Estimating the Cost of Equity Capital for Insurance Firms with Multi-period Asset Pricing Models

Transcription

1 Estimating the Cost of Equity Capital for Insurance Firms with Multi-period Asset Pricing Models Alexander Barinov 1 Steven W. Pottier 2 Jianren Xu *, 3 This version: July 29, 2016 * Corresponding author. 1 Department of Finance, School of Business Administration, University of California Riverside, 900 University Ave. Riverside, CA 92521, Tel.: , alexander.barinov@ucr.edu. 2 Department of Insurance, Legal Studies, and Real Estate, Terry College of Business, University of Georgia, 206 Brooks Hall, Athens, GA 30602, Tel.: , Fax: , spottier@uga.edu. 3 Department of Finance, Mihaylo College of Business and Economics, California State University, Fullerton, 800 N. State College Blvd., Fullerton, CA 92831, Tel.: , Fax: , jrxu@fullerton.edu.

2 Estimating the Cost of Equity Capital for Insurance Firms with Multi-period Asset Pricing Models Abstract Previous research on insurer cost of equity (COE) focuses on single-period asset pricing models, such as the CAPM and Fama-French three factor model (FF3). In reality, however, investment and consumption decisions are made over multiple periods, exposing firms to time-varying risks related to economic cycles and market volatility. We extend the literature by examining two multiperiod models the conditional CAPM (CCAPM) and the intertemporal CAPM (ICAPM). Using 29 years of data, we find that macroeconomic factors significantly influence and explain insurer stock returns. Insurers have countercyclical beta implying that their market risk increases during recessions. Further, insurers are sensitive to volatility risk, but not insurance-specific risk, liquidity risk, or coskewness after controlling for other economy-wide factors. Keywords: Cost of equity; multi-period asset pricing models; time-varying risks

3 I. Introduction Previous research on insurer cost of equity (COE) focuses on single-period asset pricing models such as the Capital Asset Pricing Model (CAPM) and Fama-French (1993) three factor model (FF3). Merton s (1973) seminal article on multi-period asset pricing demonstrates that when investment decisions are made at more than one date, additional factors are required to construct a multi-period model to obtain optimal portfolio positions because of uncertain changes in future investment opportunities. 1 In addition, firms are exposed to business and economic cycles. Multiperiod models account for the time-varying risks (factors) that create such uncertainty and reflect these cycles. In this study, we extend the insurance literature by examining two multi-period models the conditional CAPM (CCAPM) and the intertemporal CAPM (ICAPM). These two models are examined along with the single-period models studied in the prior insurance literature the CAPM and FF3. 2, 3 Our empirical analysis consists of three major parts. First, we determine the applicability of the four asset pricing models mentioned earlier (CAPM, FF3, CCAPM and ICAPM) 1 In addition to the riskless asset and market portfolio, one factor is required for each state variable needed to describe the dynamics of the investment opportunity set. This is known as the (m+2)-fund Theorem of Merton (1973). The theory does not specify the factors, but they should be related to broad economic factors capturing systematic risks that investors demand to be compensated for bearing. 2 A number of authors have examined a variety of approaches to estimating the cost of equity capital for insurers. Recent studies include Cummins and Phillips (2005) and Wen et al. (2008). Cummins and Phillips (2005) employ the CAPM and the FF3 to estimate the COE for U.S. property-liability (P/L) insurance companies. Wen et al. (2008) use the CAPM and Rubinstein-Leland model. We argue that since these models are single-period models, they do not account for the time-varying market and economic risks that insurers face. It is well-known that the insurance industry is not immune from general business cycles in the economy (i.e., economic expansions and recessions). As a result, factor betas and risk-premiums are likely to vary over time (e.g., evidence from Table 1 in Wen et al., 2008), and consequently, the relative risk of an insurance company s cash flows is likely to vary over the business cycle. In this sense, the single-period models, such as CAPM and FF3, are incomplete. In Section II, we review the related literature on insurer cost of equity capital. 3 As discussed in Section II, while re-estimating beta in the CAPM or betas in the FF3 allows for the cost of equity to vary over time, these approaches do not incorporate the covariance of factor beta(s) with economic conditions into the estimation of the beta(s). The CAPM and FF3 implicitly assume, by using the long-term averages of the market risk premium (as well as SMB and HML), that the amount of risk in the economy (as proxied, for example, by the market risk premium) is constant. Thus, in the CAPM and FF3, there is no possible covariance between the betas and the business cycle by construction. 1

