Review of Quantitative Finance and Accounting, 10 (1998): 235–267 © 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Alternative Models for Estimating the Cost of Equity Capital for Property/Casualty Insurers ALICE C. LEE KPMG Peat Marwick LLP J. DAVID CUMMINS Wharton School, University of Pennsylvania
Abstract. This paper estimates the cost of equity capital for Property/Casualty insurers by applying three alternative asset pricing models: the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), and a unified CAPM/APT model (Wei (1988)). The in-sample forecast ability of the models is evaluated by applying the mean squared error method, the Theil U2 (1966) statistic, and the Granger and Newbold (1978) conditional efficiency evaluation. Based on forecast evaluation procedures, the APT and Wei’s unified CAPM/ APT models perform better than the CAPM in estimating the cost of equity capital for the PC insurers and a combined forecast may outperform the individual forecasts. Key words: property/casualty, insurance, CAPM, APT, cost of equity capital, asset pricing
I. Introduction It is well known that the cost of capital for an industrial firm may include cost of debt, cost of preferred stock, cost of retained earnings, and cost of a new issue of common stock. Cost of equity capital is made up of cost of retained earnings and cost of a new issue of common stock. Financial theory suggests that returns of industrial firms should be a function of risk, but because the management principles of financial institutions are not necessarily identical to those of industrial firms, applications of financial theory to nonindustrial firms (such as banks, savings and loans, and insurance firms) must be theoretically and empirically investigated. What financial institutions must consider are exposure to different types of risk, the relative importance of various risks, and the existence of market imperfections and constraints such as regulation. For industrial firms, their production function and the return from their production capabilities are the main indicators of their value to investors in the market. Investments made by industrial firms are generally for the purposes of increasing and improving production capabilities. While, for financial institutions such as insurance companies, there are no production outputs to serve as an indicator of firm value. Output measurements for financial institutions are more difficult to measure because service sector output is intangible. For instance, an insurance company is in the business to manage the risk of others, as well as the firm’s own risk in financial investments.
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Specific issues of property/casualty (PC) insurance companies that differ from industrial firms include [Lee and Forbes (1980)] 1) different income and accounting measures, Generally Accepted Accounting Principles (GAAP) and Statutory Accounting Principles (SAP), which affect reported earnings and retained earnings; 2) unique borrowing-lending rate relationship; and 3) an asset portfolio comprised primarily of securities of industrial firms. An additional complicating issue for insurance companies is policyholder reserves. Reserves are the debt capital of the insurance industry, and policyholder reserves may be viewed as a source of investable funds. And there are also special issues associated with the cost of capital for regulated industries which include public utility and insurance firms. Do the regulations impose price controls or constraints which affect the profitability and valuation of firms in the industry? Because public utility prices are regulated by a “fair rate of return,” there has been much research and discussion on cost of equity capital estimation for public utilities. More recent research includes Bower et al. (1984); Bubnys (1990); Elton et al. (1994); Myers and Boruki (1994); Schink and Bower (1994). In fact, there is an entire body of literature devoted to this subject. Two books [Gordon (1974a) and Kolbe et al. (1984)] discuss this area. Similarly, the insurance industry has regulatory agencies which affect the pricing of its product. But in regulating insurance prices, regulators often use book value rates of return which are typically not accurate because cost of equity capital is a market-based value instead of a book value. Also, regulators usually do not vary the cost of capital by company or line, possibly creating excessive profits in some lines and disincentives to participate in others, and with better estimates of cost of capital for PC insurers, this would be possible. In their book, Cummins and Harrington (1987) present research on the issue of the fair rate of return in property-liability insurance. Fairley (1979), Hill (1979), Lee and Forbes (1980), and Hill and Modigliani (1987) apply the capital asset pricing model (CAPM) to derive risk-adjusted rates of return that the capital markets require of stock property-liability insurers. Based upon Ross’ (1976) arbitrage pricing theory (APT) of capital asset pricing, Kraus and Ross (1982) develop a continuous time model under uncertainty to determine the “fair” (competitive) premium and underwriting profit for a property-liability insurance contract. An empirical application of the arbitrage pricing theory (APT) in determining the cost of equity capital for PC insurers has yet to be considered.1 In addition to regulatory considerations, the insurance industry is interested in cost of capital estimation from a management prospective. Cost of capital affects an insurance company’s resource allocation and investment decisions. In project decision making, insurers have begun to use financial techniques such as financial pricing and capital budgeting but have found reliable costs of capital estimates difficult to obtain.2 Cost of capital is an important component in many insurance pricing models.3 Cummins (1990b) analyzes the two most prominent discounted cash flow (DCF) models in property-liability insurance pricing—the Myers-Cohn (MC) model [Myers and Cohn (1987)] and the National Council on Compensation Insurance (NCCI) model. Cummins points out that the two models are based on the concepts of capital budgeting, and essentially, the insurance policy is viewed as a project under consideration by the firm.
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Thus, financial pricing models can be used to determine the price for the project (in this case, the premium) that will provide a fair rate of return to the insurer, taking into account the timing and risk of the cash flows from the policy as well as the market rate of interest. Both the MC model and the NCCI model rely on cost of capital estimates for insurance pricing. In this paper, the cost of equity capital for PC insurers, during the 1988–1992 period, is estimated by applying three alternative asset pricing models: 1) the Capital Asset Pricing Model (CAPM); 2) the Arbitrage Pricing Theory (APT); and 3) a unified CAPM/ APT model [Wei (1988)]. Adjustments for nonsynchronous trading are made by applying the Scholes and Williams (1977) beta adjustment, and the errors in variables (estimation risk) problem in the second-stage regression is adjusted by the Litzenberger and Ramaswamy (1979) generalized least squares procedure. Then the in-sample forecast ability of the three models is evaluated by applying the mean squared error method, the Theil (1966) U2 statistic, and the Granger and Newbold (1978) method of testing for conditional efficiency of forecasts. Based on forecast evaluation procedures, the APT and Wei’s unified CAPM/APT models perform better than the CAPM in estimating the cost of equity capital for the PC insurers, and a combined forecast may outperform the individual forecasts. The next section discusses some of the theories and prior work in the area of cost of equity capital estimation. Section III describes the model specifications and basic estimation procedures for the CAPM, APT, and unified CAPM/APT model. In Section IV, the data set is described, and the empirical details for the asset pricing model estimations are discussed. Section V explains and applies the procedures used to evaluate the forecast quality of the cost of equity capital estimation by the various asset pricing models. Finally, Section VI contains the summary and concluding remarks.
II. Prior work Elton and Gruber (1994) point out that the estimation of a company’s cost of equity capital is one of the most important tasks because the cost of capital affects the company’s project selection, borrowing rate, and allocation of resources. They categorize and briefly describe techniques used for estimating cost of capital: (1) comparable earnings; (2) valuation models; (3) risk premium; (4) capital asset pricing models; and (5) arbitrage pricing models. Comparable earnings estimations of cost of capital use earnings on book equity for “comparable companies.” Although this technique is rarely discussed in texts, it is used in practice by regulatory agencies. Problems with this technique include difficulty in defining “comparable companies” and inaccuracy of using book values to estimate cost of capital which is a market-based value. Valuation models (also referred to as dividend growth models) define the cost of capital as the discount rate that equates expected future dividends to the current price. If we assume the dividend per share, D1, grows at a constant rate, g, forever, then the cost of capital (k) can be defined as:
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Cost of equity ~k! 5
D 1g P
(2.1)
where P represents the stock price per share. This is also called the constant-growth discounted cash-flow (DCF) formula. The DCF formula is the most widely used approach to estimate the cost of equity capital of regulated firms in the United States [Myers and Boruki (1994)] and is also used extensively in insurance regulation. Various assumptions can be made about the growth rate. The risk premium technique to estimate cost of capital requires an estimate of the extra return (a premium) on equity needed over and above the yield on some long-term bond. The estimate of the average cost of equity capital for all firms is then the premium plus the current yield to maturity on the long-term bond. A risk adjustment can then be done, on an ad hoc basis, for firms that may have a different risk from the average firm. The CAPM and APT models are related to the risk premium approach. But rather than adjusting for a firm’s risk on an ad hoc basis, the models use a set of theories and assumptions to determine the risk adjustment for firms. The CAPM states that the expected return on a security is a linear function of the riskless rate plus beta (the security’s sensitivity to the market) times the risk premium of the market portfolio over the riskless rate. The APT is similar to the CAPM in that the expected return on a security is a linear function of the riskless rate and the risk premium of different factors of systematic risk times the factor beta (the security’s sensitivity to the factor). Unlike the CAPM, the APT model allows for more than one source of systematic risk, and the factors are not predefined. While the APT model can be viewed as a generalization of the CAPM, the difficulty lies in identifying these factors. Applying a methodology similar to that used by Bubnys (1990) (which expands upon Bower et al. (1984)), this paper compares the CAPM, APT, and unified CAPM/APT model estimates of the cost of equity capital for PC insurance companies traded on the NYSE and NASDAQ during the 1988–1992 period. Bower et al. compare the CAPM model and the APT model as estimates of expected returns for utility stocks. They find that APT may be the superior model in explaining and conditionally forecasting return variations through time and across assets for public utility companies. Bubnys expands on Bower et al’s CAPM/APT comparison, concluding that neither model dominates in both forecasting and simulation. By using the Litzenberger-Ramaswamy method of adjustment, Bubnys corrects for the errors in variables problem of using firm betas in the crosssectional security market line equation. By doing so, use of large sample size in the second pass regression results in significant pricing of the riskless asset and market risk premium.
