New(ish) paper on the Shiller CAPE ratio, by J. Siegel

Published in the May/June issue of the Financial Analysts Journal (vol. 72, no. 3). I'm not sure it it's gated, so you can find a working paper (from 2013!) version here.

Abstract:

Robert Shiller’s cyclically adjusted price–earnings ratio, or CAPE ratio, has served as one of the best forecasting models for long-term future stock returns. But recent forecasts of future equity returns using the CAPE ratio may be overpessimistic because of changes in the computation of GAAP earnings (e.g., “mark-to-market” accounting) that are used in the Shiller CAPE model. When consistent earnings data, such as NIPA (national income and product account) after-tax corporate profits, are substituted for GAAP earnings, the forecasting ability of the CAPE model improves and forecasts of US equity returns increase significantly.

The gist of the paper:

"In this article, I offer an alternative explanation of the elevated CAPE ratio. The nature of the earnings series that is substituted into the CAPE model has not been consistently calculated for the long period over which Shiller has estimated his CAPE equations. Changes in accounting practices since 1990 have depressed reported earnings during economic downturns to a much greater degree than in the earlier years of Shiller’s sample."

[...]

"Companies report their earnings in two principal ways: reported earnings (or net income) and operating earnings. Reported earnings are earnings sanctioned by the Financial Accounting Standards Board (FASB), an organization founded in 1973 to establish accounting standards. Those standards—the generally accepted accounting principles, or GAAP—are used to compute the earnings that appear in annual reports and that are filed with government agencies (earnings filed with the IRS may differ from those filed elsewhere). GAAP earnings, which are the basis of the Standard & Poor’s
reported earnings series that Shiller used in computing the CAPE ratio, have undergone significant conceptual changes in recent years. 

A more generous earnings concept is operating earnings, which often exclude such “one-time” events as restructuring charges (expenses associated with a company’s closing a plant or selling a division), investment gains and losses, inventory write-offs, expenses associated with mergers and spinoffs, and depreciation or impairment of “goodwill.” But the term operating earnings is not defined by the FASB, and companies thus have some latitude in interpreting what is and what is not excluded. In certain circumstances, the same charge may be included in the operating earnings of one company and omitted from those of another. Because of these ambiguities, several versions of operating earnings are calculated."

[...]

"The definition of reported earnings has undergone substantial changes in the last two decades. In 1993, the FASB issued Statement of Financial Accounting Standards (FAS) No. 115, which stated that securities of financial institutions held for trading or “available for sale” were required to be carried at fair market value. FAS Nos. 142 and 144, issued in 2001, required that any impairments to the value of property, plant, equipment, and other intangibles (e.g., goodwill acquired by purchasing stock above book value) be marked to market.9 These new standards, which required companies to “write down” asset values regardless of whether the asset was sold, were especially severe in economic downturns, when the market prices of assets are depressed. Furthermore, companies were not allowed to write tangible fixed assets back up, even if they recovered from a previous markdown, unless they were sold and recorded as “capital gain” income."

[...]

 "A distortion related to the Standard & Poor’s methodology for computing the P/E of an index—what I call the “aggregation bias”—overestimates the effective ratio of the index when a few companies generate large losses, as happened during the financial crisis. S&P adds together the dollar profits and losses of each S&P 500 company, without regard to the weight of each company in the index, to compute the aggregate earnings of the index. This procedure would be correct if each company were a division of the same conglomerate and one wished to determine the P/E of that conglomerate"

[...]

"Because of changes in the definition of GAAP earnings, it is important to use a definition of corporate profits that has not changed over time, as in the series computed by the national income economists at the Bureau of Economic Analysis (BEA), which compiles the national income and product accounts (NIPAs)."

[...]

"In forecasting future 10-year real stock returns, the highest R squared is achieved by using NIPA profits for specifications of the CAPE regression, with either the price index portfolio or the total return portfolio."

Siegel offers alternative estimates of how over-valued the S&P is, according to each CAPE measure, as well as estimates of future returns. The CAPE that uses the NIPA profit measure produces the lowest over-valuation and the highest expected returns. I'm generally sceptical of such estimates, so I won't go into those details.

What I got from this paper is a reminder that the S&P measure of profits (and perhaps other measures that rely on reported earnings) has changed over time, due to accounting changes, so one has to be careful when using it.

QE: What if investors don't buy it?

