below — 1 percent, or 10.5 when real yields are above 6 percent. The whisker plots on the top of each bar show plus and minus one standard error around the median, estimated with the Newey-West approach to adjust for overlapping observations. The se standard errors also allow us to confirm that the median P/Es are statistically different across different intervals of real yields. For instance, the difference between the top P/E of 19.6 and the right- most P/E of 10.5 has a t-statistic of 4.10, whereas the difference between the top P/E

of 19.6 and the left-most P/E of 10.7 has a t-statistic of 3.34.

Figure 2 shows a very similar pattern between median P/Es and inflation rates. The ” sweet spot ” for the stock market lies between 2 and 3 percent inflation rates, with a median P/E of 20.3. Interestingly, the P/Es represented by the bars on the right side of the chart decline faster – as inflation rates increase – than those on the left side – as inflation rates decrea

se. This fact might come as a surprise to supporters of the thesis that stocks offer sufficient protection in an inflationary environment. The difference between the top P/E of 19.4 and the right-most P/E of 9.2 has a t -statistic of 5.74, whereas the difference between the top P/E of 20.3 and the left- most P/E of 16.0 has a t-statistic of 1.78.

Finally, Figure 3 shows the joint relationship of median P/Es with real yields and inflation – a three-dimensional (3-

D) valuation mountain. Because the peak of the 3-D chart hides some details on its far side, we also present a heat map beside it , together with the number of observations used in each bin or ” regime. ” To head off any criticism that the boundaries of the regimes were picked to enhance the results, we chose the most straightforward approach of separating them by using equal increments of two percentage points. The peak median P/E of 22.5 is observed at what are considered ” normal ” levels of inflation between one and three percent, but at a relatively high level of real yields between three and five percent.

The main point of the 3-D chart is not to pinpoint the exact location of the peak, which could be the result of a small number of unusual samples in the distribution, but to show that further reductions in real yields or inflation rates do not help boost stock prices. Moving to the next lower regime in terms of inflation – less than one percent annual inflation – reduces the median P/E by almost 20 percent to 18.5, while moving to regimes of still lower real yields reduces the median P/E all the way to 12.6 when real yields are below minus one percent.

Most of the regimes on the circumference of the matrix have very low median P/E s of about 10. It is important to emphasize that these regimes have few er observations than those in the middle of the matrix, which have levels of inflation and real interest rates that can be considered more normal. Some readers might infer that such a distribution of observations is a desirable characteristic for the U.S. economy to have because it translates in to fewer moments of low stock market valuations. We counter that periods of low valuations are wonderful for investors who are entering the stock market. ^{13}

Let’s now turn attention to our findings showing that the well-known and amply studied relationship between current P/E

s and future stock market returns can be meaningfully strengthened by incorporating information on real yields and inflation rates.

*A Better Mountain*

The charts presented thus far provide an interesting description of the data. The ir discrete nature implies, however, that small variations in real yields or inflation result in no change or, alternatively, in sudden jumps in P/Es when moving from one regime to another. The bucketing approach is also vulnerable to few samples and noisy data. In this section, we address these shortcomings by estimating the continuous function ƒ(i,π) that provides a reasonable and accurate description of P/E

s given any level of real yield and inflation. In the interest of brevity, from this point on we denote the inflation rate by π

, and the real interest rate (or yield) by i .

In our search for the most useful function we evaluated the tradeoff between simplicity and the capacity to describe the data. Polynomials score very high on the simplicity dimension, but they have one important flaw: as real yields or inflation move to extremely high or low values, P/Es tend to do the same, resulting in implausible numbers. In other words, even if we could find a polynomial that would provide a good fit to the data in our sample, it would likely fail miserably outside this domain. The solution we propose is to use a two-dimensional Gaussian (bell-shaped) function to model ln (P/E)s. Because of its technical nature, we relegate to the Appendix our method for fitting the Gaussian mountain to the P/E data. Table 2 shows the in -sample parameter estimates for the United States and Figure 4 plots curves of constant P/E. 14 A peak P/E of exp (a + b) â‰ˆ 20 can be observed at a real yield of 2.99 percent and inflation of 1.35 percent . The lowest possible P/E would be exp (a) â‰ˆ 8 at real yield and/or inflation levels far removed from these figures. The most interesting question, however, is how well this model fits the data. To find the answer, we measure the statistical fit of the model using the adjusted R-squared formula,

It is important to underscore that all of our regressions have a monthly frequency. For this reason, we report two sets of t-statistics, to correct for heteroskedasticity and serial correlation in the residuals caused by overlapping returns. The first set uses the well-known approach developed by Newey and West (1987). The second set uses the same coefficients as the original OLS regressions, but calculates their standard errors using separate regressions estimated with reduced samples, with non-overlapping data. This approach eliminates any serial correlation in the residuals and yields significantly lower t-stats, especially at longer horizons.