III. Model specification and estimation In Section III.A, first, the basic estimation procedure of the CAPM is discussed. Then, the procedure for estimating the cost of equity capital using the CAPM is explained. Similarly, in Section III.B the estimation procedure for the APT model and the APT cost of
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equity capital is discussed. And finally, in Section III.C, the unified CAPM/APT model, estimation procedure, and estimation of cost of equity capital are presented.
A. CAPM The CAPM describes the expected return on an asset as a linear function of only one variable, the asset’s systematic risk: E~Rj! 5 Rf 1 bj@E~Rm! 2 Rf# 5 5 5 5
E(Rj) Rf E(Rm) bj
(3.1)
expected return on asset j risk-free rate expected return on the market portfolio COV(RiRm)/Var(Rj), the systematic risk of asset j
Estimation of the CAPM consists of two stages. The first stage is the estimation of the systematic risk, b. This can be done by applying the time-series market model regression:4 Rjt 5 aj 1 bjRmt 1 «jt Rjt Rmt aj bj «jt
5 5 5 5 5
(3.2)
return on asset j in time t return on the market portfolio in time t intercept systematic risk of asset j residual term
From regression (3.2), bj determines how responsive the returns for the individual asset j are to the market portfolio. For each firm in the market, regressing the firm’s rate of return at time t on the market return at time t gives the estimate of the firm’s systematic risk, bj from (3.2), which is then used in the second stage cross-sectional regression:4 ¯ j 5 a 1 b 3 bj 1 µj R R¯j bj a b µj
5 5 5 5 5
(3.3)
mean return over all time periods for asset j systematic risk for asset j from first stage intercept slope residual term
If CAPM is valid, then
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(3.4)
E~bˆ! 5 E~Rm! 2 E~Rf! or E~Rm! 2 E~Rz!
(3.5)
where â is the intercept estimate, bˆ is the slope estimate, and E(Rz) is the expected return for the zero-beta portfolio. The two-stage procedure described above results in an estimate of the risk-free rate and market risk premium. After the CAPM is estimated, the betas of the PC insurance companies are estimated with regression (3.2). The CAPM cost of equity capital estimates, Kei, are computed using the following equation: Kei 5 a 1 bbi
(3.6)
where a and b are the risk-free rate and the market risk premium, respectively, estimated from regression (3.3), and bi is the beta of PC insurance firm i estimated with regression (3.2).
B. APT Similar to the CAPM, the APT (Ross (1976, 1977)) describes the expected return as a linear function of systematic risks. The CAPM predicts that security rates of return will be linearly related to a single common factor, the rate of return on the market portfolio. The APT allows the possibility of more than one systematic risk factor, and so the expected rate of return on a security can be explained by k independent influences or factors: E~Rj! 5 l0 1 l1bj1 1 l2bj2 1…1 lkbjk E(Rj) bjk
5 5
l0 lk
5 5
(3.7)
expected rate of return on asset j sensitivity of asset j’s return to an unit change in the kth factor (factor loading) common return on all zero-beta assets premium for risk associated with factor k
If there is a riskless asset with a riskless rate of return, Rf, then all bjk’s are zero and l0 is equal to Rf. bjk, similar to the CAPM bj in equation (3.1), measures how responsive returns from asset j are to the kth factor. The theory doesn’t say what factors should be used to explain the expected rates of return. These factors could be interest rates, oil prices, or other economic factors. The market portfolio, used in the CAPM, might be a factor. Thus, the APT counterpart to the CAPM beta are the sensitivity coefficients, or factor loadings, that characterize an asset. These factor loadings are estimated from the market model: Rjt 5 bj0 1 bj1I1t 1…1 bjkIkt 1 ujt
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5 5
bj0 bjk
5 5
ujt
5
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average return of portfolio j in time period t mean zero factor score at time t for factor k common to the returns of all assets estimated return on asset j when all factor loadings are zero estimated factor loading (reaction coefficient) describing the change in asset j’s return for a unit change in factor k residual term
The first-pass regression equation (3.8) is the APT analog to equation (3.2) for the CAPM, and the factor loadings, bjk, are similar to the CAPM bj in equation (3.2). For the CAPM equation (3.2), the return on the market portfolio, Rmt, is the independent variable, while for the APT, the independent variables, Ikt, are generated from factor analysis on the non-insurance assets returns. The second-pass cross-sectional regression is: ¯ 5 l 1 l b 1 l b 1…1 l b 1 u R j 0 1 j1 2 j2 k jk j R¯j l0 lk bjk uj
5 5 5 5 5
(3.9)
average return of portfolio j common return on all zero-beta assets premium for risk associated with factor k factor loading of asset j to factor k estimated from equation (3.8) residual term
Similar to the CAPM second-stage cross-sectional regression (3.3), the APT regression (3.9) estimates in the risk premium, lk for each of the k factors, and the intercept term, l0, resulting in an estimate of the APT model. After the empirical APT is obtained, the APT cost of equity capital for the PC insurance companies can be estimated. First, each individual insurance company’s rate of returns is regressed against the factor scores used in equation (3.8): Rit 5 bi0 1 bi1I1t 1…1 bikIkt 1 uit Rit Ikt
5 5
bi0 bik
5 5
uit
5
(3.10)
return on insurance firm i in time period t mean zero factor score at time t for factor k common to the returns of all assets return on insurance firm i when all factor loadings are zero factor loading (reaction coefficient) describing the change in insurance firm i’s return for a unit change in factor k residual term
Then, the cost of equity for insurance firm i, Kei, the following equation is estimated for the PC insurance firms with the following equation: Kei 5 l0 1 l1bi1 1…1 lkbik
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5 5 5 5
cost of equity capital estimate for insurance asset i common return on all zero-beta assets premium for risk associated with factor k factor loading of insurance asset i to factor k
To compute (3.11), l0 and (l1,…, lk), the risk premiums, are estimated from regression (3.9) and, (bi1,…, bik) are the factor loading of insurance firm i estimated from regression (3.10).
C. Unification of the CAPM and APT Wei (1988) develops a model which extends and integrates the CAPM and APT theories. The Wei unifying theory indicates that one only need add the market portfolio as an extra factor to the factor model in order to obtain an exact asset-pricing relationship: E~Rj! 5 l0 1 l1bj1 1 l2bj2 1…1 lkbjk 1 lmbjm
(3.12)
where (l1,…, lk) are the risk premiums associated with factor k, (bj1,…, bjk) are the factor loadings of asset j to factor k, lm is the risk premium associated with the market factor, and bjm is the market factor loading of asset j to the market factor. If there is a riskless asset with a riskless rate of return, Rf, then all bjk’s and bjm are zero and l0 is equal to Rf. Equation (3.12) is almost identical to the APT equation (3.7) except for the additional term lmbjm for the market factor. To apply Wei’s unification theory, the estimation procedure is similar to that used to estimate the APT. The factor loadings are estimated with a modified market model: Rjt 5 bj0 1 bj1I1t 1…1 bjkIkt 1 bjmrmt 1 ujt Rjt Ikt
5 5
rmt bj0 bjk
5 5 5
bjm ujt
5 5
(3.13)
average return of portfolio j in time period t mean zero factor score at time t for factor k common to the returns of all assets mean adjusted market return at time t return on asset j when all factor loadings are zero factor loading (reaction coefficient) describing the change in asset j’s return for a unit change in factor k factor loading for the market factor residual term
This first-stage regression is similar to that used for the APT estimation, regression (3.8) but with the additional factor rmt. The second-stage cross-sectional regression for the unified CAPM/APT model is:
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¯ 5 l 1 l b 1 l b 1…1 l b 1 l b 1 u R j 0 1 j1 2 j2 k jk m jm j R¯j lk lm bjk bjm uj
5 5 5 5 5 5
(3.14)
average return of portfolio j premium for risk associated with factor k premium for risk associated with market factor factor loading, estimated from (3.13), of asset j to factor k factor loading, estimated from (3.13), of asset j to market factor residual term
This is similar to the APT regression (3.9) but with the additional term, lmbim, for the market factor. The independent variables in regression (3.14) are the risk premium for each of the k 1 1 factors, and the intercept term, l0, is the risk-free rate when all bjk’s are zero. After the empirical unified CAPM/APT is obtained, the cost of equity capital for the PC insurance companies can be estimated. First, each individual insurance company’s rate of returns is regressed against the factor scores and market factor which are the same as the ones used in equation (3.13): Rit 5 bi0 1 bi1I1t 1…1 bikIkt 1 bimrmt 1 uit Rit Ikt
5 5
rmt bi0 bik
5 5 5
bim ujt
5 5
(3.15)
return on insurance asset i in time period t mean zero factor score at time t for factor k common to the returns of all assets mean adjusted market return at time t return on insurance asset i when all factor loadings are zero factor loading (reaction coefficient) describing the change in insurance asset i’s return for a unit change in factor k factor loading for market factor residual term
To estimate cost of equity capital, Kei, the following equation is computed for the PC insurance firms: Kei 5 l01l1bi1 1…1 lkbik 1 lmbim Kei lk lm bik
5 5 5 5
bim
5
(3.16)
cost of equity capital estimate for insurance firm i premium for risk, from (3.14), associated with factor k premium for risk, from (3.14), associated with market factor factor loading, from (3.15), of insurance firm i to factor k from equation factor loading, from (3.15), of insurance firm i to market factor k
For equation (3.16), l0 and the risk premiums (l1,…, lk, lm) are estimated from regression (3.14), and factor loadings (bi1,…, bik, bim) are estimated from regression (3.15).
