Alternative Models for Estimating the Cost of Equity Capital for Property/Casualty Insurers

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 EquityCapital for Property/Casualty Insurers

ALICE C. LEEKPMG Peat Marwick LLP

J. DAVID CUMMINSWharton School, University of Pennsylvania

Abstract. This paper estimates the cost of equity capital for Property/Casualty insurers by applying threealternative 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 byapplying 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 acombined 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, costof 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 ofcommon stock. Financial theory suggests that returns of industrial firms should be afunction of risk, but because the management principles of financial institutions are notnecessarily identical to those of industrial firms, applications of financial theory to non-industrial firms (such as banks, savings and loans, and insurance firms) must be theoreti-cally and empirically investigated. What financial institutions must consider are exposureto different types of risk, the relative importance of various risks, and the existence ofmarket imperfections and constraints such as regulation.

For industrial firms, their production function and the return from their productioncapabilities are the main indicators of their value to investors in the market. Investmentsmade by industrial firms are generally for the purposes of increasing and improvingproduction capabilities. While, for financial institutions such as insurance companies,there are no production outputs to serve as an indicator of firm value. Output measure-ments for financial institutions are more difficult to measure because service sector outputis intangible. For instance, an insurance company is in the business to manage the risk ofothers, as well as the firm’s own risk in financial investments.

Kluwer Journal@ats-ss5/data11/kluwer/journals/requ/v10n3art1 COMPOSED: 02/06/98 8:12 am. PG.POS. 1 SESSION: 6

Specific issues of property/casualty (PC) insurance companies that differ from indus-trial 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-lendingrate relationship; and 3) an asset portfolio comprised primarily of securities of industrialfirms. An additional complicating issue for insurance companies is policyholder reserves.Reserves are the debt capital of the insurance industry, and policyholder reserves may beviewed as a source of investable funds. And there are also special issues associated withthe 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 andvaluation of firms in the industry?

Because public utility prices are regulated by a “fair rate of return,” there has beenmuch 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 ofliterature 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 itsproduct. But in regulating insurance prices, regulators often use book value rates of returnwhich are typically not accurate because cost of equity capital is a market-based valueinstead of a book value. Also, regulators usually do not vary the cost of capital bycompany or line, possibly creating excessive profits in some lines and disincentives toparticipate in others, and with better estimates of cost of capital for PC insurers, thiswould be possible. In their book, Cummins and Harrington (1987) present research on theissue 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 pricingmodel (CAPM) to derive risk-adjusted rates of return that the capital markets require ofstock property-liability insurers. Based upon Ross’ (1976) arbitrage pricing theory (APT)of capital asset pricing, Kraus and Ross (1982) develop a continuous time model underuncertainty to determine the “fair” (competitive) premium and underwriting profit for aproperty-liability insurance contract. An empirical application of the arbitrage pricingtheory (APT) in determining the cost of equity capital for PC insurers has yet to beconsidered.1

In addition to regulatory considerations, the insurance industry is interested in cost ofcapital estimation from a management prospective. Cost of capital affects an insurancecompany’s resource allocation and investment decisions. In project decision making,insurers have begun to use financial techniques such as financial pricing and capitalbudgeting but have found reliable costs of capital estimates difficult to obtain.2 Cost ofcapital is an important component in many insurance pricing models.3

Cummins (1990b) analyzes the two most prominent discounted cash flow (DCF) mod-els in property-liability insurance pricing—the Myers-Cohn (MC) model [Myers andCohn (1987)] and the National Council on Compensation Insurance (NCCI) model. Cum-mins points out that the two models are based on the concepts of capital budgeting, andessentially, the insurance policy is viewed as a project under consideration by the firm.

236 ALICE C. LEE AND J. DAVID CUMMINS


Thus, financial pricing models can be used to determine the price for the project (in thiscase, the premium) that will provide a fair rate of return to the insurer, taking into accountthe 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 insurancepricing.

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 AssetPricing Model (CAPM); 2) the Arbitrage Pricing Theory (APT); and 3) a unified CAPM/APT model [Wei (1988)]. Adjustments for nonsynchronous trading are made by applyingthe Scholes and Williams (1977) beta adjustment, and the errors in variables (estimationrisk) problem in the second-stage regression is adjusted by the Litzenberger and Ra-maswamy (1979) generalized least squares procedure. Then the in-sample forecast abilityof 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 conditionalefficiency of forecasts. Based on forecast evaluation procedures, the APT and Wei’sunified CAPM/APT models perform better than the CAPM in estimating the cost ofequity capital for the PC insurers, and a combined forecast may outperform the individualforecasts.

The next section discusses some of the theories and prior work in the area of cost ofequity capital estimation. Section III describes the model specifications and basic estima-tion procedures for the CAPM, APT, and unified CAPM/APT model. In Section IV, thedata set is described, and the empirical details for the asset pricing model estimations arediscussed. Section V explains and applies the procedures used to evaluate the forecastquality 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 equitycapital is one of the most important tasks because the cost of capital affects the company’sproject selection, borrowing rate, and allocation of resources. They categorize and brieflydescribe techniques used for estimating cost of capital: (1) comparable earnings; (2)valuation models; (3) risk premium; (4) capital asset pricing models; and (5) arbitragepricing 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 inpractice by regulatory agencies. Problems with this technique include difficulty in defin-ing “comparable companies” and inaccuracy of using book values to estimate cost ofcapital which is a market-based value.

Valuation models (also referred to as dividend growth models) define the cost of capitalas the discount rate that equates expected future dividends to the current price. If weassume the dividend per share, D1, grows at a constant rate, g, forever, then the cost ofcapital (k) can be defined as:

ESTIMATING THE COST OF EQUITY CAPITAL 237


Cost of equity ~k! 5DP

1 g (2.1)

where P represents the stock price per share. This is also called the constant-growthdiscounted cash-flow (DCF) formula. The DCF formula is the most widely used approachto estimate the cost of equity capital of regulated firms in the United States [Myers andBoruki (1994)] and is also used extensively in insurance regulation. Various assumptionscan be made about the growth rate.

The risk premium technique to estimate cost of capital requires an estimate of the extrareturn (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 plusthe 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 thanadjusting for a firm’s risk on an ad hoc basis, the models use a set of theories andassumptions to determine the risk adjustment for firms. The CAPM states that the ex-pected return on a security is a linear function of the riskless rate plus beta (the security’ssensitivity to the market) times the risk premium of the market portfolio over the risklessrate. The APT is similar to the CAPM in that the expected return on a security is a linearfunction of the riskless rate and the risk premium of different factors of systematic risktimes the factor beta (the security’s sensitivity to the factor). Unlike the CAPM, the APTmodel allows for more than one source of systematic risk, and the factors are not pre-defined. While the APT model can be viewed as a generalization of the CAPM, thedifficulty lies in identifying these factors.

