Forecasting Equity Return

Learning Outcomes

• discuss approaches to setting expectations for equity investment market returns

• discuss risks faced by investors in emerging market equity securities

The task of forecasting equity market returns is often the central focus of setting capital market expectations. In this section, we discuss applying each of the major methodologies to equities.

Historical Statistics Approach to Equity Returns

Exhibit 3 shows the mean real return for each market portfolio centered within a 95% confidence interval. Results are also shown for a world portfolio, a world ex-US portfolio, and Europe. The portfolios are ordered from left to right on the basis of the mean return.

The means range from a low of 5.0% for Austria to a high of 9.4% in South Africa. Note that both of these values lie within the confidence interval for every country. From a statistical perspective, there is really no difference among these markets in terms of mean real return. This illustrates the fact that sample averages, even derived from seemingly long histories, are very imprecise estimates unless the volatility of the data is small relative to the mean. Clearly that is not the case for equity returns. Nonetheless, sample means are frequently cited without regard to the quality of information they convey.

Exhibit 3:

Historical Mean Returns with Confidence Intervals by Country, 1900–2017

As indicated in Section 2, shrinkage estimators can often provide more reliable estimates by combining the sample mean with a second estimate of the mean return. However, the application of a common shrinkage estimator confirms that there is no basis for believing that the true expected returns for the countries in Exhibit 3 are different.

DCF Approach to Equity Returns

Analysts have frequently used the Gordon (constant) growth model form of the dividend discount model, solved for the required rate of return, to formulate the long-term expected return of equity markets. Although this model is quite simple, it has a big advantage over using historical stock returns to project future returns. The vast majority of the “noise” in historical stock returns comes from fluctuations in the price-to-earnings ratio (P/E) and the ratio of earnings to GDP. Since the amount of earnings appears in the numerator of one ratio and the denominator of the other, the impact of these ratios tends to cancel out over time, leaving the relationship between equity market appreciation and GDP growth much more stable. And GDP growth itself, especially the real growth component, is much less volatile and hence relatively predictable.12 As an illustration, Exhibit 4 shows historical volatilities (defined as the standard deviation of percentage changes) for the S&P 500 Index return, P/E, the earnings-to-GDP ratio, real US GDP growth, and inflation for 1946–2016. The Gordon growth model allows us to take advantage of this relative stability by linking long-term equity appreciation to a more stable foundation—economic growth.

Exhibit 4:

Historical Comparison of Standard Deviations in the United States, 1946–2020

S&P 500	P/E	Earnings/GDP	Real GDP Growth	Inflation
16.09	21.75	31.1	2.4	2.8

Note: Standard deviation of % changes

In the United States and other major markets, share repurchases have become an important way for companies to distribute cash to shareholders. Grinold and Kroner (2002) provided a restatement of the Gordon growth model that takes explicit account of repurchases. Their model also provides a means for analysts to incorporate expectations of valuation levels through the familiar price-to-earnings ratio. The Grinold–Kroner model is

The Grinold–Kroner model

where E(Re) is the expected equity return, D/P is the dividend yield, %ΔE is the expected percentage change in total earnings, %ΔS is the expected percentage change in shares outstanding, and %ΔP/E is the expected percentage change in the price-to-earnings ratio. The term in parentheses, (%ΔE − %ΔS), is the growth rate of earnings per share. Net share repurchases (%ΔS < 0) imply that earnings per share grows faster than total earnings.

With a minor rearrangement of the equation, the expected return can be divided into three components:

• Expected cash flow (“income”) return: D/P − %ΔS

• Expected nominal earnings growth return: %ΔE

• Expected repricing return: %ΔP/E

The expected nominal earnings growth return and the expected repricing return constitute the expected capital gains.

