Regulator's Determination of Return on Equality in the Absence of Public Firms: The Case of Automobile Insurance in Ontario Fred Lazar (flazar@yorku.ca)
and
Eliezer Z. Prisman (eprisman@yorku.ca)
Schulich School of Business
York University
4700 Keele Street
Toronto, ON M3J 1P3
ABSTRACT In a regulated market, such as automobile insurance (AI), regulators set the return on
equity insurers are allowed to achieve. Most insurers are engaged in a variety of
insurance activities, and thus the full information beta method (FIB) is employed to
estimate the AI beta. The FIB is a two-phase method; first the beta of each insurer is
estimated and then the beta of each activity is estimated, as the beta of an insurer is a
weighted average of the betas of the activities. Theoretically, the two phases are not
separable and the estimation should be done with a one-phase method. The one-phase and
two-phase methods are compared empirically in the Ontario market of AI. Insurers are
private companies thus accounting beta is used. It is shown that a significant bias is
introduced by the two-phase method into the betas of different activities while insurers’
betas are very similar under both methods. This has a significant application to the
estimation of betas "pure players” in classic corporate finance. It implies that utilizing
multi-lines companies and the calculation of the net present values of projects in a
specific activity of companies must be done with the one-phase-method.
Introduction
In a regulated market, such as automobile insurance, a regulator sets the return on equity
(ROE) market players are allowed to achieve. Since the ROE is random and not an
assured rate its average must be above the risk free rate in the market in order to provide
a sustained environment for the insurers. Insurers must make a profit otherwise they will
not survive in the market. The ROE must be such that it compensates for the risk taken
and will attract investments. On the other hand it must be just so it does not provide
insurers with overstated profits and charge consumers with excessive premiums. Thus,
the allowed ROE must be a rate that would have prevailed in a competitive market in the
absence of regulators. Therefore there is a need to assess the risk profile of insurers to
gauge the allowed ROE for insurers so they are adequately compensated.
When the insurers are public companies determining their risk profile, e.g., their
betas utilizing the capital asset model (CAPM), and estimating an adequate ROE is an
easy task1. However, if there are only a handful of publicly listed insurers an alternative
method must be used to estimate their risk profile, and induced recommended ROE.
Moreover, most insures are engaged in a variety of insurance activities, and are not pure
players. This presents an obstacle to estimating the beta of the automobile insurance line
of business or a specific line of business. Classical corporate finance offers a solution, the
full information beta method (FIB), to the absence of pure public players, pioneered by
Ehrhardt & Bhagwat (1991) and expanded on by Kaplan and Peterson (1998).
The essence of the FIB is a property of beta. The beta of a portfolio or a firm is a
weighted average of stocks in the portfolio, or of the lines of business of the firm. Hence,
having an estimate of the beta of a firm that is engaged in automobile insurance and other
1
For an insurance company, the premium obtained for writing insurance is in fact borrowing at a risky rate
of return (depending on the unknown magnitude of the claim). Hence it can be interpreted as a short
position. For early application of the CAPM in this area, see Quirin &William (1975), Biger & Kahane
(1978), Fairley (1979), Hill (1979) and Cummins & Harrington (1985). Thus to be more precise beta
should be defined as beta equity or beta assets (unlevered beta) as in classical cooperate finance. However
for the purpose of this paper it is sufficient to address the beta equity, as all the insurers must hold the same
ratio of reserve and hence will have the same debt equity ratio. Consequently their risk profile could be
judged by the beta equity and hence we use the term beta henceforth for beta equity.
lines of business, and the percentage of the firm in each line of business, the beta can be
solved for each line of business. Specifically, a system of equations stipulating each
firm’s beta as a weighted average of the betas of the firm’s lines of business can be set.
The weight of each line of business is the percentage of the firm in each line and the
system will be solved using a second best solution, e.g. minimum least square, as
empirically the system is not consistent.