Applying a methodology similar to that used by Bubnys (1990) (which expands uponBower et al. (1984)), this paper compares the CAPM, APT, and unified CAPM/APTmodel estimates of the cost of equity capital for PC insurance companies traded on theNYSE and NASDAQ during the 1988–1992 period. Bower et al. compare the CAPMmodel and the APT model as estimates of expected returns for utility stocks. They findthat APT may be the superior model in explaining and conditionally forecasting returnvariations through time and across assets for public utility companies. Bubnys expands onBower et al’s CAPM/APT comparison, concluding that neither model dominates in bothforecasting and simulation. By using the Litzenberger-Ramaswamy method of adjustment,Bubnys corrects for the errors in variables problem of using firm betas in the cross-sectional security market line equation. By doing so, use of large sample size in thesecond pass regression results in significant pricing of the riskless asset and market riskpremium.

III. Model specification and estimation

In Section III.A, first, the basic estimation procedure of the CAPM is discussed. Then, theprocedure for estimating the cost of equity capital using the CAPM is explained. Simi-larly, in Section III.B the estimation procedure for the APT model and the APT cost of



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 onevariable, the asset’s systematic risk:

E~Rj! 5 Rf 1 bj@E~Rm! 2 Rf# (3.1)

E(Rj) 5 expected return on asset jRf 5 risk-free rateE(Rm) 5 expected return on the market portfoliobj 5 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 thesystematic risk, b. This can be done by applying the time-series market model regression:4

Rjt 5 aj 1 bjRmt 1 «jt (3.2)

Rjt 5 return on asset j in time tRmt 5 return on the market portfolio in time taj 5 interceptbj 5 systematic risk of asset j«jt 5 residual term

From regression (3.2), bj determines how responsive the returns for the individual asset jare to the market portfolio.

For each firm in the market, regressing the firm’s rate of return at time t on the marketreturn at time t gives the estimate of the firm’s systematic risk, bj from (3.2), which is thenused in the second stage cross-sectional regression:4

Rj 5 a 1 b 3 bj 1 µj (3.3)

Rj 5 mean return over all time periods for asset jbj 5 systematic risk for asset j from first stagea 5 interceptb 5 slopeµj 5 residual term

If CAPM is valid, then



E~â! 5 Rf or E~Rz! (3.4)

E~b! 5 E~Rm! 2 E~Rf! or E~Rm! 2 E~Rz! (3.5)

where â is the intercept estimate, b is the slope estimate, and E(Rz) is the expected returnfor the zero-beta portfolio. The two-stage procedure described above results in an estimateof the risk-free rate and market risk premium.

After the CAPM is estimated, the betas of the PC insurance companies are estimatedwith regression (3.2). The CAPM cost of equity capital estimates, Kei, are computed usingthe following equation:

Kei 5 a 1 bbi (3.6)

where a and b are the risk-free rate and the market risk premium, respectively, estimatedfrom 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 alinear function of systematic risks. The CAPM predicts that security rates of return will belinearly related to a single common factor, the rate of return on the market portfolio. TheAPT allows the possibility of more than one systematic risk factor, and so the expectedrate 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 (3.7)

E(Rj) 5 expected rate of return on asset jbjk 5 sensitivity of asset j’s return to an unit change in the kth factor

(factor loading)l0 5 common return on all zero-beta assetslk 5 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 responsivereturns from asset j are to the kth factor. The theory doesn’t say what factors should beused to explain the expected rates of return. These factors could be interest rates, oilprices, or other economic factors. The market portfolio, used in the CAPM, might be afactor. Thus, the APT counterpart to the CAPM beta are the sensitivity coefficients, orfactor loadings, that characterize an asset. These factor loadings are estimated from themarket model:

Rjt 5 bj0 1 bj1I1t 1…1 bjkIkt 1 ujt (3.8)



Rjt 5 average return of portfolio j in time period tIkt 5 mean zero factor score at time t for factor k common to the returns

of all assetsbj0 5 estimated return on asset j when all factor loadings are zerobjk 5 estimated factor loading (reaction coefficient) describing the

change in asset j’s return for a unit change in factor kujt 5 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 CAPMequation (3.2), the return on the market portfolio, Rmt, is the independent variable, whilefor the APT, the independent variables, Ikt, are generated from factor analysis on thenon-insurance assets returns.

The second-pass cross-sectional regression is:

Rj 5 l0 1 l1bj1 1 l2bj2 1…1 lkbjk 1 uj (3.9)

Rj 5 average return of portfolio jl0 5 common return on all zero-beta assetslk 5 premium for risk associated with factor kbjk 5 factor loading of asset j to factor k estimated from equation (3.8)uj 5 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 insurancecompanies can be estimated. First, each individual insurance company’s rate of returns isregressed against the factor scores used in equation (3.8):

Rit 5 bi0 1 bi1I1t 1…1 bikIkt 1 uit (3.10)

Rit 5 return on insurance firm i in time period tIkt 5 mean zero factor score at time t for factor k common to the returns

of all assetsbi0 5 return on insurance firm i when all factor loadings are zerobik 5 factor loading (reaction coefficient) describing the change in in-

surance firm i’s return for a unit change in factor kuit 5 residual term

Then, the cost of equity for insurance firm i, Kei, the following equation is estimated forthe PC insurance firms with the following equation:

Kei 5 l0 1 l1bi1 1…1 lkbik (3.11)



Kei 5 cost of equity capital estimate for insurance asset il0 5 common return on all zero-beta assetslk 5 premium for risk associated with factor kbik 5 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 extrafactor 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 factorloadings 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 risklessasset 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 additionalterm lmbjm for the market factor.

To apply Wei’s unification theory, the estimation procedure is similar to that used toestimate the APT. The factor loadings are estimated with a modified market model:

Rjt 5 bj0 1 bj1I1t 1…1 bjkIkt 1 bjmrmt 1 ujt (3.13)

Rjt 5 average return of portfolio j in time period tIkt 5 mean zero factor score at time t for factor k common to the returns

of all assetsrmt 5 mean adjusted market return at time tbj0 5 return on asset j when all factor loadings are zerobjk 5 factor loading (reaction coefficient) describing the change in asset

j’s return for a unit change in factor kbjm 5 factor loading for the market factorujt 5 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 theunified CAPM/APT model is:



Rj 5 l0 1 l1bj1 1 l2bj2 1…1 lkbjk 1 lmbjm 1 uj (3.14)

Rj 5 average return of portfolio jlk 5 premium for risk associated with factor klm 5 premium for risk associated with market factorbjk 5 factor loading, estimated from (3.13), of asset j to factor kbjm 5 factor loading, estimated from (3.13), of asset j to market factoruj 5 residual term

This is similar to the APT regression (3.9) but with the additional term, lmbim, for themarket factor. The independent variables in regression (3.14) are the risk premium foreach of the k 1 1 factors, and the intercept term, l0, is the risk-free rate when all bjk’s arezero.