In principle, the Grinold–Kroner model assumes an infinite horizon. In practice, the analyst typically needs to make projections for finite horizons, perhaps several horizons. In applying the model, the analyst needs to be aware of the implications of constant growth rate assumptions over different horizons. Failure to tailor growth rates to the horizon can easily lead to implausible results. As an example, suppose the P/E is currently 16.0 and the analyst believes that it will revert to a level of 20 and be stable thereafter. The P/E growth rates for various horizons that are consistent with this view are 4.56% for 5 years, 2.26% for 10 years, 0.75% for 30 years, and an arbitrarily small positive number for a truly long-term horizon. Treating, say, the 2.26% 10-year number as if it is appropriate for the “long run” would imply an ever-rising P/E rather than convergence to a plausible long-run valuation. The only very long-run assumptions that are consistent with economically plausible relationships are %ΔE = Nominal GDP growth, %ΔS = 0, and %ΔP/E = 0. The longer the (finite) horizon, the less the analyst’s projection should deviate from these values.

EXAMPLE 4

Forecasting the Equity Return Using the Grinold–Kroner Model

Cynthia Casey uses the Grinold–Kroner model in forecasting developed market equity returns. Casey makes the following forecasts:

• a 2.25% dividend yield on Canadian equities, based on the S&P/TSE Composite Index;

• a 1% rate of net share repurchases for Canadian equities;

• a long-term corporate earnings growth rate of 6% per year, based on a 1 percentage point (pp) premium for corporate earnings growth over her expected Canadian (nominal) GDP growth rate of 5%; and

• an expansion rate for P/E multiples of 0.25% per year.

1. Based on the information given, what expected rate of return on Canadian equities is implied by Casey’s assumptions?

   Solution to 1:

   The expected rate of return on Canadian equities based on Casey’s assumptions would be 9.5%, calculated as

   E(Re) ≈ 2.25% + [6.0% − (−1.0%)] + 0.25% = 9.5%.

2. Are Casey’s assumptions plausible for the long run and for a 10-year horizon?

   Solution to 2:

   Casey’s assumptions are not plausible for the very long run. The assumption that earnings will grow 1% faster than GDP implies one of two things: either an ever-rising ratio of economy-wide earnings to GDP or the earnings accruing to businesses not included in the index (e.g., private firms) continually shrinking relative to GDP. Neither is likely to persist indefinitely. Similarly, perpetual share repurchases would eventually eliminate all shares, whereas a perpetually rising P/E would lead to an arbitrarily high price per Canadian dollar of earnings per share. Based on Casey’s economic growth forecast, a more reasonable long-run expected return would be 7.25% = 2.25% + 5.0%.

   Casey’s assumptions are plausible for a 10-year horizon. Over 10 years, the ratio of earnings to GDP would rise by roughly 10.5% = (1.01)10 − 1, shares outstanding would shrink by roughly 9.6% = 1 − (0.99)10, and the P/E would rise by about 2.5% = (1.0025)10 − 1.

Most of the inputs to the Grinold–Kroner model are fairly readily available. Economic growth forecasts can easily be found in investment research publications, reports from such agencies as the IMF, the World Bank, and the OECD, and likely from the analyst firm’s own economists. Data on the rate of share repurchases are less straightforward but are likely to be tracked by sell-side firms and occasionally mentioned in research publications. The big question is how to gauge valuation of the market in order to project changes in the P/E.

The fundamental valuation metrics used in practice typically take the form of a ratio of price to some fundamental flow variable—such as earnings, cash flow, or sales—with seemingly endless variations in how the measures are defined and calculated. Whatever the metric, the implicit assumption is that it has a well-defined long-run mean value to which it will revert. In statistical terms, it is a stationary random variable. Extensive empirical evidence indicates that these valuation measures are poor predictors of short-term performance. Over multi-year horizons, however, there is a reasonably strong tendency for extreme values to be corrected. Thus, these metrics do provide guidance for projecting intermediate-term movements in valuation.

Gauging what is or is not an extreme value is complicated by the fact that all the fundamental flow variables as well as stock prices are heavily influenced by the business cycle. One method of dealing with this issue is to “cyclically adjust” the valuation measure. The most widely known metric is the cyclically adjusted P/E (CAPE). For this measure, the current price level is divided by the average level of earnings for the last 10 years (adjusted for inflation), rather than by the most current earnings. The idea is to average away cyclical variation in earnings and provide a more reliable base against which to assess the current market price.