John Authers has written another good column. It's about "what happens if rates never rise." Among many interesting things, he reminds us to watch profits (payrolls and GDP are secondary, really). He also walks us through the (increasingly plausible) scenario where the Fed doesn't raise rates at all, and  the central bank needs to resort to QE to counter an economic slowdown.

He writes, in passing, something with disturbing implications:

The risk continues to be that investors at some point give up on monetary policy and its power to make a difference — and that would be bad for stocks. So rather than plan for a continued indiscriminate rally in US stocks, it is probably better to focus on those that can show some sustained pricing power, and on those that pay a decent yield.

That first sentence (emphasis mine) entails a mind-blowing possibility. What it the emperor has no clothes?  What if the main (only?) effect of QE is through higher asset valuations? What if QE works because investors think it works and nothing else? What if investors wake up and decide that QE doesn't work?

Market crashes that never happened

Here's an article from FundReference.com on market analysts who kept predicting (incorrectly) that the stock market would crash.

(Hat tip to my old colleague and accomplished investment manager: P. M.  I'm not sure if I can say his name, since I got his pointer through LinkedIn.)

I love this table, towards the end of the article:

I'm not trying to say that bearish calls are nonsense. In fact, I'm bearish more often than bullish. But I'll stay away from specific market calls.

Forecasting long-term stock returns: the two-hour recipe (II)

Last week I started writing up a quick (?) methodology to forecast equity returns. Specifically, the question was

Forecast the real total return of U.S. equities over the next ten years. Show your work. Time: two hours.

I wrote down a decomposition of the total return into three components:

$$\frac{R_{t,t+k}}{(1+\widehat{\pi}_{t+1, t+k})^k} = \frac{V_{t+k}}{V_{t}} \frac{(1+g_F)^k} {(1+\widehat{\pi}_{t+1, t+k})^k} (1+\widehat{dy}_{t+1, t+k})^k$$

The three components are:

1) Income, which boils down to the geometric average of the dividend yield:
$$(1+\widehat{dy}_{t+1, t+k})^k$$
2) The 10-year change of a valuation ratio.
$$\frac{V_{t+k}}{V_{t}}$$
3) The real growth of the fundamental used in the construction of the valuation ratio.
$$\frac{(1+g_F)^k} {(1+\widehat{\pi}_{t+1, t+k})^k}$$
Since this is meant to be a quick estimation, I decided that I would use either the historical average or the ten-year rolling average of the relevant data to forecast each of the three components.

Pulling the Shiller long-term data set (xls) on stock prices, earnings, and dividends, I took the geometric average of the dividend yield between 2005 and 2014 as my forecast for the dividend yield over the next ten years: 2.0082%. So my forecast for \(\widehat{dy}_{t+1, t+10}\) is 0.02.

For the valuation ratio, we can calculate two from the Shiller dataset. The first one is the CAPE ("cyclically-adjusted" P/E ratio, or "Shiller's P/E"). A casual observation of the time series chart since 1880 suggests that the CAPE either experienced a shift sometime after the 1980s, or is experiencing upward drift. Today's CAPE (27.9) is significantly higher than the historical average (16.6) or the ten-year rolling average (22.6). We'll take those two values as alternative forecasts of the CAPE ten years from now. The historical CAPE implies that the ratio of valuation metrics, \(V_{t+k} / V_{t}\), is 0.595 (16.6 / 27.9). The ten-year rolling average CAPE implies a ratio of 0.81 (22.6 / 27.9).

The second valuation ratio we can compute from the Shiller dataset is the dividend yield (or rather, to fit the total return formula above, the price-to-dividend ratio):

Just like the CAPE, the price/dividend ratio seems to have experience either a shift or drift some time after the 1980s. Today's multiple (55.8) is close to the 10-year rolling average, but much higher than the historical average (27.9). As with the CAPE, we'll consider both to forecast the 10-year-ahead price/dividend ratio. Using the historical P/D, the ratio of valuation metrics, \(V_{t+k} / V_{t}\), is  0.50 (27.9 / 55.8), whereas using the 10-year rolling average, the ratio is  0.93 (51.9 / 55.8).

Growth of the fundamental

The third component of the total return is the real growth rate of the fundamental:
$$\frac{(1+g_F)^k} {(1+\widehat{\pi}_{t+1, t+k})^k}$$
Which fundamental we use is determined by the valuation ratio we pick. For the CAPE, the fundamental is the 10-year rolling average of earnings. For the price-dividend ratio, the fundamental is dividends.