After the empirical unified CAPM/APT is obtained, the cost of equity capital for the PCinsurance companies can be estimated. First, each individual insurance company’s rate ofreturns is regressed against the factor scores and market factor which are the same as theones used in equation (3.13):

Rit 5 bi0 1 bi1I1t 1…1 bikIkt 1 bimrmt 1 uit (3.15)

Rit 5 return on insurance asset i in time period tIkt 5 mean zero factor score at time t for factor k common to the returns

of all assetsrmt 5 mean adjusted market return at time tbi0 5 return on insurance asset i when all factor loadings are zerobik 5 factor loading (reaction coefficient) describing the change in in-

surance asset i’s return for a unit change in factor kbim 5 factor loading for market factorujt 5 residual term

To estimate cost of equity capital, Kei, the following equation is computed for the PCinsurance firms:

Kei 5 l01l1bi1 1…1 lkbik 1 lmbim (3.16)

Kei 5 cost of equity capital estimate for insurance firm ilk 5 premium for risk, from (3.14), associated with factor klm 5 premium for risk, from (3.14), associated with market factorbik 5 factor loading, from (3.15), of insurance firm i to factor k from

equationbim 5 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 regres-sion (3.14), and factor loadings (bi1,…, bik, bim) are estimated from regression (3.15).



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 estimationprocedures for the asset pricing models are presented. First, the procedure for estimatingthe CAPM risk-free rate and risk premium is discussed in detail, and the estimate resultsare presented. Similarly, the APT model estimates and the unified APT/CAPM modelestimates are discussed.

A. Data

Rate-of-return data for common stocks are collected from Center for Research in SecurityPrices (CRSP) tapes: 1) NYSE/AMEX (New York Stock Exchange/American Stock Ex-change) monthly files; and 2) NASDAQ (National Association of Securities Dealers’Automated Quotation system) daily files. Data is collected for a 5 year period (December1987 to December 1992). The time period is short because of the limited number of PCinsurance companies that trade continuously for an extended period.

The NYSE/AMEX data consists of 1,671 companies, excluding PC insurance compa-nies, having no missing returns during our time period. Similarly, the NASDAQ dataconsists of 1,995 companies, excluding PC insurance companies, having no missingreturns from December 1987 to December 1992. CRSP NASDAQ data is only availableas daily rate-of-return data. So, all daily rate-of-return data collected from NASDAQ tapesare converted to monthly returns using the following calculation:

Rit 5 @~1 1 ri1!~1 1 ri2!…~1 1 rin!# 2 1 (4.1)

Rit 5 monthly return on stock i in month trij 5 return on stock i on trading day jn 5 number of trading days in the month

Also, because many of the NASDAQ companies are infrequently traded with many daysof zero return (nonsynchronous trading problem), converting the data from daily tomonthly reduces the errors in variables problem. Thus, there is a total of 3,666 firms in oursample, excluding PC insurance firms. Hereafter, this sample is to be referred to asnon-insurance companies.

Data is available from the NYSE/AMEX tapes for 23 PC insurance companies (here-after, to be referred to as insurance companies) with no missing returns during the periodof December 1987 through December 1992. Multi-line insurers with at least 25% of itsbusiness in property/casualty insurance and no missing returns during the 5 year periodare also included in the insurance sample. The selection process for PC insurers from theNASDAQ tapes is the same as the selection process for NYSE/AMEX firms. For NAS-DAQ companies, there are 41 insurance companies with no missing data during the 5 year



period. Again, since only daily data is available from the CRSP tapes, all data collectedfrom 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 dis-tributions 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 sec-tional statistics of the average monthly rate of return for the firms. The values are calcu-lated 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 averagemonthly 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 returnis 0.0154 (18.52% annualized), with NASDAQ firms (20.52% annualized) averaginghigher than the NYSE/AMEX firms (16.2% annualized). The range of the firm rates ofreturns 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 broadrange of firm rate of returns is reflected in the coefficient of variation measurements whichare 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 medianlie 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 coefficientrises 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 firmmonthly rate of returns, broken down into NYSE/AMEX firms, NASDAQ firms, and allinsurance firms combined. NASDAQ insurance firms have an average monthly rate ofreturn of 0.0168 (20.16% annualized), and NYSE/AMEX firms have an average monthlyrate of return of 0.0140 (16.84% annualized). Both the standard deviation (.0130) and thecoefficient of variation (.7709) of the NASDAQ insurance firms are considerably higherthan that of the NYSE/AMEX insurance firms (0.0063 and 0.4468, respectively). Sincesmaller firms trade on NASDAQ, we expect that returns for NASDAQ companies to bemore variable than those for NYSE/AMEX companies. The overall average monthly

Table 1. Summary Statistics for Average Firm Monthly Returns

N Mean Std Dev Min Max CV Skewness Kurtosis

Non-InsuranceNYSE/AMEX 1671 0.0135 0.0141 20.0503 0.1503 1.045292 0.7004 8.0904NASDAQ 1995 0.0171 0.0183 20.0519 0.1909 1.070185 1.2846 7.9810TOTAL 3666 0.0154 0.0166 20.0519 0.1909 1.074968 1.2073 8.7420

InsuranceNYSE/AMEX 23 0.0140 0.0063 20.0005 0.0237 0.446827 20.4010 20.2928NASDAQ 41 0.0168 0.0130 20.0203 0.0670 0.770938 20.7925 1.8601TOTAL 64 0.0158 0.0111 20.0203 0.0466 0.699094 20.6331 2.5158



return for the insurance firms (19.00% annualized) is very close to that of the non-insurance firms (18.52% annualized) in the dataset. The range of average firm monthlyrate of returns for the insurance companies (224.4% to 55.9% annualized) is not as wideas that of the non-insurance companies, and the coefficient of variation for the insurancefirms is less than 1 (0.6991) which is considerably less than that of the non-insurancefirms (1.075). Skewness coefficients of the average firm monthly returns of the insurancecompanies 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 thenon-insurance companies which could result from the fact that the insurance sample sizeis much smaller than the non-insurance sample size. Also, a lower kurtosis coefficientwould 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 rateestimates 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 firmbetas 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) inmonth t(t 5 1…60). The CRSP value-weighted index, including all distributions is usedas 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 aretraded infrequently (nonsynchronous trading). Scholes and Williams (1977) discuss hownonsynchronous trading introduces the potentially serious econometric problem of errorsin variables into the market model. With errors in variables in the market model, ordinaryleast squares estimators of coefficients in the market model are both biased and incon-sistent. Estimators are asymptotically biased upward for alphas and downward for betas.Thus for the 1,671 non-insurance NASDAQ companies, the Scholes-Williams beta ad-justment is applied. To compute the Scholes-Williams estimator, bSW, the following OLSregressions 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)



