Estimating the Cost of Equity Capital of a Non-Traded Unique Canadian Entity Laurence Booth University of Toronto

In a recent paper in this journal, Patterson (1993) developed a technique for estimating the systematic risk or beta of a non-traded firm and applied the technique to Teleglobe Canada Inc. This paper was drawn from the testimony that Patterson (1991b) was asked to provide on behalf of Teleglobe, when it first went before the Canadian Radio-Television and Telecommunications Commission (CRTC) to be awarded a fair rate of return as a regulated common carrier. The problem Patterson attempted to solve is a very important problem, since the bulk of the assets of corporate Canada are either owned by non-tmded firms as divisions of foreign or Canadian companies, or are thinly traded with major control blocks. These factors are also becoming more important for regulated American utilities. Both these factors make it difficult to determine systematic risk, using either direct market data from the firm itself, or indirect market data from similar companies.

In attempting to solve this problem, Patterson is to be commended. However, there are some severe problems in Patterson’s approach, both in its theoretical development and empirical application, that render the results of doubt- ful validity. These problems are not immediately apparent in the paper published in this journal, since only a part of Patterson’s work is contained in it and none of the data is provided. However, I was asked to provide evidence on Teleglobe’s fair rate of return and critique the company’s evidence on behalf of the Ontario Ministry of Culture and Communications, and had full access to Patterson’s under- lying data. My testimony, Booth (1991a), showed that Patterson’s work consistently overestimated beta for all regulated firms, and by implication also overestimated the beta for Teleglobe. In its reasons for its decision the CRTC stated:

Address all correspondence to Laurence Booth, Faculty of Management, University of Toronto, 246 Bloor St West, Toronto, ON, Canada, M5S 1v4.

8 M A C 1993 122

The Commission notes that all the companies at the low end of Dr. Patterson’s sample are, like Teleglobe, regulated utilities. As discussed above, although the Commission considers Teleglobe to be of somewhat higher risk than the regulated telephone companies in Dr. Patterson’s sample, i t nonetheless considers that Teleglobe would be in the low risk end of the sample. In the Commission’s opinion, Dr. Patterson’s failure to account for the tendency of his model to overestimate the market betas of firms at the low risk end of his sample may result in an overestimate of Teleglobe’s market beta. (p. 133)

The CRTC decided (CRTC, p. 135) that Teleglobe’s beta was between my estimate of 0.53 and the 0.65 esti- mated by another company witness, Mr. Carmichael. It did not accept the 0.87 estimate Patterson derived from his model.

Although the CRTC is not the final arbiter of whether or notanapproachhasacademicvalue, itdoes have tomake the money decisions. Moreover, over the years the CRTC has seen a large number of expert witnesses presenting a variety of state of the art estimates. The readers of this journal may be interested intheshortcomings in Patterson’s model that led the CRTC to reject it. This note will repeat and extend part of the analysis contained in my testimony. In particular, a theoretical model is developed that will put Patterson’s model in perspective. Later, some empirical evidence is presented to show that the theoretical short- comings in Patterson’s model are of practical concern.

Theoretical Framework

In the Sharpe-Lintner capital asset pricing model (CAPM), beta measures the extent to which the holding period return on a stock is sensitive to the holding period return on the market. For an investor holding a diversified portfolio, beta measures the additional portfolio risk from adding the security to the portfolio. The use of beta as a risk

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measure in Canada is normally justified by the work of Calvet and Lefoll (1988), who concluded that “a main finding is that the hypothesis that asset returns may be generated by a factor represented by the market value weighted market index cannot be rejected” (p. 1). This weak support of the CAPM is strengthened by the absence of significant support for its maincompetitor, the arbitrage pricing theory (APT) of Ross (1976). Abeysekera and Mahjan (1987), for example, recently found that “the return premia attached to the risk factors involved [in the APT] are not significantly different from zero” (p. 193).

The acceptance of the CAPM as a model, and beta as a risk measure, is almost universal among Canadian regu- latory bodies. The problem is that even when good market data exists to estimate beta, most of the regulated compa- nies are subsidiaries of larger diversified companies. Many of these subsidiaries are in different regulated areas as well as non-regulated areas. Canadian Utilities, for example, includes two gas distribution utilities, an electricity gener- atingsubsidiary andanoil andgasdevelopmentsubsidiary. Estimating accurate betas for these subsidiaries is ex- tremely difficult. By default, betas have to be inferred Crom other sources, and the question is essentially which infer- ence involves the least estimation risk.