The real growth rate of (the 10y average) of earnings has been 1.66% per annum. The rolling 10-year counterpart fluctuates quite a bit (even though this is the rolling average growth rate of a rolling average of earnings), and is now at 3.5%.

For real dividends, the historical (10-year rolling average) growth rate is 1.34% (5.1%).

Putting everything together

I have proposed two forecasts for each of two possible valuation ratios and their corresponding fundamentals, for a total of four forecasts (the income component is the same for all four).  The following table combines the forecast components of real returns to produce the total return forecast:

	\(V_{t+k} / V_{t}\)	\(g_F\)	\((1+g_F)^k / (1+\pi)^k\)	\(dy\)	\((1+dy)^k\)	\(R_{t,t+10} / (1+\pi)^k\)	Annual real return
CAPE (historical avg.)	0.595	0.0166	1.179	0.02	1.219	0.855	-1.55%
CAPE (10y rolling avg.)	0.81	0.035	1.411	0.02	1.219	1.393	3.37%
Dividend yield (historical avg.)	0.50	0.0134	1.142	0.02	1.219	0.696	-3.56%
Dividend yield (10y rolling avg.)	0.93	0.051	1.644	0.02	1.219	1.864	6.43%

The last column shows that the forecast real return, per year, varies from -3.6% to 6.4%.

Forecasting long-term stock returns: the two-hour recipe (I)

Suppose you are given the following task:

Forecast the real total return of U.S. equities over the next ten years. Show your work. Time: two hours.

This post describes how I would go about fulfilling this assignment.

The first thing is to define total return:

$$R_{t,t+k}=\frac{P_{t+k}}{P_{t}}\prod_{s=1}^{k}(1+dy_{t+s})$$

where \(R_{t,t+k}\) is the gross total return between year \(t\) and year \(t+k\), \(P_{t}\) is the price of the stock or index at time \(t\), and \(dy_{t}\) is the income yield at time \(t\). (The income from a stock is dividends plus net repurchases by the issuer.) You can get to that equation by "rolling over" the one-period total return, as I show below.

The total return can be further decomposed into more manageable bits. If you divide the price level by a "fundamental" \(F\):

$$R_{t,t+k}=\frac{(P_{t+k} / F_{t+k})}{(P_{t} / F_{t})} \frac{F_{t+k}}{F_{t}}\prod_{s=1}^{k}(1+dy_{t+s}) = \frac{V_{t+k}}{V_{t}} (1+g_F)^k (1+\widehat{dy}_{t+1, t+k})^k$$
Now the total return is a function of three things:

1) The change of a valuation ratio \(V\).

2) The growth of a fundamental \(F\): \(g_F\).

3) The (geometric) average of income yield: \(\widehat{dy}_{t+1,t+k}\)

The question asked to forecast the real return, but for that you just need to divide through by the inflation factor \((1+\widehat{\pi}_{t+1, t+k})^k\), where \(\widehat{\pi}_{t+1, t+k}\) is the geometric average of the inflation rate between \((t+1)\) and \((t+k)\):

$$\frac{R_{t,t+k}}{(1+\widehat{\pi}_{t+1, t+k})^k} = \frac{V_{t+k}}{V_{t}} \frac{(1+g_F)^k} {(1+\widehat{\pi}_{t+1, t+k})^k} (1+\widehat{dy}_{t+1, t+k})^k$$

The "fundamental" \(F\) that goes in the valuation ratio could be anything, but you should probably pick a variable such that:

1) The resulting valuation ratio is "mean reverting" (over the relevant forecasting horizon, in this case ten years), and

2) You can forecast the growth of the "fundamental." Once you pick a particular valuation ratio, you are also committing to forecasting the growth rate of its corresponding fundamental.

Several such valuation ratios have been proposed in the finance literature. I list them in the following table:

Valuation ratio (V)	Fundamental (F)
Price/dividend	Dividend
CAPE (a.k.a. Shiller's PE)	Ten-year average of real earnings
q ratio	Net worth of corporations at market value
Market capitalization / GDP	GDP
Price/total income	Total cash flow (dividend + net repurchases)

The last ratio, price/total income, is really just a generalized version of the price/dividend ratio. I expect the two to be highly correlated, but I'll keep both for now.

Next you need a forecasting strategy, i.e. you need to put values on \(V_{t+k}\), \(\frac{(1+g_F)^k} {(1+\widehat{\pi}_{t+1, t+k})^k}\), and \((1+\widehat{dy}_{t+1, t+k})^k\) You only have two hours to do this whole thing, so you can't do a lot.

