Estimating the Cost of Equity Capital for Insurance Firms with Multi-period Asset Pricing Models Alexander Barinov a Steven W. Pottier b Jianren Xu *, c This version: October 7, 2014 * Corresponding author. a Department of Finance, A. Gary Anderson School of Business Administration, University of California Riverside, 900 University Ave. Riverside, CA 92521, Tel.: +1-951-827‑3684, [email protected]. b Department of Insurance, Legal Studies, and Real Estate, Terry College of Business, University of Georgia, 206 Brooks Hall, Athens, GA 30602, Tel.: +1-706-542-3786, Fax: +1-706-542-4295, [email protected]. c Department of Finance, Mihaylo College of Business and Economics, California State University, Fullerton, 800 N. State College Blvd., Fullerton, CA 92831, Tel.: +1-657-278-3855, Fax: +1-657-278-2161, [email protected].
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Estimating the Cost of Equity Capital for Insurance Firms
with Multi-period Asset Pricing Models
Alexander Barinova
Steven W. Pottier b
Jianren Xu*, c
This version: October 7, 2014
* Corresponding author. a Department of Finance, A. Gary Anderson School of Business Administration, University of California Riverside,
900 University Ave. Riverside, CA 92521, Tel.: +1-951-827‑3684, [email protected]. b Department of Insurance, Legal Studies, and Real Estate, Terry College of Business, University of Georgia, 206
Brooks Hall, Athens, GA 30602, Tel.: +1-706-542-3786, Fax: +1-706-542-4295, [email protected]. c Department of Finance, Mihaylo College of Business and Economics, California State University, Fullerton, 800
N. State College Blvd., Fullerton, CA 92831, Tel.: +1-657-278-3855, Fax: +1-657-278-2161, [email protected].
DEF t-1 = default spread, yield spread between Moody’s Baa and Aaa corporate
bonds,
DIVt-1 = dividend yield, the sum of dividend payments to all CRSP stocks over the
previous 12 months, divided by the current value of the CRSP value-weighted
index,
TB t-1 = risk-free rate, one-month Treasury bill rate, and
TERM t-1 = term spread, yield spread between the ten-year and one-year Treasury
bond.
Equation (8) 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. Through the CCAPM, beta and market risk premium may vary over
time, which takes into account the time-varying risks. Time-varying beta is supported by the
time-series evidence on security returns (see Table 1 in Wen et al., 2008). Consequently CCAPM
is more realistic than single-period models, such as the CAPM and FF3 model. CCAPM is
unique among asset pricing models in that its beta estimate (Equation (6)) measures the
instability of an asset’s beta over business cycles (Petkova and Zhang, 2005).
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
scarce, and hence marginal utility of consumption 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 he does not see his
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investment vanish in recessions, when he needs it the most. Such assets will be deemed as less
risky than what their market beta implies and will command lower risk premium. According to
the ICAPM, risk is the positive covariance of the asset’s returns with the news about economic
variables that are likely to be high when consumption is high and low when consumption is low.
Risk is, in addition to the market risk include in the CAPM, the decrease in security value when
bad news arrives.
The ICAPM adds longer investment horizons and time-varying investment opportunities
to the CAPM. In the ICAPM, the market portfolio serves as one factor and state variables serve
as additional factors.3 The state variables are the ones that determine how well the investor can
do in the maximization of his lifetime consumption (Cochrane, 2005).4 The additional factors
arise from investors’ demand to hedge uncertainty about future investment opportunities. They
forecast changes in the distribution of future returns or income (i.e., changes in the investment
opportunity set). Consumption and marginal utility respond to news: if a change in some variable
today signals high income in the future, then consumption rises now, and vice versa. This fact
opens the door to forecasting variables: any variable that forecasts asset returns or that forecasts
macroeconomic variables is a candidate state variable. Consumption would serve as a good state
variable; however, it is difficult to measure. Therefore, in empirical studies, researchers look for
its proxies.
In this paper, we follow a successful application of the ICAPM (see, e.g., Campbell, 1993,
1996; Lamont, 2001; Ang et al., 2006; Barinov, 2014) that uses the aggregate volatility as a state
variable which proxies for aggregate consumption. Information about future investment
opportunities and future consumption is available in changes in the expected market (i.e,
3 A range of literature including Merton (1973), Breeden (1979), Campbell (1993), Brennan et al. (2004), and
Barinov (2014) explores this model. 4 For example, current wealth can be a state variable.
Page | 15
aggregate) volatility. In Campbell (1993), an increase in aggregate volatility implies that in the
next period risks will be higher, consumption will be lower, and savings will be higher. Chen
(2002) also claims that, due to the persistency of the aggregate volatility, higher current
aggregate volatility indicates higher future aggregate volatility. Accordingly, consumers will
boost precautionary savings and lessen current consumption when they observe a surprise
increase in expected aggregate volatility. Both Campbell (1993) and Chen (2002) demonstrate
that stocks whose returns are most negatively correlated with surprise changes in expected
aggregate volatility earn a higher risk premium. These stocks are riskier because their value
declines when consumption has to be reduced to increase savings. In contrast, stocks with returns
that positively covary with the aggregate volatility risk have lower expected returns because
these stocks provide a hedge against the aggregate volatility risk, and risk-averse investors want
to hedge against changes in aggregate volatility.
