Volume 13 Issue 4 Article 3 
12-30-2011 
Adoption of projected mortality table for the Slovenian market 
using the Poisson log-bilinear model to test the minimum 
standard for valuing life annuities 
Darko Medved 
Aleš Ahčan 
Jože Sambt 
Ermanno Pitacco 
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Recommended Citation 
Medved, D., Ahčan, A., Sambt, J., & Pitacco, E. (2011). Adoption of projected mortality table for the 
Slovenian market using the Poisson log-bilinear model to test the minimum standard for valuing life 
annuities. Economic and Business Review, 13(4). https://doi.org/10.15458/2335-4216.1228 
This Original Article is brought to you for free and open access by Economic and Business Review. It has been 
accepted for inclusion in Economic and Business Review by an authorized editor of Economic and Business 
Review. 
251      ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011  |  251–272
ADOPTION OF PROJECTED MORTALITY 
TABLE FOR THE SLOVENIAN MARKET 
USING THE POISSON LOG-BILINEAR 
MODEL TO TEST THE MINIMUM 
STANDARD FOR VALUING LIFE ANNUITIES
darKo MedVed1
aleš ahČan2
JoŽe saMBt3
erManno Pitacco4
aBstract: For the best estimate of life annuity provisions, the longevity risk of the insured 
population must be estimated. In this article, we present an application of the Poisson log-
bilinear model to construct life annuity tables for the Slovenian market. As data on the 
selection effect of annuity owners are not available for the Slovenian market, we have used 
selection statistics from UK data. We then compare those tables with the German annuity 
tables DAV 1994 R, which are the current minimum standard for valuing annuity-related 
liabilities in Slovenia. It is shown that current minimum standard underestimate longevity 
risk of the insured population in Slovenia by 2–4%. 
Keywords: Solvency II, valuation of insurance liabilities, Lee-Carter, mortality projections, life expectancy 
Jel classification: G17, G23, J11
1 introduction 
Solvency II has proposed major changes to the valuation of insurance technical provi-
sions and has had a considerable impact on reserving processes. The Solvency II frame-
work requires a consistent market approach to the valuation of insurance assets and 
liabilities. In such an approach, both assets and liabilities should be valued at the amount 
for which they could be transferred, or settled, between knowledgeable and willing par-
ties (for details, see the Quantitative Impact Study 5 [QIS5]). This is a relatively new con-
1 JMD Consulting, Kamnik, Slovenia, e-mail: darko.medved@jmd-consulting.com
2 University of Ljubljana, Faculty of Economics, Ljubljana, Slovenia e-mail ales.ahcan@ef.uni-lj.si
3 University of Ljubljana, Faculty of Economics, Ljubljana, Slovenia, e-mail joze.sambt@ef.uni-lj.si
4 Università di Trieste, Facoltà di Economia, Trieste , Italy, e-mail ermanno.pitacco@econ.units.it
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011252
cept for insurance companies, as currently, according to existing insurance legislation, 
technical provisions are valued on a book-value basis (e.g., Medved, 2000).
Technical provisions represent the major part of the liability side of insurance compa-
nies’ balance sheet. Insurance companies are required to establish technical provisions 
for all future obligations arising from insurance contracts. There are three main catego-
ries of technical provisions in the insurance industry: non-life-insurance obligations, life 
insurance obligations, and health insurance obligations. 
According to Solvency II, the value of technical provisions is equal to the sum of a best 
estimate and a risk margin. The best estimate is calculated as an expected value of all fu-
ture cash-out and cash-in flows, taking into account the time value of money. Cash-flow 
projections should reflect a realistic expectation of future demographic, legal, medical, 
technological, social, and economic developments. The risk margin is a buffer above the 
best estimate of discounted cash flows, and it protects against worse-than-expected sce-
narios. The risk margin covers model risks, parameter risk, and structural uncertainty. 
For more details on how to calculate the best estimate and risk margin, see the QIS5 
technical specification.
Valuing technical provisions of life annuities depends mainly on projected demographic 
trends. A life annuity is a specific insurance contract in which one party (an insurance 
company), in exchange for payment of a premium, guarantees a series of payments until 
the death of the other party (the insured person). The projection of future mortality 
improvements has significant effects on premium calculation and reserving for life an-
nuities (see Olivieri 2001). As such, annuities are associated with longevity risk, in that 
decreasing mortality rates of the insured population lead to an increase in the number 
of annuity payments. This article addresses the stochastic projections of future demo-