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IV. Data description and cost of equity capital estimates In Section IV.A, the dataset is described. Then in Section IV.B, the details of the estimation procedures for the asset pricing models are presented. First, the procedure for estimating the CAPM risk-free rate and risk premium is discussed in detail, and the estimate results are presented. Similarly, the APT model estimates and the unified APT/CAPM model estimates are discussed.
A. Data Rate-of-return data for common stocks are collected from Center for Research in Security Prices (CRSP) tapes: 1) NYSE/AMEX (New York Stock Exchange/American Stock Exchange) monthly files; and 2) NASDAQ (National Association of Securities Dealers’ Automated Quotation system) daily files. Data is collected for a 5 year period (December 1987 to December 1992). The time period is short because of the limited number of PC insurance companies that trade continuously for an extended period. The NYSE/AMEX data consists of 1,671 companies, excluding PC insurance companies, having no missing returns during our time period. Similarly, the NASDAQ data consists of 1,995 companies, excluding PC insurance companies, having no missing returns from December 1987 to December 1992. CRSP NASDAQ data is only available as daily rate-of-return data. So, all daily rate-of-return data collected from NASDAQ tapes are converted to monthly returns using the following calculation: Rit 5 @~1 1 ri1!~1 1 ri2!…~1 1 rin!# 2 1 Rit rij n
5 5 5
(4.1)
monthly return on stock i in month t return on stock i on trading day j number of trading days in the month
Also, because many of the NASDAQ companies are infrequently traded with many days of zero return (nonsynchronous trading problem), converting the data from daily to monthly reduces the errors in variables problem. Thus, there is a total of 3,666 firms in our sample, excluding PC insurance firms. Hereafter, this sample is to be referred to as non-insurance companies. Data is available from the NYSE/AMEX tapes for 23 PC insurance companies (hereafter, to be referred to as insurance companies) with no missing returns during the period of December 1987 through December 1992. Multi-line insurers with at least 25% of its business in property/casualty insurance and no missing returns during the 5 year period are also included in the insurance sample. The selection process for PC insurers from the NASDAQ tapes is the same as the selection process for NYSE/AMEX firms. For NASDAQ companies, there are 41 insurance companies with no missing data during the 5 year
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period. Again, since only daily data is available from the CRSP tapes, all data collected from the NASDAQ tapes were converted from daily to monthly returns using equation (4.1). Data is collected for a total of 64 insurance companies. CRSP also provides whole market indices. The whole market index for the NYSE, AMEX, and NASDAQ markets combined, containing the returns and including all distributions on a value-weighted market portfolio is collected for the 5 year time period. Table 1 shows the summary statistics for the dataset. The results show the cross sectional statistics of the average monthly rate of return for the firms. The values are calculated from the average monthly rate of return for each firm during the sample period. Thus, for example, the 1,671 non-insurance firms from NYSE/AMEX have an average monthly return of 0.0135 (16.2% annualized). Also, for these 1,671 non-insurance firms, the maximum average monthly rate of return for a firm during the sample period is 0.1503 (180.36% annualized). Looking at Table 1 for the non-insurance firms, the average firm monthly rate of return is 0.0154 (18.52% annualized), with NASDAQ firms (20.52% annualized) averaging higher than the NYSE/AMEX firms (16.2% annualized). The range of the firm rates of returns for this period is quite wide. The minimum (262.28% annualized) and maximum (229.08% annualized) firm rates of return are for companies from NASDAQ. This broad range of firm rate of returns is reflected in the coefficient of variation measurements which are all over 1. Skewness coefficients for both NYSE/AMEX (0.7004) and NASDAQ (1.2846) average firm monthly returns are positive, meaning that the mode and the median lie below the mean. Kurtosis coefficients for both NYSE/AMEX (8.0904) and NASDAQ (7.9810) are similar when the data are considered separately, but the kurtosis coefficient rises to 8.7420 when the all non-insurance firms are considered together. Also shown in Table 1 is the cross-sectional data analysis of average insurance firm monthly rate of returns, broken down into NYSE/AMEX firms, NASDAQ firms, and all insurance firms combined. NASDAQ insurance firms have an average monthly rate of return of 0.0168 (20.16% annualized), and NYSE/AMEX firms have an average monthly rate of return of 0.0140 (16.84% annualized). Both the standard deviation (.0130) and the coefficient of variation (.7709) of the NASDAQ insurance firms are considerably higher than that of the NYSE/AMEX insurance firms (0.0063 and 0.4468, respectively). Since smaller firms trade on NASDAQ, we expect that returns for NASDAQ companies to be more variable than those for NYSE/AMEX companies. The overall average monthly Table 1. Summary Statistics for Average Firm Monthly Returns
Non-Insurance NYSE/AMEX NASDAQ TOTAL Insurance NYSE/AMEX NASDAQ TOTAL
N
Mean
Std Dev
Min
Max
CV
1671 1995 3666
0.0135 0.0171 0.0154
0.0141 0.0183 0.0166
20.0503 20.0519 20.0519
0.1503 0.1909 0.1909
1.045292 1.070185 1.074968
0.7004 1.2846 1.2073
8.0904 7.9810 8.7420
23 41 64
0.0140 0.0168 0.0158
0.0063 0.0130 0.0111
20.0005 20.0203 20.0203
0.0237 0.0670 0.0466
0.446827 0.770938 0.699094
20.4010 20.7925 20.6331
20.2928 1.8601 2.5158
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return for the insurance firms (19.00% annualized) is very close to that of the noninsurance firms (18.52% annualized) in the dataset. The range of average firm monthly rate of returns for the insurance companies (224.4% to 55.9% annualized) is not as wide as that of the non-insurance companies, and the coefficient of variation for the insurance firms is less than 1 (0.6991) which is considerably less than that of the non-insurance firms (1.075). Skewness coefficients of the average firm monthly returns of the insurance companies are negative which indicate that the mode and the median lie above the mean. Kurtosis coefficients for the insurance companies are much lower than those for the non-insurance companies which could result from the fact that the insurance sample size is much smaller than the non-insurance sample size. Also, a lower kurtosis coefficient would indicate that there are fewer outliers in the sample.
B. Empirical results In the next section, details of the estimation procedure for the three models (the CAPM, APT, and Wei’s unified CAPM/APT) are discussed. The risk premium and risk-free rate estimates of each model are given and also discussed. 1. CAPM estimate. For the CAPM, the first-pass time-series market model regression (3.2) is run for the 1,995 non-insurance NYSE/AMEX firms to estimate individual firm betas for the 5 year period: Rjt 5 aj 1 bjRmt 1 «jt
(3.2)
The dependent variable Rjt is the monthly rate of return for each firm (j 5 1…1,995) in month t(t 5 1…60). The CRSP value-weighted index, including all distributions is used as the market portfolio proxy for Rmt. This results in 1,995 individual firm beta estimates. For securities traded on NASDAQ, there is the concern that many securities listed are traded infrequently (nonsynchronous trading). Scholes and Williams (1977) discuss how nonsynchronous trading introduces the potentially serious econometric problem of errors in variables into the market model. With errors in variables in the market model, ordinary least squares estimators of coefficients in the market model are both biased and inconsistent. Estimators are asymptotically biased upward for alphas and downward for betas. Thus for the 1,671 non-insurance NASDAQ companies, the Scholes-Williams beta adjustment is applied. To compute the Scholes-Williams estimator, bˆ SW, the following OLS regressions are run in place of regression (3.2): Rjt 5 a21 1 b21RM,t21 1 u21,t
(4.2)
Rjt 5 a0 1 b0RM,t 1 u0,t
(4.3)
Rjt 5 a1 1 b1RM,t11 1 u1,t
(4.4)
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RM,t 5 r0 1 rMRM,t21 1 nt
(4.5)
where Rjt is the rate of return of asset j for time t, and RM,t is the market rate of return for time t. The CRSP value-weighted index, including all distributions is used as the market portfolio proxy for RM,t. The estimates from these regressions are then used to compute bˆ SW [
bˆ21 1 bˆ0 1 bˆ1 1 1 2rˆ M
.