To proxy changes or innovations in aggregate volatility, we employ changes in the VIX
index from the Chicago Board Options Exchange (CBOE).5 The VIX index measures the implied
volatility of an at-the-money option on the S&P100 index, and it represents traded options whose
prices directly reflect volatility risk. The aggregate volatility is high during recessions,
accompanied with positive changes in VIX index. The change in VIX is therefore a good proxy
for the innovations, or news, in expected aggregate volatility (see, e.g. Ang et al., 2006; Barinov,
2013; Barinov, 2014). Utilizing the VIX index, Ang et al. (2006) provide empirical support to
5 VIX is the ticker symbol for the Chicago Board Options Exchange (CBOE) Market Volatility Index. There are
two versions 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 has been 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 2003. The
reason is that the new index is constructed by backfilling only to 1990, but the VXO goes back in real time to 1986.
Ang et al. (2006) document that the correlation between the new and the “original” indexes is 98% from 1990 to
2000.
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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. In other words, firms with
less negative return sensitivity to the VIX index changes indeed earn lower expected returns than
those with more negative sensitivity to VIX changes.
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 unexplained stock return, referred to as alpha, and the
model is not essentially a capital 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
aggregate volatility risk factor, known as the FVIX factor/portfolio. It is a zero-investment
portfolio that tracks daily changes in expected aggregate volatility. By construction, the FVIX
portfolio earns positive returns when expected aggregate volatility increases, and consequently,
has a negative risk premium because it is a hedge against aggregate volatility risk. In other words,
positive FVIX betas indicate that the portfolio is a hedge against aggregate volatility risk, while
negative FVIX betas mean that the portfolio is exposed to it The ICAPM specification is as
follows:
E(Rit) – Rft = βMt [E(RMt) - Rft]+ βFt FVIXt (9)
where
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RMt = return on broad market portfolio,
Rft = return on riskless security,
FVIXt = factor-mimicking portfolio that mimics the changes in VIX index, and
βFt = asset i’s beta coefficient for FVIX factor-mimicking portfolio.
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, at a minimum we collect 25 years of
data from January 1986 to December 2010.6 All types of insurers are included by using firms
with SIC codes between 6300 and 6399. We further separate insurers into the two largest subsets,
P/L insurers (SIC code 6331) and life insurers (SIC code 6311). The insurance companies’
equal-weighted returns are calculated from stock information obtained from monthly CRSP data.
Fama-French three factors, the market return, and the risk-free rate are retrieved from Ken
French’s data library and are available from July 1926.7
To estimate the CCAPM, we collect four commonly used conditioning variables, namely
the dividend yield, the default premium, the risk-free rate, and the term premium. Dividend yield
is calculated from the data in CRSP and is available from July 1925. Data for calculating the
default spread and the term spread are obtained from FRED database at the Federal Reserve
Bank at St. Louis.8 The risk-free one-month Treasury bill rate is retrieved from Ken French’s
data library. Because 1954 is the first full year that data for all four macroeconomic variables are
available, the first month we use macroeconomic variables is January 1954.
With regard to ICAPM, we construct the factor representing aggregate volatility risk in
addition to the market factor. To measure the exposure to the aggregate volatility risk, we follow
6 One of the variables used in ICAPM, the VIX index, is only available from January 1986. 7 See http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/. 8 The Moody’s Aaa and Baa corporate bond yields are available back to January 1919, and the one-year and ten-year
Treasury bond rate are back to April 1953. See http://research.stlouisfed.org/fred2/.
the literature (see, e.g., 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. The base assets
are a small set of diversified portfolios that are sufficiently different from one another to reflect
information about different pieces of the economy. Their returns are not noisy because of their
diversified nature. The step-by-step formation of FVIX is as follows. First of all, we use the VIX
index as the proxy of aggregate/market volatility, which measures the implied volatility of the at-
the-money options on the S&P100 stock index. To measure the innovations to expected
aggregate volatility, we use daily changes in the VIX index available from the Chicago Board of
Options Exchange (CBOE). For a detailed description of VIX, see Whaley (2000) and Ang et al.
(2006). Secondly, following the literature we use the six Fama-French (1993) size and book-to-
market portfolios as our base assets, which are sorted in two groups on size and three groups on
book-to-market. The daily returns to these portfolios come from Ken French’s data library. Then
we form the factor-mimicking portfolio that tracks the daily changes in the VIX index. We
regress the VIX daily changes on the daily excess returns to our base assets. The fitted part of the
regression is the combination of the base assets with the most positive correlation with the VIX
change. Our aggregate volatility risk factor, FVIX, is the fitted part less the constant. Finally,
since the change in VIX is collected at the daily frequency to accurately proxy for the
innovations about VIX, the FVIX computed is on the daily basis as well. However, given that
daily returns are very noisy and that other variables in the models are monthly, we cumulate the
daily FVIX by adding them up to the monthly level to attain the FVIX factor.
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IV. Model Performance Comparisons
In this part, we empirically show that the insurance industry as a whole is indeed
sensitive to time-varying economic, market, and aggregate volatility risk. Then we investigate
two major insurance industry sectors separately, namely publicly traded P/L insurers and life
insurers, to determine whether their risk sensitivities differ. Moreover, we compare the
performance of four different models (CAPM, FF3, CCAPM, and ICAPM) in term of alpha
(model intercept), explanatory power (model R-squared), and statistical significance (model t-
statistics). We demonstrate that multi-period models are more appropriate for insurance firms.
The data we used are from January 1986 to December 2010; hence we obtain up to 300
months of returns on each sample firms.9 The insurer monthly returns are winsorized at 1st and
99th percentiles. This procedure results in a range of monthly returns from -33.8 to 40 for all
individual insurers. For each month, three portfolio returns are generated based on equally
weighed individual insurer returns for all insurers, P/L insurers and life insurers, respectively.