graphic trends in Slovenia as a key parameter of best-estimate valuations of Slovenian 
annuities.
There are no official projected mortality tables for the Slovenian population. To value life 
annuities, insurance companies in Slovenia must use annuity tables that are based on the 
mortality profile of populations in foreign countries. The Slovenian Insurance Supervi-
sion Agency has set the German annuity tables DAV 1994 R as the minimum standard. 
This means that insurance companies have to value their liabilities using DAV 1994 R 
annuity tables; however, they can use other tables, as long as those tables produce higher 
technical provisions than the DAV 1994 R. The result, though, is that the industry stan-
dard is to use the DAV 1994 tables for premium calculation and reserving, and in turn, 
mortality statistics from 1994 on the insured in Germany are used to value liabilities for 
annuities and pensions in Slovenia.   
The DAV 1994 tables were used in the German insurance industry until 2005, when 
the DAV 2004 R tables were introduced (see DAV, 2005). The replacement resulted in 
a 10–20% increase in premiums for deferred annuities in Germany, depending on the 
insured’s age and sex. This is a substantial increase in premium rates, and an important 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 253
question for the Slovenian insurance industry in the Solvency II framework is whether 
the DAV 1994 R tables are still sufficient or even appropriate for measuring the best 
estimate of liabilities from annuities and pensions in Slovenia. We try to answer that 
question here.
To achieve this goal, we implemented the Lee-Carter (LC) method and its extension, 
which is the current standard for actuarial modelling of future mortality. The Lee-Carter 
method is a powerful approach to mortality projections, as it combines a demographic 
model with a time-series model. In a stochastic framework, the results of LC projections 
consist of point and interval estimates. In this respect, the LC method allows for uncer-
tainty in forecasts. 
In this article, we apply both the LC model and the Poisson log-bilinear method, one of 
the latest extensions of the basic LC method, to the case of Slovenia, with mortality data 
for the period 1945–2007. Using a Poisson log-bilinear projection on Slovenian popula-
tion data, we build selection annuity mortality tables for comparison with DAV 1994 R 
and DAV 2005 R. We then use the projected mortality rates for Slovenia to test the cur-
rent minimum standard for valuing annuities. 
The structure of this article is as follows: In Section 2 we present the main features of 
the stochastic LC methodology for projecting mortality. Section 3 covers data specifica-
tion and calibration. In Section 4 we apply the LC and Poisson log-bilinear methods to 
data for Slovenia and present the results, and we also explain how kappa projections are 
calculated. With back-testing, we test the best fit for the projections. In Section 5 we 
calculate the selection effect on annuity purchasers and test the minimum standard for 
valuing annuity liabilities. Section 6 outlines our conclusions. 
2 stochastic Mortality forecasting: theoretical fraMeworK
2.1 The Lee-Carter model 
In 1992 Lee and Carter established new standard for projecting mortality. They proposed 
a simple model for describing the change of mortality as a function of time index. They 
modelled the central death rate using the log-bilinear model (Lee & Carter, 1992)
(1)
where ax describes the average age pattern of mortality over time, and bx describes the 
deviation from the average pattern when kt varies. In addition, kt explains the evolution 
of the level of mortality over time t, and ex,t  
is the error term that reflects the age-specific 
influences not captured by the model. It is assumed that the error term has a mean of 
0 and a standard deviation of σe. The basic LC assumption is that the force of mortal-
ity μx (t) and the central death rate mx (t) coincide, which is a direct consequence of the 
piecewise constant forces of assumed mortality.
The structure of this article is as follows: In Section 2 we present the main features of the 
stochasti  LC methodol gy for projecting mortality. Section 3 covers data specificati n and 
calibration. In Section 4 we apply the LC and Poisson log-bilinear methods to d ta for 
Slovenia and present the results, and we also explain how kappa projections are calculated. 
With back-testing, we test the best fit for the projections. In Section 5 we calculate the 
selection effect on annuity purchasers and test the minimum standard for valuing annuity 
liabilities. Section 6 outlines our conclusions.  
2 STOCHASTIC MORTALITY FORECASTING: THEORETICAL FRAMEWORK 
2.1 The Lee-Carter model 
In 1992 Lee and Carter established new standard for projecting mortality. They proposed a 
simple model for describing the change of mortality as a function of time index. They 
modelled the central death rate using th  log-bilinear model (Lee & Carter, 1992) 
 