(4.6)
bˆ SW are the estimates for systematic risk for NASDAQ firms. In Table 2, the cross-sectional statistics for firm beta estimates are shown. The table is broken down into non-insurance and insurance firms. Statistics are summarized for firms listed on NYSE/AMEX, firms listed on NASDAQ, and all firms combined. Betas for firms listed on NYSE/AMEX are estimated from the first-stage times-series regression (3.2), and betas for firms listed on NASDAQ are calculated using equation (4.6). The average beta estimate for non-insurance firms (NYSE/AMEX and NASDAQ firms combined) is 1.2826. NASDAQ betas have an average of 1.5673 with a standard deviation of 1.3526 while NYSE/AMEX betas average 0.9426 with a standard deviation of 0.5773. In general, betas of smaller companies tend to have more fluctuations in price and more measurement error (or estimation risk) due to nonsynchronous trading. This is evident in the high variance and wide range of NASDAQ betas. The largest beta estimated is 8.7331, and the smallest beta estimated is 24.4375. Both extreme betas are for companies listed on NASDAQ. Similarly, betas for the insurance firms are also estimated with (3.2) for NYSE/AMEX firms and (4.6) for NASDAQ firms. In Table 2, the cross-sectional summary statistics for insurance firm beta estimates are shown. For insurance firms, the average beta is 1.2751 with NYSE/AMEX betas averaging .9428 and NASDAQ betas averaging 1.4615. The standard deviation and coefficient of variation for NASDAQ insurance betas (.8372 and .5728, respectively) are considerably higher than those of NYSE/AMEX insurance betas (.3702 and .3926, respectively). Though the maximum and minimum betas of insurance betas (4.7459 and 20.1407, respectively) are not quite as extreme as those for noninsurance betas. Table 2. Summary Statistics for Firm Betas
Non-Insurance NYSE/AMEX NASDAQ TOTAL Insurance NYSE/AMEX NASDAQ TOTAL
N
Mean
Std Dev
Min
Max
CV
1671 1995 3666
0.9426 1.5673 1.2826
0.5773 1.3526 1.1154
21.6700 24.4375 24.4375
3.9581 8.7331 8.7331
.6125 .8630 .8696
0.1182 0.5817 1.0692
0.8520 2.3468 4.3662
23 41 64
0.9428 1.4615 1.2751
0.3702 0.8372 0.7455
0.2760 20.1407 20.1407
1.5415 4.7459 4.7459
.3926 .5728 .5847
20.3480 1.4212 1.7190
20.9177 4.7115 6.3220
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Next, the non-insurance beta estimates are then used in the second-stage cross-sectional regression (3.3) to estimate the risk-free rate and risk premium for the CAPM. Regression (3.3) is run, using the estimated betas from (3.2) and (4.6) of all 3,666 non-insurance companies. The dependent variable, R¯j(j 5 1,…, 3,666), is the mean monthly return over the five year period for each of the 3,666 non-insurance companies, and the independent variable, bj, is the individual firm beta estimate from equations (3.2) and (4.6). This results in the CAPM estimates for the risk-free rate, a, and the market risk premium, b. The CAPM estimates from the above procedure will be referred to as the “OLS” version of the CAPM model because the second-stage regression is an ordinary least squares regression. When taking sample statistics, there is almost always some measurement error. For regressions, the problem of inaccuracy in estimating both independent and dependent variables results in biases in the regression estimates. Even if errors of measurement are assumed to be mutually independent, independent of the true values, and have constant variance, the slope estimate will tend to be biased downward, and thus the intercept will tend to be biased upward.5 The greater the measurement error, the greater the biases. In estimating and testing the CAPM, Black, Jensen, and Scholes (1972) show that using individual firm betas may introduce significant errors in estimating beta. In the secondstage regression (3.3), the intercept (risk-free rate or zero-beta estimate) will tend to be upward biased, and the slope (the market risk premium estimate) will tend to be downward biased. A generally accepted technique for reducing the problem of errors in variables is grouping observations. When grouped, the errors of individual observations tend to be canceled out by their mutual independence, reducing measurement error effect. For CAPM formation, the grouping procedure of using portfolios betas and portfolios returns reduces the estimation error in beta. The trade-off of using portfolios instead of individual firm betas is that the number of observations in each second-stage cross-sectional regression (3.3) is greatly reduced. To resolve this dilemma, Bubnys estimates the beta of individual firms in the first-stage regression (3.2), and then, he applies the Litzenberger and Ramaswamy (1979) generalized least squares procedure (GLS) for the second-stage (3.3) run. This method takes into account possible correlation between betas and residuals in the second-stage regression (3.3). The residual standard deviations of the first-stage time series regression (3.2) are used to transform all the cross-sectional variables of the second-stage regression (3.3). The resulting equation is: ¯ /S 5 a/S 1 b3b /S 1 µ /S R j ej ej j ej j ej R¯j Sej
5 5
bj µj
5 5
(4.7)
mean monthly return over all time periods for asset j standard deviation of the residuals of firm j’s market model regression systematic risk for asset j from first stage residual term
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For regression equation (4.7), S21 ej and bj/Sej are the independent variables, and the intercept term is constrained to be zero. The coefficient estimates â and bˆ are still defined as before. â is the estimate of the expected return of a riskless or zero-beta asset, and bˆ is the estimate of the expected excess return of the market portfolio over the riskless rate or zero-beta asset rate. This version of the empirical CAPM will be referred to as the “GLS” CAPM because a generalized least squares procedure is used for the second-stage regression. A third version of the CAPM is considered where the Ibbotson Associates (1994) data are used for estimates of the risk-free rate and risk premium. For the risk-free rate estimate, the 1926–1993 average annual return of long-term government bonds (5.4%) is used, and for the risk premium estimate, the 1926–1993 average annual return of large companies minus the average annual return of long-term government bonds (12.3% 2 5.4% 5 6.9%) is used. This version of the CAPM model is referred to as the “IS” version. Ibbotson Associates estimates are also used as a comparison for the risk premium and risk-free rate estimates obtained by the OLS and GLS versions of the CAPM. To estimate the cost of equity capital for the insurance firms, equation (3.6) is calculated. Beta estimates for the firms are obtained by running regression (3.2) for NYSE/ AMEX insurance firms. NASDAQ insurance firm betas are obtained using the ScholesWilliams beta adjustment by running regressions (4.2)–(4.5) and computing equation (4.6). Estimates for the risk-free rate, â, and the risk premium, bˆ, are from Ibbotson Associates (1994) for the IS version, regression (3.3) for the OLS version, and regression (4.7) for the GLS version. Risk premium and risk-free rate estimates. Table 3 shows the annualized estimates of the market risk-free asset rate (Rf) and market risk premium (RP) for the three CAPM versions. For the CAPM IS version, a risk-free rate of 5.4% and a risk premium of 6.9% from Ibbotson Associates (1994) is used. The second CAPM version is the ordinary least squares (OLS) version of the second-stage regression. The rates shown for the OLS Table 3. CAPM and APT Second-Stage Annualized Results of Regression Coefficients 1988–1992 (t-values in parentheses below coefficients) CAPM (n 5 3666)
1
IS OLS2 GLS3
APT (n 5 40)
Rf
RP
l0
l1
l2
l3
l4
l5
.054 .1185 (24.707)* .1065 (36.031)*
.069 .0521 (14.453)* .0612 (22.964)*
.0957 (1.940)** .1062 (2.195)*
.4962 (1.181) .6400 (1.526)
1.9776 (2.914)* 1.5911 (2.421)*
.3635 (.370) 2.0941 (2.098)
2.0777 (2.087) 2.2070 (2.239)
2.1275 (2.511)* 2.5941 (3.916)*
*Significant at the 1 percent level **Significant at the 5 percent level 1 IS 5 CAPM with Ibbotson Associates risk-rate (Rf) and risk premium (RP). 2 OLS 5 CAPM or APT with ordinary least squares second stage regression. 3 GLS 5 CAPM or APT with generalized least squares second stage regression.
