Joumal of Banking and Finance 17 (1993) 483--496. North-Holland
The measurement of efficiency in
life insurance: Estimates of a mixed
normal-gamma error model
Andrew M. Yuengert*
Federal Reserve Bank 01 New York, New York, NY 10045, USA
This paper make three contributions to the literature on efficiency in financial services. First, it
presents estimates of a mixed error cost frontier in which the variances of both the normal and
gamma distributions can vary with firm size. Second, it extends the literature on life insurance
scale and product mix economies by incorporating and measuring X-inefficiency. Estimates of X­
inefficiency range from 35 to 50%; ray scale economies exist up to $15 billion in assets. Third,
comparisons of normal-gamma estimates with other methods - thick frontier, weighted least
squares, half-normal distributions - demonstrate sorne of their pitfalls.
1. Introduction
Over the past fifteen years, an extensive literature has attempted to
measure efficiency in the financial services industry. Much of the work in this
area is focussed on banking [see, e.g., Berger et al. (1987), Ferrier and Lovell
(1990), aÍtd Berger and Humphrey (1991)], although there is a less well­
developed literature in life insurance [for example: Kellner and Mathewson
(1983), Cho (1986), Fields (1988), Boose (1990), and Grace and Timme
(1992)]. From an early emphasis on the measurement of scale and product
rnix economies, researchers have in recent years shifted to a focus on the
extent of X-inefficiency [Berger and Humphrey (1991)]. This shift poses two
important econometric challenges: (1) by what method should we measure
the extent of X-inefficiency, and (2) are there systematic differences in X­
inefficiency across producers?
A series of papers have addressed the first challenge through data
envelopment analysis, econometric frontier estimation, and thick frontier
Correspondence lo: Andrew M. Yuengert, Federal Reserve Bank of New York, 33 Liberty
Street, New York, NY 10045, USA.
*The opinions expressed here do not necessarily reflect those of the Federal Reserve Bank of
New York. 1 would like to thank Allen Berger, Shawna Grosskopf, and Stephen Timme. Many
thanks to Peter Papagianakis, who read and organized the NAIC data tapes.
0378--4266/93/$06.00 1993-Elsevier Science Publishers B.V. All rights reserved
484
A.M. Yuengert, Efficiency in life insurance
A.M. Yuengert, EjJiciency in life insurance
methods.1 Some have recently begun to address the second chaJlenge, using
panel data [Schmidt and Sickles (1984), CornweJl et al. (1990), Berger
(1993)]. This paper addresses the second challenge in cross-section data,
through the estimation of a frontier cost function, making use of the
heteroskedasticity imposed on the error structure by differences in X­
inefficiency. Its conclusions have implications for the first challenge, the
L..og
1
I
differences in the mean of the gamrna distribution allow us to capture
differences in X-inefficiency. The model is estimated on a 1989 sample of life
insurance companies and groups.
The results extend the literature on life insurance costs by exammmg
evidence of X-inefficiency. Using estimates of a life insurance cost function, I
calculate scale and scope efficiency, the extent of X-inefficiency, and the
relative importance of X-inefficiency and symmetric estimation errors. Unlike
Houston and Simon (1970), Cho (1986), and Grace and Timrne (1992), I find
I
o
....
-1
choice of measurement methodology.
To measure X-inefficiency, this paper uses a mixed error model (normal­
gamma) in the tradition of Greene (1980, 1990) and Stevenson (1980).
Previous research with the normal-gamma model assumed that the moments
of both distributions were fixed. In this paper, the parameters of both normal
and gamma distributions are aJlowed to vary with firm size. Estimated
485
-2
-3 -
,; " /:;i ;'<li}.: ':,. '. " ,_ ,
-4
-5 I
12
!
!
.1
14
16
18
!!
20
22
I
24
26
L..og of Assets
Fig. 1. Life insuranee groups and eompanies: Average eost. Source: NAIC, FRBNY.
scale economies in life insurance only up to $15 billion in assets, not
cost functions. The first type is symmetric, mean O error, or white noise. It is
throughout the sample. Average X-inefficiency in the industry is on the order
of 35-50%; there are no signigicant differences in efficiency by firm size.