The average number of firms in any month during this 25-year period is 166, 67, and 52 for all
insurers, P/L insurers, and life insurers, respectively. Table 1 reports the summary statistics of
the returns to the insurance industry, market risk premium, Fama-French factors (i.e., SMB and
HML), macroeconomic variables, and FVIX. The summary statistics of the numbers of different
insurers are shown as well. From Table 1, we can see that the ranges of insurer excess returns,
market risk premium, and Fama-French factors are generally similar. The mean excess returns
for all insurers, P/L insurers, and life insurers are 0.58, 0.55, and 0.62 , respectively. The mean
monthly market risk premium is 0.57 during the sample period, almost identical to the mean for
all insurers.
9 We do not go back further in time because the VIX index is only available starting in January 1986. To compare
models over the same time period, we examine the 25-year period from January 1986 to December 2010 in the first
part of our empirical analysis (the model performance comparisons).
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Tables 2, 3 and 4 report the regression results of four different asset pricing models for all
publicly traded insurers, P/L insurers, and life insurers, respectively. For all three insurer samples,
the coefficient on the market risk premium is positive, significant, and below one for the CAPM,
FF3, and ICAPM model, implying that the market or systematic risk of insurers, on average, is
less than the market as a whole based on these three models. The beta of CCAPM cannot be
directly observed in the regression results because it is a function of the macroeconomic
(conditioning) variables, as shown in Equation (6) above. According to Tables 2 through 4, in
FF3 the coefficients of all three factors are significantly positive at the 1 percent level.
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 and economic cycles. Multi-period models, such as the CCAPM
and ICAPM, account for multiple periods and the time-varying risks. According to CCAPM, the
countercyclical beta, namely higher beta of insurers in recessions, is another source of risk in
addition to what is predicted by CAPM. Does the insurance industry have countercyclical beta?
First of all, the slopes on the products of the macroeconomic variables with the market excess
return demonstrate how the beta changes with different macroeconomic variables. Based on
Table 2 for all insurers, the CCAPM regression results indicate that the stock returns of insurance
companies significantly increase with the dividend yield and significantly decrease with the
Treasury bill rate; and are not significantly related to either the default premium or term
premium. These results show that insurers are exposed to dividend yield risk and interest rate
risk. Next, we examine whether insurers have countercyclical beta. Table 5 shows how to
determine the cyclicality of beta for companies based on modern asset pricing theories (see, e.g.,
Constantinides and Duffie, 1996; Campbell and Cochrane, 1999).
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According to the stock market predictability literature (see, e.g., Fama and French, 1988,
1989), default spread, dividend yield, and term spread are high in economic recessions and low
in expansions. In contrast, Treasury bill rate is low in recessions and high in expansions. In
Section II, Equation (8) is obtained by rearranging Equation (7), which uses the expression of
beta from Equation (6). This indicates that coefficient signs of the macroeconomic variables
should be consistent across Equation (6), (7), and (8).10 If stock returns to the insurance industry
in Equation (8) are, and hence its market beta in Equation (6) is, positively correlated with the
default spread, dividend yield, or term spread, or negatively correlated with the Treasury bill rate,
it indicates that insurers’ market beta is high in recessions and low in expansions. Thus, as a
result, the insurers have countercyclical beta, and bear extra risk—the risk exposure increases in
bad times—based on CCAPM compared to what the CAPM predicts. In essence, the CCAPM
beta is allowed to vary with economic cycles. On the contrary, if the stock returns to the
insurance industry are negatively correlated with the default spread, dividend yield, or term
spread, or positively correlated with the Treasury bill rate, the insurers have procyclical beta, and
bear less risk based on CCAPM compared to what the CAPM predicts.
From Table 2, we see that the risk of insurance industry increases in recessions because
the coefficients of the product of dividend yield and market risk premium and the product of
Treasury bill rate and market risk premium are significantly positive and negative, respectively.
Moreover, the coefficient on market risk premium is positive, and correlation coefficient of the
insurer stock excess returns and market excess returns is positive. Therefore, dividend yield and
Treasury bill rate load positively and negatively on insurer stock returns, and hence insurer
market beta, demonstrating that the insurance industry has countercyclical beta and its risk
10 For example, the coefficient of the term with DEFt-1 in Equation (8), and the coefficients of DEFt-1 in Equation (7)
and (6) are all bi1. The same situation applies to the other three macroeconomic variables.
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increases in recessions. That is to say, insurers are exposed to time-varying risks, which single-
period models, such as CAPM and FF3, do not incorporate. In this sense, CCAPM is superior to
single-period models when it comes to estimating the expected stock returns and cost of equity
for insurers.
From Table 2 we can also analyze the results of ICAPM. The negative FVIX beta of
insurance companies suggests that when volatility (VIX) increases unexpectedly, insurance firms
tend to have worse returns than firms with comparable CAPM betas, which makes insurance
companies riskier than what CAPM estimates. Even though FVIX is insignificant, the market
beta in ICAPM is smaller than the ones in CAPM, which means that FVIX shares the
explanatory power of market risk premium, and it does have impacts on insurer stock returns. In
sum, insurers are exposed to the time-varying risk, in particular, market aggregate volatility risk.
From a theoretical perspective, we claim that CCAPM and ICAPM are more appropriate
than single-period models given that they account for time-varying risks of insurers as
manifested in the significant factor loadings and the above analyses. From an empirical
perspective, are they also superior? As reported in Tables 2 to 4, alpha is not significant in any of
the four models. According to Fama and French (1993), if an asset pricing model is well-
specified, then its alpha should be indistinguishable from zero. Therefore, in regards to alpha,
CAPM, FF3, CCAPM, and ICAPM are equivalent in explaining insurer stock returns. Moreover,
for the regressions that include all insurers, Table 2, the adjusted R-squareds are 0.61, 0.78, 0.66,
and 0.61 for CAPM, FF3, CCAPM, and ICAPM, respectively. Even though FF3 has the highest
adjusted R-squared, all four models have relatively high explanatory power. 11 The pattern of the
11 Lewellen, Nagel, and Shanken (2010) point out that researchers should not rely too heavily on R-squared in asset
pricing tests, and suggest to evaluate in combination with other important tests, particularly with theoretical
guidance. Therefore, the highest adjusted R-squared of FF3 should not be regarded as the evidence that it is superior
to the others.