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 fortime t. The CRSP value-weighted index, including all distributions is used as the marketportfolio proxy for RM,t. The estimates from these regressions are then used to compute

bSW [b21 1 b0 1 b1

1 1 2rM

. (4.6)

bSW 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 firmslisted on NYSE/AMEX, firms listed on NASDAQ, and all firms combined. Betas for firmslisted 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 firmscombined) is 1.2826. NASDAQ betas have an average of 1.5673 with a standard deviationof 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 moremeasurement error (or estimation risk) due to nonsynchronous trading. This is evident inthe 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 listedon NASDAQ.

Similarly, betas for the insurance firms are also estimated with (3.2) for NYSE/AMEXfirms and (4.6) for NASDAQ firms. In Table 2, the cross-sectional summary statistics forinsurance firm beta estimates are shown. For insurance firms, the average beta is 1.2751with NYSE/AMEX betas averaging .9428 and NASDAQ betas averaging 1.4615. Thestandard 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 insurancebetas (4.7459 and 20.1407, respectively) are not quite as extreme as those for non-insurance betas.

Table 2. Summary Statistics for Firm Betas

N Mean Std Dev Min Max CV Skewness Kurtosis

Non-InsuranceNYSE/AMEX 1671 0.9426 0.5773 21.6700 3.9581 .6125 0.1182 0.8520NASDAQ 1995 1.5673 1.3526 24.4375 8.7331 .8630 0.5817 2.3468TOTAL 3666 1.2826 1.1154 24.4375 8.7331 .8696 1.0692 4.3662

InsuranceNYSE/AMEX 23 0.9428 0.3702 0.2760 1.5415 .3926 20.3480 20.9177NASDAQ 41 1.4615 0.8372 20.1407 4.7459 .5728 1.4212 4.7115TOTAL 64 1.2751 0.7455 20.1407 4.7459 .5847 1.7190 6.3220



Next, the non-insurance beta estimates are then used in the second-stage cross-sectionalregression (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-insurancecompanies. The dependent variable, Rj(j 5 1,…, 3,666), is the mean monthly return overthe five year period for each of the 3,666 non-insurance companies, and the independentvariable, bj, is the individual firm beta estimate from equations (3.2) and (4.6). This resultsin the CAPM estimates for the risk-free rate, a, and the market risk premium, b. TheCAPM estimates from the above procedure will be referred to as the “OLS” version of theCAPM model because the second-stage regression is an ordinary least squares regression.

When taking sample statistics, there is almost always some measurement error. Forregressions, the problem of inaccuracy in estimating both independent and dependentvariables results in biases in the regression estimates. Even if errors of measurement areassumed to be mutually independent, independent of the true values, and have constantvariance, the slope estimate will tend to be biased downward, and thus the intercept willtend to be biased upward.5 The greater the measurement error, the greater the biases. Inestimating and testing the CAPM, Black, Jensen, and Scholes (1972) show that usingindividual firm betas may introduce significant errors in estimating beta. In the second-stage regression (3.3), the intercept (risk-free rate or zero-beta estimate) will tend to beupward biased, and the slope (the market risk premium estimate) will tend to be down-ward biased. A generally accepted technique for reducing the problem of errors in vari-ables is grouping observations. When grouped, the errors of individual observations tendto be canceled out by their mutual independence, reducing measurement error effect. ForCAPM formation, the grouping procedure of using portfolios betas and portfolios returnsreduces the estimation error in beta. The trade-off of using portfolios instead of individualfirm betas is that the number of observations in each second-stage cross-sectional regres-sion (3.3) is greatly reduced.

To resolve this dilemma, Bubnys estimates the beta of individual firms in the first-stageregression (3.2), and then, he applies the Litzenberger and Ramaswamy (1979) general-ized least squares procedure (GLS) for the second-stage (3.3) run. This method takes intoaccount 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) areused to transform all the cross-sectional variables of the second-stage regression (3.3).The resulting equation is:

Rj/Sej 5 a/Sej 1 b3bj/Sej 1 µj/Sej (4.7)

Rj 5 mean monthly return over all time periods for asset jSej 5 standard deviation of the residuals of firm j’s market model re-

gressionbj 5 systematic risk for asset j from first stageµj 5 residual term



For regression equation (4.7), Sej21 and bj/Sej are the independent variables, and the inter-

cept term is constrained to be zero. The coefficient estimates â and b are still defined asbefore. â is the estimate of the expected return of a riskless or zero-beta asset, and b is theestimate of the expected excess return of the market portfolio over the riskless rate orzero-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 regres-sion.

A third version of the CAPM is considered where the Ibbotson Associates (1994) dataare used for estimates of the risk-free rate and risk premium. For the risk-free rateestimate, the 1926–1993 average annual return of long-term government bonds (5.4%) isused, and for the risk premium estimate, the 1926–1993 average annual return of largecompanies minus the average annual return of long-term government bonds (12.3% 25.4% 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 andrisk-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 calcu-lated. 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 Scholes-Williams 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, b, are from IbbotsonAssociates (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 ofthe market risk-free asset rate (Rf) and market risk premium (RP) for the three CAPMversions. 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 leastsquares (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 inparentheses below coefficients)

CAPM (n 5 3666) APT (n 5 40)

Rf RP l0 l1 l2 l3 l4 l5

IS1 .054 .069OLS2 .1185 .0521 .0957 .4962 1.9776 .3635 2.0777 2.1275

(24.707)* (14.453)* (1.940)** (1.181) (2.914)* (.370) (2.087) (2.511)*GLS3 .1065 .0612 .1062 .6400 1.5911 2.0941 2.2070 2.5941

(36.031)* (22.964)* (2.195)* (1.526) (2.421)* (2.098) (2.239) (3.916)*

*Significant at the 1 percent level**Significant at the 5 percent level1IS 5 CAPM with Ibbotson Associates risk-rate (Rf) and risk premium (RP).2OLS 5 CAPM or APT with ordinary least squares second stage regression.3GLS 5 CAPM or APT with generalized least squares second stage regression.



version are the estimates obtained from running the second-stage cross-sectional regres-sion (3.3) before applying the Litzenberger and Ramaswamy adjustment. Estimate valuesof 11.5% for the risk-free rate and 5.21% for the risk premium are both significant at the0.5 percent level. The third CAPM version is the intercept constrained GLS regression(equation (4.8)), using the Litzenberger and Ramaswamy adjustment. The estimates are10.65% for the risk-free rate and 6.12% for the risk-premium, both significant the 0.5percent level.