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version are the estimates obtained from running the second-stage cross-sectional regression (3.3) before applying the Litzenberger and Ramaswamy adjustment. Estimate values of 11.5% for the risk-free rate and 5.21% for the risk premium are both significant at the 0.5 percent level. The third CAPM version is the intercept constrained GLS regression (equation (4.8)), using the Litzenberger and Ramaswamy adjustment. The estimates are 10.65% for the risk-free rate and 6.12% for the risk-premium, both significant the 0.5 percent level. Although the OLS estimate of the CAPM Rf for the five year period, 11.85%, is much higher than historical and IS estimates of riskless rate, the estimate is significant at the 0.5 percent level. One possible interpretation is that this the not the riskless rate but the zero-beta asset rate. Also to be considered is the fact that NASDAQ companies are used in the model estimates. As can be seen from Table 1, the mean monthly return of the non-insurance NASDAQ companies (20.52% annualized) is considerably higher than that of the non-insurance NYSE/AMEX companies (16.25% annualized), and the overall average (18.48% annualized) is also considerably higher than the Ibbotson Associates average annual return of large company stocks (12.3%). The NASDAQ mean increases the overall mean used to estimate the risk-free rate and directly affects the estimate of the regression intercept. This can be explained as follows. The dependent variables of regression (3.3) used to estimate the risk-free rate (the intercept) are the monthly returns of the non-insurance firms. In general, estimation of the intercept, a, is a 5 y¯ 2 bx¯
(4.8)
where y¯ is the mean of the dependent variable, b is the slope estimate, and x¯ is the mean of the independent variable. The NASDAQ returns increase the overall mean of the dependent variable, y¯, which increases the intercept estimate. Another possible reason for high risk-free rate and low risk premium estimates, by historical standards, is that small company betas have more measurement error (or estimation risk). The standard deviation and coefficient of variation of the non-insurance NASDAQ betas (1.3526 and .8630, respectively) are much higher than that of NYSE/ AMEX betas (.5773 and .6125, respectively). With higher estimation risk from NASDAQ betas, the second-stage regression will tend to underestimate the slope (risk premium) and overestimate the intercept (risk-free rate). Also, another explanation for a high risk-free rate estimate is that there are missing explanatory variables. Fama and French (1992) find that two easily measured variables, firms size and book-to-market equity, need to be considered in the CAPM model in addition to the market betas. In the GLS version of the CAPM, the Litzenberger-Ramaswamy method (regression (4.7)) is applied to correct for overestimation of the intercept and underestimation of the slope due to measurement error. In Table 3, the GLS version estimate of Rf and RP are 10.65% and 6.12%, respectively. Both estimates are significant at the 0.5% level. It appears that the Litzenberger-Ramaswamy adjustment has corrected for some of the measurement error problems present in the OLS method by lowering the risk-free rate
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estimate, increasing the risk premium estimate, and increasing the significance in both estimates. The OLS and GLS CAPM estimates for the risk-free rate and risk premium differ from the historical and the IS estimates. This can be partly attributed to the fact that NASDAQ companies (i.e. small companies) are used in the sample to estimate the CAPM. Smaller companies tend to have higher average returns and more measurement error than larger firms, and both factors contribute to the overestimation of the risk-free rate and underestimation of the risk premium. Or, it could be that these estimates are consistent with this time period and these companies.
2. APT estimate. To estimate the APT model, first, factor analysis is used to estimate the factor scores for the market, the 3,666 non-insurance companies. Factor analysis is a body of multivariate data-analysis techniques which provide “analysis of interdependence” of a given set of variables or responses [Churchill (1983), Morrison (1990)]. For the APT model, we apply factor analysis to the monthly returns of the 3,666 non-insurance stocks to generate k common factor loadings for all the stocks. These factors, in a sense, summarize and capture the unobserved, underlying interdependencies of the monthly returns of the 3,666 non-insurance stocks. Because the number of variables (3,666 companies) is much greater than the number of observations (60 months for each company), we form portfolios from the 3,666 companies to reduce the number of variables. First, the 3,666 non-insurance companies are grouped alphabetically into 40 equally-weighted portfolios (14 portfolios of 91 companies and 26 portfolios of 92 companies). The equal-weighted average returns for each month of each portfolio are then used in a maximum likelihood estimation (MLE) method of factor analysis, varimax rotation of factors, to generate the factor scores (Ikt in equation (3.8)).6 A varimax rotation is basically an orthogonal (angle-preserving) rotation of the original factor axes for simplification of the columns or factors (Churchill (1983), p. 634, Morrison (1990), pp. 370–372). Rotations are done to facilitate the isolation and identification of the factors underlying a set of observed variables (the monthly returns of the 3,666 non-insurance stocks). A five factor (k 5 1…5) model is used (Roll and Ross (1980)), and this results in 5 factor scores for each observed month t(t 5 1…60). These five factor scores are then input into regression equation (3.8) as independent variables, resulting in the estimated factor loading, bjk, of asset j to factor k. The dependent variables of this regression, Rjt, are the monthly average return of each portfolio j(51…40) for the t(51…60) months. Rjt 5 bj0 1 bj1I1t 1…1 bjkIkt 1 ujt
(3.8)
The market model (3.8) is the APT analog to market model equation (3.2) for the CAPM. This results estimates of factor loadings (bj1,…, bj5) for each of the 40 portfolios. Next, the second pass cross-sectional regression (3.9) is run to estimate the risk premium associated with each factor k in the APT model (equation (3.7)). The dependent variables are the average return for the 5 year period for each of the 40 portfolios, and the
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independent variables are the estimated 5 factor loadings, bik, for each portfolio from the first-stage regression (3.8): R¯j 5 l0 1 l1bj1 1 l2bj2 1…1 l5bj5 1 uj
(3.9)
This results in the risk premium, lk, associated with each factor of the five factors and l0 which is the risk-free rate when all bjk’s are zero. The APT model estimate from the above described procedure will be referred to as the “OLS APT” version. Like the CAPM analysis, to adjust for possible sampling error in the estimation of the APT factor loadings in (3.9), the Litzenberger and Ramaswamy procedure is applied. The resulting regression is: R¯j/Sej 5 l0/Sej 1 l1~bj1/Sej! 1 l2~bj2/Sej! 1…1 lk~bjk/Sej! 1 uj/Sej
(4.9)
where all variables are defined as before in regression (3.9) and Sej is the standard deviation of the residuals of firm j’s market model regression (3.8). The intercept term of regression (4.9) is constrained to zero. (1/Sej) and (bjk/Sej) are the independent variables. Equation (4.9) for the APT is analogous to equation (3.8) for the CAPM. The APT estimates resulting from regression (4.9) will be referred to as the GLS APT version. Next, to estimate the cost of equity capital, each individual insurance company’s rate of returns are regressed against the five factor scores obtained from the MLE factor analysis. Regression (3.10) is run to obtain the estimated factor loadings (bi1,…, bi5) for the 64 insurance companies: Rit 5 bi0 1 bi1I1t 1…1 bi5I5t 1 uit
(3.10)
To estimate APT cost of equity capital, Kei, use the equation Kei 5 l0 1 l1bi1 1 l2bi2 1 l3bi3 1 l4bi4 1 l5bi5
(3.11)
Insurance factor loading estimates, bik, from (3.10), and l0 and the risk premium estimates, lk, from either (3.9) or (4.9) are used in equation (3.11) to compute the APT cost of equity capital. Risk premium estimates. Table 3 shows the risk premium estimates (lambdas) of the OLS version (regression (3.9)) and the intercept constrained GLS version of APT (regression (4.9)). The factors of the APT are difficult to identify. l0 is the estimate of the zero-loading return. The OLS APT estimate of l0 is 9.57% and is significant at the 5 percent level. For the GLS APT version, the estimate of l0 is 10.62% and significant at the 1 percent level. The APT estimates of l0 are similar to the risk-free rate estimates of the CAPM. The first factor of the APT, l1, is considered highly correlated with the market index of the CAPM. Estimates of the first factor are .4962 and .6400 for the OLS and GLS versions, respectively, but these estimates are not highly significant.
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Overall, for the APT OLS version, two factors are significant at the 1 percent level, and the zero-loading return is significant at the 5 percent level for the APT OLS version. And, for the APT GLS version, the Litzenberger-Ramaswamy adjustment improved the statistical significance of the some of the factors. l0, l2, and l5 are all significant at the 1 percent level. After applying the Litzenberger and Ramaswamy adjustment, the t-values of l0, l1, l4, and l5 increase.