Thus the beta of each line of business is recovered using a two-phase method. In
the first phase, the beta of each firm is estimated and the second phase solves the beta for
each line of business. However, theoretically if the beta of each line of business is the
same and independent of the firm in which this line of business resides the two phases
above are not separable. Employing the two-phase method causes losses of information
and so the two phases should have been combined. The beta of each firm should have
been replaced by the weighed average of the beta of the line of business in the firm, and
the estimation problems of the beta of each firm should have been combined into one
estimation problem.
This one-phase estimation problem might not be practical for general-purpose
beta estimation. It requires a partition of the groups of all the firms in the market into
subgroups so that sets of line of business in each subgroup are mutually disjoint.
Furthermore, the different lines of business might not be very homogenous, e.g., software
companies may be classified as in the same line of business but they could be quite
different from one another. In contrast, lines of business practised by insurance
companies are much more homogenous and customers are getting much of the same
product from each of the companies. It might be impractical to combine the data needed
to estimate the beta of each of the public firms in a certain group into one set of data in
order to solve the one-phase estimation problem. After all, it is common that, the beta of
each firm is obtained from a third party and only the second phase problem is solved inhouse. However, in a situation where there are not enough publicly traded companies, an
alternate choice would be to use an accounting beta. When the lines of business are
homogenous, as in the case of the insurance companies, the one-phase method should be
used.
In this paper we empirically demonstrate the use of the one-phase method for the
case of insurance companies that operate in Ontario with the aim of estimating the beta
(accounting) of automobile insurance in Ontario. We compare it to the results obtained
when the two-phase method is used and report some by-product results concerning the
riskiness of automobile insurance in Ontario relative to automobile insurance in the rest
of Canada. As well the beta of other lines of Canada-wide businesses such as aircraft
insurance, credit protection insurance, hail insurance, etc. The rest of the paper is
organized as follows. The next section is devoted to the background and modelling. It
motivates the use of accounting beta as an alternative when there are not enough publicly
traded companies and formulates the FIB estimation problems for the two-phase and the
one-phase method. It is followed by a section in which the empirical results are
presented and compared. Conclusions are offered in the last section.
Background and Modeling
Traditionally, the CAPM or another factor model is applied to publicly traded companies
and the returns to investors in each of these companies are regressed against the market
return to estimate the betas. However, there are only a handful of insurers in Canada that
are publicly traded on the Toronto Stock Exchange (TSX) and none of them is a “pure
player” in the automobile insurance market in Ontario. Consequently, even if the betas
were to be estimated using publicly traded companies, the betas for auto insurance in
Ontario would need to be derived from the aggregate company betas using the FIB.
We use the CAPM for illustration, another model like the Fama and French
(1992, 1996) three factor model (FF3F) can be used2 and the FIB would then be modified
as in Cummins & Phillips (2005). There is a debate in the academic literature about the
FF3F model. Some researchers claim that the “superior” results of the FF3F model are
the result of "data snooping" whose bias could be immense, Lo and MacKinlay (1990)
and/or a selection (survival) bias, Kim (1995), Kothari, Richard & Sloan, 1995 and Jagannathab and Wang (1996). The academic debate regarding the FF3F model has
been summarized informatively and neutrally by Cochrane (1999).
One can also argue that FF3F’s added factors of size and book value to market
2
In that case, the estimation problem we suggest, equation (7), will no longer be quadratic.
value will help insurers that are considered “under financial distress suppliers” survive in
the market. However, presumably the additional ROE allowed for these insurers would
translate into higher premiums that these insurers would be allowed to charge their
customers. The customers on the other hand are not getting a substantially different
service from small growth or under-financial-distress insurers. Thus, equilibrium and
competitiveness in the market will influence customers to move to the “cheaper” and
larger suppliers. Consequently, the survival of small or under-financial-distress insurers
that are small or under financial distress will again be questionable.
In the last two decades the allowed ROE for insurers in Ontario was virtually
constant, set at 12.5 percent and changed to 12 percent after a decade. The change was
uniform across insurers regardless of size. Yet, we see that insurers of different sizes
survived in the market for a long period. Furthermore, the risk premium for the added
factors used in the FF3F model should be calculated for the Canadian market, which
poses empirical difficulties. Consequently, the use of the FF3F in this context might not
be optimal and/or practical. Hence, our empirical study is based on the CAPM.