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adjusted R-squared is similar in Tables 3 and 4 for P/L and life insurers; that is, the FF3 model
yields the highest and CCAPM the second highest with the CAPM and ICAPM having very
similar ones.
The CCAPM results for P/L insurers in Table 3 report that the coefficient for the product
of dividend yield and market excess returns is significantly positive. As true for the sample of all
types of insurers, these results suggest the countercyclicality of P/L insurers and that their risk
increases in recessions. For P/L insurers, ICAPM regression estimates indicate that FVIX loads
significantly negatively at the 5 percent level. This implies that when the stock market volatility
index (VIX) increases unexpectedly and when investors have to cut current consumptions and
save for the future, the portfolio of P/L insurance firms erodes further on the limited
consumption and spares even less for precautionary savings. They tend to have worse returns
than firms with comparable CAPM beta, which makes them riskier than what CAPM predicts.
Therefore, P/L insurer stock returns are sensitive to aggregate volatility risk. These additional
risks reflected in the CCAPM and ICAPM are ones that the single-period models do not
explicitly include.
Table 4 reports the regression results for the four asset pricing models for life insurers.
Again, the loading for market risk premium is significantly positive for CAPM, and those for
Fama-French three factors are significantly positive for FF3. It is worth noting that the market
betas for CAPM, FF3, and ICAPM are higher for life insurers than for P/L insurers, or all
insurers, indicating the life insurers face more market risk than other types of insurers based on
these all models. With regard to CCAPM, we observe that the coefficients for the products of the
market excess returns and the default spread, dividend yield, Treasury bill rate, and term spread
are significantly positive, positive, negative, and negative, respectively. According to Table 5
Page | 24
and earlier discussion of CCAPM, the coefficient signs on the default spread, dividend yield, and
Treasury bill rate imply that the risk of life insurers increases in recessions, while the coefficient
sign on the term spread suggests that the risk of life insurers decreases in recessions. Since the
magnitude and significance of the coefficients on the first three macroeconomic variables are
much larger than that on the term spread, we conclude that the life insurers, like P/L insurers,
have countercyclical beta, and that their risk increases in recessions. When looking at the
regression results of the ICAPM, we find that FVIX loads insignificantly negatively on the life
insurer stock returns. Even though FVIX is insignificant, the market beta in ICAPM is smaller
than the ones in CAPM and FF3, which means that FVIX shares part of the explanatory power of
the market risk premium, and it indeed impacts life insurer stock returns indirectly. In sum, life
insurers are exposed to the time-varying risk, an additional risk source recognized in the
CCAPM and ICAPM that the single-period models do not explicitly include.
V. Cost of Equity Estimation
Estimation Data and Methodologies
In the study, we estimate the insurer cost of equity for 11 years from 2000 to 2010. Our
general methodology closely follows Cummins and Phillips (2005). Slope coefficients, or betas,
for each of the four asset pricing models (CAPM, FF3, CCAPM and ICAPM) are estimated in
Ordinary Least Squares regressions over the past 60 months for each sample insurer. We use
insurance company monthly stock returns from July 1995 to June 2010 from the CRSP database
in the beta estimation regressions; the 60 month periods end in June of 2000, 2001, 2002, 2003,
2004, 2005, 2006, 2007, 2008, 2009 and, 2010, resulting in 11 different estimation periods. Like
Cummins and Phillips (2005), we require a minimum of 36 consecutive months of stock returns
for each sample firm. Estimated beta coefficients are winsorized at an absolute value of 5 (see,
Page | 25
e.g., Cummins and Phillips, 2005). We perform estimates for all types of publicly traded insurers
combined, and separately for P/L insurers and life insurers.
Once slope estimates are obtained based on regressions over 60 month periods, consistent
with existing literature, the cost of equity is estimated by multiplying the slope estimates with the
long-run average of the factor risk premiums.12 For all models, the market risk premium (market
return less risk-free rate) is averaged from July 1926 to the final month of each estimation period
(such as June 2008 for 2008 COE estimation), and the risk-free rate is the 60-month average
ending the final month of each estimation period. For FF3, averages on SMB and HML are
calculated from July 1926 to the end of each estimation period. For the CCAPM, the four
macroeconomic return variables are averaged over the 60 months ending in June of each
estimation period for purposes of calculating the CCAPM market beta as shown in Equation
(6).13 For the ICAPM, the FVIX factor is calculated from January 1986 to the final month of
each estimation period.14
Cost of Equity Estimation Results
We compare the COE estimates for the eleven years of estimation results individually
and in the aggregate. The CAPM is used as a benchmark model because it is the earliest of the
“modern” asset pricing in the field of financial economics, it is a single-period and one-factor
model, it is the most widely investigated cost of equity model in the academic literature, it is
12 Longer time periods are used to estimate risk premiums than regression coefficients, consistent with prior finance
and insurance literature on the cost of equity. The reason for this approach is that factor risk-premiums are expected
to be more stable than individual firm factor risk sensitivities. 13 Note that Equation (8) is first estimated in the firm-level regressions using monthly values of macroeconomic
variables during the past 60 months. The estimated parameters are then inserted back into Equation (6) to calculate
the CCAPM market beta. Finally, the CCAPM cost of equity is obtained by multiplying the estimated market beta
with the long-run market risk premium as in Equation (4). 14 To control for potential biases resulted from infrequent trading and for robustness check, following Cummins and
Phillips (2005), the betas of all four asset pricing models (CAPM, FF3, CCAPM, and ICAPM) are also estimated
using the widely accepted sum-beta approach. The estimated sum-beta coefficients are obtained by adding the
contemporaneous and lagged beta estimates from Equation (1), (2), (7), (8), and (9). The results of COE estimates
after the sum-beta approach adjustment do not qualitatively change.