The estimates presented here have implications for the use of other
efficiency measurement methods in life insurance. In particular, they highlight
costly adjustment to changing technology and markets, may push firms
above the cost frontier. This type of error is always non-negative.
a weakness of thick frontier methods: they are not robust with respect to
heteroskedasticity in estimation errors arising from either X-inefficiency or
white noise; also, the assumption that the variance of symmetric, white noise
error is small relative to X-inefficiency is violated in the life insurance sample.
Because of the presence of substantial X-inefficiency in the data, estimates of
scale efficie'ncy generated by the weighted least squares approach are biased
upward, relative to the normal-gamma estimates. Estimates from data
. envelopment analysis, while not calculated here, may be inappropriate, given
the magnitude of the variance of the symmetric, white noise errors. The
estimates suggest that half-normal specifications are not l1exible enough to
capture substantial X-ineffiency.
2. Heteroskedasticity and the measurement of cost efficiency
Assume that there are two types of error in the measurement of frontier
1 For a review of the literature on data envelopment analysis and eeonometrie frontier
estimation, and reeent examples, see the OetoberfNovember 1990 volume of the Journal of
Econometrics. For a look at thiek frontier approaehes, see Berger and Humphrey (1991).
often attributed to measurement error, or left-out variables. The second type
of error is X-inefficiency. Differences in the quality of management, and
Both types of error may in theory be heteroskedastic. Indeed, if we are
looking for. systematic differences in X-inefficiency across firms, we are
looking for heteroskedasticity. Heteroskedasticity in either type of error can
pose a problem for the measurement of all types of cost efficiency (scale,
product mix, and X-efficiency). It is difficult to sort out scale and product
mix economies from differences in error term distributions.
Data from the life insurance industry clearly illustrate this difficulty. Fig. 1
is a scatterplot of log average costs (equal to total costs, to be defined in
section 3, divided by total assets) and log assets for a sample of life insurance
companies and groups. At first glance, it seems apparent that there are
economies of scale in this industry - there is a general downward trend in
average costs as assets increase. In the presence of what appear to be
heteroskedastic errors, however, things are not so simple.
There are a number of small firms whose average costs are as low as the
costs of the larger firms in the sample. In addition, the variance of average
costs among the smaJler firms is obviously larger than the variance for the
larger firms. It is important for reseaI:chers to know which type of error is
heteroskedastic. Because of the presence of a mean-positive error, misspecifi-
486
A.M. Yuengert, Ejficiency in life illsurallce
A.M. Yuengert, Efjiciency in life insurance
cation of the error structure can lead to both inconsistent and inefficient
3. Estimation strategy and data
estimates of scale, scope, and X-inefficiency.
To take an extreme example, one might assume that there were no X­
3.1. The mixed distribution mode!
487
The cost function to be estimated is the familiar translog form of eq. 1:2
inefficiencies (as do traditional life insurance cost studies), and that the white
noise errors are heteroskedastic. Weighted least squares (WLS) estimates of
the relationship in fig. 1 would probably yield a downward sloping cost
function (economies of scale): small, low-cost firms would fall below the
estimated frontier, and would be treated as negative realizations, drawn from
the white noise distribution. If the assumption were wrong (if there is in fact
X-inefficiency, and it is heteroskedastic) then WLS estimates will be biased in
favor of economies of scale, and will, of course, have incorrectly assumed
m
m
m
In Ci = o: L {3jg( Y;) + L
j=!
j=!
p
p
L {3ji$( Y;)g( Y;d
k=!
p
+ L Yjlnwij+ L L Yjklnwijlnwik
j=!
away X-inefficiencies where the y exist.
On the other extreme, one might assume that all estimation error is X­
inefficiency - that there is no white noise error. An appropriate estimation
method would then be data envelopment analysis (DEA), which would
presumably yield a relatively flat line in fig. 1 (no economies of scale). The
low-cost small firms would be on or elose to the estimated frontier, and
small firms on average would exhibit greater X-inefficiency (larger errors)
than large ones. If the assumption of heteroskedastic X-inefficiency were
wrong, and instead white noise errors were present and heteroskedastic, DEA
estimates would be biased against economies of scale, and would overstate
m
j=! k=l
p
q
+ L L pjf${Y;)lnwik+ L c5jXij+Bi•
j=!
k=!
( 1)
j=!
Ci is total costs (general expenses plus commissions), the Y;jS are output
variables, the wijs are proxies for factor prices, and the Xijs are some other
variables. g( Y) is defined as follows:
the X-inefficiency among the smaller firms.