One approach is to try to specify theoretically the sources of risk. For example, the general DCF model can be used to specify that the price of a security is a function of the expected dividend, the future discount rate, and the market’s expectations of the subsequent stream of divi- dends. Functionally, this can be represented as P = P[d,,k,g], where d, is the expected dividend, k the discount rate, and g an average expected growth rate in the uncertain stream of future dividends. Imposing assumptions ong would then generate different dividend discount models. However, the exact functional form need not be specified. Assume in- stead that there are three orthogonal factors in the economy. One factor is the level of aggregate demand that affects the cash flows to the firm -call this Y for income. The second factor affects the level of the discount rate - call this I for interest rates. The third factor affects market expectations and the risk premium effect on the discount rate - call this S for market sentiment. These three factors are general at the moment, but more structure will be imposed later.

With these three market factors the uncertainty in the holding period return (R) can be approximately expressed as

Equation (1) is anapproximation because R, will dependon the particular dividend discount model assumed, and the extent to which a linear approximation is accurate. The interprelation of Equation (1) is straightforward. The un-

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certainty in the rate of return depends on the uncertainty in the three market factors (E), and theelasticity, orsensitivity coefficients (q ), for a particular stock, subscripted j , with respect to the common market factors. Further structure can be imposed byeither specifying the linkages of the cash flow or dividend to income or the linkage of capitalisation uncertainty to interest rates and market sentiment.

Patterson’s model is based on Chung’s work, 1989, which essentially states that

where the overall income elasticity is further broken down into the income elasticity of demand, subscripted r for revenues, the production elasticity, subscripted p, and the financing elasticity, subscripted f. This is a commondevel- opment, where with further restrictive assumptions the latter two elasticities can be reduced to the familiar degrees of operating and financial leverage from the introductory finance course.

If the distribution of the holding period returns on all stocks is multivariate normal ,then the CAPM continues to hold even in a multi-factor world. In this case, Sharpe (1977) showed that beta can be determined as a weighted average of the sensitivities in Equations (1) and (2). For example, if the return on all stocks follow Equation (l), then so too will the return on the market, since that is simply a value weighted average of each individual return. If this definition of the market return is substituted into the definition of beta, it follows that,

% Pi = rly, -- Pym + rl, U -SIm +

where (7 denotes variance. If the three factors explain most of the uncertainty in the market return, the final term can be ignored. This leaves the market beta of any security as a weighted averageof thestock’s sensitivities with respect to the three market factors, where the weights are common to all securities.

Alternatively, since the return on the security also follows Equation (l), and the market factors areorthogonal, the further substitution of Equation (1) into Equation (3) gives

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which states that the beta for each security is a function of the uncertainty in each of the three underlying factors, and the sensitivities of the particular security and the market to each of the three factors. Equivalent to Equation (3), the security’s beta is still an average of its sensitivity coeffi- cients. Equation (4) is perfectly general, and can be ex- tended to more than three factors. However, three factors are enough to bring out the intuition, and the theoretical shortcomings of Patterson’s model.

This approach leads to a straightfonvard procedure. First, estimate the factor sensitivities inaconventional time series regression using Equation (1). Second, estimate the market coefficients in Equation (4) from a cross sectional regression using actual betas and the estimated factor sensitivities from the first series of regressions. The betas estimated indirectly in this manner can then be compared with the directly estimated betas to consider, for example, whether knowledge of the economic fundamentals re- quires that the direct beta estimates need to be adjusted.

The reason why this approach has not been used is that it does not help in estimating betas for non-traded firms. This is because if betas can not be directly estimated, then neithercan the factor sensitivities in Equation (1). Patterson finesses this problem somewhat by implicitly focusing on the first term in Equations (1) and (4) as expanded in Equation (2). Patterson uses the firm’s revenues to deter- mine the first elasticity in Equation (2), and then uses empirical proxies to approximate the production and fi- nancingelasticities. Patterson ignores the other two market factors in Equation (4). This is the theoretical basis for his Equation (12). However, there are several theoretical and empirical problems with this model.

