Sources of Return

Building Blocks

Roger Ibbotson has proposed several building blocks methods that decompose observed stock market returns (for a summary, see Ibbotson 2001). These models begin with an identity and then decompose this relationship into equivalent measures. The basic identity states that nominal stock returns are the product of inflation, the real risk-free rate (on long-term Treasuries) and the equity risk premium:

In the analysis that follows, we will assume that inflation is measured as the year-over-year percentage change in the consumer price index and that the risk-free rate is the nominal yield on the 10-year Treasury. The equity risk premium is defined as follows:

and where the real risk-free rate is defined by:

Substituting and into the first equation establishes the identity.

Using Shiller's annual data spanning 1871 to 2009, I estimate the average long-term return on the 10-year nominal Treasury to be 4.66 percent and a 2.26 percent average inflation rate. The average real return to the S&P 500 over this period was 8.0 percent. Therefore, the nominal stock return is approximated by (1.08)∗(1.0226) – 1 = 0.104, or 10.4 percent. Using the formula for , I then solve for a 5.5 percent annual equity risk premium. This model says that the observed long-run average nominal return to stocks can be decomposed into a 2.26 percent inflation rate, a 2.35 percent real yield on the 10-year Treasury, and a 5.5 percent risk premium. Some analysts stop here; if they have views on forward inflation, real bond yields, and the risk premium, they can use these to get a rough estimate of the forward return on equity.

Earnings Model

Ibottson and Chen break the equity return into real capital gains (rcg) and income (inc), rewriting the first identity as follows:

The last term is reinvested income (Ibottson assigns this a value of 0.2 percent) whereas income itself is assumed to come in the form of dividends. We therefore use the real dividend yield in place of inc. The real capital gain can be written in terms of p/e growth and earnings growth as follows:

This definition can be expanded upon by substituting earnings:

Therefore, real capital gains are the product of growth rates in the price-to-earnings ratio and real earnings per share. Substituting this into the identity gives us the following expression:

We could evaluate this expression a couple of ways: If we simply substitute for annual p/e and real earnings growth plus the real dividend yield along with the reinvestment rate on income of 0.2 percent, we find from Shiller's data that p/e growth is 4.03 percent over this period while real earnings growth was 4.61 percent and the growth in real dividend yield has been 1.87 percent. Together, these imply a nominal return to equity equal to 13.3 percent annually. Shiller smoothes the p/e ratio by dividing the current index price by a 10-year moving average of trailing earnings. The growth rate of this series is 1.75 percent. Substituting this for the 4.03 percent produces a nominal equity return of 10.9 percent. This is slightly higher than the 10.4 percent measure derived from the building blocks model.

Let's assume for the moment that Rinv is fixed at 0.2 percent. Then, again, given our views on inflation and the growth rates of p/e, real earnings, and real dividends, we could come up with an equity return forecast.

Forecasting Earnings

Let me propose a simple structural model to forecast earnings. Using the Shiller data, suppose we estimated a vector autoregression (VAR) with a single lag selecting the variables used in the earnings model, that is, inflation and growth in both Shiller's 10-year smoothed earnings and smoothed p/e along with the growth rate in the real dividend yield. We forecast out 10 years (VARs are by definition mean-reverting). These forecasted paths represent the model's long-run forecasted conditional means for the variables. This model and method (VAR) is conceptual and intended to illustrate how analysts might use more disciplined approaches to pricing that are based on our understanding of the structural economy and how earnings behave, conditional on that structure. It is therefore not a normative model of earnings.

The average growth rates for this 10-year forward window are 1.92 percent for real dividends, 3.2 percent for the smoothed p/e multiple, 0.2 percent for real smoothed earnings, and 2.34 percent for inflation. Geometrically linking these growth rates and adding in 0.2 percent reinvestment of income produces an expected nominal equity return equal to 7.88 percent. This forecast is illustrated in the following building blocks diagram.

Figure 4.4 Building Blocks

We could substitute inflation expectations using TIPS or CPI swap spreads. The squishy component in the forecast is the price adjustment variable (p/e). It should be clear that while building blocks help isolate attention to the components of returns, it does not provide the forecast; rather, forecasts are generally extrapolated from history (a very tenuous process as discussed earlier in the section on trend analysis) or from forward markets such as the TIPS or CPI spreads, forward p/e ratios, and so forth. The advantage of the VAR approach is that its structural form gives us the ability to produce confidence intervals for forecasts, analyze how forecasts respond to unanticipated shocks (impulse responses), and conduct sensitivity analysis.