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 10year 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 tenyear rolling average of the relevant data to forecast each of the three components.
Pulling the Shiller longterm 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 ("cyclicallyadjusted" 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 tenyear 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 tenyear 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 pricetodividend 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 10year rolling average, but much higher than the historical average (27.9). As with the CAPE, we'll consider both to forecast the 10yearahead 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 10year 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 10year rolling average of earnings. For the pricedividend ratio, the fundamental is dividends.
The real growth rate of (the 10y average) of earnings has been 1.66% per annum. The rolling 10year 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 (10year 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%.