3. Unified CAPM/APT Estimate. The procedure to estimate Wei’s unified CAPM/APT is almost identical to the procedure to estimate the APT model except that an additional factor, the market factor, is added. Factor scores are estimated for the market by using the monthly returns of the 3,666 non-insurance firms. The equal-weighted average returns for each month of 40 equally-weighted portfolios are used in a MLE method of factor analysis, varimax rotation of factors, to generate the factor scores (Ikt in equation (3.13)). These factor scores are the same as those used in the APT (Ikt in equation (3.8)). 5 (k 5 5) factor scores are generated, and an additional factor, rmt, is calculated for the Wei model. The market factor, rmt, is calculated is the mean adjusted market return and is calculated by rmt 5 Rmt 2 Em
(4.10)
where Rmt is the market index and Em is the mean of Rmt. The first-stage times-series regression (3.13) estimates the factor loadings (bj1,…, bj5, bjm) which are then used as independent variables in the second-stage cross-sectional regression (3.14). This results in estimates for l0 and the risk premiums (l1,…, l5, lm) associated with each factor which is similar to the APT second-stage cross-sectional regression (3.9) but with the additional term, lmbjm, for the market factor. The resulting estimates for this model are referred to as the “WEI OLS” version. Similar to the CAPM and APT analysis, to adjust for possible sampling error in the estimation of the factor loadings in (3.14), the Litzenberger and Ramaswamy procedure is applied. The resulting regression is R¯j/Sej 5 l0/Sej 1 l1~bj1/Sej! 1 l2~bj2/Sej! 1…1 l5~bj5/Sej! 1 lm~bjm/Sej! 1 uj/Sej (4.11) where the variables are defined are before in regression (3.14), and Sej is the standard deviation of the residuals of firm j’s market model regression (3.13). The intercept term of regression (4.11) is constrained to zero. (1/Sej) and (bjk/Sej) are the independent variables. The estimates resulting from regression (4.11) will be referred to as the “WEI GLS” version. To obtain the estimated factor loadings, bi1…bi5, for the insurance companies, each individual insurance company’s rate of return are regressed against the same five factor scores and the market factor which are used in regression (3.13):
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254 Rit 5 bi0 1 bi1I1t 1…1 bikI5t 1 bimrmt 1 uit
(3.15)
To estimate Wei cost of equity capital, Kei, equation (3.22) is calculated. Kei 5 l0 1 l1bi1 1 l2bi2 1 l3bi3 1 l4bi4 1 l5bi5 1 lmbim
(3.16)
Insurance factor loading estimates from (3.15) and risk premium estimates from either (3.14) (OLS version) or (4.11) (GLS version) are used in equation (3.16) to compute the cost of equity capital of Wei’s model. Risk premium estimates. Table 4 shows the results for the OLS and GLS versions of Wei’s unified CAPM/APT model. For the OLS version, two factor premiums, l2 and l5, are statistically significant at the 1 percent level, and l0 and lm, are statistically significant at the 5 percent level. The same factor premiums are significant for the GLS version, and in addition, l1 is significant at the 5 percent level. Comparing estimates of the Wei model with those of the APT model, the risk premium estimates are similar in magnitude for l0 through l5. And basically the same factors are statistically significant for both the Wei model and the APT model except for l1 which is statistically significant at the 5 percent level for the GLS Wei version. The risk premium for the market model, lm, is significant at the 5 percent level for both the OLS and GLS versions of the Wei model and seems to improve the specification of the APT model without detracting from the significance of the other factors.
V. Evaluations of simulations and estimates In this study, the CAPM, APT, and Wei models are used to do in-sample forecasting of the cost of equity capital for non-life insurance companies. In any forecasting study, it is important to evaluate the quality of predictions made. Lee et al. (1986) discusses procedures for evaluating forecasts. Some of the forecast evaluating procedures considered in Lee et al. are used to measure the quality of the cost of equity capital estimations from the Table 4. WETs UNIFIED CAPM/APT Second-Stage Annualized Results of Regression Coefficients 1988–1992 (t-values in parentheses below coefficients) WETs Model (n 5 40)
1
OLS
GLS2
l0
l1
l2
l3
l4
l5
lm
.0887 (1.789)** .0992 (2.090)**
.5781 (1.360) .6750 (1.688)**
2.0047 (2.962)* 1.7068 (2.642)*
.4471 (.456) .02916 (.031)
.0188 (0.21) 2.1307 (2.154)
2.2453 (2.639)* 2.5950 (4.025)*
.0885 (1.849)** .0870 (1.860)**
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*Significant at the 1 percent level **Significant at the 5 percent level 1 OLS 5 Wei model with ordinary least squares second stage regression. 2 GLS 5 Wei model with generalized least squares second stage regression.
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various versions of the three asset pricing models. The Theil mean squared error decomposition (MSE) and the Theil U2 statistic enables us to get measures of the forecast performance of the models. Also, to evaluate Granger and Newbold (1973) conditional efficiency of the asset pricing models, the Bates and Granger method (1969) is applied. Several models for evaluating forecasts are considered because no one model can accurately assess the performance of the cost of equity capital estimation models. Each evaluation procedure provides different information about forecast quality.
A. MSE method Let Ft(t 5 1…N) be a set of forecasts in time t, and Xt(t 5 1…N) be the corresponding true values at time t. Calculation of forecast mean squared error (MSE) (Theil (1958)) is computed by the following equation which can be decomposed into the terms on in the right-hand side of the equation: N
MSE 5
( ~Xt 2 Ft!2 t51 N
¯ !2 1 ~S 2 rS !2 1 ~1 2 r2!S2 5 ~F¯ 2 X F x x
(5.1)
where F¯ and X¯ are the sample means, SF and Sx are the sample standard deviations, and r is the sample correlation between actual and predicted values. Mincer and Zarnowitz (1969) refer to the three components on the right-hand side of equation (5.1) as representing the contributions to forecast mean squared error due to bias, inefficiency, and random error, respectively. Calculation of MSE for the CAPM, APT, and the Wei model forecasts gives us measures, and the decomposition of these measures, with which we can compare the performance of the various forecasts by comparing the measures. Applying (5.1) to the cost of equity capital estimates: 64
MSE 5 R¯t Kei ¯ ¯ R, K
e
SR, Sk r
( ~R¯i 2 Kei!2 i51 64 5 5 5 5 5
¯ 2 R¯!2 1 ~S 2 rS !2 1 ~1 2 r2!S2 5 ~K e K R K
average rate of return for insurance firm i cost of equity capital estimate for insurance firm i sample means of R¯t and Kei, respectively sample standard deviation for R¯t and Kei, respectively sample correlation between average rate of return and cost of equity capital estimate
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256 B. Theil U2
The quality of cost of equity capital estimates can also be assessed by using the Theil measure (1966), U2: U2 5
(64i51 ~R¯i 2 Kei!2 (64i51 ~R¯i 2 R¯!2
R¯t Kei
5 5
R¯
5
(5.2)
average return during 1988–1992 period for insurance asset i cost of equity capital estimate for 1988–1992 for insurance asset i average return for all insurance companies
The smaller the U2 ratio, the better the model estimate relative to the naive model. If the U2 ratio is equal to 1, then the model estimates are no better than using the sample average. If the U2 ratio is greater (less) than 1, then the model estimates are worse (better) than the naive model. Notice that the numerators of the MSE and the U2 ratio are the same, but for the U2 ratio, the denominator makes the ratio into a evaluation of the forecasts relative to the naive model. Thus, with the U2 ratio, not only can we evaluate performance by comparing the ratio of various models, the U2 ratio also provides us with a benchmark (the naive model) with which we can evaluate a model’s forecast performance.
C. Conditional efficiency method 1. Bates and Granger model. Bates and Granger (1969) propose a model that allows for the possibility of a composite, or combined, forecast which is a weighted average of two individual forecasts and which may be superior to either forecast individually in terms of predictive ability. Consider the following regression: Xt 5 kF1t 1 ~1 2 k!F2t 1 Ut
(5.3)
or equivalently, Xt 2 F2t 5 k~F1t 2 F2t! 1 Ut Xt F1t F2t Ut
5 5 5 5
(5.4)
true values in time t(t 5 1, 2,…, N) set of forecasts alternative set of forecasts random error term
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In terms of forecast evaluation, if the predictions F2t contain no useful information about the Xt which is not already incorporated in F1t, then the optimal value of k is one. Granger and Newbold (1973) define F1t as conditionally efficient with respect to F2t if k, the composite predictor of equation (5.3), is one. A test for conditional efficiency applied by fitting equation (5.4) by ordinary least squares and testing the hypothesis that k 5 1 through the usual t-test. Given a pair of forecasts, not only can we determine their conditional efficiency, it is possible to use this model to find a combined forecast which outperforms each individual forecast by the mean squared error criterion. 2. Davidson and Mackinnon’s model. Chen (1983) assumes that the CAPM and APT models are nonnested for the purposes of comparing and testing the two models. In an attempt to compare the two models, Chen (1983) points out the problems with simply doing a regression with the CAPM beta and the APT factor loadings: (1) this specification is not justified on theoretical grounds, and the two models are nonnested; and (2) because the CAPM beta and APT factor loadings are measures of risk, such a regression would have the problem of a high degree of multicollinearity between the independent variables. Davidson and MacKinnon (1981) proposed several procedures to test the validity of regression models of nonnested alternative hypotheses. Using, the procedures suggested by Davidson and Mackinnon, Chen suggests the following regression in which a is estimated ri 5 arˆi,APT 1 ~1 2 a!rˆi,CAPM 1 ei
(5.5)
where rˆAPT and rˆCAPM are the expected returns predicted by the APT and CAPM, respectively. From (5.5), if the APT is the correct model relative to the CAPM, we would expect a to be close to 1. Thus, regression (5.5) suggested by Chen (1983), based upon the Davidson and Mackinnon (1981) procedures, is basically the same as the Granger and Newbold (1973) test for conditional efficiency which uses the Bates and Granger (1969) model. Thus, we apply the Granger and Newbold test of conditional efficiency to evaluate the performance of the CAPM and APT forecasts of cost of equity capital. This test of conditional efficiency allows us to compare two models’ forecasts performance simultaneously, unlike the MSE and U2 ratio in which each model’s forecast performance is tested separately.