,
ln ( )
x x x t x t
m t α β κ ε= + +  (1) 
where α  describes the average age pat ern of mortality over time, and 
x
 describes the 
deviation from the average pattern when 
t
κ  varies. In addition, 
t
κ
 
explains the evolution of 
the level of mortality over time t , and 
,x t
ε
 
is the error term that reflects the age-specific 
influences not captured by the model. It is assumed that the error term has a mean of 0 and a 
standard deviation of 
ε
σ . The basic LC assumption is that the force of mortality ( )
x
tµ  and 
the central death rate ( )
x
m t  coincide, which is a direct consequence of the piecewise constant 
forces of assumed mortality. 
 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011254
The usual approach to estimating the parameters is to use the least-squares method. 
Furthermore, one must impose additional constraints to obtain a unique solution. The 
usual approach is to assume the following:
(2)
which forces ax  to be an average of the log of central death rates over calendar years. Once parameters ax , bx  and kt  are estimated (denoted by ax, bx and k t), we can forecast mortality by modelling the values of k t in the future with a time-series approach (e.g., 
as a random walk with a drift or an autoregressive integrated moving average - ARIMA 
model).
2.2 The Poisson log-bilinear model 
As several authors have noted (e.g., Brouhns et al., 2002; Sithole et al., 2000, Lee, 2000), 
the Lee-Carter method assumes that random errors are homoscedastic. That is, the 
error terms are assumed to have finite variance, and with the assumption of normal-
ity, they share the same underlying probability density function. In most cases, this 
assumption is violated because the logarithm of the observed mortality rate has much 
greater variability at older ages than at younger ages. It is therefore sensible to assume 
that the number of deaths follows the Poisson law with parameter (Brouhns et al., 
2002):
(3)
where Dx,t  is the number of observed deaths of persons aged x  in year t, ETRx,t is the 
central number of persons exposed to risk, and μx (t)  is the force of mortality. The log of the force of mortality is ln μx (t) = ax + bxkt, as in the LC model. The parameters have the 
same meaning as in the LC model. 
The estimates of parameters ax, bx and kt are denoted with ax, bx  and k t, obtained by maximising the log-likelihood in model , which is given by the following:
(4)
Because of the presence of the bilinear term bxkt, it is impossible to estimate the proposed 
model with commercial statistical packages that implement the Poisson regression. An 
option for obtaining the estimates is to use the method proposed by Goodman (1979), 
who suggested the iterative method for estimating log-linear models with bilinear terms. 
With this approach, we define the starting values for parameters as ax0 = 0, bx0 = 0 and 
k t0 = 0. The parameters are then estimated using the following iteration:
The usual approach to estimating the parameters is to use the least-squares method. 
Furthermore, one must impose additional constraints to obtain a unique solution. The usual 
approach is t  assume the f llowing: 
 0
t
t
κ =∑  and  1x
x
β =∑  (2) 
which forces 
x
α
 
to be an average of the log of central death rates over calendar years. Once 
parameters
x
α , 
x
β
 
and 
t
κ
 
are estimated (denoted by 
x
α , 
)
 
and 
t
κ
)
), we can forecast 
mortality by modelling the values of 
t
κ
 
in the future with a time-series approach (e.g., as a 
random walk with a drift or an autoregressive integrated moving average (ARIMA) model. 
2.2 The Poisson log-bilinear model  
As several authors have noted (e.g., Brouhns et al., 2002; Sithole et al., 2000, Lee, 2000), the 
Lee-Carter method assumes that random errors are homoscedastic. That is, the error terms are 
assumed to have finite variance, and with the assumption of normality, they share the same 
underlying probability density function. In most cases, this assumption is violated because the 
logarithm of the observed mortality rate as much greater variability at older ages than at 
younger ages. It is therefore sensible to assume that the number of deaths follows the Poisson 
law with parameter (Brouhns et al., 2002): 
 
, ,
( ( ))
x t x t x
Poisson ETR tµ⋅  (3) 
where 
,x t
 is the number of observed deaths of persons aged x  in year t , 
,x t
TR  is the 
central number of persons exposed to risk, and ( )
x
tµ
 
is the force of mortality. The log of the 
force of mortality is ln ( )
x x x t
tµ α β κ= + , as in the LC model. The parameters have the same 
meaning as in the LC model.  
 