Similar to DEA, but not as extreme in its assumptions about white noise
g(Y) =ln (Y),
Y>O,
=0
Y=0.3
Bi equals the sum of two errors, Ui and Vi' ui is a white noise error,
distributed N(O, (2). Vi' representing X-inefficiency, must be non-negative. For
this error some researchers choose the half-normal distribution. This distri­
bution is somewhat inflexible in that it has a mode of O, and may be unable
error, is the thick frontier approach (TFA) of Berger and Humphrey (1991). In
TFA, only firms in the lowest cost quartile by size elass are ineluded in least
squares estimates of the cost function. TFA estimates are sensitive to
heteroskedasticity in both white noise and X-inefficiency errors. In the life
insurance example, heteroskedasticity in the white noise error will lead to an
upward bias on the TFA estimate of the slope of the cost function (a bias
against economies of scale). Heteroskedasticity in X-inefficiency willlead to a
downward bias on the slope coefficient (a bias in favor of economies of
scale). Unlike the biases in the first two methods, which arose because of
overly restrictive assumptions on the error terms, the biases in TFA estimates
result from the use of the dependent variable, costs, in choosing the sample.
The question which arises from fig. 1 is: do small firms have on average
higher average costs because they are scale inefficient, or X-inefficient?
Without fuller knowledge of the nature of the error structure, it is very
to capture X-inefficiency when its magnitude is large. While I inelude
estimates of a normal-half normal specification for comparison, the central
results of this paper assign the more flexible gamma distribution to Vi' If we
assume that Vi is distributed T(0:, (J), the mixed error Bi has mean 0:(J-! and
variance a(J - 2 + a2•
The likelihood function, from Stevenson (1980), has no elosed form, except
when o: takes integer values. Stevenson (1980) solves this problem by fixing o:
at 1, 2, and 3, and estimating (J. Greene (1990) and Beckers and Hammond
(1987) use various approximations to the likelihood for noninteger values of
o: in order to estimate both o: and (J over their full range. Since, as a practical
21nput share equations, expressed as functions of the above parameters, and derived by the
difficult to sort out the different types of efficiency. Because both white noise
and X-inefficiency errors may play significant roles in the generation of
observed costs, they should both playaroIe in estimation. In what follows, I
model the error term as the sum of normal and gamma distributed
heteroskedastic en.-ors. This specification will allow us to sort out scale,
Cox transformation of the output variables. I see no theoretical or practical reasons to prefer a
Box-Cox function over the modified log function, g( Y). Since there are no observations in the
data with output less than or equal to one, g( Yl is in practice monotonic. In addition, I do not
product mix, and X-inefficiencies.
need the derivative of g(Y) at zero.
application of Shephard's Lemma to the cost function, are not incIuded in this system. loint
estimates of the cost function and the share equations may generate more eflicient estimates.
3Caves et al. (1980), Fields and Murphy (1989), and Grace and Timme (1992) suggest a Box­
l
A.M. Yuengert, Efficiency in life insurance
A.M. Yuengert, Efficiency in life insurance
488
sioners Life Insurance database for 1989. Data for the various companies are
consolidated into groups, using A.M. Best's Company Reports, Life/Health
1990. This eliminates any problems that arise from the division oi sales, data
processing, and management tasks within a life insurance group, which may
distort reported costs for the member companies. To date, most research into
life insurance costs has used unconsolidated data [Fields (1988) is an
matter, it is unlikely that a model in which both C( and (j were allowed to
vary with firm size would be estimable, the simpler technique of Stevenson is
more appropriate for this problem. It should be noted that the gamma
distribution with C( fixed is still quite flexible. The estimates presented here fix
C( at 2.4
Previous studies have assumed homoskedastic normal and gamma errors.
exception].
Since we are interested in the systematic differences in these distributions by
firm size, we allow both (J and B to be functions of a vector Di of firm size
Costs are defined as general expenses plus commissions, net of reinsurance.
To proxy for factor prices, I collected data on average yearly pay by state
(from the Statistical Abstract of the United States, 1991) and the average
replacement cost of non-residential capital by state, from F.W. Dodge's
Construction Potentials Bulletin. An insurance company was assigned the
factor prices for the state in which its head office is located (which is not
necessarily where it is chartered); a consolidated factor price for a Life group
is calculated as the asset-weighted average of the member companies' factor
dummies:
7
Bi= I bJ>ij,
(2)
j=l
7
(Ji= I J1J>ij'
489
(3)
j= l
prices.