Page | 26
widely used in practice, and it is commonly used as the benchmark model to which other models
are compared. We do, however, make some general comparisons of all four models.
The mean COE estimates for our sample of all publicly-traded insurers are presented in
Table 6 for each of the 11 years (2000-2010) and 11 years combined. We also report t-test and
Wilcoxon signed-rank test statistics comparing the FF3 model, CCAPM, and ICAPM to the
CAPM. We present cost of equity estimates winsorized at the 5th and 95th percentiles. The FF3
model produces highest mean COE estimates for each of the eleven years and for all eleven years
combined. It gives a mean COE for all eleven years of 13.72 percent, almost 450 basis points
higher than the ICAPM which yields the second highest 11-year average of 9.24. The FF3 model
mean (median) COE estimates are significantly higher than the CAPM for each year and for all
years combined at the 0.01 level. The CCAPM generated a mean COE estimate of 8.82, or
around 23 basis points higher than the CAPM, a difference significant at the 0.01 level. The
median COE estimates from CCAPM are significantly higher than those from CAPM by 31 basis
points. Looking at individual years, the CCAPM COE estimates are significantly higher (lower)
than the CAPM COE estimates in four (two) of the eleven years. The ICAPM produces a mean
(median) COE of 9.24 (8.90) percent for the entire 11-year period, and significantly higher COE
estimates than the CAPM in six (nine) of the eleven years based on comparisons of means
(medians). According to untabulated results, the difference between the maximum and minimum
mean COE estimates by model across the years ranges from about 350 basis points for the
CCAPM to 690 for the FF3 model. The mean differences among the CAPM, CCAPM and
ICAPM for all eleven years are below 100 basis points, but exceed 200 basis points in some
years. Therefore, for the all insurer sample, while the differences are statistically significant
Page | 27
between any two of the models, the substantially higher estimates from the FF3 are striking from
an economic perspective as well.
As mentioned in the Introduction, FF3 lacks ample theoretical foundation and the COE
estimates are short of unambiguous risk-based explanations. Moreover, FF3 is a single-period
model that does not explicitly recognize time-varying risks. Hence, we advocate that COE
estimates from multi-period models are more appropriate. However, company stakeholders can
still use FF3 estimates as a reference to make an informed decision. In Section IV we find that
insurance industry as a whole, as well as major sub-industries of P/L and life insurers, has
countercyclical beta. This is an additional risk to market risk embedded in CAPM, which
indicates that the risk exposure of insurers increases in bad times when bearing risk is especially
painful for investors. As a compensation for the undesirable behavior of such stocks, investors
demand an extra risk premium. This explains why our COE estimates from CCAPM are
significantly higher than those from CAPM in general. It is consistent with the theoretical
prediction in Equation (3). In Table 2 we find that FVIX factor does not load significantly on
stock returns of all publicly-traded insurer sample. However, FVIX does share explanatory
power from market risk factor. Therefore, we still suggest using ICAPM estimates as a reference
when making COE-related decisions for insurers.
The cost of equity estimates for the subsamples of P/L and life insurers are presented in
Tables 7 and 8, respectively. As with the all insurer sample, the FF3 model produces estimates
significantly higher than the other three models for each of the eleven years. An interesting
contrast, however, is that COE estimates for P/L insurers tend to be lower on average than for
life insurers. The mean COE estimates of the whole 11-year period for P/L insurers versus life
insurers are as follows: 8.01 v. 9.54 percent for the CAPM; 12.56 v. 15.03 percent for the FF3
Page | 28
model; 8.27 v. 9.32 percent for the CCAPM; and 8.70 v. 10.13 percent for the ICAPM. These
COE estimates suggest that the required rate of return for life insurers is higher than that of P/L
insurers, which in turn, suggests that life insurer stocks are riskier. These estimated COE
differences between P/L and life insurers are all significantly different at the 0.01 level using
unpaired t-test assuming heteroskedasticity.
For P/L insurers, mean and median COE estimates from the CCAPM and ICAPM are
significantly higher than the CAPM for all years combined. The mean COE spreads of CCAPM
and ICAPM compared with CAPM are 26 and 69 basis points, while the median spreads are 18
and 71 basis points. Among these years, CCAPM produces significantly higher (lower) mean
COE estimates in 4 years (1 year) than CAPM. It also generates significantly higher median
COE estimates in 5 years and no significantly lower estimates in any year than CAPM. Similar
to the explanations to the all insurer sample, these results reflect the risk premium for
countercyclical beta of P/L insurers recognized in CCAPM. With regards to ICAPM, it produces
significantly higher (lower) mean COE estimates in 6 years (2 year), and significantly higher
(lower) median COE estimates in 7 years (2 year) than CAPM. As shown in Table 3, P/L insurer
stock returns are significantly associated with FVIX factor, which means that P/L insurers are
sensitive to aggregate volatility risk. The significant higher COE estimates by ICAPM compared
to CAPM are due to the risk premium of exposure to aggregate volatility risk of P/L insurers. It
is the compensation for the risk that when bad news (surprise increase in aggregate volatility)
arrives the value of P/L insurance portfolio drops. Since, based on empirical results in Table 3,
both CCAPM and ICCAPM work well for P/L insurers, we suggest P/L insurer stakeholders
using COE estimates from both models when making related decisions.