The estimates of parameters 
x
α ,
x
β
 
and 
t
κ
 
are denoted with 
x
α , 
 
and 
t
κ
)
, obtained by 
maxi ising the log-likelihood in model (3), which is given by the following: 
 
, ,
,
( , , ) ( ) exp( ))
x t x x t x t x x t
x t
L D ETRα β κ α β κ α β κ= + − +∑
v
v v
 (4) 
Because of the presence of the bilinear term
x t
κ , it is impossible to estimate the proposed 
model with commercial statistical packages that implement the Poisson regression. An option 
for obtaining the estimates is to use the method prop sed by Goodman (1979), who suggested 
the iterative method for estimating log-linear models with bilinear terms. With this approach, 
we define the starting values for parameters as 
0
0
x
α =
)
, 
0
0
x
β =
)
 
and 
0
0
t
κ =
)
. The parameters 
are then estimated using the following iteration: 
 
, ,
1 1 1
,
1 1
, ,
2 1 2 1 2 1
1 2
,
2 2
, ,
3 2
2 2 2
,
( )
, ,
( )
, ,
( )
( )
,
( )
n
x t x t
n n n n n nt
x x x x t tn
x t
t
n n
x t x t x
n n n n n nt
t t x x x x
n n
x x t
t
n n
x t x t t
n n t
x x n n
t x t
t
D D
D
D D
D
D D
D
α α β β κ κ
β
κ κ β β α α
β
κ
β β κ
κ
+ + +
+ +
+ + + + + +
+
+ +
+ +
+ +
−
= + = =
−
= + = =
−
= +
∑
∑
∑
∑
∑
∑
)
) )
) ) ) )
)
))
) )
) ) ) )
) )
)
)
) )
)
)
)
3 2 3 2
,
n n n n
t t x x
κ α α
+ + + +
= =
) ) )
 (5) 
2.3 Projecting future mortality 
To obtain estimates of future mortality, one must estimate the dynamics of kappa for both 
men and women (Lee-Carter, 1992;). As several authors have noted (e.g., Carter, 1996; Lee, 
2000), 
t
κ
 
can be regarded as a stochastic process, modelled by fitting an ARIMA(p,d,q) 
model. The dynamics of 
t
κ  can thus be described as follows: 
The usual approach to estimating the parameters is to use the least-squares method. 
Furthermore, one must impose additional constraints to obtain a unique solution. The usual 
approach is to assume the following: 
 0
t
t
κ =∑  and  1x
x
β =∑  (2) 
which forces 
x
α
 
to be an average of the log of central death rates over calendar years. Once 
paramete s
x
α , 
x
β
 
and 
t
κ
 
are estimated (denoted by 
x
α , 
)
 
a d 
t
κ
)
), we can forecast 
mortality by modelling the values of 
t
κ
 
in the future with a time-series approach (e.g., as a 
random walk with a drift or an autoregressive integrated moving average (ARIMA) model. 
2.2 The Poisson lo -bilinear m del  
As several authors have noted (e.g., Brouhns et al., 2002; Sithole et al., 2000, Lee, 2000), the 
Lee-Carter method assum s that random errors are homoscedastic. That is, the error terms are
assumed to have finite variance, and with the assumption of normality, they share the same 
underlying probability density function. In most cases, this assumption is violated because the 
logarithm of the observed mortality rate has much greater variability at older ages than at 
younger ages. It is therefore sensible to assume that the number of deaths follows the Poisson 
law with parameter (Brouhns et al., 2002): 
 
, ,
( ( ))
x t x t x
D Poisson ETR tµ⋅  (3) 
where 
,x t
 is the number of observed deaths of persons aged x  in year t , 
,x t
TR  is the 
central number of pers ns exposed to risk, and ( )
x
tµ
 
is the force of mortality. The log of the 
force of mortality is ln ( )
x x x t
tµ α β κ= + , as in the LC model. The parameters have the same 
meaning as in the LC model.  
∼
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 255
(5)
2.3 Projecting future mortality
To obtain estimates of future mortality, one must estimate the dynamics of kappa for 
both men and women (Lee-Carter, 1992). As several authors have noted (e.g., Carter, 
1996; Lee, 2000), kt can be regarded as a stochastic process, modelled by fitting an 
ARIMA(p,d,q) model. The dynamics of kt can thus be described as follows:
(6)
where ϕp ≠ 0, ψq ≠ 0, and ξt is a Gaussian white-noise process, such that σξ2 > 0. In most 
instances, the appropriate time-series model takes a simpler form, such as kt = kt–1 + c + 
ξt + ψξt–1 derived from model ARIMA(0,1,1). The constant term c  indicates the average 
annual change of tk  and presents the forecasts of the long-term change in mortality. On 
the basis of the results of the time-series model, we can obtain forecasts of future mortal-
ity and its moments from the following:
(7)
3 data
Population mortality data by age and sex for the period 1971–2008 were provided by the 
Statistical Office of the Republic of Slovenia. For the 1971–1980 period, there are some 
minor discrepancies between the data used and the official cumulative data for the same 
years, especially for 1972 and 1973. The discrepancies are in the range of 10 persons, 
which is negligible. 
Again, we denote by ETRx,t the exposure to risk at age x on the last birthday during year t. 
The exposure to risk refers to the total number of person-years in a given population over 
a calendar year, and it is estimated by the number of the population aged x in the middle 
of the calendar year (1 July of each year); that is, those who reached age x between 1 July 
of the previous year and 30 June of the observing year. By Dx,t  we denote the number of 
 
