A New Valuation Measure for the Stock Market
AA NOTE ON NEW VALUATION MEASURESFOR STANDARD & POOR COMPOSITE INDEX RETURNS
TARAN GROVE, MICHAEL REYES, ANDREY SARANTSEV
Abstract.
Financial theory tells that long-run total real returns of the stock market areapproximately equal to long-run earnings growth plus average dividend yield. Thus the totalreal returns minus real earnings growth must be stable in the long run. If this difference isabnormally high in the last few years, then we consider the market to be overheated andheaded for a crash. A measure of such heat is (detrended) cumulative sum of differences. Weregress future total real returns upon current heat measure. To make sure all residuals arenormal i.i.d., we move in three-year steps. We use Bayesian inference with a non-informativeprior. After verifying goodness-of-fit, we simulate future returns for horizons of 9, 15, 30years, starting from current market conditions. We verify the conventional wisdom thatfuture long-run stock market returns are likely to be lower than the historical averages. Introduction
The
P/E ratio , or price-to-earnings ratio , for a stock is computed as the current price of astock divided by its earnings per share over the last year. For the market is a whole (measuredby the Standard & Poor 500 Index or any other index) this P/E ratio is computed as thesum of P/E ratios of individual stocks, weighted by market capitalizations (total marketvalue) of stocks; or, equivalently, the ratio fo the total market value of all these stocks totheir total earnings over the last year.Robert Shiller and his collaborators noted that when the P/E ratio of the stock market ishigh, the future expected returns of the stock market (over the next 5 or 10 years) are low.Since the seminal work by Robert Shiller [5, 10], the price-to-earnings (PE) ratio approachto investing has captured the attention of academics and practitioners in finance. See alsoa discussion in [11, Chapter 11], and a related concept of value investing which applies thesame concept to individual stocks, [11, Chapter 12].However, the traditional P/E ratio can be significantly influenced by the high volatility ofnet income. For example, in 2008, the P/E ratio was very high because multiple companiesin the financial sector reported negative incomes. This, in turn, greatly reduced the totalnet income for the stock market as a whole. We compute the P/E ratio of the stock market(measured by a benchmark such as the Standard & Poor 500 index) by summing marketcapitalization (market value) of all stocks in this index and dividing it by the sum of thenet incomes of all these companies in the last year. This, however, suffers from a drawback.Namely, the negative income of one company does not cancel the positive income of anothercompany. This is called aggregation bias .Shiller considered the trailing 10-year P/E ratio, where the earnings are averaged over thelast 10 years. This quantity is called the Shiller P/E ratio. Shiller found that future 10-yearand 20-year returns are highly negatively correlated with the Shiller P/E ratio. However, he
Mathematics Subject Classification.
Key words and phrases.
Linear regression, total returns, Jeffrey’s prior, Bayesian inference, earnings yield. a r X i v : . [ q -f i n . S T ] A p r TARAN GROVE, MICHAEL REYES, ANDREY SARANTSEV (a)
Index (b)
Earnings (c)
Dividends
Figure 1.
Annual nominal S&P index, earnings and dividends per sharewas unable to use standard linear regression tools since these 10-year periods overlap and thecorresponding residuals are not independent. Instead, he devised a new statistical approach,which we shall not utilize here. It is impossible to survey all existing literature; instead, werefer the reader to the articles cited above, as well as [1, 7, 8, 9].In this article, we consider the inverse of the P/E ratio: earnings yield , defined as lastyear’s total net income of all stocks in the benchmark index, divided by the current sum ofthe market capitalizations of all these companies.We take annual data 1946–2019. By that time, the SEC was already created, GreatDepression and the Second World War were in the past, and the U.S. Federal Governmentadopted an activist monetary and fiscal policy. Since then, there were no stock marketcrashes comparable to the Great Depression. Indeed, consider the events in 2008–2009:the Federal Reserve and the Federal Goverment stepped in to arrest the collapse of theeconomy. Yet, the resulting drop in stock prices and incomes was small compared to theGreat Depression. Another example is the Black Monday of 1987. This stock market crashwas every bit as terrifying as October 1929. However, the Federal Government cut interestrates, earnings growth did not slow down, and markets soon recovered.According to the financial theory, long-run total real returns must be equal to long-runreal earnings growth plus long-term average dividend yield. Thus we can view annual totalreal return minus annual real earnings growth as implied dividend yield. Companies candistribute earnings to shareholders using buybacks instead of payouts, or reinvest theminto business, which raises future earnings and dividends. Recently, corporate payout andbuyback policy was incorporated into the classic CAPE research, [4, 6]. Thus arguably thisimplied dividend yield is a more comprehensive measure than actual dividend yield.In our research, we find that the intrinsic long-run average for this implied dividend yieldis 3 . EW VALUATION FOR INDEX RETURNS 3 (a)
Real Earnings Growth G ( t ) (b) Real Return ˆ R ( t ) Figure 2.