4 to insurance firms by verifying that insurance firms are exposed to the risks measured by these models. Our analysis is conducted on all U.S. publicly traded insurers, and the two subgroups, property-liability (P/L) insurers and life insurers. In this first part of empirical analysis, using 29 years of data, we find that in addition to the CAPM market factor and FF3 factors, insurers are sensitive to volatility risk. More specifically, insurance portfolio values drop when current consumption has to be cut in response to surprise increases in expected market volatility. Further, insurers have countercyclical beta, which means their risk exposure (market beta) increases in recessions, when bearing risk is more costly, and decreases in expansions, when investors have a greater appetite for risk. Therefore, insurers are riskier and consequently should have a higher expected return than what the CAPM predicts because of the undesirable pattern of beta changes. We further explore FF3 augmented with the volatility risk factor and conditional FF3 models. In addition, we examine liquidity risk and coskewness because previous literature documents these factors important determinants of COE; however, these additional factors are economically insignificant once we include the market, FF3, and the volatility risk factor. FF3 is also often regarded as an ICAPM-type model with SMB and HML acting as placeholders for yet unidentified risk. 4 While a number of papers have tried to identify the risks behind these factors (Liew and Vassalou, 2000; Petkova and Zhang, 2005; Zhang, 2005; Petkova, 2006; Campbell, Polk, and Vuolteenaho, 2010), the consensus as to which business cycle variables are behind SMB and HML still has not emerged. Even more, a number of papers have contested the assumption that SMB and HML are driven by risk and argued that they represent mispricing and market sentiment swings (La Porta, Lakonishok, Shleifer, Vishny, 1997; Daniel and Titman, 1997; Ali, Hwang, and Trombley, 2003; Baker and Wurgler, 2006). 4 For example, this is the view Fama and French took in their original paper, Fama and French (1993), as well in subsequent papers like Fama and French (1995) and Fama and French (1996). 2

5 Company stakeholders might not only want to know the COE of their firm or projects, but also the reasons behind a certain rate, namely, what risk sources result in a high or low COE. Without valid risk-based explanations stakeholders might feel uncomfortable in accepting a COE estimate. The additional alternative that the factors can be picking up market-wide mispricing makes the decision process even more complicated. For example, our analysis reveals that insurance companies tend to be value firms. 5 If we believe the mispricing hypothesis that HML is picking up the systematic underpricing of value firms followed by superior returns as the mispricing is corrected, should we benchmark insurers COE against other value firms, thus asking them to deliver a higher return than their risk profile warrants and thus abandoning some positive NPV projects? (This is what using FF3 in COE estimation suggests). Or should we exercise all positive NPV projects, effectively ignoring the positive HML beta if we think HML is mispricing? Theory-based multi-period models, such as the CCAPM and ICAPM, considered in our paper, are immune to both problems. First, they identify the risks they are talking about ( insurance companies lose more than average when market volatility increases, the market beta of insurance companies increases when deflation occurs ). Second, they are only picking up risk-based effects in expected returns/coe, and one does not have to worry about mispricing. In the second major part of our empirical analysis, we also consider for potential inclusion in the CCAPM and ICAPM, underwriting cycle/insurance-specific variables, in addition to the standard business cycle variables from the finance literature. While changes to underwriting cycle/insurance-specific variables clearly affect the value of insurance firms, it is not clear a priori that they will be related to expected returns, because all their effects can be on the cash flow side. 5 The Fama-French model (FF3) reveals that insurance companies are value firms because HML loads positively and significantly on insurer excess returns. More details on page 19. 3

6 The finance theory suggests (see, e.g., Cochrane, 2007) that only the variables that are related to expected market risk premium and thus to marginal utility of consumption should be included in any asset pricing models (either CCAPM or ICAPM in this study). We check the existence of such relation between several underwriting cycle variables (such as average combined ratio or total catastrophic losses in a quarter) and find none: the underwriting cycle variables cannot predict market returns. That is probably not surprising: gains and losses in the insurance industry are unlikely to materially affect the aggregate consumption in the US economy. Consistent with the lack of relation between underwriting cycle/insurance-specific variables and the expected market risk premium, we find that inclusion of these variables in either the CCAPM or ICAPM does not materially impact our COE estimates. That happens even though some underwriting cycle variables seem to be related to the beta/realized returns of insurance companies; in the finance language, the underwriting cycle variables earn zero risk premium (controlling for other economy-wide factors), because underwriting cycle variables are unrelated to the marginal utility of consumption for the marginal investor. 6 In the third major part of our empirical analysis, we apply the four models (CAPM, FF3, CCAPM and ICAPM) to estimate cost of equity for all U.S. publicly traded insurers, and the two subgroups, P/L insurers and life insurers over an 18 year period. 7 As implied by the notion that additional time-varying risks demand greater rewards, we find that on average ICAPM generate COE estimates that are significantly higher than CAPM COE and marginally higher than FF3 COE. If we add the volatility risk factor to FF3 (turning it into a four-factor model, FF4), we observe 6 The irrelevance of underwriting cycle/insurance-specific variables as candidate CCAPM/ICAPM goes beyond the application at hand suggesting that insurance-specific factors should not be used to measure expected return to insurance firms. Even if such factors are correlated to insurance companies realized returns, they will not contribute to expected returns due to being unrelated to the economy as a whole and thus having zero alphas (unexplained portion of portfolio excess returns) controlling for other market-wide risk factors. 7 In COE estimation, we lose 10 years as the initial estimation period for CCAPM, for which we need 120 months to estimate six parameters with enough precision. 4