The essence of the classical (two-phase) FIB, as explained above, is concisely
stipulated by the optimization problems below. In the first phase the beta of each insurer
is estimated (obtained from a third party) by the N regressions stipulated in (2):
Min ηi'ηi
i = 1,..., N
s.t. Ri = α i + βi Rm + ηi
(1)
where
N
is the number of publicly traded insurers
Ri
is the vector of the rate of return on the stock of the i th insurer
Rm
is the vector of the rate of return on the market
ηi
th
is the vector of the error term of the i regression.
In the second phase the optimization problem in (2) is solved
N
Min∑ ε i2
i=1
K
βi = ∑ wij β + ε i , i = 1,.., N
j
j=1
(2)
where
βi
is the beta of the i th insurer, given exogenously or solved for in (1), the first
phase .
βj
is the beta of the j th line of business
wij
th
is the percentage of business of type j of the i insurer
K
is the number of lines of business across the companies, and
εi
th
is a scalar error term of the i equation.
If the betas of automobile insurance companies in Ontario are to be estimated using
equation (2), utilizing companies that are publicly traded in the Canadian market (i.e. are
listed on the TSX), N is likely to be smaller than K . The number of companies is likely
to be smaller than the number of separate lines of insurance. Thus the N equations
comprising the regression in (2) could have many possible solutions. Consequently, its
solution would not be reliable and very sensitive to errors or deviations from some
external reasons.
Some of the insurers in the Ontario market operate also in a foreign market
(mostly the US and Europe) and are publicly traded in these markets. Expanding the
number of companies in the sample to estimate equation (2) is possible by adding
companies that operate in the US and the European markets. These additional companies
however may mask the results and add some difficulties:
•
The possibility of different conditions in different markets may subject companies
to different risk profiles. There is some evidence that suggests differences in the
Canadian and US markets.
•
It may not be appropriate to compare beta estimates of foreign companies that are
based on a foreign market index (or risk premium) and treat them comparably to
the domestic betas.3
•
Adding foreign markets raises the issue of exchange-rate risk and/or hedging.
3
This is done in some of the Canadian studies, thus generating a crude estimation, for example Fox (2011).
•
Even in the relatively liquid American market, Cummins & Phillips (2005) have
argued that trading in Property Casualty (P&C) stocks is not frequent and thus a
measure should be taken to remove biases due to stale prices.
In the absence of a sufficient number of publicly traded, Canadian P&C
companies to generate statistically reliable estimates, the accounting beta approach can be
used. While the accounting beta may suffer from some caveats like creative accounting4
and biases due to unrealized gains, it may be worth the tradeoff. The results would be
specific to the Ontario market; be deduced from a comparison to a Canadian index, and
hence the risk will be priced as per the local market; and the sample of companies should
be sufficiently large enough to generate statistically reliable estimates. Therefore, we
estimate the beta of automobile insurance in Ontario using accounting data and the
accounting data methodology.
Traditionally the beta of automobile insurers using accounting data is estimated in
the following way. The book equity value is substituted for the market value of the
shares; the rate of return on the shares can be replaced by the rate of return based on the
book equity value utilizing net income (before taxes). Once the beta ( βi ) has been
estimated for each insurer, the FIB procedure as in equation (2) can be applied to estimate
the beta for the automobile insurance line in Ontario. Such a study was carried out by
Cummins & Harrington (1985) for the American market and by Zhang & Nielson (2009)
for the Canadian market.
The traditional FIB two-phase method is a relaxation of the actual constraints the
beta of each line and the insurers’ equity betas must satisfy. If the beta of each P&C
insurer is a weighted average of the betas of its various lines, and if the beta of a specific
line is invariant across insurers (the fundamental assumption in the FIB model), some
information will be lost in the traditional FIB model. If the company beta is a weighted
average of the betas of its separate lines, then this constraint should be used in the first
phase of the traditional FIB model where the company beta is calculated. Hence instead
of estimating the betas for each insurer separately, βi , is to be used as an input to the
4
With the recent introduction of fair value accounting (FVA), market values are used more often reducing
this effect.
second phase. As stipulated by equation (2), the two phases should be combined into one.