I chose seven categories of output for the analysis: individual life insur­
Eqs. (1)-(3) are estimated by maximum likelihood. The estimates of eq. (1)
generate estimates of economies of scale and product mix economies.
Estimates of eq. (2) will capture any differences in X-inefficiency by firm size,
and (2) and (3) together give. us an idea of the relative importance of X­
ance, group life insurance, individual annuities, group annuities, deposit
funds (including GICs), accident and health insurance, and other. As with all
service sectors, output measurement in insurance is problematic. There are
two common measures of output in the Life insurance efficiency literature:
inefficiency and white noise estimation error.
reserve levels and premiums. A third is suggested here: additions to reserves.
Life insurers are required to set aside reserves to meet future obligations:
they are calculated as the present value of future liabilities (from a certain
policy or annuity) minus the present value of future premiums and consider­
ations. Because they are a stock, and not a flow, reserves capture only the
amount of business an insurer must carry from year to year. These services
to 'old' business are surely a non-trivial determinant of costs, but are not as
important as the costs of new business. Premium revenue (including annuity
considerations and deposit funds) attempts to capture the flow of services,
but it measures price times output, not output. Systematic difTerences in price
across large and small firms may lead to misleading inferences about average
As Greene (1980) and Bauer (1990) point out, the normal and gamrna
errors may be correlated. These estimates do not take into account any
possible correlation between the two error terms; any correlation between the
two will result in a loss in efficiency.
For comparison, I also estimate eq. (1) using WLS, TFA, and a half-normal
specification for X-inefficiency. For WLS, I model the variance of Ui as an
exponential function of log assets and log assets squared. For TFA, I divide
the life insurance sample into seven asset size classes [corresponding to the
asset size dummies in eqs. (2 and 3)J, and include only companies in the
lowest quartile of average costs by size class in the analysis.
Because the normal-gamma model is a more flexible error specification
than competing efficiency measurement methods, in the discussion of the
results I will treat the normal-gamma estimates as the benchmark against
which qualitative judgements of biases in the other methods are made. If the
measurement errors are in fact not normally distributed, or the X-inefficiency
component is not gamma distributed, these judgements will, of course, be
costs if premiums are used as an output proxy.
A third proxy for output, additions to reserves, may more closely capture
output. Additions to reserves equals reserves set up for new business, new
deposit funds, and new reserves set up as old policies age. Additions to
reserves are not immune to differences in prices across firms, because
premiums enter negatively into their calculation. Insurance actuaries calcu­
late additions to reserves in a systematic way, using conservative mortality
and interest rate assumptions. There are some potential drawbacks to this
measure: the most important is that insurers can choose among different
reserving methods to calculate changes in reserves. In addition, an actuary
may choose one of several interest rates for the reserve calculation, as long as
inexact.
3.2. The life insurance sample
The data are drawn from the National Association of Insurance Commis4Also eSlimated was a model with ex = 1. The estimates are available upon request.
I
!
..l
A.M. Yuengert, Efficiency ilJ life ilJsurance
A.M. YuelJgert, EfficielJcy ilJ life ilJsurance
490
it is below the (usually low) regulatory maximum.5 Additions to reserves
Table 1
Descriptive statistics, life insurance sample, 1989.
are defined net of reinsurance.
For each output category except accident and health insurance, I chose
additions to reserves as a measure of output. Since companies do not report
additions to reserves for accident and health insurance, the level of reserves is
used instead. In practice, additions to reserves track closely with total
reserves, so the lack of a flow measure for accident and health is probably
unimportant. Total reserves in each category were initially added to the
model, to capture the carrying costs of insurance and annuity policies; these
additions caused multicollinearity problems, however, and the reserve level
variables were discarded in favor of a more general stock variable, total
Variable
Mean
Costs (bil. $)
Assets (bil. $)
Year1y wage (thous. $)
173.1
1,587.0
7.933.0
and annuity lines which come from first-year and single premium business
are added to capture the increased costs of new business.6 Other variables,
describing ownership structure (mutual/stock dummies) and the growth rate
of assets, are not included in this study; their role in explaining X-inefficiency
merits further research [see Gardner and Grace (1993)].
