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.

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