Empirical Problems

First, since Sharpe and Cooper (1972) pointed out, actual betas are determinedalmostentirely by pricechanges, to the extent that dividends can be ignored in most beta estimation procedures. This means that Equation (2) is certainly the least important linkage between rates of return and market factors. This is not to say that aggregate income effects are not important. Aggregate income can also affect the rate of return through price effects as Equation (1) indicates. However, this linkage is not estimated by the use of Equation (2). The most important empirical linkage is between rates of return and capitalisation changes, that is the second two factors in Equation (l), as well as the impact of aggregate income on price changes. Simply put, Patterson’s model is not focusing on the most important empirical relationships.

Second, betas estimated according to Patterson’s Equa- tion (12) may be biased, since that equation only’uses one of the three sensitivity coefficients. From Equation (4),

there are two missing variables, apart from the inadequa- cies of Equation (2) as a proxy for the impact of aggregate income. Patterson mentions (page 118) that he explored the use of PE ratios, but dropped the idea due to negative values. The link to interest rates and market sentiment variables that drives the PE ratio was neverdiscllssed. This treatment is also apparent in Patterson’s empirical results, where the intercept is highly significant in Equations (14) and (15). This confirms the missing variable problem, since in Equation (4) these should be no intercept.

If the theoretical validity of Patterson’s model is questioned, then resort must be made to positive arguments that it works better than other models. The definition of working better should be based on both the explained variance of the cross-sectional estimation of the coeffi- cients inEquation (4), and the absence ofbias. Amodel that explains very little of the actual cross-sectional variation in betas is obviously of little use, since it indicates that the elasticities estimated in Equation (1) are not accurate. This could be due either to normal estimation problems, or the more serious missing variables problem mentioned above. If the explained variance is high, the second test is that the model does not consistently over or under estimate betas for certain types of firms. Again, a full specification using Equation (1) by definition avoids the problem of bias.

On explained variance, Patterson’s model works very well. His 40% explained variance in his cross-sectional regression exceeds the 19% of Chung’s original work using American firms. Moreover, 40% explained variance in cross-sectional work is usually regarded as very good, considering the difficulties in capturing capitalisation un- certainty. Rosenberg and Rudd (1987) reviewed most of the literature using accounting data to estimate betas instrumentally, and the best models used at least four independent variables to obtain similar values of explained variance. To obtain this value of explained variance using only one independent variable is impressive.

However, impressive as it is, there is still the question of bias. From Equation (4) there are obvious missing variables in the cross-sectional regression. Moreover, Rosenberg and Rudd found pronounced industry effects in most of the studies they reviewed. This indicates that even with four or more accounting variables, there are still missing variables, causing bias. Partly the problem is inevitable, given the incomplete speciFication of the factors inEquation (1) and the particularproblems incapturing the influence of market sentiment. However, additional ex- planatory variables are available that can proxy for these variables.

For example, for a given level of demand uncertainty, Booth (1981) showed that most firms with market power were less risky than were similar competitive firms. Con- sequently, monopolies and firms with market power have lowerbetas. Booth(1991)alsoshowed that firms withhigh

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labour-capital ratios tended to be more risky than firms with low capital-labour ratios. This indicates that capital intensive firms have lower betas. Gordonand Gould (1978) and Myers (1977) have also argued that growth is per- ceived to be risky, because of the option nature of growth opportunities. This result also follows from the higher equity duration of growth stocks.

These three theoretical results all affect regulated utilities and the example used by Patterson. Regulated utilities are natural monopolies that Booth’s work (1981) indicates should have lower betas. Moreover, regulated monopolies have relatively large capital-labour ratios, which Booth (1991) also showed results in lower betas. Finally, since rate of return regulation is supposed to remove excess earnings, regulated utilities should have relatively low forecast growth rates and low equity duration. This also implies lower betas. All of these three factors indicate that regulated utilities like Teleglobe have lower betas than non-regulated firms. Moreover, these factors are largely independent of the single independent variable used by Patterson. Finally, the very act of regulation shields regu- lated utilities from some of the factors influencing market sentiment, because investors know that most regulators protect the regulated company as much as the consumer. This is partly the reason why utilities arc regarded as defensive stocks. For example, in the early 198Os, TCPL was protected from contract errors caused by the collapse of energy prices and the existence of take or pay contracts. The result was TOPGAS, whereby the payments made by TCPL under take or pay contracts, instead of hitting the shareholder, were, in substance, rolled into the rate base with the financing costs passed onto consumers.