Off the top of my head, I would say you can either:

1) Use historical averages, using the entire history of data available.
2) Use the historical averages from a recent subset of the data available.

The strategy should depend on (a) how long are your historical time series, and (b) whether you suspect structural changes that shifted those averages over time, or make them drift.

Where should you get the data for this exercise? A lot of people use the Shiller's time series that go back to 1870. You won't get q-ratios or total cash flow or total market capitalization from Shiller's spreadsheet, so you would be limited to the CAPE and the dividend yield as valuation ratios. For today that will suffice.

The income component

Let's start with the last component of the return, the dividend yield: \( (1+\widehat{dy}_{t+1, t+k})^k\).

A cursory inspection of the time series suggests the dividend yield has declined over its entire history, but it seems relatively stable since the late 1990s. I would then use the most recent ten years to forecast the dividend yield over the next ten.

The Shiller dataset is monthly. For each December, I take the 12-month trailing average of the dividend series (column C), and I divide it by the price (column B). That's my estimated dividend yield for the year ended in December. Next I calculate the geometric average of the dividend yield between 2005 and 2014, which is 2.0082%. So my forecast for \(\widehat{dy}_{t+1, t+10}\) is 0.02.

The valuation ratio

Next, the valuation ratio. Let's start with the CAPE (cyclically-adjusted PE ratio). The chart below shows that the historical average (in green, 16.6) is much below today's CAPE (27.85) and also below today's ten-year rolling average (in red, 22.6). It does seem like the CAPE shifted upward sometime in the 1980s or 1990s, but we don't know whether that shift is permanent. We can use both the historical average and the ten-year rolling average to come up with alternative forecasts of the CAPE ten years from now.

[I ran out of blogging time today! I will continue next time.]
........................................................................................................................................................
Derivation of the multi-period total return formula:

The one-period total return is given by
$$R_{t,t+1} = \frac{P_{t+1}+D_{t+1}}{P_{t}}$$
If you reinvest the income \(D_{t+1}\) into the stock, that will buy you \(D_{t+1} / P_{t+1} \) additional stock units, for a total return of
$$R_{t,t+1} = \frac{P_{t+1}(1+D_{t+1} / P_{t+1})}{P_{t}} = \frac{P_{t+1}(1+dy_{t+1})}{P_{t}}$$
Next period you do the same thing, reinvesting \(D_{t+2}\) at price \(P_{t+2}\), for a total return
$$R_{t,t+2} = \frac{P_{t+2}(1+dy_{t+1})(1+D_{t+2} / P_{t+2})}{P_{t}} = \frac{P_{t+2}(1+dy_{t+1})(1+dy_{t+2})}{P_{t}}$$
When you generalize to \(k\) periods you get
$$R_{t,t+k}=\frac{P_{t+k}}{P_{t}}\prod_{s=1}^{k}(1+dy_{t+s})$$

Anticipating the Fed's language

Tim Duy wonders whether predicting the Fed's change of language over the next six months will be as easy as looking at the statements from 2004. On Jan. 28, 2004 the FOMC statement introduced the word "patient", replacing the "considerable period" phrase. The Fed kept "patient" in the March 16 statement, and then in May 4 it replaced the key word "patient" with "measured." The first hike would come on June 30.

What did the markets do over that period?

    (Click on the chart to enlarge.)

	10y yield (b.p.)	3m yield (b.p.)	S&P 500 (%)	VIX (%)
Oct. 28, 2003-Dec. 9, 2003	+9	-3	1.28	4.82
Dec. 9, 2003-Jan. 28, 2003	-10	+1	6.15	-4.18
Jan. 28, 2003-Mar. 16, 2004	-52	+2	-1.58	21.22
Mar. 16, 2004-May 4, 2004	+86	+4	0.80	-18.63
May 4, 2004-Jun. 30, 2004	+6	+33	1.90	-13.35

In the bond market the largest yield change was between March 16 and May 4, but it appears that at least part of it was a correction of a previous 50 b.p. decline. Treasury bill yields only moved significantly between May 4 and June 30, when the Fed actually raised rates. The stock market seemed even less unfazed by the Fed's changing statement.

The bottom line is: the markets seemed to be affected by other news much more than by the change in language in the FOMC statements, perhaps because they correctly anticipated the Fed's subtle changes of stance (or lack thereof). (Of course, a better way to do this is to look at minute-by-minute market prices, before and after the FOMC announcements.)