The estimates of parameters 
x
α , 
x
β
 
and 
t
κ
 
are denoted with 
x
α , 
 
and 
t
κ
)
, obtained by 
maximising the log-likelihood in model (3), which is given by the following: 
 
, ,
,
( , , ) ( ) exp( ))
x t x x t x t x x t
x t
L D ETRα β κ α β κ α β κ= + − +∑
v
v v
 (4) 
Because of the presence of the bilinear term
x t
κ , it is impossible to estimate the proposed 
model with commercial statistical packages that implement the Poisson regression. An option 
for obtaining the estimates is to use the method proposed by Goodman (1979), who suggested 
the iterative method for estimating log-linear models with bilinear terms. With this approach, 
we define the starting values for parameters as 
0
0
x
α =
)
, 
0
0
x
β =
)
 
and 
0
0
t
κ =
)
. The parameters 
are then estimated using the following iteration: 
 
, ,
1 1 1
,
1 1
, ,
2 1 2 1 2 1
1 2
,
2 2
, ,
3 2
2 2 2
,
( )
, ,
( )
, ,
( )
( )
,
( )
n
x t x t
n n n n n nt
x x x x t tn
x t
t
n n
x t x t x
n n n n n nt
t t x x x x
n n
x x t
t
n n
x t x t t
n n t
x x n n
t x t
t
D D
D
D D
D
D D
D
α α β β κ κ
β
κ κ β β α α
β
κ
β β κ
κ
+ + +
+ +
+ + + + + +
+
+ +
+ +
+ +
−
= + = =
−
= + = =
−
= +
∑
∑
∑
∑
∑
∑
)
) )
) ) ) )
)
))
) )
) ) ) )
) )
)
)
) )
)
)
)
3 2 3 2
,
n n n n
t t x x
κ α α
+ + + +
= =
) ) )
 (5) 
2.3 Projecting future mortality 
To obtain estimates of future mortality, one must estimate the dynamics of kappa for both 
men and women (Lee-Carter, 1992;). As several authors have noted (e.g., Carter, 1996; Lee, 
2000), 
t
κ
 
can be regarded as a stochastic process, modelled by fitting an ARIMA(p,d,q) 
model. The dynamics of 
t
κ  can thus be described as follows: 
 
1 1 1
...
d d d
t t p t t t q t q
κ ϕ κ ϕ κ ξ ψ ξ ψ ξ
− −
∇ = ∇ + + ∇ + + +  (6) 
wher  0
p
ϕ ≠ , 0
q
ψ ≠ , and 
t
ξ is a Gaussian white-noise process, such that 
2
0
ξ
σ > . In most 
instances, the appropriate time-series model takes a simpler form, such as 
1 1t t t t
cκ κ ξ ψξ
− −
= + + +  derived from model ARIMA(0,1,1). The constant term c  indicates 
th  average annual change of 
t
κ  and pres nts the f r casts of t e long-term change in 
mortality. On the basis of the results of the time-series model, we can obtain forecasts of 
future mortality and its moments from the follo ing: 
 ( ) exp( )exp( )
x x x t n
t nµ α β κ
+
+ =  (7) 
3 DATA 
Population mortality data by age and sex for the period 1971–2008 were provided by the 
Statistical Office of the Republic of Slovenia. For the 1971–1980 period, there are some 
minor discrepancies between the data used and the official cumulative data for the same years, 
especially f r 1972 and 1973. The discrepancies are in the range of 10 persons, which is 
negligible.  
 