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For life insurers, mean and median COE estimates from the ICAPM are significantly
higher than the CAPM for all years combined. The mean and median spreads are 59 and 21 basis
points, respectively. Similar to all insurers and P/L insurers, ICAPM yields significantly greater
COE estimates than CAPM in most of the individual years. With regards to CCAPM, for some
of the single years it produces higher COE estimates than CAPM, while for some other years
lower. Further, there is not a significant difference between COE estimates based on the CCAPM
and CAPM for all years combined.
In general, our results with respect to multi-period models are consistent with the notion
that investors are rewarded extra risk premium for bearing time-varying risks that are not
explicitly recognized in single-period models. Note that in some individual years, multi-period
models do not yield strictly greater COE estimates than the CAPM. We summarize several
reasons as follows. First of all, models are not perfect, and any model is an approximation to
reality. The interesting question is how accurate a model is (see, Jagannathan and Wang, 1996).
Throughout this paper, we argue that multi-period models are closer to the reality, and more
exact and more appropriate in estimating COE. Second, model parameters, including slope
coefficients (i.e., betas) and factor risk premiums are estimates, and as such, subject to
measurement error. For example, neither the true beta nor the true expected market risk premium
is observable. Third, while our approach is consistent with the existing literature, historical betas
and risk premiums vary over different time periods, and such variations may reflect short-term
market behavior rather than long-run investor expectations. Further, the sample composition for
the 25 years of performance comparison regression is different from that for the 11 years of COE
estimates. It may result in some discrepancies. Lastly, the beta estimation in the performance
comparison regression is based on the past 25 years, while that in the COE estimates for each
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year is based on the past 5 years. Especially for CCAPM, whether the past 5 years capture
variations in economic conditions is a question. However, in order to make comparable and
consistent comparison with CAPM and FF3 whose COE estimation method is well established in
the literature (see, e.g., Fama and French, 1997; Cummins and Phillips, 2005), we use the past 5
years.
VI. Summary and Conclusions
The importance of cost of equity to insurance company owners, managers and regulators
is widely recognized by academics and practitioners. We extend prior academic literature by
exploring multi-period asset pricing models and comparing them to single-period models. We
find that insurer stock returns are sensitive to time-varying macroeconomic and aggregate
volatility risks. More precisely, insurers’ risk exposure is higher in recessions as characterized by
higher default spreads, higher dividend yields, and higher term spreads, and low Treasury bill
rates, when bearing risk is especially costly. In other words, insurance company market betas are
countercyclical. Moreover, P/L insurance portfolio values decrease when current consumption
has to be cut in response to surprise increase in expected aggregate volatility. Therefore, P/L
insurers are riskier than what the CAPM predicts because their risk exposure rises when bad
news arrives. The two multi-period models examined (CCAPM and ICAPM) account for
additional time-varying systematic risks not accounted for in single-period models, such the
CAPM and FF3.
Our empirical evidence supports the value of using multi-period models to explain
insurer stock returns and to estimate insurer COE. Specifically, using 25 years of data we find
that the macroeconomic factors mentioned above have significant impacts on the ex post stock
returns of insurers, and insurers have countercyclical beta. Further, insurers are sensitive to the
Page | 31
aggregate volatility risk. While our empirical results indicate that the CCAPM and FF3 model
are comparable based on statistical criteria (alpha, explanatory power, and statistical
significance), economic theory gives more credence to the CCAPM because it explicitly
recognizes time-varying risks, which may be proxied for by the FF3 factors. Further, the
CCAPM risk factors have a natural economic interpretation. In addition, our findings suggest
that life insurer stock returns are more sensitive to macroeconomic risk factors than P/L insurer
stock returns. Both the FF3 model and the CCAPM explain more of the variation in life insurer
stock returns than in P/L insurer stock returns.
In relation to cost of equity estimates, based on an 11 year window, the FF3 model
generates the highest values, averaging around 400 basis points higher than the next highest
model. 11 year-average COE estimates from the other three models (CAPM, CCAPM and
ICAPM) are within 100 basis points of one another. Consistent with the notion that additional
time-varying risks require greater rewards, the average COE estimates from the CCAPM and
ICAPM are significantly higher than the ones from the CAPM. Each of the four asset pricing
models yield higher average COE estimates for life insurers than P/L insurers, suggesting that
life insurers are riskier, and consequently that investors require higher returns from life insurers.
Our results indicate that the two prevalent models (CAPM and FF3) used in the extant literature
produce widely different COE estimates. Specifically, on average, COE estimates based on
CAPM are the lowest and those based on FF3 are highest, with those based on the CCAPM or
ICAPM in the middle. As a result, in addition to the stronger theoretical appeal of the multi-
period models, they may also enable decision makers to more confidently estimate COE in a
narrower range, and at a level closer to the traditional CAPM rather than the FF3 model.
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Our study adds to the existing literature in three ways. First, it is the first to examine
multi-period asset pricing models for insurance firms. Second, it provides empirical evidence in
support of the pricing of time-varying macroeconomic and aggregate volatility risks for insurers.
Third, it provides evidence of meaningful economic and statistical differences between single-
period and multi-period models. And, lastly, it demonstrates the practical importance of
generating COE estimates from a variety of models. These differences in COE estimates, at the
margin, determine whether investing and financing decisions insurers make increase or decrease
firm value.
References
Ali, A., L. Hwang, and, M. A. Trombley, 2003, Arbitrage Risk and the Book-to-Market Anomaly,
Journal of Financial Economics, 69: 355-373.