Real Earnings Growth and Real Total Return in 3-Year StepsJudging by the quantile-quantile plots or standard normality tests, these regressions willnot have i.i.d. normal residuals if we take data in one-year steps. To make residuals normal,we must switch to 3-year steps. This leaves us with only a few data points: 24 three-yearsteps. We account for the uncertainty in the parameter estimate using a Bayesian framework.We put a non-informative Jeffrey’s prior on all regression parameters, and utilize well-known,explicit formulae for the posterior: multivariate normal-inverse χ distribution.Our article is organized as follows. In Section 2, we describe the data. In Section 3, weformulate our main model. In Section 4, we fit the model, discuss the point estimates forparameters, and goodness-of-fit. Section 5 is devoted to Bayesian inference and simulationmethod. This section also contains a review of Bayesian linear regression inference withJeffrey’s prior, taken from [2, pp.46-47]. Results of simulations are given in Section 6.Sections 7 and 8 are devoted to discussion and conclusions, respectively. Our data and codeare available at github.com/asarantsev/BayesianLT .2. Data
Our benchmark index is the Standard & Poor 500, created in 1957, which contains the500 largest companies in the United States of America. Our benchmark also contains itspredecessor the Standard & Poor 90, created in 1926. Below, we simply refer to this indexas S&P. We take the beginning-of-month level of this index from Yahoo Finance. To adjustfor inflation, we use the monthly Consumer Price Index (CPI) data. We also use annualearnings and dividends data for the S&P share. The data can be found in the book [12] and(with updates) Robert Shiller’s Yale University web site.Denote by S ( t ), the S&P index level at the beginning of the year t , for t = 0 , . . . D ( t ) and E ( t ) the dividends and net income per S&P share, respectively;and denote by C ( t ) the CPI level in January of the year t , with t = 0 corresponding to 1946,and t = 74 to 2019. We define total real returns during the year t by(1) R ( t ) = ln S ( t ) + D ( t ) S ( t − − ln C ( t ) C ( t − , for t = 1 , . . . , . TARAN GROVE, MICHAEL REYES, ANDREY SARANTSEV (a)
Heat ∆(0) + . . . + ∆( t − (b) Deviation ∆( t ) Figure 3.
Deviation for each 3-year step and Heat without detrendingFor 3-year windows (there are T = (cid:98) / (cid:99) = 24 of them), we getˆ R ( t ) := R (3 t −
2) + R (3 t −
1) + R (3 t ) , t = 1 , . . . , T. Total earnings for 3-year windows areˆ E ( t ) := E (3 t −
2) + E (3 t −
1) + E (3 t ) , t = 1 , . . . , T. Real earnings growth from previous to next window is(2) G ( t ) = ln ˆ E ( t + 1)ˆ E ( t ) − ln C (3 t + 3) C (3 t ) , for t = 1 , . . . , T − . We use logarithmic (geometric) returns in (1) and (2), instead of standard arithmeticreturns, to avoid the complication of compound interest.3.
Main Model
The implied dividend yield is defined by(3) ∆( t ) := ˆ R ( t + 1) − G ( t ) , t = 1 , . . . , T − . Assume its intrinsic average is c . Then the heat measure for year t is H ( t ) = t − (cid:88) k =1 (∆( k ) − c ) = t − (cid:88) k =1 ∆( k ) − c ( t − . We now model G ( t ) for t = 1 , . . . , T − G ( t ) ∼ N ( m G , v G ) i.i.d.The QQ plot from Figure 4 shows this is reasonable. Next, we model∆( t ) = α + βH ( t ) + γG ( t ) + ε ( t )= α + β t − (cid:88) k =1 ∆( k ) − cβ ( t −
1) + γG ( t ) + ε ( t ) , ε ( t ) ∼ N (0 , v ∆ ) i.i.d.(4) EW VALUATION FOR INDEX RETURNS 5 (a)
QQ Real Earnings Growth G ( t ) (b) QQ Regression Residuals ε ( t ) Figure 4.