Chart of the week

Admit it: you keep an eye on the World Cup while at work. So do stock traders, apparently.

Bloomberg Businessweek relays the results of a paper by researchers at the European Central Bank, showing that stock market trading volume plunged, at least in some countries, during matches of the 2010 World Cup.

Here's the abstract of the paper, by Michael Ehrmann and David Jan-Jansen (the chart above is from Table 3b, page 28):

At  the 2010 FIFA World Cup in South Africa, many soccer matches were played during stock market  trading  hours,  providing  us with  a  natural  experiment  to analyze  fluctuations in  investor  attention.  Using  minute‐by‐minute  trading  data for  fifteen  international  stock exchanges, we present three key findings. First, when the national team was playing, the number of  trades  dropped  by  45%, while volumes  were  55%  lower.  Second,  market activity  was influenced by match events. For instance, a goal caused an additional drop in trading activity by 5%. The magnitude of this reduction resembles what is observed during lunchtime, and as such might not be indicative for shifts in attention. However, our third finding is that the comovement between  national  and  global  stock  market  returns decreased  by over  20%  during World  Cup matches, whereas no comparable decoupling can be  found during lunchtime. We conclude that stock markets were following developments on  the soccer pitch rather  than in  the  trading pit, leading to a changed price formation  process.

Aside from these findings, I found something else noteworthy. On pages 18-19, Ehrmann and Jan-Jansen report that the standard deviation of minute-by-minute stock returns across individual stocks declined significantly during matches. They interpret this result as that limited attention makes traders process less firm-specific information, relative to market- and sector-wide information.

If you don't believe in market efficiency, World Cup games might be a good time to trade.

What caught my eye

1. The Bank of Japan's balance sheet is about to get much, much bigger, by Sober Look, via the excellent MacroDigest.

Here is why. Credit Suisse for example projects that Japan's inflation rate has peaked and is about to begin declining. In fact CS researchers see a complete divergence between the BoJ's own projection of inflation and reality. A number of other researchers (for example Scotiabank) agree.

2. Big Ideas in Macroeconomics, by Kartik Athreya. Noah Smith's excellent, three-part review makes me want to read it.

3. Is this an example of financial repression?

Federal Reserve officials have discussed whether regulators should impose exit fees on bond funds to avert a potential run by investors, underlining concern about the vulnerability of the $10tn corporate bond market.
The article is somewhat ambiguous on whether the exit fees would apply only to corporate-bond funds or to government-bond funds as well.

4. An organization I didn't know about: OMFIF, or Official Monetary and Financial Institutions Forum. They put together commentary, analysis, surveys, conferences, etc. around central banking. This week they were in the news because they published a report showing that "public-sector institutions" (including central banks, public pension funds, and sovereign funds) are buying more and more equities. The report is not available online.

5. Speaking about central banks, I just signed up for the "Grand Central" newsletter, the WSJ's blogging service about, well, central banking.

6. Blog recommendation. A mysterious Jesse Livermore writes (mostly) about the stock market on Philosophical Economics. I particularly enjoyed this post, but I would say everything he writes is worth reading.

7. The macroeconomic effects of asset purchases, by Martin Weale and Tomasz Wieladek on VOX EU. I am skeptical of the VARs (how are shocks identified?), but here it is anyways.

Our results suggest that an asset-purchase shock that results in an announcement worth 1% of nominal GDP leads to a rise in real GDP of about 0.36% in the US and 0.18% in the UK; and to a rise in the CPI of 0.38% in the US and 0.3% in the UK. These findings are encouraging, because they suggest that asset purchases can be effective in stabilising output and prices. The implied UK Phillips curve is steeper than in the US, meaning that the same change in output would have a relatively greater impact on UK inflation. Quantitatively, monetary easing leading to a 1% rise in output results in a 1% rise in the US CPI, whereas in the UK the CPI rises by 1.5%. These estimates of the inflation–output trade-off are similar to those that previous studies reported for conventional (interest rate-based) monetary policy. Table 1 compares the implied effect on output and prices with that reported in previous studies of unconventional monetary policy. For real GDP, our reported figures are very similar to those reported in previous studies. For the US, we also find a similar effect on the CPI, but for the UK, our results suggest that the impact on the CPI is almost three times as large as the effect reported in Baumeister and Benati (2013) and Kapetanios et al. (2012).

8. A note on Piketty and diminishing returns to capital. Highly recommended by Tyler Cowen.