What the above remarks indicate is that the missing variables problem that theoretically exists with Patterson’s model almost certainly exists in practice. In Table 1 is my replication of Patterson’s model using his data. The first equation u.es LPROD1, which is Patterson’s (1991b) term for the directly estimate elasticity of Equation (2). The second equation uses LPROD2, which is Patterson’s term for the same equation with the degree of financial leverage estimated in the conventional way. By way of comparison, Patterson (1991) estimated the following equations:

LBMKT = -.344 t 0.222 LPRODl

(6.20)

LBMKT = -.326 + 0.204 LPROD2

(5.76)

The first equation is equivalent to Equation (15) in Patterson (1993) and the first equation in Table 1. The minor differences indicate changes in Patterson’s underly-

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ing data since his testimony and rounding differences. A simple comparison between these equations and those in Patterson (1991, 1993) indicate that the equations are substantially identical. Below them, in Table 1, are the predictionerrors for all the regulated utilities in Patterson’s sample. The prediction errors are defined as the difference between the betas predicted from Patterson’s model and the actual betas used by Patterson in estimating the model.

Casual inspection of the prediction errors indicates that Patterson’s model overpredicts for each of the ten regulated utilities in his sample. Moreover, the average prediction error is 0.19 using the LPRODl variable and 0.22 using the LPROD2 variable. Given the typical beta estimate for a regulated utility is 0.4 to 0.5, these are very significant prediction errors. The bias in the empirical application of Patterson’s model indicates that the missing variables problem is significant. To remedy the bias, there are two alternatives. One approach is to try to specify empirically the missing variables in Equation (4). How- ever, given the problems in estimating the market factors affecting capitalisation uncertainty, this approach is un- likely to differ significantly from the second approach. The second approach is to use any and all accounting and economic data to try and specify the model as fully as possible. This is essentially an atheoretical approach, arid is the approach adopted in the literature reviewed by Rosenberg and Rudd (1987).

For the problem at hand the independent variables, necessary to specify the cross sectional regression, were not available. However, one simple approach to remedy some of the deficiencies of Patterson’s model is to enter a dummy variable for each of the regulated firms. This is common practise in analysing regulated and unregulated firms. Bradley, Jarrell, and Kim (1984), for example, used dummy variables to capture the effects of regulation, when analysing the empirical determinants of capital structure. In analysing regulated firms it also recognises that market power, capital intensity, and growth prospects are likely to be highly collinear independent variables. Table 2gives the same two equations as Table 1, except that a dummy variable has been included for each of the ten regulated firms in Patterson’s sample.

Three features of Table 2 are important. First, if the explained variance is regarded as important, i t increases very significantly, from 40% to 68%, for the first regres- sion, and from 36% to 69% for the second regression. Second, the coefficient on the dummy variable is highly significant for both equations, in fact it is much more significant than the coefficient on Patterson’s LPROD variable. Moreover, it indicates that, all else constant, betas are about 0.67 lower for regulated than for non-regulated firms. Finally, the standard deviation of the prediction errors, marked deviation in Tables 1 and 2, is lower for the dummy variable adjusted model, indicating more accurate

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estimates. Clearly, the relationship between betas and the LPROD variable used by Patterson is different between regulated and non-regulated firms. This empirical result confirms the theoretical result that Patterson’s model is misspecified.

Estimation of the cost of equity capital is particularly important for regulated utilities, because it is used both for setting the firm’s revenue requirement and ultimately the prices for its regulated services, as well as for evaluating incremental investment. The fact that Patterson’s model is biased in the case of regulated utilities makes it inappropri- ate for either function. Moreover, if the regulated firm’s equity cost is overestimated, it follows that the cost for unregulated firms is underestimated, since the prediction errors by definition sum to zero over the full sample. This makes it unreliable for unregulated firms as well.