Although the OLS estimate of the CAPM Rf for the five year period, 11.85%, is muchhigher than historical and IS estimates of riskless rate, the estimate is significant at the 0.5percent level. One possible interpretation is that this the not the riskless rate but thezero-beta asset rate. Also to be considered is the fact that NASDAQ companies are usedin the model estimates. As can be seen from Table 1, the mean monthly return of thenon-insurance NASDAQ companies (20.52% annualized) is considerably higher than thatof the non-insurance NYSE/AMEX companies (16.25% annualized), and the overallaverage (18.48% annualized) is also considerably higher than the Ibbotson Associatesaverage annual return of large company stocks (12.3%). The NASDAQ mean increases theoverall mean used to estimate the risk-free rate and directly affects the estimate of theregression intercept. This can be explained as follows. The dependent variables of regres-sion (3.3) used to estimate the risk-free rate (the intercept) are the monthly returns of thenon-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 meanof the independent variable. The NASDAQ returns increase the overall mean of thedependent variable, y, which increases the intercept estimate.

Another possible reason for high risk-free rate and low risk premium estimates, byhistorical standards, is that small company betas have more measurement error (or esti-mation risk). The standard deviation and coefficient of variation of the non-insuranceNASDAQ betas (1.3526 and .8630, respectively) are much higher than that of NYSE/AMEX betas (.5773 and .6125, respectively). With higher estimation risk from NASDAQbetas, the second-stage regression will tend to underestimate the slope (risk premium) andoverestimate the intercept (risk-free rate).

Also, another explanation for a high risk-free rate estimate is that there are missingexplanatory 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 inaddition 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 theslope due to measurement error. In Table 3, the GLS version estimate of Rf and RP are10.65% and 6.12%, respectively. Both estimates are significant at the 0.5% level. Itappears that the Litzenberger-Ramaswamy adjustment has corrected for some of themeasurement error problems present in the OLS method by lowering the risk-free rate



estimate, increasing the risk premium estimate, and increasing the significance in bothestimates.

The OLS and GLS CAPM estimates for the risk-free rate and risk premium differ fromthe historical and the IS estimates. This can be partly attributed to the fact that NASDAQcompanies (i.e. small companies) are used in the sample to estimate the CAPM. Smallercompanies tend to have higher average returns and more measurement error than largerfirms, and both factors contribute to the overestimation of the risk-free rate and underes-timation of the risk premium. Or, it could be that these estimates are consistent with thistime period and these companies.

2. APT estimate. To estimate the APT model, first, factor analysis is used to estimate thefactor scores for the market, the 3,666 non-insurance companies. Factor analysis is a bodyof multivariate data-analysis techniques which provide “analysis of interdependence” of agiven set of variables or responses [Churchill (1983), Morrison (1990)]. For the APTmodel, we apply factor analysis to the monthly returns of the 3,666 non-insurance stocksto generate k common factor loadings for all the stocks. These factors, in a sense, sum-marize and capture the unobserved, underlying interdependencies of the monthly returnsof the 3,666 non-insurance stocks.

Because the number of variables (3,666 companies) is much greater than the number ofobservations (60 months for each company), we form portfolios from the 3,666 compa-nies to reduce the number of variables. First, the 3,666 non-insurance companies aregrouped alphabetically into 40 equally-weighted portfolios (14 portfolios of 91 companiesand 26 portfolios of 92 companies). The equal-weighted average returns for each monthof each portfolio are then used in a maximum likelihood estimation (MLE) method offactor 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 theoriginal 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 identi-fication of the factors underlying a set of observed variables (the monthly returns of the3,666 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 independentvariables, resulting in the estimated factor loading, bjk, of asset j to factor k. The depen-dent variables of this regression, Rjt, are the monthly average return of each portfolioj(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 pre-mium associated with each factor k in the APT model (equation (3.7)). The dependentvariables are the average return for the 5 year period for each of the 40 portfolios, and the



independent variables are the estimated 5 factor loadings, bik, for each portfolio from thefirst-stage regression (3.8):

Rj 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 abovedescribed procedure will be referred to as the “OLS APT” version.

Like the CAPM analysis, to adjust for possible sampling error in the estimation of theAPT factor loadings in (3.9), the Litzenberger and Ramaswamy procedure is applied. Theresulting regression is:

Rj/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 standarddeviation of the residuals of firm j’s market model regression (3.8). The intercept term ofregression (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 APTestimates 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 ofreturns 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 64insurance 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 esti-mates, lk, from either (3.9) or (4.9) are used in equation (3.11) to compute the APT costof equity capital.

Risk premium estimates. Table 3 shows the risk premium estimates (lambdas) of theOLS version (regression (3.9)) and the intercept constrained GLS version of APT (re-gression (4.9)). The factors of the APT are difficult to identify. l0 is the estimate of thezero-loading return. The OLS APT estimate of l0 is 9.57% and is significant at the 5percent level. For the GLS APT version, the estimate of l0 is 10.62% and significant at the1 percent level. The APT estimates of l0 are similar to the risk-free rate estimates of theCAPM. The first factor of the APT, l1, is considered highly correlated with the marketindex of the CAPM. Estimates of the first factor are .4962 and .6400 for the OLS and GLSversions, respectively, but these estimates are not highly significant.



Overall, for the APT OLS version, two factors are significant at the 1 percent level, andthe 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 statis-tical significance of the some of the factors. l0, l2, and l5 are all significant at the 1percent level. After applying the Litzenberger and Ramaswamy adjustment, the t-values ofl0, l1, l4, and l5 increase.