Data on 1,147 companies and groups were read from the NAIC tapes for
1989. Very small and inactive companies (additions to reserves < $250,000)
were excluded from the sample, leaving 805. In addition, the sample was
divided into seven asset size categories. Within each category, a standard
outlier rule (exclude all observations whose average costs are more than two
interquartile ranges from the first and third quartiles) left 765 firms,
accounting for 92.5% of the life insurance industry's assets. Computational
difficulties forced the further exclusion of eight small firms before conver­
achieved.7 Within each size
category, those firms in the lowest quartile of average costs were identified
for the TFA analysis. Table 1 contains descriptive statistics for the sample; the
next section presents the results of estimation.
Max
Min
2,792.0
134,400.0
0.07
0.4
17.4
2.1
65.3
25.9
17.4
12.7
130.3
46.6
61.7
Additions to reserves (mil. $)
Indiv. Iife
Group life
Indiv. annuities
Group annuities
Depositt funds
Additional variables were added to capture the increased cost burden of
new insurance sales. The proportion of premiums written in the insurance
Standard
deviation
51.7
Property cost
assets.
gence of the normal-gamma model was
491
266.1
18.6
4,083.0
138.7
44.8
2,627.0
186.5
O
20.9
1,875.0
2,146.0
38.9
137.6
367.1
198.4
O
3,807.0
O
O
O
3.5
13.7
188.8
O
61.5
Other
Acc/health reserves
New business share
AI! ins and anns.
Indiv.life
Group Iife
Indiv. annuities
Group annuities
0.48
0.36
0.24
0.27
0.10
0.24
0.41
0.28
0.42
0.32
0.09
Other
0.38
6,066.0
1.0
1.0
1.0
1.0
1.0
1.0
O
O
O
O
O
O
O
757
n
Source: NAIC, FRBNY.
Yuengert (1992).8 Table 2 uses the estimates to calculate the magnitude of
the white noise and X-inefficiency errors across asset categories. For
comparison, mixed normal-half normal, WLS, and TFA estimates also appear.
Column one in the panel A of the table presents the normal-gamma
estimates of a. The estimates are large: for the smaller firms, the standard
deviation of the white noise error is equal to 60% of total costs, declining to
21% for large firms. The estimates of the half-normal model in column 2 are
similar. The weighted least squares estimates in column 3 are smaller, and
show a more gradual deciine.
4. Estimation results
Eqs. (IH3) are estimated by maximum likelihood; the estimates appear in
5For the WLS and TFA methods. the choice of output proxy did not affect early estimation
results. They were not tried in the mixed error models.
6Insurers may not accrue commissions; as a result. rapidly growing companies. with a larger
new business share, report higher expenses than companies with established business.
7The normal cumulative distribution function used in the maximization routine was unable to
return precise enough estimates for the large arguments of the eight excluded firms. The
excluded observations were not influential in the WLS and TFA estimates, and were among the
smal!est firms in the sample
Column 1 in panel B of table 2 presents estimates of average X­
inefficiency. Each estimate is significantly different from zero. The mixed
error estimates of X-inefficiency range from 34% for the largest companies
(assets > $1 billion) to 49% for the smallest companies (assets < $3 million).
A likelihood ratio tests fails to reject the equality of the X-inefficiency
estimates across assets classes [X2( 6) = 3.373]. Although significant (both
8The coefficients are general!y of the expected sign. The estimated cost function is convex in
the factor prices. Of the new business share variables, only the overal! new business share and
the group life share were significant and positive, but only in the mixed error estimates.
A.M. Yuengert, Efficienc)' in life insurance
492
i
A.M. Yuengert, Efficieııcy iıı life iıısw'aııce
i
i
Table 2
11
!
MLE,
Weighted
Assets
Normal-
Normal-half-
least
(biL. $)
gamma
normal
squares
(%)
(%)
(%)
<$0.003
58.8
(6.61)
60.5
(5.64)
42.6 (2.76)
62.7
(8.09)
65.0
(8.40)
38.9 (2.06)
0.008-0.020
62.3
(9.58)
63.6
(8.89)
36.3 (1.66)
0.020-0.050
67.3
(9.25)
71.1
(8.64)
33.3 (1.35)
0.050-0.180
64.8 (10.3)
65.3 (10.1)
30.4 (1.24)
0.180-1.1
41.4
(6.81)
42.1
(6.05)
26.7 (1.36)
>1.1
20.9
(4.74)
20.5
(3.93)
21.4 (1.75)
economies. by estimation method (standard errors).'
f
Assets
normal-
normal-half-
least
f
(biL. $)
gamma
normal
squares
ı
0.003-0.008
Life insurance companies and groups, 1989. Estimates of ray scale and product mix
i
A. Standard deviation (<I) of normal error (/ of minimum cost).
MLE,
Table 3
i
Life insurance companies and groups, 1989. Estimates of inefficiency and <I, by estimation method (t-statistics).