It remains to consider whether the bias is significant for the Teleglobe example used by Patterson. Using the independent values for Teleglobe indicates a beta of 0.44 for the dummy variable adjusted model, not the 0.87 estimated by Patterson. Even the 0.44 estimate may not be the best estimate for Teleglobe’s beta. My own estimate of Teleglobe’s beta was 0.53, using data from other regulated companies, and a traditional instrumental beta estimation model. This is not the point of this paper, however, but the discrepancy between the two estimates, which indicates the problems involved in implementing a model in the face of obvious missing variable problems.

Table 1 Beta Estimation Using Patterson’s Model

Table 2 Beta Estimation Using Dummy Variable Model

LBMKT = -.34237 + .22084 * LPRODl R2 = 39.6% (6.223)

LBMKT = -.32382 + .20285 * LPROD2 R2 = 36.1% (5.774)

Prediction errors1

TRANSALTA QUEBEC TEL NEWTEL M T&T FORTIS CANADIAN UTILITIES BRUNCOR BC TEL BC GAS BCE INC

AVERAGE DEVIATION

LPRODl LPROD2

.287

.164

.216

.093

.241

.192

.088

.196

.186

.237

.190

.0595

.324

.128

.253

.166

.292

.197

.116

.236

.221

.264

.220

.0648

LBMKT = -.14184 + .06925 * LPRODl - .666 * D (2.068) (7.192)

R2 = 68.1%

LBMKT = -.13982 + .07284 * LPROD2 - 6699 * D (2.447) (7.830)

R2 = 68.9%

Prediction Errors’

LPRODl LPROD2

TRANSALTA QUEBEC TEL N E W E L M T&T FORTIS CANADIAN UTILITIES BRUNCOR BC TEL BC GAS BCE INC

.065

.011 (.045)

.117

.052

.055

.022 (.035) .031

(-04)

.068 ( * O W (.041) ( . O W .121 .045 .05 1 .025

(.032) .031

____ _______ ~~~ ~ ~~

1. Prediction errors are the differences between the values predicted by myestimation of Patterson’s model and theactualvalua. Positivevalues indicate that the predicted values exceed actual values.

AVERAGE DEVIATION

.023

.0497 .023 .0489

1. Predictionerrorsare thedifferences between thevalues predicted by the dummy variable model and actual values.

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References

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Booth, L. (1981). Market structure, uncertainty and the cost of equity capital. Journal of Banking and Finance, 5(4), 467- 482.

Booth, L. (1991a). Evidence by the Ministry of Culture and Communications for the Government of Ontario on the fair rate of return on common equity for Teleglobe Canada Inc. Prepared for submission to the CRTC.

Booth, L. (1991b). The influence of production technology on risk and the cost of capital. Journal of Financial and Quantitative Analysis, 26(1), 109-127.

Bradley, M., Jarrell, G., &Kim, E.H. (1984). On theexistence of an optimal capital structure: Theory andempirical evidence. Journal of Finance, 39(2), 857-878.

Calvet, A., & Lefoll, J. (1988). Risk and return on Canadian capital markets. Canadian Journal of Administrative Sci- ences, 5(1), 1-12.

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CKTC. Teleglobe Canada 1nc.-Regulation after the Transitional Period. Telecom decision CRTC 91-21 (December 19, 1991), Ottawa.

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Gordon, M.J., & Gould, L. (1978) The cost of equity capital: A reconsideration. Journal of Finance, 33(3), 849-861.

Myers, S. (1977). Determinants of corporate borrowing. Journal of Financial Economics, 4(3), 147-176.

Patterson, C. (1991). Memorandum re: Fair rate of return on Teleglobeequity investment. Prepared for submission to the CRTC.

Patterson, C. (1993). The cost of equity capital of a non-traded unique entity: a Canadian study. Canadian Journal of Ad- ministrative Sciences, 10(2), 115-121.

Rosenberg, B., & Rudd, A.(1987). Thecorporateusesof beta. In J. Stern & D. Chew (Eds.), The revolution in corporate finance (pp. 38-68). Oxford, England: Blackwell.

Ross, S. (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13,343-362.

Sharpe, W.(1977). Thecapital asset pricingmodel: Amulti-beta interpretation. In H. Levy & M. Sarnat (Eds.), Financial decision making under uncertainty (pp. 127-136). New York: Academic Press.

Sharpe, W ., & Cooper, G. (1972). Risk-return classes of New York Stock Exchange common stocks: 1931-1967. Finan- cial Analysts Journal, 28 (March-April), 46-52.

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