3. Unified CAPM/APT Estimate. The procedure to estimate Wei’s unified CAPM/APT isalmost identical to the procedure to estimate the APT model except that an additionalfactor, the market factor, is added. Factor scores are estimated for the market by using themonthly returns of the 3,666 non-insurance firms. The equal-weighted average returns foreach month of 40 equally-weighted portfolios are used in a MLE method of factoranalysis, 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 55) factor scores are generated, and an additional factor, rmt, is calculated for the Weimodel. The market factor, rmt, is calculated is the mean adjusted market return and iscalculated 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-sectionalregression (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-sectionalregression (3.9) but with the additional term, lmbjm, for the market factor. The resultingestimates 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 theestimation of the factor loadings in (3.14), the Litzenberger and Ramaswamy procedure isapplied. The resulting regression is

Rj/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 standarddeviation of the residuals of firm j’s market model regression (3.13). The intercept termof regression (4.11) is constrained to zero. (1/Sej) and (bjk/Sej) are the independent vari-ables. 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, eachindividual insurance company’s rate of return are regressed against the same five factorscores and the market factor which are used in regression (3.13):



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 thecost of equity capital of Wei’s model.

Risk premium estimates. Table 4 shows the results for the OLS and GLS versions ofWei’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 significantat the 5 percent level. The same factor premiums are significant for the GLS version, andin addition, l1 is significant at the 5 percent level.

Comparing estimates of the Wei model with those of the APT model, the risk premiumestimates are similar in magnitude for l0 through l5. And basically the same factors arestatistically significant for both the Wei model and the APT model except for l1 which isstatistically significant at the 5 percent level for the GLS Wei version. The risk premiumfor the market model, lm, is significant at the 5 percent level for both the OLS and GLSversions of the Wei model and seems to improve the specification of the APT modelwithout 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 thecost of equity capital for non-life insurance companies. In any forecasting study, it isimportant to evaluate the quality of predictions made. Lee et al. (1986) discusses proce-dures for evaluating forecasts. Some of the forecast evaluating procedures considered inLee 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)

l0 l1 l2 l3 l4 l5 lm

OLS1 .0887 .5781 2.0047 .4471 .0188 2.2453 .0885(1.789)** (1.360) (2.962)* (.456) (0.21) (2.639)* (1.849)**

GLS2 .0992 .6750 1.7068 .02916 2.1307 2.5950 .0870(2.090)** (1.688)** (2.642)* (.031) (2.154) (4.025)* (1.860)**

*Significant at the 1 percent level**Significant at the 5 percent level1OLS 5 Wei model with ordinary least squares second stage regression.2GLS 5 Wei model with generalized least squares second stage regression.



various versions of the three asset pricing models. The Theil mean squared error decom-position (MSE) and the Theil U2 statistic enables us to get measures of the forecastperformance of the models. Also, to evaluate Granger and Newbold (1973) conditionalefficiency 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 accu-rately assess the performance of the cost of equity capital estimation models. Each evalu-ation 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 correspondingtrue values at time t. Calculation of forecast mean squared error (MSE) (Theil (1958)) iscomputed by the following equation which can be decomposed into the terms on in theright-hand side of the equation:

MSE 5(t51

N

~Xt 2 Ft!2

N5 ~F 2 X!2 1 ~SF 2 rSx!

2 1 ~1 2 r2!Sx2 (5.1)

where F and X are the sample means, SF and Sx are the sample standard deviations, andr 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 repre-senting the contributions to forecast mean squared error due to bias, inefficiency, andrandom error, respectively. Calculation of MSE for the CAPM, APT, and the Wei modelforecasts gives us measures, and the decomposition of these measures, with which we cancompare the performance of the various forecasts by comparing the measures.

Applying (5.1) to the cost of equity capital estimates:

MSE 5(i51

64

~Ri 2 Kei!2

645 ~Ke 2 R

¯!2 1 ~SK 2 rSR!2 1 ~1 2 r2!SK

2

Rt 5 average rate of return for insurance firm iKei 5 cost of equity capital estimate for insurance firm i

R¯, Ke 5 sample means of Rt and Kei, respectively

SR, Sk 5 sample standard deviation for Rt and Kei, respectivelyr 5 sample correlation between average rate of return and cost of

equity capital estimate



B. Theil U2

The quality of cost of equity capital estimates can also be assessed by using the Theilmeasure (1966), U2:

U2 5(i51

64 ~Ri 2 Kei!2

(i5164 ~Ri 2 R!2

(5.2)

Rt 5 average return during 1988–1992 period for insurance asset iKei 5 cost of equity capital estimate for 1988–1992 for insurance asset

iR 5 average return for all insurance companies

The smaller the U2 ratio, the better the model estimate relative to the naive model. If theU2 ratio is equal to 1, then the model estimates are no better than using the sampleaverage. 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 thesame, but for the U2 ratio, the denominator makes the ratio into a evaluation of theforecasts relative to the naive model. Thus, with the U2 ratio, not only can we evaluateperformance by comparing the ratio of various models, the U2 ratio also provides us witha benchmark (the naive model) with which we can evaluate a model’s forecast perfor-mance.

C. Conditional efficiency method

1. Bates and Granger model. Bates and Granger (1969) propose a model that allows forthe possibility of a composite, or combined, forecast which is a weighted average of twoindividual forecasts and which may be superior to either forecast individually in terms ofpredictive 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 (5.4)

Xt 5 true values in time t(t 5 1, 2,…, N)F1t 5 set of forecastsF2t 5 alternative set of forecastsUt 5 random error term



In terms of forecast evaluation, if the predictions F2t contain no useful information aboutthe Xt which is not already incorporated in F1t, then the optimal value of k is one. Grangerand Newbold (1973) define F1t as conditionally efficient with respect to F2t if k, thecomposite predictor of equation (5.3), is one. A test for conditional efficiency applied byfitting equation (5.4) by ordinary least squares and testing the hypothesis that k 5 1through the usual t-test. Given a pair of forecasts, not only can we determine theirconditional efficiency, it is possible to use this model to find a combined forecast whichoutperforms each individual forecast by the mean squared error criterion.

2. Davidson and Mackinnon’s model. Chen (1983) assumes that the CAPM and APTmodels are nonnested for the purposes of comparing and testing the two models. In anattempt to compare the two models, Chen (1983) points out the problems with simplydoing a regression with the CAPM beta and the APT factor loadings: (1) this specificationis not justified on theoretical grounds, and the two models are nonnested; and (2) becausethe CAPM beta and APT factor loadings are measures of risk, such a regression wouldhave the problem of a high degree of multicollinearity between the independent variables.Davidson and MacKinnon (1981) proposed several procedures to test the validity ofregression models of nonnested alternative hypotheses. Using, the procedures suggestedby Davidson and Mackinnon, Chen suggests the following regression in which a isestimated

ri 5 ari,APT 1 ~1 2 a!ri,CAPM 1 ei (5.5)

where rAPT and rCAPM are the expected returns predicted by the APT and CAPM, respec-tively. From (5.5), if the APT is the correct model relative to the CAPM, we would expecta to be close to 1.

