return-on-equity (ROE) analyses. More specifically, the question centers on the choice of dividend and bond yields in cost-of-equity models.
The discounted cash flow (DCF) model follows this general form:
Required Return = DIVIDEND YIELD + Expected Dividend Growth
To estimate the required ROE with this model, the analyst must select a dividend yield as well as a growth rate.
Similarly, the risk premium model involves selecting a bond yield and an expected risk premium estimate:
Required Return = BOND YIELD + Expected Risk Premium
Ideally, when using either model, it would be helpful to know what the dividend or bond yield will be in the near future, since regulators are setting rates that will be in effect for the indefinite future. Unfortunately, professional forecasts of financial variables are notoriously unreliable and appear to be getting worse, not better, over time.1 In keeping with these financial research findings, I develop yield estimates based on actual rather than forecasted data.
There are two basic choices for the yield: 1) averages of historic yields (such as a 12-month average), and 2) the current or spot yield (such as today's dividend yield).
Statistical Characteristics of Dividend and Bond Yields
Financial economists determined long ago that forecasts based on spot data were in many cases better predictors of future financial variables than forecasts based on any average of historic yields. As time went on, researchers further noted that certain financial data are generated by what are known as nonstationary processes: A nonstationary series does not tend to revert to a fixed mean or average over time. In their pioneering work on time series analysis in the late 60's and early 70's, Box and Jenkins noted that the data used in business applications is especially likely to exhibit this tendency:
"[F]orecasting has been of particular importance in industry, business, and economics, where many time series are often better represented as nonstationary, and in particular, as having no natural mean."2
The absence of any natural mean for the nonstationary series used in many business applications proves that historic averages provide no useful information about future values of dividends.
Just to be clear, while many data series used in business applications are nonstationary, one should not assume that all data series exhibit this characteristic. Many do not. For example, to estimate the likely dividend payout ratios for utilities, averages of past ratios may be very helpful in predicting future values.3
The "random walk" model (a nonstationary, non-mean-reverting process) is often used to approximate the behavior of bond yields and dividend yields:4
Yieldt+1 = Yieldt + et+1
Elegant in its simplicity, the model indicates that the forecast of tomorrow's yield is simply today's yield. But since we are dealing with series subject to random variation, we need to add the error term. For forecasting purposes we assume that error is zero. Only once the forecast period is over can we
determine the direction and the magnitude of the error. This is a characteristic of any forecasting model.
Note that the only datum in developing the forecasted yield under the random walk approach is the current yield; there are no terms to reflect data prior to the current period. Murphy gives a simple working definition of the random walk model for bond yields:
"The direction of interest rates [bond yields] cannot be predicted any better than by the flip of a coin."5
Put another way, if bond yields today have fallen below the market yield of six months ago, the random walk model tells us they are as likely to decline further as they are to return to prior levels. With a 50-percent chance that yields will increase and a 50-percent chance that they will decrease, forecast error tends to be minimized by using the current yield as the forecast. Because the model contains error, it produces a forecast that is far from perfect, but over time it outperforms any other model.
Using averages of historical yields in the situation described above incorrectly assumes that future yields are more likely to revert to prior levels than they are to decline. Financial research reveals that yield data simply do not behave in this manner. Unfortunately, the idea that bond and dividend yields tend to revert to the mean over time is as intuitively appealing as it is incorrect. That makes it difficult to dislodge from financial practice.
So, despite strong theoretical support for the use of spot yields in estimating cost of capital, many rate-of-return analysts regularly use an average of historic yields. These analysts often defend the use of averages on statistical grounds, such as "avoiding the luck of a draw." However, using an average of yields over time rather than the current spot provides no insulation from the random nature of financial time series. In fact, using averages not only increases estimation error in cost-of-capital models, but also increases the volatility of the errors. The longer the averaging period, the greater the average error and the greater its volatility.
We can demonstrate these points using actual data.
Empirical Support for Using Spot Yields
The research design for the following empirical analysis consists of data on utility bond yields and electric utility dividend yields from Moody's Public Utility Manual. The data set runs from January 1954 through July 1993. This amounts to 475 observations for each series.
Table 1 shows how the historic and the forecast periods were defined. The first historic period is the 12 months comprising calendar year 1954. Our
forecast target is the average yield over the next 12 months. The process was repeated by rolling the analysis forward one month at a time.
The forecasted yields were then compared to the actual yields over the forecast period. The forecasts were evaluated against four accuracy criteria and two volatility criteria taken from forecast evaluation literature.6 The accuracy criteria are 1) minimize average absolute error, 2) minimize average absolute percentage error, 3) minimize mean squared error, and 4) maximize coefficient of determination. The volatility criteria are 1) minimize standard error, and 2) minimize coefficient of variation.
Bond Yields. Aa-rated utility bonds yields were used in this analysis. Table 2 lists the performance of the models. The results are overwhelming and, not surprisingly, consistent with the earlier research. By all accuracy measures, the spot forecast outperforms the forecasts based on historic averages. The spot forecast is also dominant in terms of volatility reduction. And we see clearly that the longer the averaging period, the worse the forecasting method by any measure.
Dividend Yields. For the dividend yield series, we used the yield of Moody's Electric Utility Stock Index. Table 3 lists the results of this analysis, which are qualitatively identical to those found in the bond yield analysis. This should leave little doubt as to the superiority of spot yields as predictors of future yields.
An Intuitive Explanation
of the Superiority of Spot Forecasts
Some intuitive thinkers might be asking, "How can one data point be a better predictor of the future than an average of 12 data points?" The answer is that the single data point is more than it appears to be. Spot yields are not developed in a vacuum. They actually contain all useful information from the past as well as the most up-to-date expectations of the future. To the extent that prior yields form a reference point for expectations of future yields, the information content of historic yields is already included in the current spot yield. Thus, to average the historic yield with the spot yield simply double counts any relevant historic information and leads us away from rather than toward the actual future yield.
Note also that by averaging historical data we introduce more distant data into the analysis. This forces us to put less weight on the current spot yield, so that we can consider yields estimated in a period where market participants knew less about next year than they do today. This simply does not make sense.
Rarely in any analysis does one method so clearly dominate the others. Those who understand the nature of nonstationary statistical processes, however, should not find these results a surprise. Given the overwhelming superiority of spot yields, it will be interesting to see whether some rate of return analysts continue to rely on average yields. t
Steven G. Kihm is a senior financial and pricing analyst with the natural gas division of the Public Service Commission of Wisconsin. Prior to joining the PSC, Mr. Kihm worked as a consultant for MSB Energy Associates in Madison, WI. Mr. Kihm, a chartered financial analyst,
has master's degrees in finance and quantitative analysis from the University of Wisconsin-Madison, and a
bachelor's degree in economics from the University of
1. Financial Analysis Journal, May/June 1995, p. 30.
2. Box and Jenkins, Time Series Analysis: Forecasting and Control, 1976, p. 7.
3. Those interested in determining whether a particular time series is stationary or nonstationary should refer to the literature for a discussion of the statistical testing procedures. See, e.g., Cryer, Time Series Analysis, 1986.
4. Malkiel, A. Random Walk Down Street, 1985.
5. Murphy, Stock Market Probability, 1994, p. 124.
6. Makridakis and Wheelwright, The Handbook of Forecasting, 1987, pp. 509-512.
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