Quantile-quantile plotsRegression residuals are also i.i.d. normal, uncorrelated with G ( t ), because one covariateof this regression is G ( t ). From (3), total real return for the t th 3-year window is given byˆ R ( t ) = G ( t −
1) + ∆( t − t = 2 , . . . , T . The model (4) can be considered as time seriesARIMA(1, 1, 0). We can rewrite it as a simple autoregression in terms of H ( t ): H ( t ) − H ( t −
1) = α + c + γm G + β · H ( t −
1) + δ ( t ) ,δ ( t ) := γ ( G ( t ) − m G ) + ε ( t ) ∼ N (0 , v δ ) , v δ := v ∆ + γ v ε . (5)As we see later, β <
0. It follows that, as t → ∞ ,(6) H ( t ) → h := − β − ( α + c + γm G ) , ∆( t ) → . Taking expectation in (5), we get: E H ( t ) − E H ( t −
1) = β · ( E H ( t − − h ). Thus E H ( t ) = h + ( H (0) − h ) β t → h . As we see, this model exhibits mean reversion. Later, we see thatwe can reject the unit root hypothesis. The limit h of H ( t ) is positive from (6); thus, theaverage total real return is13 t [ G (0) + . . . + G ( t −
1) + ∆(0) + . . . + ∆( t − t [ G (0) + . . . + G ( t − c H ( t )3 t . This has expectation ( m G + c + t − E H ( t )) / ≈ ( m G + c + t − h − t − β t h ) /
3. As t → ∞ ,this becomes closer to the long-run growth rate ( m G + c ) / . h (1 − β t ) / t is positive but decreases to 0 as t increases. Note that c is not equal to themean of all deviation terms ∆( t ), as explained in the Introduction.4. Regression Fitting
We have: ˆ m G = 0 . v G = 0 . . Thus the annualized real earnings growth overthe long-run is ˆ m G / . −
3% per year.Next, ˆ α = 0 . β = − . γ = − . β := − β · c we have: ˆ β = − . v ∆ = 0 . . More detailed statistics is given below. TARAN GROVE, MICHAEL REYES, ANDREY SARANTSEV
Coefficient estimate stderr t -value p -value [0.025 0.975] α β -0.3982 0.174 -2.289 0.034 -0.762 -0.034 β γ -0.3827 0.271 -1.412 0.174 -0.950 0.184As we see, the confidence interval for β does not contain zero, and the p -value is less than5%. Thus we can reject the null hypothesis β = 0, which is equivalent to having so-calledunit root. In other words, we can reject the hypothesis that this autoregression with respectto H is not mean-reverting but integrated.The estimate for c is ˆ c = ˆ β / ˆ β = 0 . . .
3% average dividend yieldover the postwar period, but far away from the current 2% dividend yield. Surprisingly,the market was overvalued in the postwar period most of the time. Indeed, see the plots of∆( t ) − c and H ( t ) in figure 6. Almost every year, this measure of heat was above 0.The QQ plots are shown in Figure 4. We applied two classic normality tests for 3-yearreal earnings growth rates, and for residuals of the main regression. These are Shapiro-Wilkand Jarque-Bera tests. For regression residuals, Shapiro-Wilk gives us p = 0 .
95 and Jarque-Bera gives us p = 0 .
92. For real earnings growth rates, Shapiro-Wilk gives us p = 0 .
74 andJarque-Bera gives us p = 0 .
70. For regression, raw and adjusted R are 0 .
316 and 0 . h from (6) is equal toˆ h = 0 . . − . · . . . . Bayesian Inference and Simulation
Normal sample.