Thus, regression (5.5) suggested by Chen (1983), based upon the Davidson andMackinnon (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, weapply the Granger and Newbold test of conditional efficiency to evaluate the performanceof the CAPM and APT forecasts of cost of equity capital. This test of conditional effi-ciency allows us to compare two models’ forecasts performance simultaneously, unlikethe 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 squareserror decomposition (MSE), the Theil U2 ratio, and the conditional efficiency (Grangerand Newbold) methods are applied to evaluate forecast performance of the CAPM, APT,



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 yearperiod 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 calculatedfrom annualized instead of monthly rates of returns so that the magnitudes of the MSEvalues would not be too small. Equation (5.2) is applied to the average returns and cost ofequity 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 ofall 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 merelyindicates 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 wouldnot necessarily imply that a model provides poor overall forecasts. The largest inefficiencyvalues are also those of the APT OLS and Wei OLS models (.00068 and .00075, respec-tively). The inefficiency term is an indication of difference between the sample and theforecast standard deviations.

The terms which have the most impact on the total MSE value are the random errorterms which are a function of the sample correlation between the predicted and actualvalues. And despite have the largest biases and inefficiency values, the APT OLS and theWei OLS have the lowest random error values (.00169 and .00201, respectively) which areone order of magnitude smaller than that of all other models, resulting in the lowest totalMSE values (.00852 and .00891, respectively) for the APT OLS and the Wei OLS. TheCAPM IS has the highest total MSE of .01996. Thus, using the MSE method to evaluateforecast performance, the APT OLS and the Wei OLS models are considerably bettermodels 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 allthe models.

Table 5. Mean Squared Error* Decompositions for Forecasts1

CAPM APT WEI

IS OLS GLS OLS GLS OLS GLS

Bias .00231 .00003 .00003 .00615 .00032 .00615 .0003Inefficiency .00066 .00017 .00040 .00068 .00030 .00075 .00012Random error .01670 .01700 .01700 .00169 .01662 .00201 .01615Total MSE. .01996 .01720 .01743 .00852 .01724 .00891 .01659

*MSE 5(i51

64

~Ri 2 Kei!2

645 ~Ke 2 R

¯!2 1 ~SK 2 rSR!2 1 ~1 2 r2!SK

2

5 Bias 1 Inefficiency 1 Random Error1IS 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.



Because there is no scale or basis to which we can compare the MSE values, it isdifficult to determine exactly how much better the APT OLS and Wei OLS models arethan the other models. The Theil decomposition of the MSE gives us an indication of therelative strengths and weaknesses of a model’s forecasts in terms of bias, inefficiency, andrandom error.

Next, consider the Theil U2 statistic (equation (5.2)) for the cost of equity capitalforecasts shown in Table 6. Because the U2 ratio and the MSE equation have the samenumerators, it is likely that both methods will rank the models in terms of forecastperformance similarly. The additional information about forecast performance that the U2

statistic provides is that the statistic compares a model’s forecast performance with thenaive model (using the overall mean as a predictor). A U2 ratio equal to 1 means that theforecast performance is equivalent to the naive model, and a ratio value . (,) 1 meansthat 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 returnof the insurance companies (19.00%, annualized) or the market average of 18.52% (of the3666 non-insurance companies). But because the difference between the insurance com-pany averages and the market average is not large, the U2 are not likely to differ much. Inall cases of all model versions, the U2 ratio goes down when the market average is usedin the denominator instead of the insurance average.

All U2 ratios are less than 1, except for the IS CAPM model, implying that almost allmodels have forecast performances which are better than that of the naive model. Thehighest U2 ratios and the only ratios that are larger than 1 are those of the IS CAPM modelwhich is consistent with the MSE evaluation of the IS CAPM having the poorest forecastperformance. In fact, U2 ratio rankings of model performance is the same as the MSErankings, 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 easilycalculated for comparison, they do not give a complete picture of forecast quality. Withthese two methods, it is not possible to compare the forecast quality of by consideringthem 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 doesnot necessarily imply that the one set of forecasts is more efficient than the other. It maybe the case that F2t may contain useful information not captured by F1t. Thus as anadditional measure of forecast performance is applied. The Granger and Newbold (1973)definition of conditional efficiency is tested for by applying the Bates and Granger re-gression (5.5). Conditional efficiency of forecasts F1t with respect to F2t is defined byGranger 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 CAPMversions.

Considering first the APT and CAPM comparisons (regressions 1, 2, and 3 in Table 7),it would be logical expect the k of the APT OLS model to be very close to 1 whenevaluated with other models because of its dramatically lower MSE and U2 values com-pared to those of the other models, but this is not the case. In fact, k for APT OLS



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regressions with each of the CAPM versions are all approximately .65 and statisticallydifferent from 1. Table 7 shows that the t-statistics (k 5 1) of regressions 1, 2, and 3 areall statistically significant at the 1 percent level. This implies that the APT OLS model isnot conditionally efficient with respect to the CAPM versions and that the CAPM versionscontain useful information not present in the APT OLS forecasts. Although the k’s forregressions 1, 2, and 3 are statistically different from 1, the t-statistics for k 5 0 are allalso significant at the 1 percent level, and the R2 values for regressions 1, 2, and 3 are allaround .7. This suggests that perhaps a combined forecast of the APT OLS and an CAPMversion would outperform each individual forecast by the MSE and/or U2 criteria.

The highest k for the APT and CAPM regressions is .8366 with t-statistic (k 5 1) of2.632 for the APT GLS and CAPM IS regression (regression 4). Thus, the null hypothesisof conditional efficiency (k 5 1) is not rejected. Also, the t-statistic (k 5 0) for regression4 is significant at the 1 percent level. This implies that k is not statistically different from1 and that the APT GLS forecasts may be considered conditionally efficient with respectto the CAPM IS forecasts. Regressions 5 and 6 which test the APT GLS with respect tothe CAPM OLS and IS, respectively, also do not reject the null hypothesis of k 5 1. Thecorresponding 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 respectto the CAPM versions, the R2s of the regressions are quite low. Thus, a combined forecastof the APT GLS with any of the CAPM versions is unlikely to provide a forecast that issuperior to each individual forecast.