Table 4 shows that all coefficients are negative; lower valuation multiples (P/Es) indicate depressed prices, which in turn signal subsequent higher returns. Our results confirm findings in the existing literature (e.g., Cochrane 2008) that the statistical power of return forecasting in terms of both

t-stats as well as R-squareds increases with horizon. For instance, going from one month to 10 years raises the R-squared from less than 1 percent to more than 30 percent and the magnitude of the t-stat from 1.79 to 3.12.^{15}

How can we use the information we have gained thus far to enhance stock market forecasts? We propose a simple idea. Traditional forecasting regressions, such as Equation (7),compare the current

P/E with a single full-sample historical average. Given that inflation and real yields provide valuable information about the median P/E, we should be able to enhance forecasting power by comparing the current P/E with the Gaussian model P/E, conditional on current levels of inflation and real yields, instead of merely comparing the current P/E with the long-term average P/E. For instance, when real yields and inflation are both very low, as in the lower left corner of the diagram in Figure 4, a relatively low average P/E of about 10 might be normal. Therefore, if the current P/E were above 10, this would suggest a lower future return

. By contrast, in a traditional regression, if the current P/E is below the full-sample average, we

would erroneously expect a higher future return. Translating this simple logic into a regression

equation (and lagging the earnings by three months to reflect actual practice), we have

where ƒ(i,π) denotes the natural logarithm of P/E, predicted by the Gaussian function stated in Equation (9) in the Appendix, conditional on the current levels of inflation and real yield.

Table 5 shows the results of using Equation (8) to forecast returns. The most notable improvements come at the short-horizon forecasts for which the coefficients almost double from those in Table 4, followed by an increase in the magnitudes of t-stats, from borderline significance in Table 4 to high significance in Table 5.

Interestingly, the results are reversed at longer horizons; coefficients are reduced, especially for 10-year results, and the statistical significance of P/E disappears. These patterns are unsurprising: the current real yield and inflation contain significant information about what P/E should be in the near-term. Because the real yield and inflation change over time, today’s levels are likely to differ substantially from those a decade into the future. Hence, conditioning our long-term forecasts on current

inflation or real rates is a serious mistake. The crossover point for the coefficient is between 3 and 5 years in the future (for statistical significance, it is even shorter, between 1 and 3 years in the future). In short, conditioning P/E on these two current macroeconomic measures would appear to be ill advised for spans of 3 years or more, beyond which the basic Shiller P/E shows its impressive merit.

To further investigate our claim that current macroeconomic conditions are not very important for long-term forecasts because the real yield and inflation change significantly over time, we adopted a slightly different specification for the regression in Equation (8). Imagine that a genie informs us of future real yields and inflation rates, but not of future P/Es. If we substitute our expected P/E at the end of the forecasting window, ƒ(i_{t+k},π_{t+k}), for our expected P/E at the beginning of the forecasting window, ƒ(i_{t},π_{t}), we obtain significantly higher statistical significance and fit across all forecasting horizons-but especially at 5- and 10-year windows. In the interest of space-and because these results are utterly nsurprising-we do not report them here, but they are available from the authors on request.

How does this method fare “out of sample,” when we apply it outside of the United States? The results for global developed markets are similar to the U.S. results, with a few small differences due to data availability. In the interest of space we comment only briefly on the similarities and discuss the differences in greater detail. Throughout the discussion of our results, keep in mind that we pooled all available data starting in 1972, the first year with availability of data for at least four countries.

