## Friday, March 20, 2015

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