7 that this augmented version of FF3 produces even higher COE estimates than either ICAPM or FF3. Furthermore, we document that all models consistently yield higher COE estimates for life insurers than P/L insurers, suggesting that life insurer stocks are riskier. We also apply a novel estimation technique for deriving COE from CCAPM by predicting, using business cycle variables, both the market beta of insurance firms and the expected market risk premium. The resulting COE series reflects well the risk shifts during our sample period: for example, in , during the aftermath of the Great Recession, the CCAPM s COE is higher than COE estimate from any other model. The average level of COE from CCAPM in is relatively low, due to the fact that the expected market risk premium is estimated at about 3% per annum (in contrast to the long-term average of 6% per annum used in other models) before the Great Recession. This low level of the market risk premium is, however, consistent with alternative market risk premium estimates in Claus and Thomas (2001) and Fama and French (2002), who find that before the Great Recession investors deemed the market risk as historically low. We also find that if one plugs the 3% market risk premium in the standard CAPM (as Claus and Thomas, 2001, and Fama and French, 2002, suggested for the pre-crisis years), the standard CAPM will produce significantly lower average COE than the CCAPM, consistent with the CCAPM finding more risk in insurance companies. The remainder of this paper is organized as follows. Section II provides a literature review and background on the four asset pricing models (CAPM, FF3, CCAPM, and ICAPM). Section III describes the data and variables. Section IV applies the four main models to the insurance industry and two subgroups (P/L and life insurers) and studies the insurers risk sensitivities. We also discuss the augmented Fama-French models, and liquidity and coskewness factors. Section V discusses the potential inclusion of underwriting cycle variables into the CCAPM and ICAPM. 5

8 Following that, the cost of equity is estimated based on each of the models and the results are discussed in Section VI. Section VII summarizes and concludes. II. Literature and Asset Pricing Models CAPM The CAPM is a single-period asset pricing model, in which the beta and the expected market risk premium are assumed to be constant over the estimation period. In COE applications, betas are periodically updated, but the market risk premium is always fixed at its long-term average level. Thus, the CAPM implicitly assumes that beta is unrelated to expected market risk premium and the state of the economy, as the CAPM does not recognize that the market risk premium can be different in expansions versus recessions. The shortcomings of the CAPM are well documented. The most striking weakness of the CAPM is its inability to explain the cross-sectional variation in average stock returns. Fama and French (1992) show that historical market betas are unrelated to expected returns in the crosssection and thus cannot explain why size and book-to-market appear to predict expected returns. Since then, the list of variables that predict expected returns controlling for beta and of implied trading strategies (also called anomalies) earning significant CAPM alphas has expanded to include dozens of variables. McLean and Pontiff (2016), Harvey, Liu, and Zhu (2016), Hou, Xue, and Zhang (2015) provide the (largely overlapping) lists of violations of the CAPM documented as of today. Fama-French Three Factor Model In response to actual and perceived weaknesses in the CAPM, Fama and French (1992 and 1993) developed a multifactor asset pricing model that has become the most widely used 6

9 alternative to the CAPM. The Fama-French (FF3) model, like the CAPM, is a single-period model that has the following specification: Ri RF = βi (RM-RF) + si SMB + hi HML + ε (1) where Ri = return for asset i, RM = return on market portfolio, RF= return on riskless security, βi = asset i s market beta, SMB = difference in returns to portfolios of small and large stocks, si = asset i s size (SMB) beta, HML = difference in returns to portfolios of high and low book-to-market (value and growth) stocks, hi = asset i s HML beta, and ε is the error term. Fama and French (1992 and 1993) provide evidence that size and book-to-market equity factors combine to better capture the cross-sectional variation in average stock returns associated with market beta, size, leverage, and earnings-price ratios. Fama and French (1995) argue that the FF3 is an equilibrium pricing model and a threefactor version of Merton s (1973) intertemporal CAPM or Ross (1976) arbitrage pricing theory model. Fama and French claim that the size and book-to-market factors mimic combinations of underlying risk factors or state variables of special hedging concern to investors. The exact nature of the state variables behind SMB and HML remains elusive despite 20 years of ongoing research. Some candidate state variables include GDP growth (Liew and Vassalou, 2000), investment (Zhang, 2005; Cooper, 2006), default risk (Vassalou and Xing, 2004), changes in the slope of the yield curve (Hahn and Lee, 2006; Petkova, 2006). Another strand of research, started by Lakonishok, Shleifer, and Vishny (1994) and Daniel and Titman (1997), argues that SMB and HML represent market-wide mispricing. If this is the case, the use of FF3 in the COE estimation becomes ambiguous. Imagine, for example, that we are talking about a value firm that loads positively on HML. Using FF3 for COE estimation is likely to yield higher-than-average COE, reflecting the fact that value firms have high average returns. 7

10 If the manager feels the need to beat the peers (also value firms), he/she will use the COE from FF3. However, if the value effect is mispricing and value firms have higher returns than warranted by their risk, using COE from FF3 will imply turning down some positive NPV projects (which earn more than what their risk warrants, but less than an average value firm makes). Conditional CAPM The CCAPM assumes that the conditional version of the CAPM holds, that is, the expected return on an asset based on the information available at any given point in time is linear in its conditional beta. The CCAPM allows a stock s market beta and the expected market risk premium to differ under different economic conditions. First, CCAPM recognizes that expected market risk premiums are lower during economic expansions or good times, and higher during economic recessions or bad times, as empirical studies in finance find (Fama and French, 1989; Keim and Stambaugh, 1986; Jagannathan and Wang, 1996; Campbell et al., 1997; Cochrane, 2005; Cochrane, 2007). In bad economic times, investor wealth is lower and the marginal utility of wealth is higher. Consequently, investors willingness to bear risk is lower in bad economic times, and the required risk premium is higher. Second, the risk of some stocks also varies with economic conditions. During a recession, for example, financial leverage of firms in weak financial condition may increase dramatically relative to other firms, causing their stock betas to rise (Hamada, 1972). The CCAPM states that the unconditional expected risk premium of a particular stock can be computed as follows, assuming both beta and the market risk premium are random variables: E(Ri-RF) = E[βi (RM-RF)] = E(βi) E(RM-RF) + Cov[βi, (RM-RF)] (2) Equation (2) follows directly from definition of covariance: Cov(X, Y)=E[(X - E(X)) (Y - E(Y))]=E(X Y) - E(X) E(Y) (3) 8