Figure 5, remarkably similar to Figure 1, shows that the developed countries of the world exhibit the same tent-shaped relationship between P/E and real yield that we find in the U.S. stock market. The peak median P/E of 22.0 is reached in the range of real yields between 2 and 3 percent. The univariate relationship between P/E and inflation, illustrated in Figure 6, differs in minor ways from the relationship of P/E and inflation in the U.S. stock market depicted in Figure 2. The reason for the weaker tent-shaped relationship in the global sample can be traced to fewer episodes of very low inflation in international countries after 1972. The majority of inflation rate observations lie in the range of zero percent or higher. Furthermore, because in the late 1980s Japan experienced a stock market bubble with extremely high P/Es-in excess of 100!-concurrent

ly with very low inflation, the middle-to-left sections of the chart in Figure 7 might be overstated. Even using

ln (P/E), these outliers can dominate the analysis. Confirming our observation of few periods of low inflation, the bottom half of Figure 8 shows that the data did not provide any episodes of very low inflation and very low real yields. Nevertheless, it is still possible to see a pattern similar to that represented in Figure 3, with peak median P/Es falling in the moderate range of real yields and inflation around zero.

Using the developed country sample to estimate the parameters in the minimization function applied to the Gaussian P/E model in Equation (10) in the Appendix results in numbers very similar to those obtained for the United States. Table 6 shows a marginally higher parameter and an almost identical parameter b, jointly indicating a slightly higher peak P/E of about 24. The location of the international peak P/E is almost identical to the location of the U.S. peak P/E, at a real yield and inflation rate of 3.11 percent and 1.24 percent, respectively. (The U.S. peak is at 2.99 percent and 1.35 percent, respectively.) While the near-identical location

of the peak is likely to be coincidence, we believe that the broad similarity of the peaks is not.

These developed country data provide a powerful out-of-sample ratification of the U.S. findings.Figure 8 also confirms the similarity between the level sets obtained from the two distinct samples.

The Gaussian model does not perform as well in terms of statistical fit for the international sample as for the U.S. market. Table 7 shows the 29.9 percent adjusted R-squared of the developed market sample, which is smaller than the 50.7 percent adjusted R-squared of the U.S. market. We are not troubled by this, however, as a single model faces a significant challenge in its ability to fit the multinational data

in the international sample: it has to explain P/Es in more than 20 countries as different, for example, as Canada and Japan, with different accounting standards, different investor risk aversion, and so forth.

When using linear regressions to explain P/Es, we observe some similar results but also some differences. Unlike the U.S. sample, the coefficient on real yields has the expected negative sign, but the statistical fit of less than 1 percent is very poor. The picture is reversed when inflation is included in the regressions. The R-squareds jump to 23 percent (univariate) and 25 percent (multivariate), twice the numbers in the U.S. sample. These relatively higher statistical fits can be easily explained by the lack of observations occurring in low inflation periods. With a reduced number of observations on the left half of the chart, the regression does a good job in fitting only on the right half.

Finally, Tables 8 and 9 report results for return forecasting regressions using Equations (7) and (8), respectively. Similar to the U.S. sample, we observe a significant increase in forecasting power when we include the information provided by inflation and real yields. The short-horizon t-stats move from being marginally significant to strongly significant, and the R-squareds very nearly double. The improvements are especially important at the one-month horizon, with the coefficient moving from -0.10

to -0.21 and the R-squared from 0.5 to 1.1 percent. In this case, however, long-horizon coefficients and R-squareds are very similar across the two regressions.

Our work on the relationship between stock prices and the levels of inflation and of the real

interest rate has been gratifying on two levels. First, along with many others, we have long

believed that valuation is an important determinant of future real asset class returns. But the

poor linkage with short-term returns has always been an Achilles’ heel for valuation measures. The Shiller

P/E has customarily been a favorite of the valuation community because of its demonstrated correlation with long-horizon real returns for U.S. and international stock markets. We are pleased to see that conditioning the “normal” P/E, toward which mean reversion gravitates, on current macroeconomic conditions can lead to an impressive step-up in its efficacy as a predictor of near-term capital market returns. Valuations do matter-and not just in the long term, but also in the short term. We need only recognize that on a short-term basis, depending on current macroeconomic conditions, a stock’s valuation may gravitate toward a level that is

different from its long-term historical average.