Ang, A., R. Hodrick, Y. Xing, and X. Zhang, 2006, The Cross-Section of Volatility and Expected Returns,
Journal of Finance, 61: 259-299.
Barinov, A., 2013, Analyst Disagreement and Aggregate Volatility Risk, Journal of Financial and
Quantitative Analysis, 48(6): 1877-1900.
Barinov, A., 2014, Turnover: Liquidity or Uncertainty? Management Science, forthcoming.
Berk, J. B., 1995, A Critique of Size-Related Anomalies, Review of Financial Studies, 8: 275-
286.
Breeden, D.T., 1979, An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment
Opportunities, Journal of Financial Economics, 7, 265 – 296.
Breeden, D.T., M.R. Gibbons, and R.H. Litzenberger, 1989, Empirical Tests of the Consumption-
Oriented CAPM, Journal of Finance, 44: 231-262.
Brennan, M. J., A. W. Wang, and Y. Xia, 2004, Estimation and Test of a Simple Model of Intertemporal
Capital Asset Pricing, Journal of Finance, 59: 1743-1775.
Campbell, J. Y., 1987, Stock Returns and the Term Structure, Journal of Financial Economics, 18: 373–
399.
Campbell, J. Y., 1993, Intertemporal Asset Pricing without Consumption Data, American Economic
Review, 83: 487-512.
Campbell, J. Y., 1996, Understanding Risk and Return, Journal of Political Economy, 104: 298-345.
Campbell, J. Y., and J. H. Cochrane, 1999, By Force of Habit: A Consumption-based Explanation of
Aggregate Stock Market Behavior, Journal of Political Economy, 107 (2): 205–251.
Campbell, J. Y., A. W. Lo, and A. C. MacKinlay, 1997, The Econometrics of Financial Markets,
Princeton University Press.
Campbell, J. Y., C. Polk, and T. Vuolteenaho, 2010, Growth or Glamour? Fundamentals and
Systematic Risk in Stock Returns, Review of Financial Studies, 23: 305-344.
Chen, J., 2002, Intertemporal CAPM and the Cross-Section of Stock Returns, Working Paper, University
of Southern California.
Chen, N., R. Roll, and S. A. Ross, 1986, Economic Forces and the Stock Market, Journal of Business, 59:
383 – 403.
Page | 33
Cochrane, J. H., 2005, Asset Pricing, Princeton University Press, Princeton, NJ.
Cochrane, J. H., 2007, Financial Markets and the Real Economy, International Library of Critical
Writings in Financial Economics, London: Edward Elgar.
Constantinides, G. M., and D. Duffie, 1996, Asset Pricing with Heterogeneous Consumers, Journal of
Political Economy, 104 (2): 219–240.
Cummins, J. D., and R. D. Phillips, 2005, Estimating the Cost of Equity Capital for Property-Liability
Insurers, Journal of Risk and Insurance, 72 (3): 441-478.
Davis, J. L., E. F. Fama, and K. R. French, 2000, Characteristics, Covariances, and Average Returns:
1929-1997, Journal of Finance, 55: 389-406.
Eckbo, B. E., R. W. Masulis, and O. Norli, 2000, Seasoned Public Offerings: Resolution of the New
Issues Puzzle, Journal of Financial Economics, 56: 251-291.
Fama, E. F., 1981, Stock Returns, Real Activity, Inflation, and Money, American Economic Review, 71:
545–565.
Fama, E. F., and K. R. French, 1988, Dividend Yields and Expected Stock Returns, Journal of Financial
Economics, 22: 3–25.
Fama, E. F., and K. R. French, 1989, Business Conditions and Expected Returns on Stocks and
Bonds, Journal of Financial Economics, 25: 23-49.
Fama, E. F., and K. R. French, 1992, The Cross-Section of Expected Stock Returns, Journal of
Finance, 47 (2): 427–465.
Fama, E. F., and K. R. French, 1993, Common Risk Factors in the Returns on Stocks and Bonds, Journal
of Financial Economics, 33 (1): 3–56.
Fama, E. F., and K. R. French, 1995, Size and Book-to-Market Factors in Earnings and Returns, Journal
of Finance, 50 (1): 131-155.
Fama, E. F., and K. R. French, 1997, Industry Costs of Equity, Journal of Financial Economics, 43: 153-
193.
Fama, E. F., and G. W. Schwert, 1977, Asset Returns and Inflation, Journal of Financial Economics, 5:
115-146.
Jagannathan, R., and Z. Wang, 1996, The Conditional CAPM and the Cross-Section of Expected Returns,
Journal of Finance: 51: 3-54.
Keim, D. B., and Stambaugh, R. F., 1986, Predicting Returns in the Stock and Bond Markets, Journal of
All Years 1667 8.5905 13.7190 8.8231 9.2423 -5.1284*** -5.1000*** -0.2325*** -0.3079*** -0.6518*** -0.7862***
Note: This table shows average cost of equity estimates for all publicly traded insurance companies based on CAPM, FF3 model, CCAPM, and ICAPM with
FVIX. Cost of equity is calculated for each firm according to Equations (1), (2), (7), (8), and (9). The 30-day Treasury bill rate observed at the beginning of each
month is used as risk-free rate, and the market return is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks. N is the number of firms. In FF3,
SMB is the difference in the return on a portfolio of small stocks and the return on a portfolio of large stocks, and HML is the difference in the return on a
portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks. We use four macroeconomic variables as conditioning
variables in CCAPM, which include dividend yield (DIV), defined as the sum of dividend payments to all CRSP stocks over the previous 12 months divided by
the current value of the CRSP value-weighted index, default spread (DEF), defined as the yield spread between Moody’s Baa and Aaa corporate bonds, risk-free
rate (TB), which is 30-day T-bill rate, and term spread (TERM), defined as the yield spread between the ten-year and one-year T-bond. In the ICAPM, FVIX is
the factor-mimicking portfolio of stocks that mimics the changes in the VIX index, which measures the implied volatility of the S&P100 index options. Values of
VIX are obtained from the CBOE. The market return, SMB, HML, and the risk-free rate are retrieved from Ken French’s data library. The data on the dividend
yield are retrieved from CRSP, while those on the default spread and the term spread are obtained from FRED database at the Federal Reserve Bank at St. Louis.