Thus, there is some merit in estimating the equity beta of the auto insurance
industry not by estimating the equity betas of each insurer separately but collectively,
enforcing the FIB relation. After all, since the insurers are very homogenous, the equity
beta of each line for each insurer should be the same and the equity beta of each insurer
should satisfy the relation as in equation (2). An alternative would be to solve one
optimization problem with the regression constraints for each insurer and the FIB
equation.
We develop this method as an alternative to the traditional FIB approach; one that
we believe is more efficient and avoids the need to first estimate the aggregate company
beta. In applying two-phase FIB procedures wij is defined, in the context of insurers and
different lines of business, to be the percentage of premiums obtained by insurer i from
line of business j . The wij varies from year to year and thus wij is, in fact, an average of
the years used in the study. The one-phase FIB, utilizing the average of the wij , gives
rise to the optimization problem:
N
Minα ,...,α
1
N ,β
1
,..., β k
∑ δ 'δ
i
i
i=1
⎛ K
⎞
Ri = α i + ⎜ ∑ wij β j ⎟ Rm + δ i , i = 1,..., N
⎝ j=1
⎠
(3)
where 5
N is the number of insurers in the database
Rm is the vector of the rate of return on the market (the dimension of Rm and of the
vectors defined below are the number of years in the data base)
Ri is defined to be the vector of insurer's i yearly rate of return:
Yearly Net Income (before taxes)
(4)
Yearly Average Equity
wij is, as before, the average premium obtained by insurer i from line of business j ,
δ i is the vector of deviations associated with the i th insurer.
5
For simplicity we use the same symbols for either the number of insurers in the database or publicly
traded, and for the rate of return on the stock of the insurers and the rate of return estimated from
accounting data.
However, our database contains the value of wij for each year and there is no
need to approximate it by the yearly average. Hence our approach of the one-phase FIU is
defined by the optimization below6:
L
Minα ,...,α
1
N ,β
1
,...,β
k
N
∑ ∑δ
2
iy
y=1 i=1
⎛ K
⎞
Riy = α i + ⎜ ∑ wijy β j ⎟ Rmy + δ iy , i = 1,..., N, y = 1,..., L
⎝ j=1
⎠
(5)
when
L is the number of years in the data base
Riy is the year y component of the vector Ri of insurers’ i rate of return
Rmy is the yth component of the vector Rm - the market rate of retune in year y
wijy is the premium from line of business j for insurers i year y
Since the wijy are known and so are the Rmy we define
ηijy = wijy Rmy
(6)
and solve
L
Minα ,...,α
1
N ,β
1
,...,β
k
N
∑ ∑δ
2
iy
y=1 i=1
(7)
K
Riy = α i + ∑ ηijy β + δ iy , i = 1,..., N, y = 1,..., L
j
j=1
Data Base and Empirical Results
MSA Research Inc.7 is a Canadian-owned, independent analytical research firm
that is focused on the Canadian insurance industry. The data used for the estimation of
the accounting beta is taken from their database. The decision to use annual data was
made because automobile insurers’ quarterly data may include seasonality. The use of
annual data also avoids the need to make assumptions for certain required percentages
that are not reported on a quarterly basis.
We used annual data of the last available ten years, i.e. 2002-2011. We are not
6
For ease of presentation we assumed that all the insurers worked from year 1 to
in general but the modification is trivial.
7
see http://www.msaresearch.com.
L . This may not the case
concerned by the lack of sufficient data as our estimates are supported by Forty-seven 8
insurers in Ontario. Hence altogether we have 470 annual observations of the rate of
return of the insurers. It is important to note that one insurer that was active during the
period 2003-2011, and was not included in the sample, was a pure player in the Ontario
automobile insurance line of business. It was left out so its beta, calculated separately,
can help gauge the difference in the estimation utilizing the two-phase FIB method and
the one phase FIB method suggested in this paper.