Risk Premium Approaches to Equity Returns

The Grinold–Kroner model and similar models are sometimes said to reflect the “supply” of equity returns since they outline the sources of return. In contrast, risk premiums reflect “demand” for returns.

Defining and Forecasting the Equity Premium

The term “equity premium” is most frequently used to describe the amount by which the expected return on equities exceeds the riskless rate (“equity versus bills”). However, the same term is sometimes used to refer to the amount by which the expected return on equities exceeds the expected return on default-free bonds (“equity versus bonds”). From the discussion of fixed-income building blocks in Sections 3 and 4, we know that the difference between these two definitions is the term premium built into the expected return on default-free bonds. The equity-versus-bonds premium reflects an incremental/building block approach to developing expected equity returns, whereas the equity-versus-bills premium reflects a single composite premium for the risk of equity investment.

Exhibit 5 shows historical averages for both of these equity premium concepts by country for the period 1900–2020.14 For each country, the bottom portion of the column is the realized term premium (i.e., bonds minus bills) and the top segment is the realized equity-versus-bonds premium. The whole column represents the equity-versus-bills premium. The equity-versus-bills premiums range from 3.0% to 6.3%, the equity-versus-bonds premiums range from 1.8% to 5.2%, and the term premiums range from −0.6% to 2.9%.

Exhibit 5:

Worldwide Annualized Bonds vs. Bills and Equity vs. Bonds Premium (%), 1900–2020

Worldwide Annualized Bonds vs. Bills and Equity vs. Bonds Premium (%), 1900–2020

Notes: Germany excludes 1922–1923. Austria excludes 1921–1922. Returns are shown in percentages.

Source: Dimson et al. (2021, Chapter 2, Tables 8 and 9).

As with the mean equity returns in Exhibit 3, these historical premiums are subject to substantial estimation error. Statistically, there is no meaningful difference among them. Thus, the long-run cross section of returns/premiums provides virtually no reliable information with which to differentiate among countries.

Since equity returns are much more volatile than returns on either bills or bonds, forecasting either definition of the equity premium is just as difficult as projecting the absolute level of equity returns. That is, simply shifting to focus on risk premiums provides little, if any, specific insight with which to improve forecasts. The analyst must, therefore, use the other modes of analysis discussed here to forecast equity returns/premiums.

An Equilibrium Approach

There are various global/international extensions of the familiar capital asset pricing model (CAPM). We will discuss a version proposed by Singer and Terhaar (1997) that is intended to capture the impact of incomplete integration of global markets.

The Singer–Terhaar model is actually a combination of two underlying CAPM models. The first assumes that all global markets and asset classes are fully integrated. The full integration assumption allows the use of a single global market portfolio to determine equity-versus-bills risk premiums for all assets. The second underlying CAPM assumes complete segmentation of markets such that each asset class in each country is priced without regard to any other country/asset class. For example, the markets for German equities and German bonds are completely segmented. Clearly, this is a very extreme assumption.

A superscript “G” has been added on the asset’s risk premium to indicate that it reflects the global equilibrium. The term in parentheses on the far right is the Sharpe ratio for the global market portfolio, the risk premium per unit of global market risk.

Now consider the case of completely segmented markets. In this case, the risk premium for each asset will be determined in isolation without regard to other markets or opportunities for diversification. The risk premium will be whatever is required to induce investors with access to that market/asset to hold the existing supply. In terms of the CAPM framework, this implies treating each asset as its own “market portfolio.” Formally, we can simply set β equal to 1 and ρ equal to 1 in the previous equations since each asset is perfectly correlated with itself. Using a superscript “S” to denote the segmented market equilibrium and replacing the global market portfolio with asset i itself in Equation 4, the segmented market equilibrium risk premium for asset i is

This is the second component of the Singer–Terhaar model. Note that the first equality in Equation 5 is an identity; it conveys no information. It reflects the fact that in a completely segmented market, the required risk premium could take any value. The second equality is more useful because it breaks the risk premium into two parts: the risk of the asset (σi) and the Sharpe ratio (i.e., compensation per unit of risk) in the segmented market.15

The final Singer–Terhaar risk premium estimate for asset i is a weighted average of the two component estimates

To implement the model, the analyst must supply values for the Sharpe ratios in the globally integrated market and the asset’s segmented market; the degree to which the asset is globally integrated, denoted by φ; the asset’s volatility; and the asset’s β with respect to the global market portfolio. A pragmatic approach to specifying the Sharpe ratios for each asset under complete integration is to assume that compensation for non-diversifiable risk (i.e., “market risk”) is the same in every market. That is, assume all the Sharpe ratios equal the global Sharpe ratio.