11 Thus, in the CCAPM the risk premium on a stock equals to the product of the stock beta and the market risk premium plus the beta-premium sensitivity covariance term, as shown in Equation (2). The standard CAPM misses the covariance term. In most COE applications, the CAPM assumes that expected market risk premium is constant at its long-term average, thus effectively setting the covariance term to zero even if the betas are allowed to change from one estimation period to another. The economic meaning of the covariance term is that stocks with countercyclical betas (higher in bad times) are riskier than their average (over expansions and recessions) beta (namely, CAPM beta) would imply. For these stocks, the covariance piece in Equation (2) will be positive, because expected market risk premium, E(RM), is also higher in recession. Higher risk and higher beta in recessions are undesirable, because marginal utility of consumption is higher during recessions and potential losses (coming from higher risk) are more painful. Also, a firm with higher risk in recession will witness a larger increase in discount rate applied to its cash flows and therefore a sharper drop in value, and losses in bad times are more painful for investors. The fact that the covariance piece is the difference between the CAPM and CCAPM also guides our choice of conditioning variables that will be assumed to be driving the beta. These variables need to be related to the expected market risk premium (i.e., they have to predict the market return). If the beta is related to a variable that does not predict the market return, controlling for this relation will not affect our estimate of the covariance term and thus will not create extra difference between expected return/coe estimates from the CAPM and CCAPM. For this study we select four commonly used conditioning variables, namely, dividend yield (DIV), default spread (DEF), Treasury bill rate (TB), and term spread (TERM) that are long 9

12 known to predict the market return. 8,9 Our choice of conditioning variables is standard for the CCAPM literature (Petkova and Zhang, 2005, O Doherty, 2012, Boguth et al., 2011). Following Petkova and Zhang (2005), we assume that the market beta is a linear function of the four macroeconomic variables above: E(β it ) = b i0 + b i1 DEF t 1 + b i2 DIV t 1 + b i3 TB t 1 + b i4 TERM t 1 (4) If we substitute Equation (4) into the standard CAPM equation, we get R it RF t = α i + (b i0 + b i1 DEF t 1 + b i2 DIV t 1 + b i3 TB t 1 +b i4 TERM t 1 ) (RM t RF t ) + ε (5) or, after rearranging, R it RF t = α i + b i0 (RM t RF t ) + b i1 DEF t 1 (RM t RF t ) + b i2 DIV t 1 (RM t RF t ) +b i3 TB t 1 (RM t RF t ) + b i4 TERM t 1 (RM t RF t ) + ε (6) where DEF t-1 = default spread, yield spread between Moody s Baa and Aaa corporate bonds, DIVt- 1 = dividend yield, the sum of dividend yield (dividend divided by last year price) to all CRSP stocks over the previous 12 months, TB t-1 = risk-free rate, 30-day Treasury bill rate, TERM t-1 = term spread, yield spread between the ten-year and one-year Treasury bond, and ε is the error term. Equation (6) means the insurer stock returns are regressed not only on the excess market return, as in CAPM, but also on the products of the excess market return with the macroeconomic variables. Since TB is on average low in bad times, and DEF, DIV, and TERM are high, a negative loading on TB t 1 (RM t RF t ) product and a positive loading on all other products implies higher beta during recessions and hence higher expected return/coe than what the CAPM predicts. Intertemporal CAPM ICAPM is a multi-period model. From ICAPM s point of view, investors try to smooth their consumption over time by trying to push more wealth to the periods when consumption is 8 The definitions of DIV, DEF, TB, and TERM are in Section III. 9 Fama and French (1988) document that dividend yield predicts market returns. Keim and Stambaugh (1986) and Fama and Schwert (1977) find similar evidence for DEF and TB. Campbell (1987) and Fama and French (1989) find that the TERM spread is related to expected market risk premium. 10

13 scarcer, and hence marginal utility of consumption is higher. Therefore, investors will value the assets that pay them well when bad news arrives. These assets have the ability to transfer wealth from thriving periods to floundering periods: one invests in them and does not see one s investment vanish in recessions, when it is needed the most. Such assets will be deemed less risky than what their market beta implies and will command lower risk premium. Risk is, in addition to the market risk included in the CAPM, the decrease in security value when bad news arrives. 10 In this paper, we follow a successful application of the ICAPM (Ang et al., 2006; Barinov, 2014) that uses market volatility as a state variable. Investors care about changes in volatility for two reasons. In Campbell (1993), an increase in volatility implies that in the next period risks will be higher, consumption will be lower, and savings in the current period have to be higher at the expense of lower current consumption to compensate for future consumption shortfall. Chen (2002) also claims that, due to the persistency of the volatility, higher current volatility indicates higher future volatility. Accordingly, consumers will boost precautionary savings and lessen current consumption when they observe a surprise increase in expected volatility. Both Campbell (1993) and Chen (2002) demonstrate that stocks whose returns are most negatively correlated with surprise changes in expected market volatility are riskier because their value declines when consumption has to be reduced to increase savings. To proxy for shocks to market volatility, we employ changes in the VIX index from the Chicago Board Options Exchange (CBOE). 11 The VIX index measures the implied volatility of at- 10 A range of literature including Merton (1973), Breeden (1979), Campbell (1993), Brennan et al. (2004), and Barinov (2013, 2014) explores this model. 11 VIX is the ticker symbol for the Chicago Board Options Exchange (CBOE) Market Volatility Index. There are two versions of the VIX index. The original VIX index is based on trading of S&P 100 options. It has a price history dating back to 1986, and the method and formula of calculation have remained the same ever since. On September 22, 2003, CBOE introduced a new VIX index that is based on prices of S&P 500 options. Ever since then, the original VIX index has changed its name to VXO to avoid duplication of name with the new VIX index. Following Ang et al. (2006), we use the original VIX index, known as the ticker VXO since September The reason is that the new 11