A. Ray scale economies.
0.002
0.004
0.010
0.026
0.065
0.16
0.40
1.0
2.5
6.0
B. 1nefficiency (% over minimum cost).
15
38
MLE,
MLE,
0.818 (0.101)**
0.829 (0.089)**
0.840 (0.077)*
0.852 (0.066)*
0.863 (0.055)*
0.874 (0.045)*
0.886 (0.036)*
0.897 (0.031)*
0.909 (0.030)*
0.920 (0.034)*
0.931 (0.042)**
0.943 (0.051)
0.954 (0.062)
0.960 (0.067)
0.835
0.846
0.858
0.869
0.881
0.893
0.904
0.916
0.928
0.939
0.951
0.957
Weighted
0.811 (0.110)**
0.823 (0.097)**
(0.085)**
(0.072)*
(0.061)*
(0.050)*
(0.040)*
(0.034)*
(0.032)*
(0.036)*
(0.044)**
(0.054)
(0.066)
(0.072)
0.786 (0.085)*
0.797 (0.075)*
0.808 (0.065)*
0.819 (0.056)*
0.830 (0.047)*
0.841 (0.039)*
0.852 (0.033)*
0.863 (0.030)*
0.875 (0.031)*
0.886 (0.035)*
0.897 (0.042)*
0.908 (0.050)**
0.919 (0.059)
0.925 (0.064)
Thick
frontier
0.619 (0.113)*
0.655 (0.100)*
0.690 (0.087)*
0.726 (0.075)*
0.761 (0.065)*
0.797 (0.056)*
0.833 (0.052)*
0.868 (0.051)*
0.904 (0.055)**
0.939 (0.062)
0.975 (0.072)
1.011 (0.084)
1.046 (0.096)
1.064 (0.103)
MLS,
MLE,
Assets
Normal-
Normal-half-
(biL. $)
gamma
normal
Thick
frontier
(%)
(%)
(%)
49.5 (2.15)
40.0 (1.45)
114.8
37.4 (2.01)
27.5 (1.18)
103.4
MLE,
MLE,
37.6 (2.52)
29.7 (1.54)
120.1
normal-
Weighted
normal-half-
46.8 (4.06)
38.7 (2.29)
121.7
least
gamma
normal
39.4 (3.11)
114.8
squares
45.0 (4.34)
Thick
frontier
40.6 (4.40)
36.5 (3.40)
109.6
0.162 (0.126)
0.146 (0.127)
0.160 (0.133)
0.049 (0.245)
34.5 (4.76)
32.4 (4.68)
59.4
<0.003
0.003-0.008
0.008-0.020
0.020-0.050
0.050-0.180
0.180-1.1
>1.1
95
150
B. Scope economies.
a*: Significantly different from 1 al 5% 1evel of significance.
**: Significantly differenl from 1 at 10";' level of significance.
statisticaIly and economically) the errors in X-ineffıciency are small compared
to the white noise component a, whose standard deviation is roughly 40%
larger.
For
493
comparison,
column
2
presents
X-ineffıciency
estimates
for
the
normal-half-normal modeL. Perhaps because the half-normal distribution is
less flexible, the estimates of X-ineffıciency are 10-20% smaIler. In estimates
not reported here, a mixed exponential-normal model generated much
smaller estimates of X-ineffıciency, suggesting that it is less f1exible even than
the half-normal. Column 3 presents TFA estimates of average X-ineffıciency
by size category. The estimates accord with the predicted direction of bias in
section 2: because of the heteroskedastic white noise errors, TFA overstates
the X-ineffıciency of the smaIler firms relative to the large firms. Because the
variance of the white noise errors is large relative to the X-ineffıciency errors,
TFA overstates the average X-ineffıciency of every size category.