In practice, the analyst must make a judgment about the degree of integration/segmentation—that is, the value of φ in the Singer–Terhaar model. With that in mind, some representative values that can serve as starting points for refinement can be helpful. Developed market equities and bonds are highly integrated, so a range of 0.75–0.90 would be reasonable for φ. Emerging markets are noticeably less integrated, especially during stressful periods, and there are likely to be greater differences among these markets, so a range of 0.50–0.75 would be reasonable for emerging market equities and bonds. Real estate market integration is increasing but remains far behind developed market financial assets, perhaps on par with emerging market stocks and bonds overall. In general, relative real estate market integration is likely to reflect the relative integration of the associated financial markets. Commodities for which there are actively traded, high-volume futures contracts should be on the higher end of the integration scale.

To illustrate the Singer–Terhaar model, suppose that an investor has developed the following projections for German shares and bonds.

The risk-free rate is 1.0%, and the investor’s estimate of the global Sharpe ratio is 0.30. Note that the investor expects compensation for undiversifiable risk to be higher in the German stock market and lower in the German bond market under full segmentation. The following are the fully integrated risk premiums for each of the assets (from Equation 4):

Equities: 0.70 × 17.0% × 0.30 = 3.57%.

Bonds: 0.50 × 7.0% × 0.30 = 1.05%.

The following are the fully segmented risk premiums (from Equation 5):

Equities: 17.0% × 0.35 = 5.95%.

Bonds: 7.0% × 0.25 = 1.75%.

Based on 85% integration (φ = 0.85), the final risk estimates (from Equation 6) would be as follows:

Equities: (0.85 × 3.57%) + (1 − 0.85) × 5.95% = 3.93%.

Bonds: (0.85 × 1.05%) + (1 − 0.85) × 1.75% = 1.16%.

Adding in the risk-free rate, the expected returns for German shares and bonds would be 4.93% and 2.16%, respectively.

The expected return for the German bonds should be 2.16% (1%+1.16%).

Virtually all equilibrium models implicitly assume perfectly liquid markets. Thus, the analyst should assess the actual liquidity of each asset class and add appropriate liquidity premiums. Although market segmentation and market liquidity are conceptually distinct, in practice they are likely to be related. Highly integrated markets are likely to be relatively liquid, and illiquidity is one reason that a market may remain segmented.

EXAMPLE 5

Using the Singer–Terhaar Model

1. Stacy Adkins believes the equity market in one of the emerging markets that she models has become more fully integrated with the global market. As a result, she expects it to be more highly correlated with the global market. However, she thinks its overall volatility will decline. Her old and new estimates are as follows:

   	Previous Data	New Data
   Volatility (σi)	22.0%	18.0%
   Correlation with global market (ρi,M)	0.50	0.70
   Degree of integration (φ)	0.55	0.75
   Sharpe ratio (global and segmented markets)	0.30	0.30

   If she uses the Singer–Terhaar model, what will the net impact of these changes be on her risk premium estimate for this market?

Solution:

The segmented market risk premium will decline from 6.6% (calculated as 22.0% × 0.30 = 6.6%) to 5.4% (= 18% × 0.30). (volatility * sharpe ratio)

The fully integrated risk premium will increase from 3.30% (= 0.50 × 22.0% × 0.30) to 3.78% (= 0.70 × 18.0% × 0.30). (correlation * volatility * sharpe ratio)

The weighted average premium will decline from 4.79% [= (0.55 × 3.30%) + (0.45 × 6.60%)] to 4.19% [= (0.75 × 3.78%) + (0.25 × 5.40%)], so the net effect is a decline of 60 bps.