14 the-money options on the S&P100 index, and thus derives volatility expectations from option prices, effectively using all the information the traders have. Market volatility is high during recessions, accompanied with positive changes in VIX index. Utilizing the VIX index, Ang et al. (2006) provide empirical support to the hypotheses of Campbell (1993) and Chen (2002). They document that the highest quintile portfolio sorted on return sensitivity to the innovations in the VIX index indeed earn about 1 percent per month less on average than the lowest quintile portfolio, confirming that negative loadings on VIX changes mean risk. If one adds the change in VIX to the right-hand side of the CAPM equation to explain the firm returns, the intercept is no longer the abnormal return, referred to as alpha, and the model is not essentially an asset pricing model. In fact, it does not have any economic interpretation, since the market return is measured in percent and the VIX change in VIX unit, which is inconsistent. Therefore, a factor-mimicking portfolio, i.e., a portfolio of stocks with the highest possible correlation with the VIX change, is needed. In addition, constructing the factor-mimicking portfolio from stock returns will allow us to keep the return-relevant portion of the VIX change and discard the noise and irrelevant information (Barinov, 2013). Following Breeden et al. (1989), Ang et al. (2006) and Barinov (2014), we form a portfolio that mimics the volatility risk factor, known as the FVIX factor/portfolio. 12 It is a zero-investment portfolio that tracks daily changes in expected volatility. By construction, the FVIX portfolio earns positive returns when expected volatility increases, and consequently, has a negative risk premium because it is a hedge against volatility risk. In other words, positive FVIX betas indicate that the portfolio is a hedge against index is constructed by backfilling only to 1990, but the VXO goes back in real time to Ang et al. (2006) document that the correlation between the new and the original indexes is 98% from 1990 to The detailed description of the factor-mimicking procedure that creates FVIX is in the next section. 12

15 volatility risk, while negative FVIX betas mean that the portfolio is exposed to it. The ICAPM specification is as follows: Ri RF = βi (RM - RF) + βf FVIX + ε (7) where RM = return on broad market portfolio, RF = return on riskless security, FVIX = factormimicking portfolio that mimics the changes in VIX index, βf = asset i s FVIX beta, and ε is the error term. Prior COE Studies in the Insurance Literature Cummins and Phillips (2005) estimate cost of equity by line using the full-information industry beta (FIB) method in conjunction with the CAPM and FF3. In the FIB method, the estimated regression coefficients from the CAPM or FF3 are regressed on line of business weights. The estimated coefficients on the line of business weights are interpreted as line of business betas to obtain COE estimates by line. They find that the estimated COE is significantly different across sectors of the insurance industry and across lines within property-liability (P/L) insurance. They estimate that the COE of life insurers is approximately 200 basis points higher than P/L insurers. Lastly, the FF3 generates significantly higher COE estimate than the CAPM, with an average COE for the whole sample period of 17.2 and 10.6 percent, respectively, suggesting the importance of using a multifactor asset pricing model when estimating COE for P/L insurers. These authors attribute the large difference between FF3 and CAPM model estimates to risk premia for size and book-to-market equity risk factors. We follow the approach of Cummins and Phillips (2005) in estimating CAPM and FF3 betas and related risk premia; that is, we use a time-series regression to obtain beta estimates and we use a longer time period to obtain the market, SMB and HML risk premia. 13

16 Wen et al. (2008) compare CAPM COE estimates of the U.S. P/L insurance companies to COE estimates from what the authors denote as the Rubinstein-Leland (RL) model. The RL model is derived from the more general asset pricing model of Rubinstein (1976) and incorporated into a CAPM framework by Leland (1999). The RL model requires that assumptions be made about the degree of risk aversion. The authors find that while COE estimates are not significantly different for the full sample period, the estimates are significantly different in certain sub-periods. In addition, the COE estimates are significantly different for insurers with more skewed returns, less normal returns, and smaller insurers. They also find that alphas (unexplained excess returns) are significantly smaller for the RL model compared to those of the CAPM for insurers with returns that are highly skewed distributed and for smaller insurers. However, for the full sample period and full sample, the alphas are not significantly different. The authors obtain CAPM betas using a time series regression, as we do. In general, the method we use to obtain COE estimates is most similar to the two preceding papers. There are some other studies of insurer cost of equity in the insurance literature that follow approaches that differ from Cummins and Phillips (2005), Wen et al. (2008) and the present study. Cummins and Lee (1998) estimate CAPM and APT (multi-factor) betas in a time-series regression, then use the estimated betas in a second stage cross sectional regression to estimate the risk premia, and then compare the estimated risk premia to the average realized risk premia over time. Cummins and Lamm-Tennant (1994) use Value Line betas to estimate insurer COE. They identify insurer characteristics that help explain the cross-sectional variation in Value Line betas, such as financial leverage. Bajtelsmit et al. (2015) estimate upside and downside betas, coskewness and cokurtosis in time series regressions. Then they use these and other factors in cross-sectional regressions to explain realized insurer returns on equity. They find that downside risk is statistically and 14