Take i.i.d. x i ∼ N ( m, v ), i = 1 , . . . , n . Frequentist statistics gives uspoint estimates: x and s , respectively. Impose Jeffrey’s non-informative prior: π ( m, v ) ∝ v − . With this prior, there is no pre-existing information about m and v . Conditioned upon v , it is uniform in m . This prior is actually improper , since the total integral of this densitywith respect to m ∈ R and v > p ( m, v ), computed below,is a proper probability measure, namely the normal-inverse χ : The marginal posterior of v is the inverse χ distribution(7) p ( v ) ∼ Ξ( n − , s ) . The inverse χ distribution Ξ( ν, c ) with ν degrees of freedom and scale c is defined as thedistribution on (0 , ∞ ) with Lebesgue density1Γ( ν/ (cid:16) ν (cid:17) ν/ c ν x − ν/ − exp (cid:16) − νc x (cid:17) . Thus ξ ∼ Ξ( ν, c ) if c/ξ ∼ χ ν . And the conditional posterior distribution of m given v is p ( m | v ) = N (cid:16) ˆ x, vn (cid:17) . Therefore, the marginal distribution of m is Student (Gosset) t -distribution, with n − m has a heavy-tailed distribution. EW VALUATION FOR INDEX RETURNS 7
Multiple linear regression.
Assume we have n data points and d factors x , . . . , x d : y i = k + k x i + . . . + k d x di + ε i , ε i ∼ N (0 , v ) i.i.d. i = 1 , . . . , n. Classical theory gives us point estimates for standard error v and vector of regression coef-ficients k := [ k , k , . . . , k d ]:ˆ k := ( X T X ) − X y , ˆ v := 1 n − d − y − X k ) T ( y − X k ); X := [ x ij ] i =0 ,...,d , x j := 1; y := [ y , . . . , y n ] T . (8)Impose a non-informative Jeffrey’s prior, similarly to the previous subsection: π ( v ) = v − .This means we do not have any pre-existing information about regression coefficients andthe standard error v . Conditioned upon v , it is uniform with respect to the coefficients. Thisprior is actually improper , since the total integral of this density with respect to parametersspace k ∈ R d +1 and v > π is not a true probability measure. However, theposterior p is a proper probability measure. It is computed using the likelihood L ( x, y | k , v ),a product of normal densities: L ( x, y | k , v ) = n (cid:89) i =1 ϕ ( y i − k − k x i − . . . − k d x di , v ) ,ϕ ( z, v ) := 1 √ πv exp (cid:18) − z v (cid:19) ,p ( k , v ) ∝ L ( x, y | k , v ) π ( k , v ) . (9)The result is the following multivariate normal-inverse χ -distribution: p ( v ) = Ξ( n − d − , s ) , p ( k | v ) ∼ N d +1 (ˆ k , ( X T X ) − v ) . Applications for our simulations.
We take Jeffrey’s prior: π ( m G , v G ) ∝ v − G . Thenposterior will be multivariate normal-inverse χ , as shown in the Appendix. Independentlyof real earnings growth G ( t ), we take Jeffrey’s prior for α, β, β := − βc, γ, v ∆ : π ( α, β, β , γ, v ∆ ) ∝ v − . We compute the posterior for d = 3, using results from the Appendix: multivariate normal-inverse χ distribution. As discussed in the Appendix, each of these priors is non-informative(no pre-existing belief or information), and improper (total integral over all parameter spaceis infinite). The posterior is a proper probability measure: a multivariate normal-inverse χ distribution. We are interested in Value at Risk: x α for level α is defined by P ( X ≥ x α ) = α .For both the current value of heat and value 0, we perform 10000 simulations. We considertwo cases: 15 and 30 years, that is, 5 and 10 three-year steps. For each simulation of N number of steps and 3 N years, N = 5 and N = 10, we do the following steps:(a) Simulate parameters: first standard errors, then coefficients.(b) Simulate N three-year time steps.(c) Compute the average real return over these 3 N years as the sum of these N totalreal returns, divided by 3 N .(d) Next, for each of the two cases, we have 10000 results. Compute the mean, standarddeviation, and 90% and 95% Values at Risk. TARAN GROVE, MICHAEL REYES, ANDREY SARANTSEV (a) (b)
15 Years (c)
30 Years
Figure 5.
Histograms of Annualized Average Total Real Returns6.
Simulation Results
It is well-known that the long-run average total real return of the stock market is 6 − . c used to detrend heat (one can think of c as the averagedeviation, or, in other words, implied dividend yield) is 3 . .
53% + 3 .
88% = 6 . H (0) = 3 . − · c = 0 . Discussion
The results above show that given current (as of end of 2019) market conditions, futurelong-term returns are substantially lower than the long-term 6 .