To sum up the error structure: estimates of X-ineffıciency vary from 35 to
50%, but the differences across firm size are statisticaIly insignificant. The
symmetric, white noise error variance is large relative to X-ineffıciency, is
heteroskedastic, and decreases in firm size. The presence of X-ineffıciency
violates the assumptions of the WLS modeL. Because TFA relies on the
dominance of X-ineffıciency over white noise errors, it is not appropriate in
this sample. Moreover, heteroskedastic white noise errors will bias TFA
effıciency estimates. The half-normal model understates the magnitude of
X-ineffıciency.
Table 3 calculates scale and scope economies. Ray scale economies are the
percent change in costs from a 1% change in all outputs, maintaining output
mix at the mean of the sample. ır the estimate is lessjgreater than 1, ray scale
economiesjdiseconomies exist. Scope economies are the increase in costs
when afirm, producing the sample mean output, is broken up into separate
single-product firms. A positive estimate implies economies of scope.
The mixed normal-gamma estimates of column one suggest ray scale
economies up to $15 billion doIlars in assets; beyond that, the estimates are
insignificantly different from ı. The half-normal estimates of column 2 are
nearly identical. The WLS estimates of column 3 imply a greater range of ray
494
A.lvI. YlIengert, Efficiency in life inslIrance
scale economies, up to $38 billion. The TFA estimates of column three imply
a smaller range of economies, up to $1 bil\ion. Again, this understatement of
economies of scale is a result of TFA's failure to account for the presence of
heteroskedastic white noise error. The ditTerences in these estimates are
economical1y significant. According to the normal-gamma, WLS and TFA
estimates, roughly 60%, 30%, and 95% of industry assets are controlled by
scale efficient firms, respectively. None of the estimates of product mix
economies, in panel B of the table, are significant.
Of the two kinds of estimation error discussed here (white noise and X­
inefficiency), X-inefficiency has received most of the recent research attention;
the results presented here suggest that research should not ignore the
potentially significant role of good old-fashioned white noise error in the
generation of observed costs.
Not only do white noise errors play a significant role in observed costs;
they may persist over time. The standard deviation of log average costs in
each of the years 1987-1989 (1.17,1.14,1.18), is roughly six times the average
within-firm standard deviation across all three years (0.16). These persistent
errors could be white noise or X-inefficiency. Future research should explore
the time dimensions of efficiency [e.g., Berger (1993)].
A summary of the efficiency results for life insurance, which are of interest
apart from their implications for the measurement of efficiency, is in order.
There is evidence of ray scale economies up to $15 bil1ion in assets in the life
industry. There is no evidence of any product mix economies. There is a
substantial amount of X-inefficiency in the industry, but the ditTerences
across firm size are insignificant; cost are on average 35-50% above the cost
frontier. These estimates resolve the question: do small firms have higher
costs because of economies of scale, or X-inefficiency? The answer appears to
be: economies of scale.
Several researchers have found significant economies of scale in the life
industry. Houston and Simon (1970), Praetz (1980), Cho (1986), and Grace
and Timme (1992) found economies of scale throughout the sample. Both
Geehan (1977) and Kellner and Mathewson (1983) emphasize unmeasured
ditTerences in the output of large and small insurers. By control1ing for these
ditTerences (Geehan uses a unit cost weighting scheme on an extensive array
of insurance outputs, and Kellner and Mathewson emphasize the role of
sales and advertising), both papers found less evidence of economies of scale.
The results presented here suggest economies of scale, but not throughout
the entire sample. The estimates of weaker scale economies may be due to
the use of additions to reserves, instead of conventional output proxies such
as premiums or total reserves. As evidenced by the stronger scale economy
estimates of the weighted least squares (WLS) methodology, however, the
weaker scale economy estimates are due at least partially to the richer error
specification of the normal-gamma model.
AJv!. Yuengerl, Efficiency in iife inslIrance
I
11
i
I
I
II
495
5. Further challenges
This line of research can proceed in severa! directions. There are important
implications here for firm-level estimates of inefficiency [see Jondrow et aI.
(1982) and Battese and Coel1i (1988)]. Accurate firm-level estimates must
incorporate the systematic differences in both white noise and X-inefficiency
across firms.
The persistence of the estimation errors over time suggests an important
role for panel data. With it, better estimates of the extent of X-inefficiency,
and the role of management and mergers in its determination, can perhaps
be gleaned. Panel data will not, however, obviate the need to distinguish
white noise from X-inefficiency, since white noise errors are as likely to be as
persistent as X-inefficiency. Cornwell et aI. (1990), and Berger (1993) have
explored the possibilities of panel data.