17 economically significant. Nissim (2013) and Berry-Stölzle and Xu (2016) use implied cost of capital method to estimate COE for insurance companies, but that method follows a different set of assumptions and is based on the dividend discount model (rather than the capital asset pricing model), which derives from a different set of theories. Among the insurance studies discussed thus far, only Cummins and Phillips (2005), actually estimate insurer CAPM and FF3 COE following an equilibrium pricing approach; that is, as discussed in Section I, an approach that first uses a risk factor that is priced for the market as a whole in contrast to a risk factor or firm characteristic that helps explain realized returns but may either be diversifiable or proxy for a risk captured by FF3 factors or other market wide, and hence, nondiversifiable risk factors. 13 Ammar et al. (2015), similar in spirit and method to Bajtelsmit et al. (2015), use the twostage Fama-Macbeth (1973) method to identify risk factors and insurer characteristics that help explain the cross-sectional variation in insurer stock returns. However, they study which factors are priced using a cross-section of returns of one industry (insurance industry) only (rather than the market). A shortcoming of such an approach is as follows. Cross-sectionally all insurance companies, or P/L insurers (the sample in Ammar et al., 2015), may have very similar exposure to risk X (which is priced for the market, systematic/nondiversifiable risk) so that risk X will probably not explain the cross-sectional variation in insurers realized returns (not enough variation crosssecitonally). But, risk X, may very well matter in the cross-section of all firms, and in the timeseries we may see that insurance firms are significantly exposed to risk X. In short, our study does not take the approach of looking at the insurance industry only to conclude whether a risk is priced for the market; rather, we draw upon finance theory and the asset pricing literature generally by examining the market. 13 In asset pricing, a risk factor is priced if it has non-zero alpha in the baseline model (such as CAPM or FF3). If a factor is priced, it means it is priced for the whole market. There is no such thing as a factor is priced for industry X. 15

18 However, we do examine some industry-specific factors and find that they are not priced. Ammar et al. (2015) do consider a wide range of market-wide risk factors and insurer-specific characteristic, but they do not estimate COE in a manner consistent with Cummins and Phillips (2005), Fama and French (1993), or papers that begin by asking, does a risk factor reflect economy-wide risk, that is, nondiversifiable risk, or only risk related to one industry (diversifiable risk)? In addition to a fundamentally different approach to identifying priced risk factors, we estimate COE using the CCAPM and using ICAPM factors, such as FVIX and liquidity that the Ammar et al. (2015) study does not. A general problem with studies that attempt to identify industry-specific risks for asset pricing or COE purposes is that they are based on a false premise, namely that the risk premium for a particular risk, whether it be the market risk premium, SMB, HML or some other market-wide risk, is different for a particular industry. In equilibrium asset pricing models, such as the CAPM, CCAPM and ICAPM, it is only the factor beta of an individual asset or industry portfolio that may differ from other individual assets or other industry portfolios. III. Data and Variables We collect monthly data on insurer stock returns and risk factors over differing time spans depending on data availability. For any data series, we collect 29 years of data from January 1986 to December All types of insurers are included and we further separate them into seven major subsectors. 15 We perform major analysis on all insurers and two largest subsets of P/L and life insurers. The insurance companies value-weighted returns are calculated from stock information obtained from monthly CRSP data (market cap for value-weighting is lagged by one 14 We do not go back further in time because the VIX index is only available starting in January To compare models over the same time period, we examine the 29-year period from January 1986 to December All insurers are firms with SIC codes between 6300 and The seven subsectors are life insurance ( ), accident and health insurance ( ), property-liability insurance ( ), surety insurance ( ), title insurance ( ), pension, health, welfare funds ( ), and other insurance carriers (insurers falling into none of the above categories). 16

19 month). Fama-French three factors, the market return, and the risk-free rate are retrieved from Ken French s data library. 16 To estimate the CCAPM, we collect four commonly used conditioning variables, namely the dividend yield (DIV), the default spread (DEF), the Treasury bill rate (TB), and the term spread (TERM). DIV is the sum, over the previous 12 months, of dividend yield (dividend divided by last year price) to all CRSP stocks. DIV is obtained from CRSP as the difference between cum-dividend and ex-dividend market return. TB is the one-month Treasury bill rate from Ken French s data library. DEF is the yield spread between Moody s Baa and Aaa corporate bonds, and TERM is the yield spread between the ten-year and one-year Treasury bond. Data for calculating DEF and TERM are obtained from FRED database at the Federal Reserve Bank at St. Louis. 17 To measure the exposure to volatility risk in the ICAPM, we follow the literature (Breeden et al., 1989; Ang et al., 2006; Barinov, 2014) and create a factor-mimicking portfolio, FVIX, using a set of base assets such that it is the portfolio of assets whose returns are maximally correlated with realized innovations in market volatility. We use the VIX index from the Chicago Board of Options Exchange (CBOE) as a proxy of expected market volatility, and its change as a proxy for innovations. VIX measures implied volatility of the at-the-money options on the S&P100 stock index (for a detailed description of VIX, see Whaley, 2000 and Ang et al., 2006). FVIX index is constructed by regressing changes in the state variable, the VIX index, on the daily excess returns to the five portfolios (base assets) sorted on past sensitivity to VIX changes: ΔVIX t = γ 0 + γ 1 (VIX1 t RF t ) + γ 2 (VIX2 t RF t ) + γ 3 (VIX3 t RF t ) + γ 4 (VIX4 t RF t ) + γ 5 (VIX5 t RF t ) + ε (8) 16 See 17 See 17