D. Comparison of alternative testing results To obtain the cost of equity capital estimates, Kei, of the various versions of the CAPM, APT, and Wei models, equations (3.6), (3.11), and (3.16) are used. The Theil mean squares error decomposition (MSE), the Theil U2 ratio, and the conditional efficiency (Granger and Newbold) methods are applied to evaluate forecast performance of the CAPM, APT,
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and Wei models in estimating the cost of equity capital for the 64 insurance companies. For all evaluation methods, the average return for the insurance company during the 5 year period is considered the actual value, and Kei is the forecasted (or predicted) value. First, consider the MSE decomposition presented in Table 5. The terms are calculated from annualized instead of monthly rates of returns so that the magnitudes of the MSE values would not be too small. Equation (5.2) is applied to the average returns and cost of equity capital estimates of the insurance companies. The CAPM versions (OLS and GLS) have biases (.00003) which are at least one order of magnitude smaller than the biases of all other models. The APT OLS and Wei OLS models have bias values (.00615 for both) which are by far the largest of any of the other models. But a large bias value merely indicates that the true values and the predicted values do not have similar sample means. As long as the difference of the samples means are not overwhelmingly large, this would not necessarily imply that a model provides poor overall forecasts. The largest inefficiency values are also those of the APT OLS and Wei OLS models (.00068 and .00075, respectively). The inefficiency term is an indication of difference between the sample and the forecast standard deviations. The terms which have the most impact on the total MSE value are the random error terms which are a function of the sample correlation between the predicted and actual values. And despite have the largest biases and inefficiency values, the APT OLS and the Wei OLS have the lowest random error values (.00169 and .00201, respectively) which are one order of magnitude smaller than that of all other models, resulting in the lowest total MSE values (.00852 and .00891, respectively) for the APT OLS and the Wei OLS. The CAPM IS has the highest total MSE of .01996. Thus, using the MSE method to evaluate forecast performance, the APT OLS and the Wei OLS models are considerably better models in estimating the cost of equity capital for the sample of PC insurance companies, and the CAPM IS model can be considered to have the worse forecast performance of all the models.
Table 5. Mean Squared Error* Decompositions for Forecasts1 CAPM
Bias Inefficiency Random error Total MSE.
APT
WEI
IS
OLS
GLS
OLS
GLS
OLS
GLS
.00231 .00066 .01670 .01996
.00003 .00017 .01700 .01720
.00003 .00040 .01700 .01743
.00615 .00068 .00169 .00852
.00032 .00030 .01662 .01724
.00615 .00075 .00201 .00891
.0003 .00012 .01615 .01659
64
*MSE 5
( ~R¯i 2 Kei!2 i51 64
¯ 2 R¯¯!2 1 ~S 2 rS !2 1 ~1 2 r2!S2 5 ~K e K R K
5 Bias 1 Inefficiency 1 Random Error IS 5 CAPM with Ibbotson Associates risk-rate (Rf) and risk premium (RP). OLS 5 CAPM or APT or Wei model with ordinary least squares second stage regression. GLS 5 CAPM or APT or Wei model with generalized least squares second stage regression.
1
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Because there is no scale or basis to which we can compare the MSE values, it is difficult to determine exactly how much better the APT OLS and Wei OLS models are than the other models. The Theil decomposition of the MSE gives us an indication of the relative strengths and weaknesses of a model’s forecasts in terms of bias, inefficiency, and random error. Next, consider the Theil U2 statistic (equation (5.2)) for the cost of equity capital forecasts shown in Table 6. Because the U2 ratio and the MSE equation have the same numerators, it is likely that both methods will rank the models in terms of forecast performance similarly. The additional information about forecast performance that the U2 statistic provides is that the statistic compares a model’s forecast performance with the naive model (using the overall mean as a predictor). A U2 ratio equal to 1 means that the forecast performance is equivalent to the naive model, and a ratio value . (,) 1 means that the forecast performance is worse (better) than the naive model. The “grand mean”, R¯, used in the denominator for U2 can be either the average return of the insurance companies (19.00%, annualized) or the market average of 18.52% (of the 3666 non-insurance companies). But because the difference between the insurance company averages and the market average is not large, the U2 are not likely to differ much. In all cases of all model versions, the U2 ratio goes down when the market average is used in the denominator instead of the insurance average. All U2 ratios are less than 1, except for the IS CAPM model, implying that almost all models have forecast performances which are better than that of the naive model. The highest U2 ratios and the only ratios that are larger than 1 are those of the IS CAPM model which is consistent with the MSE evaluation of the IS CAPM having the poorest forecast performance. In fact, U2 ratio rankings of model performance is the same as the MSE rankings, with the APT OLS model having both the lowest MSE value and U2 ratio. Although both the MSE decomposition and the Theil U2 statistic values can be easily calculated for comparison, they do not give a complete picture of forecast quality. With these two methods, it is not possible to compare the forecast quality of by considering them simultaneously. And as Granger and Newbold (1973) and Lee et al. (1986) point out, simply because one set of forecasts, F1t, outperform another set of forecasts, F2t, it does not necessarily imply that the one set of forecasts is more efficient than the other. It may be the case that F2t may contain useful information not captured by F1t. Thus as an additional measure of forecast performance is applied. The Granger and Newbold (1973) definition of conditional efficiency is tested for by applying the Bates and Granger regression (5.5). Conditional efficiency of forecasts F1t with respect to F2t is defined by Granger and Newbold to be (k 5 1) for regression (5.4). The Granger and Newbold (1973) definition of conditional efficiency is tested for by using the Bates and Granger regression (5.4) or, equivalently, regression (5.5). Results of the regression are shown in Table 7, comparing APT versions with CAPM versions and Wei model versions with CAPM versions. Considering first the APT and CAPM comparisons (regressions 1, 2, and 3 in Table 7), it would be logical expect the kˆ of the APT OLS model to be very close to 1 when evaluated with other models because of its dramatically lower MSE and U2 values compared to those of the other models, but this is not the case. In fact, kˆ for APT OLS
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(64i51 ~R¯i 2 Kei!2 (64i51 ~R¯i 2 R¯!2
*U2 5
1.1313 0.9724 0.9853
Market R¯ 5 .1852 0.4883 0.9774
Insurance R¯ 5 .1900
APT
0.4876 0.9761
Market R¯ 5 .1852
1
IS 5 CAPM with Ibbotson Associates risk-rate (Rf) and risk premium (RP). OLS 5 CAPM or APT or Wei model with ordinary least squares second stage regression. GLS 5 CAPM or APT or Wei model with generalized least squares second stage regression.
1.1328 0.9737 0.9866
IS OLS GLS
Insurance R¯ 5 .1900
CAPM
Table 6. Theil U2 Ratios* for CAPM and APT model versions1
0.6539 0.9439
Insurance R¯ 5 .1900
WEI
0.6530 0.9427
Market R¯ 5 .1852
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Table 7. Granger and Newbold’s Conditional Efficiency Estimated Weights of the Expected Return from the CAPM, APT, and Wei model1 Rt 5 kF1t 1 (1 2 k)F2t 1 Ut F1t
F2t
1.APT OLS 2.APT OLS 3.APT OLS 4.APT GLS 5.APT GLS 6.APT GLS 7.WEI OLS 8.WEI OLS 9.WEI OLS 10.WEI GLS 11.WEI GLS 12.WEI GLS
CAPM CAPM CAPM CAPM CAPM CAPM CAPM CAPM CAPM CAPM CAPM CAPM
IS OLS GLS IS OLS GLS IS OLS GLS IS OLS GLS
kˆ
t-statistic (k 5 0)
t-statistic (k 5 1)
R2
.65196 .65138 .65541 .83660 .48762 .52743 .59992 .58792 .59514 .78071 .57418 .59837
15.666* 12.086* 12.069* 3.237* 1.549 1.796** 14.178* 10.712* 10.688* 3.731* 2.142** 2.320**
28.363* 26.468* 26.345* 2.632 21.627 21.585 29.455* 27.508* 27.380* 21.048 21.589 21.557
.7925 .6939 .6933 .1290 .0214 .0322 .7576 .6400 .6389 .1680 .0531 .0641
*Significant at the 1 percent level **Significant at the 5 percent level IS 5 CAPM with Ibbotson Associates risk-rate (Rf) and risk premium (RP). OLS 5 CAPM or APT or Wei model with ordinary least squares second stage regression. GLS 5 CAPM or APT or Wei model with generalized least squares second stage regression.