The results of the conditionally efficiency of the Wei forecasts with respect to theCAPM 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

Table 7. Granger and Newbold’s Conditional Efficiency Estimated Weights of the Expected Return from theCAPM, APT, and Wei model1 Rt 5 kF1t 1 (1 2 k)F2t 1 Ut

F1t F2t k

t-statistic

(k 5 0)

t-statistic

(k 5 1) R2

1.APT OLS CAPM IS .65196 15.666* 28.363* .79252.APT OLS CAPM OLS .65138 12.086* 26.468* .69393.APT OLS CAPM GLS .65541 12.069* 26.345* .69334.APT GLS CAPM IS .83660 3.237* 2.632 .12905.APT GLS CAPM OLS .48762 1.549 21.627 .02146.APT GLS CAPM GLS .52743 1.796** 21.585 .03227.WEI OLS CAPM IS .59992 14.178* 29.455* .75768.WEI OLS CAPM OLS .58792 10.712* 27.508* .64009.WEI OLS CAPM GLS .59514 10.688* 27.380* .638910.WEI GLS CAPM IS .78071 3.731* 21.048 .168011.WEI GLS CAPM OLS .57418 2.142** 21.589 .053112.WEI GLS CAPM GLS .59837 2.320** 21.557 .0641

*Significant at the 1 percent level**Significant at the 5 percent levelIS 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.



OLS forecasts are not conditionally efficient with respect to the CAPM forecasts (regres-sions 7, 8, and 9). The k estimates for these regressions are approximately .6 witht-statistics which are significant at the 1 percent level for both the (k 5 0) and the (k 51) hypotheses. This implies that the CAPM forecasts may contain information not cap-tured by the WEI OLS forecasts and that a combined forecast may outperform eachindividual forecast by the MSE and the U2 criteria. Also the high R2 values of regressions7, 8, and 9 suggest that a combined forecast should be considered.

Regression 10 which considers the WEI GLS forecasts with respect to the CAPM ISforecasts has a k value of .78071 and a t-statistic (k 5 1) of 21.048, so it is possible toconsider the conditional efficiency of the WEI GLS forecasts with respect to the CAPMIS forecasts. And the hypothesis of conditional efficiency of the WEI GLS forecasts withrespect to the CAPM OLS or the CAPM GLS (regressions 11 and 12, respectively) cannot be rejected. But as with the APT GLS, the WEI GLS combined with the CAPMversions have low R2 values which indicate that it is unlikely that a combined forecast ofthe WEI GLS with any of the CAPM versions would be superior to that of a combinedforecast 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 equitycapital for 64 non-life insurance companies during the 5 year period 1988–1992. 3,666non-insurance companies were used to estimate CAPM and APT characteristic line equa-tions. Three versions of the CAPM cost of equity capital estimates (IS, OLS, GLS) wereconsidered. The IS version used Ibbotson and Sinquefield estimates of the risk-free rateand the risk premium for the CAPM. The OLS version used the risk-free rate and the riskpremium estimates from an second-stage ordinary least squares regression. To adjust forthe problem of errors in variables, the Litzenberger-Ramswamy generalized least squaresprocedure was used for the second-stage regression, resulting in the GLS version. Forboth, the OLS and GLS versions, the Scholes-Williams beta adjustment was applied toNASDAQ stocks for the problem of nonsynchronous trading. Two versions of the APTcost of equity capital estimates were considered (OLS and GLS). Similar to those of theCAPM model, the OLS version resulted from a second-stage ordinary least squaresregression, and the GLS version resulted from a second-stage generalized least squaresregression to adjust for the problem of error in variables. Finally, a model unifying theCAPM and APT models, developed by Wei, was also used to estimate cost of equitycapital for the insurance companies. An OLS and GLS version of the Wei model, similarto those of the APT model were, were considered.

For the CAPM model, the OLS and GLS estimates of the risk-free rate and riskpremium were higher and lower, respectively, than the historical and the IS values. Al-though, the GLS version seemed to somewhat correct for the errors in variables problemof underestimating the slope and overestimating the intercept, perhaps other errors invariables estimation methods of adjusting the estimates should be considered in futureresearch. In addition, higher average returns and increased estimation risk associated with



NASDAQ companies also contributed to the high risk-free rate estimates and the low riskpremium estimates. But, because many of the insurance companies trade on NASDAQ,using NASDAQ companies to estimate the asset pricing models should improve theforecast 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 interceptestimates and low slope estimates.

The APT and Wei’s unified CAPM/APT model had similar factor risk premium esti-mates. The additional market factor of the Wei model was statistically significant in boththe OLS and GLS version. It is intuitively appealing to have a model which combines theAPT and CAPM. Though, results from this research do not empirically show that theadditional market factor significantly improves the cost of equity capital estimation.

The MSE and U2 methods of evaluating forecast quality found the APT OLS and theWei OLS models to be the best estimators of cost of equity capital for the insurancecompanies during the 5 year period tested, but the tests of conditional efficiency suggestedthat a weighted combination of either the APT OLS or the Wei OLS with the CAPM mayprovide forecasts that are better than each individual forecast.

Overall, the results show that the APT and the Wei model, which is basically a modifiedAPT, perform better than the CAPM in estimating the cost of equity capital for insurancecompanies in our sample. The methods presented here are straight-forward in their imple-mentation and application, but additional techniques for adjusting for problems such aserrors in variables should be explored.7 The insurance industry is very much interested inapplying asset pricing models to estimate cost of equity capital for project decisionmaking and regulation purposes. These results show that the APT and Wei model, andperhaps a combination of either with the CAPM, can be used as more reliable estimatesof 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 Pennsyl-vania. I am grateful to my dissertation committee members [J. David Cummins (Chair),Franklin Allen, Randy Beatty, Neil Doherty, Donald Morrison] for their guidance andsupport. In addition, I would like to thank seminar participants at the University ofPennsylvania, Indiana University, Northern Illinois University, University of Georgia,University of Illinois at Urbana-Champaign, and Syracuse University for their commentsand 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 industryincludes Cummins and Nye (1972); Forbes (1971); Forbes (1972); Haugen and Kroncke (1971); Launie(1971, 1972); Lee and Forbes (1980); and Quirin and Waters (1975).



2. A 1993 conference on “Key Issues in Financial and Risk Management in the Insurance Industry,” sponsoredby the Wharton Financial Institutions Center, brought together consultants, executives, and regulators fromthe insurance industry, and academicians. Those from the insurance industry pointed out areas of concernfor the industry and areas for research. Among the areas mentioned was the estimation of cost of capital andthe 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 pricinginsurance contracts and proposes some new alternatives.

4. Fama and French (1992) have questioned the validity of the CAPM. However, Kim (1995) and Jagannathanand 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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Alternative Models for Estimating the Cost of Equity Capital for Property/Casualty Insurers

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