20 where VIX1t,, and VIX5t are the VIX sensitivity quintiles described below, with VIX1t being the quintile with the most negative sensitivity. The VIX sensitivity quintiles in month t are formed using information from month t-1 and are rebalanced monthly. The fitted part of the regression above less the constant is our volatility risk factor (FVIX factor). The daily returns to FVIX are then cumulated within each month to get the monthly return to FVIX used in the paper. The return sensitivity to VIX changes (γδvix) we use to form the base assets is measured separately for each firm-month by regressing daily stock excess returns in the past month on daily market excess returns and the VIX index change using daily data (at least 15 non-missing returns are required): R i,t 1 RF t 1 = α + β M (RM t 1 RF t 1 ) + γ ΔVIX ΔVIX t 1 + ε (9) We also hand-collect a bunch of underwriting cycle/insurance-specific variables as candidate CCAPM conditioning variables and candidate ICAPM additional factors. 18 IV. Insurer Risk Sensitivities, Augmented Fama-French Models, and Liquidity and Coskewness Factors Model Applicability and Insurer Risk Sensitivities In this part, we empirically show that the insurance industry as a whole is indeed sensitive to time-varying economic, market, and volatility risk. We also investigate two major insurance industry sectors separately, namely publicly traded P/L insurers and life insurers, to determine whether their risk sensitivities differ. The data we used are from January 1986 to December 2014; hence we obtain up to 348 months of returns on each sample firms. Panel A of Table 1 reports the summary statistics of the 18 The variables include CatLoss (catastrophic losses) and CombRat (combined ratio) from 1986 to 2014 and Surplus, PremW (premiums written), PremE (premiums earned), NetInvInc (net investment income), and CapGain (net realized capital gains) from 1987 to CatLoss, CombRat, Surplus, PremW, PremE, NetInvInc, and CapGain are collected from the Insurance Services Office Inc. (ISO) quarterly publication Property-Casualty Insurance Industry Financial Results. CatLoss is for property catastrophes only and the ISO obtains it from the Property Claim Services Company. 18

21 returns to the insurance industry, market risk premium, Fama-French factors (i.e., SMB and HML), macroeconomic variables, and FVIX. The mean monthly excess returns for all insurers, P/L insurers, and life insurers are close at 0.62, 0.56, and 0.76, respectively, suggesting that P/L (life) insurers have somewhat lower (visibly higher) risk than an average insurance company. The mean monthly market risk premium is 0.66 during the sample period, very close to the mean for all insurers. The summary statistics of firm numbers of different insurance sub-sectors are shown in Panel B of Table 1. For our sample period, there are 61,042 firm-month observations in total, out of which 25,798 belong to P/L insurers and 18,486 belong to life insurers. It makes these two subsectors the largest among all seven categories. The average number of firms in any month is 167, 71, and 50 for all insurers, P/L insurers, and life insurers, respectively. Table 2 reports the regression results of four different asset pricing models for all publicly traded insurers, P/L insurers, and life insurers in Panels A, B, and C, respectively. 19 We observe that while the insurance industry as a whole seems less risky than the market (its market beta is 0.87, more than two standard errors below 1), life insurers are significantly more risky than the market (with β = 1.20) and P/L insurers (β = 0.73) are less risky than an average insurer. The betas also align well with the average excess returns in Panel A of Table 1. The Fama-French model (FF3) additionally reveals that all insurance companies are value firms (see their positive and significant HML betas), and, if one views HML as a risk factor, are 19 Our all insurers include firms with different risks and operating characteristics, which arguably are very different from the P/L and life insurers usually considered to represent the insurance industry. For robustness tests, we run analysis on other insurance subsectors in addition to P/L and life insurers. As we can see from Panel B of Table 1, the monthly average numbers of surety insurers, title insurers, pension, health, welfare funds, and other insurance carriers are so small that we put these insurers together as a combined category (other insurers). Therefore, for regression purposes, we divide the insurance industry into four major categories, namely, property-liability (P/L) insurers, life insurers, accident and health (A/H) insurers, and other insurers. We run analysis based on Tables 2 and 3 for A/H and other insurers, and we find that the results are consistent with those of all, P/L, and life insurers. A/H and other insurers have countercyclical beta and they are exposed to volatility risk. 19