There are sixteen different lines of business that are practiced by these insurers.
One of these lines of business is automobile insurance but in order to impute the beta of
automobile insurance in Ontario this line had to be split into two parts; automobile
insurance in Ontario and automobile insurance in the rest of Canada (excluding Ontario).
Thus in total there were seventeen lines of business as stipulated in Table 1 below.
Hence, optimization problem (7), the one-phase method, has 470 constraints, as the
number of aggregate annual observations on the rate of return, 47 alpha-type variables
and 17 beta-type variables. Thus in summary the optimization problem (7) has 470
constraints and 64 variables (the delta-type variables are redundant variables introduced
only for the ease of the presentation). The first phase of the two-phase method, the
regressions in (1), are solved to estimate the beta of each insurer. The market rate of
return used was the TSXS&P total return index as stipulated in Table 1 below.
8
In total there are 76 insurers in Ontario but we excluded 24 insurers due to the limited amount of business
that they conduct in this area and/or due to the fact that they operated for a very short period of time or
had other irregularities. This leaves 52 insurers. Forty-seven of them were active for the full duration and
these are the ones used for the study. One insurer was active for the period 2003-2011 one from 2004-2011,
one from 2005-2011 and two from 2008-2011. The results are virtually the same if the fifty-two insurers
were analyzed.
Table 1: One Year Rate of Return of the TSXS&P during 2002-2011
Date
02-01-02
02-01-03
02-01-04
04-01-05
03-01-06
02-01-07
02-01-08
02-01-09
04-01-10
04-01-11
Risk Free
Rate
2.30%
2.86
2.52
2.69
3.74
4.05
3.66
0.89
0.74
1.39
The second phase of the two-phase method, the optimization problem in (2), has 47
constraints and 17 variables. The three optimization problems (1), (2) and (7) were
solved numerically using the interface of the optimization package in Maple that utilizes
the quadratic Numerical Algorithms Group (NAG) procedure9. This procedure takes
advantage of the quadratic nature of the problems. The objective function value of the
one-phase method, problem (7) was 7.55 and that of the second phase of the two-phase
method, problem (2) was 1.00, so both problems had a good fit. The beta estimates of
the line of business based on the one-phase method and the two-phase method are
stipulated in Table (2) below.
9
Maple and Maplesoft are trademarks of Waterloo Maple Inc. see http://www.mapsoft.com/. NAG and the
NAG logo are registered trademarks of The Numerical Algorithms Group Ltd http://www.nag.com/.
Table 2: Line of Business and betas of the one-phase and two-phase methods
Lines of Business
One-Phase Beta
Two-Phase Beta
Automobile Insurance in Ontario
0.320981487
0.3067422
Automobile Insurance- rest of Canada 0.305585122
0.07024669
Property
0.241679915
0.473945802
Aircraft
2.753648743
-4.893442779
Boiler and Machinery
-1.97220059
-7.431367487
Credit
-32.12753313
-71.87961573
Credit Protection
4352.985227
-9067.398786
Fidelity
-0.861457333
-1.508346249
Hail
57.70547731
26.32582278
Legal Expense
-16.14983628
-35.84887776
Liability
-0.025141814
-0.002031996
Mortgage
0
0
Other Approved Products
0
-1124229.329
Surety
-0.19093064
0.461434699
Title
0
0
Marine
-3.803367899
-3.065778103
Accident and Sickness
1.486384688
5.746214615
Looking casually at the beta estimates of the two methods it is apparent that some
estimates could be very different. When calculated using both methods, Ontario’s
automobile insurance companies have very similar betas that are also consistent with the
betas of the pure players. In contrast, the betas of automobile insurers in the rest of
Canada as well as property and aircraft insurers are quite different when calculated using
both methods. The latter has positive beta under one method and negative under the
other. Indeed the Euclidian norm of the difference between the two vector of estimates
(the first and second column in table 2) is “large” being 1124309.43. Hence, using the
one-phase FIB suggested in this paper versus the classical two-phase FIB might give very
different results. And since theoretically the one-phase method should be used, the
approximation of the one-phase method might have a very biases on the results of the
individual lines of busies beta.