41% market returns mentionedabove. This confirms the conventional wisdom. However, the long-run returns: 9, 15, or 30years, are still positive. Note that the longer the time horizon is, the larger are the averagereturns, and the smaller the standard deviation is. Indeed, over time, heat reverts to themean, and the effects of overheated market wane out.Also, the market is more stable over the long run than over the short run - another wellknown observation. This stability is reflected in VaR too: For a longer time horizon, 90%and 95% VaR are higher than for the short term.However, due to small sample, there are large uncertainty in parameter estmates: Standarderrors are large, and the 90% and 95% Values at Risk are far away from the averages in allsimulations. Thus we cannot really be sure that our conclusions above (future returns givencurrent conditions are lower than long-term returns of 6 − EW VALUATION FOR INDEX RETURNS 9 (a)
Heat H ( t ) (b) Deviation ∆( t ) − c Figure 6.
Deviation for each 3-year step and Heat8.
Conclusion
The conventional wisdom, as of end of year 2019, states that the stock market is overvalued,and the market is due to a correction. We confirm this wisdom by constructing a novelmeasure of “market heat” which takes into account both dividends and capital appreciation(price increase) and compares them to earnings growth. Our model fits the data well,although small amount of data leads to large uncertainty in point estimates. To account forthis, we use Bayesian inference with a non-informative Jeffrey’s prior for parameters, withmultivariate-normal inverse- χ posterior distribution.The future returns over the long run are positive but lower than they would be if themarket was perfectly neutral (not overvalued and not undervalued). The longer the run, thecloser the average returns to the long-run 6 − Acknowledgements.
We are thankful to other students at the University of Nevada,Reno, who took part in the work: Chyna Metz-Bannister (who collected the data), LissaCallahan (who tried fitting factors using AR( p ) and ARMA( p, q ), but this did not leadto normally distributed residuals); Akram Reshad for help with coding. The third authorthanks Tran Nhat for useful discussion. We thank the Department of Mathematics andStatistics for welcoming atmosphere for collaborative research.8.2. Conflict of interest.
The authors declare no conflict of interest.8.3.
Financial support.
The authors did not have external support for this work.8.4.
Roles.
Methodology, A.S.; Coding, all authors; Data management, A.S.; first drafts,all authors; final version, A.S.
References [1]
Robert D. Arnott, Denis B. Chaves, Tzee-man Chow (2017). King of the Mountain: The ShillerP/E and Macroeconomic Conditions.
The Journal of Portfolio Management (1), 55–68.[2] Biliana S. Bagasheva, Frank J. Fabozzi, John S. J. Hsu, Svetlozar T. Rachev (2008).
Bayesian Methods in Finance . Wiley.[3]
Alicia Barrett, Peter Rappoport (2011). Price-Earnings Investing.
JP Morgan Asset Manage-ment, Reality in Returns
November 2011 (1), 1–12.[4]
Oliver D. Bunn, Robert J. Shiller (2014). Changing Times, Changing Values: A Historical Anal-ysis of Sectors within the US Stock Market 1872–2013. NBER Working Paper 20370.[5]
John Y. Campbell, Robert J. Shiller (1998). Valuation Ratios and the Long-Run Stock MarketOutlook.
The Journal of Portfolio Management (2), 11–26.[6] Farouk Jivrai, Robert J. Shiller (2017). The Many Colours of CAPE. Yale ICF Working PaperNo. 2018-22.[7]
Jane A. Ou, Stephen H. Penman (1989). Accounting Measurement, Price-Earnings Ratio, and theInformation Content of Security Prices.
Journal of Accounting Research , 111–144.[8] Thomas Philips, Cenk Ural (2016). Uncloaking Campbell and Shillers CAPE: A ComprehensiveGuide to Its Construction and Use.
The Journal of Portfolio Management (1), 109–125.[9] Pu Shen (2000). The P/E Ratio and Stock Market Performance.
Federal Reserve Bank of Kansas CityEcomonic Review Q (4), 23–36.[10] Robert J. Shiller (2015).
Irrational Exuberance.
Princeton University Press, 3rd edition.[11]
Jeremy G. Siegel (2014).
Stocks for the Long Run.
McGraw-Hill, 5th edition.[12]
Robert J. Shiller (1992).
Market Volatility . MIT University Press.
University of Nevada in Reno, Department of Mathematics and Statistics
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