22 more risky than their market betas suggest. Likewise, SMB betas suggest that insurance companies, with the exception of life insurers, are big firms, and thus somewhat less risky. As discussed earlier, CAPM and FF3 are single-period models. However, in reality investment and consumption decisions are made over multiple periods, and the insurance industry is exposed to business cycles. The ICAPM column adds FVIX, the volatility risk factor mimicking the changes in VIX, the expected market volatility. The negative FVIX beta of insurance companies suggests that when expected market volatility (VIX) increases unexpectedly, insurance firms tend to have worse returns than firms with comparable CAPM betas, which makes insurance companies riskier than what the CAPM estimates. 20 This is true for all insurance companies, including property-liability and life insurers, though we observe some cross-section here: life insurers have the lowest exposure to volatility risk, much lower than the average for all insurers, and P/L insurers have the highest volatility risk exposure (the most negative FVIX beta). The pattern in FVIX beta is opposite to the pattern in the CAPM betas (βlife insurers > βall insurers > βp/l insurers), which suggest that life insurers are riskier and P/L insurers are safer than an average insurer. According to CCAPM, the countercyclical beta, namely higher beta of insurers in recessions, is another source of risk in addition to what is captured by the CAPM. The dependence of the beta on the business cycle variables is captured in Table 2 by the slopes on the products of the market return and the business cycle variable. The CCAPM regression results in Panel A 20 Thanks to an anonymous referee, we also study how the other industries do in terms of volatility risk exposure compared to insurers. In untabulated results, we investigate the 48 industry portfolios from Fama and French (1997). The portfolios span the whole economy and include insurance and other financial services industries. We find that while negative FVIX betas dominate our sample (higher volatility is generally bad for every industry), roughly a third of FVIX betas are positive, and the average FVIX beta across all 48 industries is only (compared to for the insurance industry). We also document that the FVIX beta of the insurance industry is the 5 th most negative (behind Food Products, Candy & Soda, Beer & Liquor, and Tobacco Products). Therefore, the insurance industry does differ from an average industry, and this is an interesting result from the paper. For the definitions of the 48 industry portfolios, refer to 20

23 indicate that the beta of insurance companies significantly increases with the dividend yield (DIV) and significantly decreases with the Treasury bill rate (TB); and is not significantly related to either the default spread (DEF) or term spread (TERM). Since dividend yield is higher in recessions, and Treasury bill rate is lower, both significant coefficients indicate that the beta of insurance companies is countercyclical, which makes them riskier than what the CAPM would suggest. The same is true about Panel B, in which dividend yield stays a significant driver of the risk of property-liability companies, and Treasury bill rate loses significance, but keeps its sign. Panel C is somewhat more complicated, because it suggests that the beta of life insurers is related to all four business cycle variables, and the sign on TERM contradicts the other three, suggesting that the beta of life insurers can be procyclical. The term spread, which measures the slope of the yield curve, is high in recessions, and the negative slope on TERM*(RM - RF) suggests the beta of life insurers is lower in recessions (The positive sign on DEF*(RM - RF) suggests the opposite, because the default spread is higher in recessions). How do we conclude whether the beta is countercyclical or not, if some slopes disagree? (One can also notice that in Panels A and B the TERM*(RM - RF) slope also contradicted the others, but was statistically insignificant). An easy test is looking at the alphas. Effectively, all assetpricing models, including the CAPM and CCAPM, partition, in-sample, the average left-hand-side return (in our case, average return to insurance companies in ) into expected return (cost of equity, the risk-based part), which is the factor loadings times factor risk premiums, and the alpha (i.e., the average abnormal return, the unexplained part). If we are looking at the same sample, a decrease in the alpha as one goes from one model to another mechanically implies an increase in the expected return (cost of equity, risk) part, as the alpha and the risk-based part have to sum up to the same average return in the same sample. 21

24 Comparing the alpha in the CAPM and CCAPM column, we observe that it decreases by economically non-negligible 8-12 basis points (bp) per month (1-1.5% per year) as we go from the CAPM to CCAPM. Hence, the CCAPM discovers more risk in insurance companies than CAPM, and for that to be true, the beta of the insurance companies has to be countercyclical. 21 A more formal test of whether the beta of insurance companies is countercyclical is presented in Table 3. In this table we follow Petkova and Zhang (2005) in reporting the average beta in expansions and recessions and testing if their difference is zero using the standard difference-in-means test. Following Petkova and Zhang, we first define expansions and recessions as the periods with low and high expected market risk premium, respectively. This definition is superior to defining expansions and recessions using statistical measures of business activity, because it goes to the heart of things: it looks at whether investors have high marginal utility of consumption and demand a high risk premium because of that. We measure expected market risk premium as the fitted part of the regression predicting the market return. Following the seminal papers of Fama and Schwert (1977) and Fama and French (1989), the predictive regression includes the same four macroeconomic variables we use as conditioning variables in the CCAPM: 22 RM t RF t = b i0 + b i1 DEF t 1 + b i2 DIV t 1 + b i3 TERM t 1 + b i4 TB t 1 + ε (10) In each month t, we substitute the values of the four business cycle variables from month t-1 in Equation (10), estimate the predicted/expected market risk premium, and label the month as 21 Another way to come to the same conclusion is to look at Equation (2) and observe that in order for the difference in expected return according to the CCAPM and CAPM to be positive, the covariance between the beta and expected market return (which is this difference) has to be positive, that is, the beta has to be high when expected market return is high, that is, in recessions. 22 In fact, the well-documented predictability of the market return by those four variables was the evidence that guided our choice of DIV, DEF, TB, and TERM as conditioning variables in the beta Equation (4), because, as Equation (2) suggests, the difference between the CCAPM and CAPM is driven by the common variation in the beta and the expected market risk premium. 22