The data period for each year ends on June 30. Estimates are calculated using the previous 60-months of returns. If some returns are missing for individual firms,
at least 36 consecutive months are required. The risk-free rate used to estimate the cost of equity is the average 30-day T-bill rate from the first to the last month
of each 60-month estimation period. The market risk premium, SMB, and HML are calculated from July 1926 to the final month of each estimation period. The
four macroeconomic variables are averaged over the 60 months ending in the last month of each estimation period. FVIX is calculated from January 1986 to the
final month of each estimation period.
Page | 39
Table 7: Cost of Equity Estimates for Property-Liability Insurers
All Years 720 8.0070 12.5602 8.2679 8.7008 -4.5532*** -3.8059*** -0.2609*** -0.1837*** -0.6938*** -0.7086***
Note: This table shows average cost of equity estimates for property-liability insurance companies based on CAPM, FF3 model, CCAPM, and ICAPM with
FVIX. Cost of equity is calculated for each firm according to Equations (1), (2), (7), (8), and (9). The 30-day Treasury bill rate observed at the beginning of each
month is used as risk-free rate, and the market return is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks. N is the number of firms. In FF3,
SMB is the difference in the return on a portfolio of small stocks and the return on a portfolio of large stocks, and HML is the difference in the return on a
portfolio of high book-to-market stocks and the return on a portfolio of low book-to-market stocks. We use four macroeconomic variables as conditioning
variables in CCAPM, which include dividend yield (DIV), defined as the sum of dividend payments to all CRSP stocks over the previous 12 months divided by
the current value of the CRSP value-weighted index, default spread (DEF), defined as the yield spread between Moody’s Baa and Aaa corporate bonds, risk-free
rate (TB), which is 30-day T-bill rate, and term spread (TERM), defined as the yield spread between the ten-year and one-year T-bond. In the ICAPM, FVIX is
the factor-mimicking portfolio of stocks that mimics the changes in the VIX index, which measures the implied volatility of the S&P100 index options. Values of
VIX are obtained from the CBOE. The market return, SMB, HML, and the risk-free rate are retrieved from Ken French’s data library. The data on the dividend
yield are retrieved from CRSP, while those on the default spread and the term spread are obtained from FRED database at the Federal Reserve Bank at St. Louis.
The data period for each year ends on June 30. Estimates are calculated using the previous 60-months of returns. If some returns are missing for individual firms,
at least 36 consecutive months are required. The risk-free rate used to estimate the cost of equity is the average 30-day T-bill rate from the first to the last month
of each 60-month estimation period. The market risk premium, SMB, and HML are calculated from July 1926 to the final month of each estimation period. The
four macroeconomic variables are averaged over the 60 months ending in the last month of each estimation period. FVIX is calculated from January 1986 to the
final month of each estimation period.
Page | 40
Table 8: Cost of Equity Estimates for Life Insurers
All Years 373 9.5430 15.0297 9.3158 10.1283 -5.4867*** -6.1671*** 0.2271 -0.2221 -0.5853*** -0.2122***
Note: This table shows average cost of equity estimates for life insurance companies based on CAPM, FF3 model, CCAPM, and ICAPM with FVIX. Cost of
equity is calculated for each firm according to Equations (1), (2), (7), (8), and (9). The 30-day Treasury bill rate observed at the beginning of each month is used
as risk-free rate, and the market return is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks. N is the number of firms. In FF3, SMB is the
difference in the return on a portfolio of small stocks and the return on a portfolio of large stocks, and HML is the difference in the return on a portfolio of high
book-to-market stocks and the return on a portfolio of low book-to-market stocks. We use four macroeconomic variables as conditioning variables in CCAPM,
which include dividend yield (DIV), defined as the sum of dividend payments to all CRSP stocks over the previous 12 months divided by the current value of the
CRSP value-weighted index, default spread (DEF), defined as the yield spread between Moody’s Baa and Aaa corporate bonds, risk-free rate (TB), which is 30-
day T-bill rate, and term spread (TERM), defined as the yield spread between the ten-year and one-year T-bond. In the ICAPM, FVIX is the factor-mimicking
portfolio of stocks that mimics the changes in the VIX index, which measures the implied volatility of the S&P100 index options. Values of VIX are obtained
from the CBOE. The market return, SMB, HML, and the risk-free rate are retrieved from Ken French’s data library. The data on the dividend yield are retrieved
from CRSP, while those on the default spread and the term spread are obtained from FRED database at the Federal Reserve Bank at St. Louis. The data period
for each year ends on June 30. Estimates are calculated using the previous 60-months of returns. If some returns are missing for individual firms, at least 36
consecutive months are required. The risk-free rate used to estimate the cost of equity is the average 30-day T-bill rate from the first to the last month of each 60-
month estimation period. The market risk premium, SMB, and HML are calculated from July 1926 to the final month of each estimation period. The four
macroeconomic variables are averaged over the 60 months ending in the last month of each estimation period. FVIX is calculated from January 1986 to the final