regressions with each of the CAPM versions are all approximately .65 and statistically different from 1. Table 7 shows that the t-statistics (k 5 1) of regressions 1, 2, and 3 are all statistically significant at the 1 percent level. This implies that the APT OLS model is not conditionally efficient with respect to the CAPM versions and that the CAPM versions contain useful information not present in the APT OLS forecasts. Although the kˆ’s for regressions 1, 2, and 3 are statistically different from 1, the t-statistics for k 5 0 are all also significant at the 1 percent level, and the R2 values for regressions 1, 2, and 3 are all around .7. This suggests that perhaps a combined forecast of the APT OLS and an CAPM version would outperform each individual forecast by the MSE and/or U2 criteria. The highest kˆ for the APT and CAPM regressions is .8366 with t-statistic (k 5 1) of 2.632 for the APT GLS and CAPM IS regression (regression 4). Thus, the null hypothesis of conditional efficiency (k 5 1) is not rejected. Also, the t-statistic (k 5 0) for regression 4 is significant at the 1 percent level. This implies that kˆ is not statistically different from 1 and that the APT GLS forecasts may be considered conditionally efficient with respect to the CAPM IS forecasts. Regressions 5 and 6 which test the APT GLS with respect to the CAPM OLS and IS, respectively, also do not reject the null hypothesis of k 5 1. The corresponding t-statistics are 21.627 and 21.585, respectively. Although regressions 4, 5, and 6 do not reject the hypothesis of conditional efficiency of the APT GLS with respect to the CAPM versions, the R2s of the regressions are quite low. Thus, a combined forecast of the APT GLS with any of the CAPM versions is unlikely to provide a forecast that is superior to each individual forecast. The results of the conditionally efficiency of the Wei forecasts with respect to the CAPM forecasts parallel those for the APT forecasts with respect to the CAPM forecasts. Despite having low MSE and U2 values compared to those of CAPM forecasts, the WEI
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OLS forecasts are not conditionally efficient with respect to the CAPM forecasts (regressions 7, 8, and 9). The kˆ estimates for these regressions are approximately .6 with t-statistics which are significant at the 1 percent level for both the (k 5 0) and the (k 5 1) hypotheses. This implies that the CAPM forecasts may contain information not captured by the WEI OLS forecasts and that a combined forecast may outperform each individual forecast by the MSE and the U2 criteria. Also the high R2 values of regressions 7, 8, and 9 suggest that a combined forecast should be considered. Regression 10 which considers the WEI GLS forecasts with respect to the CAPM IS forecasts has a kˆ value of .78071 and a t-statistic (k 5 1) of 21.048, so it is possible to consider the conditional efficiency of the WEI GLS forecasts with respect to the CAPM IS forecasts. And the hypothesis of conditional efficiency of the WEI GLS forecasts with respect to the CAPM OLS or the CAPM GLS (regressions 11 and 12, respectively) can not be rejected. But as with the APT GLS, the WEI GLS combined with the CAPM versions have low R2 values which indicate that it is unlikely that a combined forecast of the WEI GLS with any of the CAPM versions would be superior to that of a combined forecast of either the APT OLS or the WEI OLS with the CAPM versions.
VI. Summary and concluding remarks This paper applies three alternative asset pricing models to estimate the of cost of equity capital for 64 non-life insurance companies during the 5 year period 1988–1992. 3,666 non-insurance companies were used to estimate CAPM and APT characteristic line equations. Three versions of the CAPM cost of equity capital estimates (IS, OLS, GLS) were considered. The IS version used Ibbotson and Sinquefield estimates of the risk-free rate and the risk premium for the CAPM. The OLS version used the risk-free rate and the risk premium estimates from an second-stage ordinary least squares regression. To adjust for the problem of errors in variables, the Litzenberger-Ramswamy generalized least squares procedure was used for the second-stage regression, resulting in the GLS version. For both, the OLS and GLS versions, the Scholes-Williams beta adjustment was applied to NASDAQ stocks for the problem of nonsynchronous trading. Two versions of the APT cost of equity capital estimates were considered (OLS and GLS). Similar to those of the CAPM model, the OLS version resulted from a second-stage ordinary least squares regression, and the GLS version resulted from a second-stage generalized least squares regression to adjust for the problem of error in variables. Finally, a model unifying the CAPM and APT models, developed by Wei, was also used to estimate cost of equity capital for the insurance companies. An OLS and GLS version of the Wei model, similar to those of the APT model were, were considered. For the CAPM model, the OLS and GLS estimates of the risk-free rate and risk premium were higher and lower, respectively, than the historical and the IS values. Although, the GLS version seemed to somewhat correct for the errors in variables problem of underestimating the slope and overestimating the intercept, perhaps other errors in variables estimation methods of adjusting the estimates should be considered in future research. In addition, higher average returns and increased estimation risk associated with
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NASDAQ companies also contributed to the high risk-free rate estimates and the low risk premium estimates. But, because many of the insurance companies trade on NASDAQ, using NASDAQ companies to estimate the asset pricing models should improve the forecast ability of the models in estimating the cost of equity capital for insurance firms. Also, a misspecification of the CAPM (missing variables) could cause the high intercept estimates and low slope estimates. The APT and Wei’s unified CAPM/APT model had similar factor risk premium estimates. The additional market factor of the Wei model was statistically significant in both the OLS and GLS version. It is intuitively appealing to have a model which combines the APT and CAPM. Though, results from this research do not empirically show that the additional market factor significantly improves the cost of equity capital estimation. The MSE and U2 methods of evaluating forecast quality found the APT OLS and the Wei OLS models to be the best estimators of cost of equity capital for the insurance companies during the 5 year period tested, but the tests of conditional efficiency suggested that a weighted combination of either the APT OLS or the Wei OLS with the CAPM may provide forecasts that are better than each individual forecast. Overall, the results show that the APT and the Wei model, which is basically a modified APT, perform better than the CAPM in estimating the cost of equity capital for insurance companies in our sample. The methods presented here are straight-forward in their implementation and application, but additional techniques for adjusting for problems such as errors in variables should be explored.7 The insurance industry is very much interested in applying asset pricing models to estimate cost of equity capital for project decision making and regulation purposes. These results show that the APT and Wei model, and perhaps a combination of either with the CAPM, can be used as more reliable estimates of cost of equity capital than techniques currently being used.
Acknowledgment This is the first essay of the dissertation of the first author at the University of Pennsylvania. I am grateful to my dissertation committee members [J. David Cummins (Chair), Franklin Allen, Randy Beatty, Neil Doherty, Donald Morrison] for their guidance and support. In addition, I would like to thank seminar participants at the University of Pennsylvania, Indiana University, Northern Illinois University, University of Georgia, University of Illinois at Urbana-Champaign, and Syracuse University for their comments and suggestions. As always, any errors are the sole responsibility of the authors.
Notes 1. Additional early academic research on cost of capital estimates for the property-liability insurance industry includes Cummins and Nye (1972); Forbes (1971); Forbes (1972); Haugen and Kroncke (1971); Launie (1971, 1972); Lee and Forbes (1980); and Quirin and Waters (1975).
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2. A 1993 conference on “Key Issues in Financial and Risk Management in the Insurance Industry,” sponsored by the Wharton Financial Institutions Center, brought together consultants, executives, and regulators from the insurance industry, and academicians. Those from the insurance industry pointed out areas of concern for the industry and areas for research. Among the areas mentioned was the estimation of cost of capital and the applicability of models such as the CAPM and APT. 3. D’Arcy and Doherty (1988) review financial pricing models proposed for property-liability contracts. Cummins (1992) also provides a review of principal financial models that have been proposed for pricing insurance contracts and proposes some new alternatives. 4. Fama and French (1992) have questioned the validity of the CAPM. However, Kim (1995) and Jagannathan and Wang (1996) have shown that Fama and French results are misleading. 5. Detailed discussions of the measurement error issue can be found in Greene (1997) and Judge et al. (1980). 6. Alternative methods for the selection of factors is dicussed in detail by Campbell et al. (1997). 7. Kim (1995) uses Judge et al.’s (1980, pp. 521–531) errors in variables method to estimate the CAPM as defined in Equation (3.3). Kim’s method can be used to estimate the cost of capital in term of the CAPM.
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Kluwer Journal @ats-ss5/data11/kluwer/journals/requ/v10n3art1
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