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

U.S. equities in 2013: Highest Sharpe ratio since 1995

This is by Gavyn Davies at the Financial Times. In his article, Davies ponders whether the bull market can last another year.

I am a little alarmed by this:

"...there is an unusually strong consensus in analysts’ forecasts for next year, with almost everyone expecting stronger global GDP growth, dovish central banks and further rises in equity markets. As John says, this “cozy consensus” borders on complacency."

Such a widespread bullish sentiment has often be a preamble to short-term corrections. Although,

"Past returns do not help us very much to predict the out-turn for 2014."

Besides, Davies states:

"Nor are valuation signals are very helpful in picking the top of a bull market, except when they are at extremes.That is not the case at present."

Even more,

"Research on identifying bubbles by new econometric techniques (which I will write more about in the near future) is also re-assuring. It suggests that the probability that the US equity market is currently in a bubble is less than 20 per cent."

In the rest of the article Davies turns to estimations of the output gap, and to the monetary policy outlook.

Well worth reading

1. Forward interest rates and monetary tightening, by Jim Hamilton at Econbrowser. This is a nice refresher of forward interest rates, and an application of the Gurkaynak-Sack-Wright data set (xls).

2. A summary and critique of Summers' "great stagnation" hypothesis, by Stephen King at the FT. King makes an important distinction between "supply side" and "demand side" theories of the stagnation. Tyler Cowen already pointed the discussion in that direction a few days ago.

3. NBER working paper by David Dollar and Benjamin Jones, on the link between China's institutions and its "unusual" macroeconomic performance.

China presents several macroeconomic patterns that appear inconsistent with standard stylized facts about economic development and hence inconsistent with the standard neoclassical growth model. We show that Chinese macroeconomic patterns instead appear consistent with an environment where state control of factor markets can promote aggressive output goals. We consider the micro-institutional features that can sustain this behavior, emphasizing the hukou system and state control over capital allocation, and present a simple model built on these features. The model can explain several puzzling facts about the Chinese economy, including its unusually low labor share and unusually high saving and investment rates. Interestingly, the model also shows that free-market reforms can initially take the economy further from global macroeconomic norms.

4. John Hussman on his usual crusade to show that we should expect meager returns from U.S. equities. I find his charts persuasive--until he starts trying to show that the Philips curve is incorrectly specified, or that there is a tiny correlation between unemployment and the stock market. (I don't disagree with the theses, but with the methods.)

Is the profit share a trending variable now?

Gavyn Davies writes a piece on corporate profits and the equity market. He observes that some investors are concerned that "the profit share in the U.S. economy is currently abnormally high." Because investors assume that the profit share mean reverts, he explains, sooner or later corporate earnings and profit margins will decline, and equities will tumble. But then Davies goes on to undermine this assumption, and ventures that the profit share might not mean revert at all "in the foreseeable future."

The following chart shows the ratio of a national accounting measure of profits (corporate profits after tax with inventory valuation adjustment and capital consumption adjustment) to nominal GDP. The share of income that goes to corporations (the profit share) lacked a clear trend until the early 1990s. The profit share appeared to hover around a long-term median of 5%-6%, with large cyclical fluctuations. Then each subsequent cyclical peak of the profit share (late 1990s, mid- to late-2000s, and 2012-2013) got higher and higher, and there appears to be an underlying, long-term rising trend in the profit share. The question of whether we are witnessing a rising trend profit share is of the utmost importance, and not only for the valuation of equities. But framing the question, as Davies does, as an "either/or" proposition, I think, obscures  projections for the medium term.

I am willing to accept some arguments for a rising trend of the profit margin. Globalization and the decline of unions are two of them. But that is still compatible with the view, which I subscribe, that the profit share exhibits cyclical fluctuations. The only thing that may have changed since the early 1990s is that, instead of fluctuating around a fixed number, say 5%, now the profit exhibits cyclical fluctuations around a rising trend. There is tremendous uncertainty about what this trend level might be right now, or even about whether this trend exists at all, but if we are still on the same long-term trend that started in the early 1990s, the trend value of the profit share might be now between 7.5% and 9%, based on a simple linear extrapolation. That means that the profit share right now is between 0.5 and 2 percentage points above trend.

Data on corporate earnings going back to the 1870s [xls] seems to suggest that earnings do revert. In most cycles, earnings peak at a median 2 months before the onset of recession, and bottom out during recessions. If you think that this time the cyclical behavior of profits is going to be different, I would like to know why.

If Davis, on the other hand, is referring to the effect of a rising trend profit share on market valuations, while allowing for (potentially deep) cyclical fluctuations, then I agree with him. Still, I think we need to understand more about what is driving this trend before we make long-term projections.