Recall the in the first phase of the two-phase FIB of the beta of each insurer is
estimated based on the average percentage of each line of business over the ten years. In
one-phase method the beta of each insurer is not estimated but the method solves directly
for the beta of each line of business based on the percentage of business in each line for
each insurer for every year (rather than the average percentage of each line of business
for each insurer over the ten years.) However, given the beta of each line of business,
based on the one-phase method, it is possible to reconstruct the beta of each insurer using
the average percentage of each line over the ten years for each insurer. The estimates of
the beta of each insurer based on the two-phase method and the reconstruction of the beta
of each insurer based on the two–phase method is present in Table 3 below:
Table 3: Betas of each insurer based on the one-phase and two-phase methods
2 phase
1 phase
0.668365708
0.297426887
0.032021285
0.183357656
-0.063621997
0.232123979
-0.104399724
-0.020130533
0.436811529
0.285160581
0.296705752
0.315233404
0.188159473
0.042295811
0.835321421
0.281759798
0.151689206
0.263865053
0.141574306
0.297974856
0.108048463
0.144557175
0.030543765
0.257346959
0.245797834
0.270052811
0.30810132
0.244134726
0.395969961
0.237156709
0.527292904
0.254403202
0.342932068
0.185486529
0.157394434
0.244402598
-0.058758187
-0.067045364
0.413612282
0.228476475
0.24073484
0.2380656
0.042846671
0.068308808
0.424016584
0.789395023
0.598368744
0.295221388
0.14944381
0.154502741
0.006522738
0.303633616
0.396006249
0.317076469
0.191129065
0.265082396
-0.083185403
0.299341177
0.111221845
0.303887657
0.232966566
0.119394611
-0.183139364
0.22440812
0.108036431
0.297735173
0.198055996
0.319261962
0.200053039
0.31230893
0.317412228
0.298416491
0.100340015
0.299804544
0.017492777
0.283672748
0.931167149
0.298618183
0.156001717
0.300754895
0.387768764
0.301329113
0.18911969
0.311952644
0.09664733
0.305033366
0.374212632
0.298317213
0.364693663
0.313388219
0.297324031
0.466572336
0.632082111
0.30171322
Casually inspecting these two estimates it seems that they are vey similar. Indeed the
Euclidian norm of the difference between the two vectors of estimates (the first and
second column in Table 3) is “small” being 1.53.
Under the two-phase methods the beta of each line is assumed (implicitly by the
optimization problem) as being invariant with respect to the years. However, since the
percentage of business in each line of business varies across the year the beta of each
insurer also varies across the years. The beta of each insurer, under the two-phase
method is a different weighted average of the same quantities (the beta of each line which
is indeed fixed across the year). In fact under the two-phase method it is possible to
calculate the beta of each insurer for each year, based on the percentage of business done
in each year in each line of business. Of course, under the two-phase method, for each
insurer, the average of the betas of each year is equal to the beta of each insurer that is
calculated based on the average percentage of each line of business over the ten years.
Conclusions
The paper motivates the concept that theoretically the proper optimization problem to
estimate betas of different lines of business is the one-phase method. It empirically
compares the classical FIB two-phase method to the one-phase method suggested here.
The results indicate that the beta of the different lines of business should be estimated
using the two-phase method. Theoretically, even the beta of each company should have
been estimated utilizing the one-phase method suggested here. This would have required
portioning the set of all companies in the market to subsets so that the set of lines of
business practiced in each subset of companies were disjoint. Estimating the beta of a
company would have to be carried out by solving the one-phase method for the group
including the company at hand. However, based on the empirical results presented here,
the approximation using the two-phase method is not very biased. Yet, the result
presented suggests that the ROE, for a line of business, even in the case where there are
publicly traded companies, should be estimated utilizing the one-phase method. This
observation has implications not only to regulators seeking an estimate of ROE but also
to multi-line companies when estimating the net preset value of a project in a line of
business.
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