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%.
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