Second, our work suggests a rich path for further research. Many macroeconomic and market measures have been found to be linked to near-term capital market returns, including the price-to-dividend ratio, price-to-book ratio, corporate issuances, investment-to-capital ratio, and consumption-wealth-income (CAY) ratio.18 Our technique demonstrates that there are far more powerful ways to integrate macroeconomic measures with stock valuation methods than the usual linear combinations that predominate in the quantitative community. As in the example we have presented, the efficacy of valuation is increased (in this specific case, doubled) by assuming

that the equilibrium level for P/E is non-static, varying with the changing macroeconomic state. The result is that we need no longer rely on long-term historical averages to infer near-term mean-reversion targets. This result suggests that valuation can become useful tool for assessing short-term market prospects, no longer merely a powerful tool for the long-term investor. We are hopeful that our work opens the door to others who will explore new ways to think about valuation measures and the manner in which we can use them.

Arnott, Robert D., and Peter L. Bernstein. 2002. “What Risk Premium is ‘Normal’?” Financial Analysts Journal, vol. 58, no. 2 (March/April):64-85.

Arnott, Robert D., and Denis B. Chaves. 2012. “Demographic Changes, Financial Markets, and the Economy.”Financial Analysts Journal, vol. 68, no. 1 (January/February):23–46.

Asness, Clifford S. 2003. “Fight the Fed Model.” Journal of Portfolio Management , vol. 30, no. 1 (Fall):11–24.

Bernanke, Ben S. 2010. “What the Fed Did and Why: Supporting the Recovery and Sustaining Price Stability.” Washington Post (November 4). Campbell, John Y., and Robert J. Shiller. 1988. “Stock Prices, Earnings and Expected Dividends.” Journal of Finance, vol. 43, no. 3 (July):661-676.

———————————–. 1998. “Valuation Ratios and the Long-Run Stock Market Outlook.” Journal of Portfolio Management, vol. 24, no. 2 (Winter):11-26.

Campbell, John Y., and Tuomo Vuolteenaho. 2004. “Inflation Illusion and Stock Prices.” American Economic Review, vol. 94, no. 2 (May):19–23.

Cochrane, John H. 2008. “The Dog That Did Not Bark: A Defense of Return Predictability.”Review of Financial Studies, vol. 21, no. 4 (July):1533–1575.

Goyal, Amit, and Ivo Welch. 2008. “A Comprehensive Look at the Empirical Performance of Equity Premium Prediction.” Review of Financial Studies, vol. 21, no. 4 (July):1455–1508.

Graham, Benjamin, and David L. Dodd. 2008. Security Analysis, 6th ed. Foreword by Warren E. Buffett. New York: McGraw-Hill Education.

Leibowitz, Martin L., and Anthony Bova. (2007)”P/Es and Pension Funding Ratios.”Financial Analysts Journal, vol. 63, no. 1 (January/February):84–96.

Lettau, Martin, and Stijn Van Nieuwerburgh. (2008) “Reconciling the Return Predictability Evidence.” Review of Financial Studies, vol. 21, no. 4 (July):1607–1652.

Modigliani, Franco, and Richard A. Cohn. 1979.”Inflation, Rational Valuation and the Market.” Financial Analysts Journal, vol. 35, no. 2 (March/April):24–44.

Newey, Whitney K., and Kenneth D. West. 1987.”A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix.”Econometrica, vol.55, no. 3 (May):703–708.

Pettenuzzo, Davide, and Allan Timmermann. 2011. “Predictability of Stock Returns and Asset Allocation under Structural Breaks.” Journal of Econometrics, vol. 164, no. 1 (September):60–78.

Ritter, Jay R., and Richard S. Warr. 2002. “The Decline of Inflation and the Bull Market of 1982–1999.” Journal of Financial and Quantitative Analysis, vol. 37, no. 1 (March):29–61.

Shiller, Robert J. 2015. Irrational Exuberance. 3^{rd} ed. Princeton, NJ: Princeton University Press.

^{1} In their seminal book, Security Analysis, Graham and Dodd actually say, “In other words, the market is not a weighing machine, in which the value of each issue is registered by an exact and impersonal mechanism, in accordance with its specific qualities. Rather we should say that the market is a voting machine, whereon countless individuals register choices which are partly the product of reason and partly the product of emotion.” Graham and Dodd (2008), page 70.

^{2} John Y. Campbell and Robert Shiller used 10-year average earnings in the earnings-to-price ratio in Campbell and Shiller (1988) and tested its predictive strength in Campbell and Shiller (1998). Shiller adopted the term CAPE in the third edition of Irrational Exuberance (Shiller 2015, page xv).