Market Condition	Old Estimate	New Estimate	Calculation
Segmented Market Risk Premium	6.6%	5.4%	= Volatility * Sharpe Ratio
Volatility	22.0%	18.0%
Sharpe Ratio	0.30	0.30
Fully Integrated Risk Premium	3.30%	3.78%	= Correlation * Volatility * Sharpe Ratio
Correlation	0.50	0.70
Weighted Average Premium	4.79%	4.19%	= (Weight A * Fully Integrated) + (Weight B * Segmented)
Weight A (Fully Integrated)	0.55	0.75
Weight B (Segmented)	0.45	0.25

Risks in Emerging Market Equities

Most of the issues underlying the risks of emerging market (and “frontier market” if they are classified as such) bonds also present risks for emerging market equities: more fragile economies, lower degree of informational efficiency, less stable political and policy frameworks, and weaker legal protections. However, the risks take somewhat different forms because of the different nature of equity and debt claims. Again, note that emerging markets are a very heterogeneous group. The political, legal, and economic issues that are often associated with emerging markets may not, in fact, apply to a particular market or country being analyzed.

There has been a debate about the relative importance of “country” versus “industry” risk factors in global equity markets for over 40 years. The empirical evidence has been summarized quite accurately as “vast and contradictory.”16 Both matter, but on the whole, country effects still tend to be more important than (global) industry effects. This is particularly true for emerging markets. Emerging markets are generally less fully integrated into the global economy and the global markets. Hence, local economic and market factors exert greater influence on risk and return in these markets than in developed markets.

Political, legal, and regulatory weaknesses—in the form of weak standards and/or weak enforcement—affect emerging market equity investors in various ways. The standards of corporate governance may allow interested parties to manipulate the capital structure of companies and to misuse business assets. Accounting standards may allow management and other insiders to hide or misstate important information. Weak disclosure rules may also impede transparency and favor insiders. Inadequate property rights laws, lack of enforcement, and weak checks and balances on governmental actions may permit seizure of property, nationalization of companies, and prejudicial and unpredictable regulatory actions.

Whereas the emerging market debt investor needs to focus on ability and willingness to pay specific obligations, emerging market equity investors need to focus on variety of risks beyond the traditional credit and counterparty risks, especially in times of macroeconomic and political distress.

EXAMPLE 6

Emerging Market Equity Risks

Bill Dwight has been discussing investment opportunities in Belvia with his colleague, Peter Valt (see Example 3). He is aware that Valt declined to buy the recently issued government bond, but he believes the country’s equities may be attractive. He notes the rapid growth, substantial investment spending, free trade agreement, deregulation, and strong capital inflows as factors favoring a strong equity market. In addition, solid global growth has been boosting demand for Belvia’s natural resources. Roughly half of the public equity market is represented by companies in the natural resources sector. The other half is a reasonably diversified mix of other industries. Many of these firms remain closely held, having floated a minority stake on the local exchange in the last few years. Listed firms are required to have published two years of financial statements conforming to standards set by the Belvia Public Accounting Board, which is made up of the heads of the three largest domestic accounting firms. With the help of a local broker, Dwight has identified a diversified basket of stocks that he intends to buy.

Discuss the risks Dwight might be overlooking.

Guideline answer:

Dwight might be overlooking several risks. He is almost certainly underestimating the vulnerability of the local economy and the vulnerability of the equity market to local developments. The economy’s rapid growth is being driven by a large and growing fiscal deficit, in particular, rapidly rising redistributive social payments, and investment spending financed by foreign capital. Appreciation of the currency has made industries other than natural resources less competitive, so the free trade agreement provides little support for the economy. When the government is forced to tighten fiscal policy or capital flows shrink, the domestic economy is likely to be hit hard. Political risk is also a concern. A return to the prior regime is likely to result in a less pro-growth, less business-friendly environment, which would most likely result in attempts by foreign investors to repatriate their capital. Dwight should also have serious concerns about corporate governance, given that most listed companies are closely held, with dominant shareholders posing expropriation risk. He should also be concerned about transparency (e.g., limited history available) and accounting standards (local standards set by the auditing firms themselves).