The large white noise variance for the smaller firms in the life insurance
sample suggests that we do not understand very well the processes generat­
ing their costs. There may be some information in the institutional details of
the smaller firms. Management quality may be the most important left-out
variable. The costs or adjustment to new, efficient technologies may also
explain the higher costs or smaller firms, more or whom may be Iiquidity­
and credit-constrained than the larger insurers.
References
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generaHzed frontier production function and panel data, Journal of Econometrics 38,
387-399.
Bauer, P.W., 1990, Recent developments in the econometric estimation of frontiers, Journal of
Econometrics 46, 39-56.
Beckers, O. and C. Hammond, 1987, A tractable likelihood function for the normal-ganuna
stochastic frontier model, Economics Letters 24, 33-38.
Berger, A.N., 1993, 'Oistribution free' estimates of efficiency of the U.S. banking industry and
tests of the standard distributional assumptions, Journal of Productivity Analysis 4
(forthcoming).
Berger, A.N. and O.B. Humphrey, 1991, The dominance of inefficiencies over scale and product
mix economies in banking, Journal of Monetary Economics 28, 117-148.
Berger, A.N., G.A. Hanweck and O.B. Humphrey, 1987, Competitive viability in banking: Scale,
scope, and product mix economies, Journal of Monetary Economics 20, 501-520.
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X-Efficiency in the US life lnSUrance
industry*
Lisa A. Gardner
Old Dominion University, Norfolk, VA 23529, USA
141-163.
Martin F. Grace
assurance, Econometrica 38, 856-864.
Georgia State University, P.O. Box 4036, Atlanta, GA 30302-4036, USA
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inefficiency in the stochastic frontier production model, Journal of Econometrics 19, 233-238.
Kellner, S. and F.G. Mathewson, 1983, Entry, size distribution, scale, and scope economies in
the life insurance industry, Journal of Business 56, 25--44.
Praetz, P., 1980, Returns to scale in the life insurance industry, Journal of Risk and Insurance
47, 525-532.
Schmidt, P. and R.C Sickles, 1984, Production frontiers and panel data, Journal of Business and
Economic Statistics 2, 367-374.
Stevenson, R.E., 1980, Likelihood functions for generalized stochastic frontier estimation, Journal
of Econometrics 13, 57-66.
Yuengert, A.M., 1992, The measurement of efficiency in life insurance: Estimates of a mixed
normal-gamma error model, Research paper no. 9216, Federal Reserve Bank of New York.
Using six years of data, 1985-1990, we estimate hybrid translog cost functions for 561 life
insurers. We examine the resuiting residuals to determine the relative efficiency of insurers in the
sample. We then test the residuals to see if they are related to so-called X-efficiencies because of
internal and external monitoring, or to other factors related to rent-seeking. Results show a
large degree of persistent inefficiency seems to exist among sample firms, the inefficiencies relate
to some internal or external monitoring, and rent-seeking may be occurring.
1. Introduction
The competitive environment faced by US Iife insurers and other financiaI
services providers has undergone a myriad of changes during thepast
decade.Such changes were faciIitated, in part, by financial services deregula­
tion, entry by non-US companies into the US market, interest rate volatility,
and technologicaI advances in information processing. The Iife insurance
industry's responses incIude innovations in product design, a movement from
protection to investment-oriented product writings (i.e., from whole Iife
insurance products to annuities), a heightened IeveI of merger and acquisition
activity, the demutualizations of a few Iarge insurers, and the serious
financiaI problems and insolvencies of others.
Researchers are examining insurer cost structure and industry performance
[Grace and Timme (1991) and Geehan (1986)]. TraditionaI research focuses
Correspondence to: Professor Martin F. Grace, Research Associate, Center for Risk Manage­
ment and Insurance Research, College of Business Administration, Georgia State University,
P.O. Box 4030, Atianta, GA 30302-4036, USA.
*The authors would like to thank the Center for Risk Management and Insurance Research
for support for this project and to Ms. Lori1ee Medders for her research assistance. The authors
would a1so like to thank Eric Anderson, Neil Doherty, Shawna Grosskopf, Harris, Schlesinger,
and Stephen Timme for their valuable comments. All mistakes, errors or omissions remain the
authors' responsibility.
0378--4266/93/$06.00 1993-EIsevier Science Publishers B.V. All rights reserved