^{3} A precise notation would be âˆ†E_{tâ†’t+36} and π_{tâ†’t+36} . To minimize clutter, we simplify this to

Î”E_{t} and Ï€_{t}.

^{4} Standard errors are corrected for heteroskedasticity and time-series correlation in the residuals using the methodology of Newey-West. See the Data section for sources and details.

^{5}One of the authors [as a courtesy to the referees, we leave the name out … please insert the lead author’s name here] has frequently posed a thought experiment: suppose the “equity risk premium” had been labeled as a “fear premium” from the early days of the concept. After all, there’s little empirical evidence of the correct linkage between objective measures of risk-such as volatility or beta-and return. And, investors in an inefficient market may demand a higher reward, hence a lower starting price, for investing where others fear to tread. Had finance theory begun with a fear premium, rather than a risk premium, many of the anomalies of modern finance would have been unsurprising, even expected.

^{6} This is analogous to the method used in Arnott and Chaves (2012) in finding polynomial linkages between demographic profiles and capital market returns.

^{7} It bears mention that the “Fed model” was developed based on market data from the 1960s to the 1990s. Before the 1960s and after the 1990s, the model fails. Data from before the 1960s was readily available when the “Fed model” was in its heyday, but was conveniently ignored. It has subsequently failed miserably post-2000, yet the model retains many adherents.

^{8} The frequent central bank assertions that quantitative easing is not contributing to the much-vaunted wealth gap are at odds with this stated intent for quantitative easing. Who has assets? Overwhelmingly, the affluent. Therefore, seeking to create a wealth effect, ipso facto, drives wealth inequality-but we digress.

^{9} Following the replacement of Ben Bernanke with Janet Yellen as Federal Reserve chairman, we see little evidence that investors or the Fed have changed their belief that extremely low interest rates can artificially maintain high stock market valuations.

^{10} We use a three-year inflation window when calculating the real yield in order to reduce the risk of having two variables-inflation and real yields-that are simple mirror images of each other during periods of stable nominal interest rates. In addition, Arnott and Bernstein (2002) found that long-term bond yields were better correlated with three-year inflation than with longer or shorter spans.

^{11} This ratio has formally been known as the cyclically adjusted price-earnings ratio, or CAPE. Now that Professor Robert Shiller has garnered Nobel recognition for his work on market volatility, the ratio is increasingly known by its already popular shorthand name, the Shiller P/E. As is the case with Bill Sharpe and the Sharpe ratio, Bob Shiller has earned this honor!

^{12}We focus our attention on median P/Es to reduce the influence of outliers, especially in international markets, but our results are qualitatively the same if we use average P/Es.

^{13} We find the cheerleading for bull markets to be interesting; as Arnott and Bernstein (1997) observed, bull markets are good for those who are about to sell and bear markets are good for those who are still accumulating and investing.

^{14} The model is defined on ln (P/E), but figure 4 plots P/E directly as P/E ratio is most commonly quoted in linear numbers.

^{15} Low R-squareds on short-term results are deceptive; they’re often much more useful than they seem. For instance, the R-squared of 33 percent on 120-month real returns implies a correlation of 0.57, while the R-squared of 4.7 percent on 12-month real returns implies a correlation of 0.22. A very crude analogy is that the P/E is approximately 57 percent as useful as a genie telling us the exact 10-year future real return for the stock market or 22 percent as useful as perfect foresight on the 1-year future real return for stocks. Given a choice, would you rather have 57 percent correlation with perfect 10-year foresight on the real return, or 10 snapshots, each with 22 percent correlation with 1-year perfect foresight? We would probably choose the latter, but it would be a tough call. In other words, the 33 percent R-squared on 10-year real returns is not necessarily better than the 4.7 percent R-squared on 1-year real returns.

^{16} One possible concern is that our Gaussian function is estimated using the full sample, but this is also a concern for regular forecasting regressions because they use the full sample to estimate averages of dependent and independent variables.

^{17} We have also applied the Gaussian model to smaller samples including Europe or Asia (including Australia and New Zealand, but excluding Japan) with success. These results are available from the authors by request. The case of Japan is interesting; the Gaussian model fails to identify a peak given the extremely high valuations at times of very low inflation.

^{18} See Goyal and Welch (2008) for a review and summary of those variables.