Again, we denote by 
,x t
TR  the exposure to risk at age x on the last birthday during year t. 
The exposure to risk refers to the total number of person-years in a given population over a 
calendar year, and it is estimated by the number of the population aged x in the middle of the 
calendar year (1 July of each year); that is, those who reached age x between 1 July of the 
previous year and 30 June of the observing year. By 
,x t
 we denote the number of deaths 
recorded at age x last birthday during calendar year t. Then, the maximum likelihood 
estimator for ( )
x
xµ
)
(force of mortality) equals 
 
1 1 1
...
d d d
t t p t t t q t q
κ ϕ κ ϕ κ ξ ψ ξ ψ ξ
− −
∇ = ∇ + + ∇ + + +  (6) 
where 0
p
ϕ ≠ , 0
q
ψ ≠ , and 
t
ξ is a Gaussian white-noise process, such that 
2
0
ξ
σ > . In most 
instances, the appropriate time-series model takes a simpler form, such as 
1 1t t t t
cκ κ ξ ψξ
− −
= + + +  derived from model ARIMA(0,1,1). The constant term c  indicates 
the average annual change of 
t
κ  and presents the forecasts of the long-term change in 
mortality. On the basis of the results of the time-series model, we can obtain forecasts of 
future mortality and its moments from the following: 
 ( ) exp( )exp( )
x x x t n
t nµ α β κ
+
+ =  (7) 
3 DATA 
Population mortality data by age and sex for the period 1971–2008 were provided by the 
Statistical Office of the Republic of Slovenia. For the 1971–1980 period, there are some 
minor discrepancies between the data used and the official cumulative data for the same years, 
especially for 1972 and 1973. The discrepancies are in the range of 10 persons, which is 
negligible.  
 
Again, we denote by 
,x t
TR  the exposure to risk at age x on the last birthday during year t. 
The exposure to risk refers to the total number of person-years in a given population over a 
calendar year, and it is estimated by the number of the population aged x in the middle of the 
calendar year (1 July of each year); that is, those who reached age x between 1 July of the 
previous year and 30 June of the observing year. By 
,x t
 we denote the number of deaths 
recorded at age x last birthday during calendar year t. Then, the maximum likelihood 
estimator for ( )
x
xµ
)
(force of mortality) equals 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011256
deaths recorded at age x last birthday during calendar year t. Then, the maximum likeli-
hood estimator for μx(x) (force of mortality) equals
(8)
Assuming a constant force of mortality for noninteger years, we have μx(t) = mx(t) (see 
Pitacco et al., 2009). With this assumption, we can construct Slovenian mortality data 
for the period 1971–2008.
For our analysis, we also needed mortality data for the Slovenian population before 1971. 
The Human Mortality Database (HMD) contains average central mortality rates for 
1930–1933, 1948–1952, 1952–1954, and 1960–1962. We used this information to interpo-
late mx(t) for the years 1945–1970. We calculated a log regression line between 1932 and 
1985 and made an interpolation between 1945 and 1970 using a 95% confidence interval. 
We chose 1945 as a starting year for our analysis because the generation born before this 
year is highly likely to have already retired.  
Slovenian mortality data have some irregularities that had to be adjusted before we could 
use them for forecasting. For example, at very old ages, the data have very low risk ex-
posures, which leads to large sampling errors and highly volatile crude death rates. For 
example, risk exposures for men varied between 567 in 1971 and 1300 in 2007. For the 
1971–1980 period, no data are available for age groups older than 85. Therefore, we need-
ed a method to extrapolate a survival function at very old ages, without requiring  accu-
rate mortality data for that part of the population. Recent mortality studies suggest that 
the force of mortality is slowly increasing at very old ages and approaching a relatively 
flat shape (e.g., Pitacco et al., 2009). In other words, the exponential rate of the mortality 
increase at very old ages is not constant but declines.
We apply the method that Denuit and Goderniaux (2005) proposed to extrapolate death 
rates at very old ages. Following this approach, the death rates for very old ages were 
estimated according to the logistic formula proposed here. Parameters were chosen in 
such a way as to maximise the fit. 
The log-quadratic regression model is defined as follows:
(9)
where the one-year death probability at time t with ex,1 is independent and normally 
distributed, with a mean of 0 and variance of σ2. If w  is an age limit, then we have a con-
straint: qw(t) = 1. To ensure the concave behaviour of ln q̂x(t), we implemented a second 
constraint:
(10)
 
,
,
( )
x t
x
x t
D
t
ETR
µ =
)
 (8) 
Assuming a constant force of mortality for noninteger years, we have ( ) ( )
x x
t m tµ =
) )
 
(see 
Pitacco et al., 2009). With this assumption, we can construct Slovenian mortality data for the 
period 1971–2008. 
 
For our analysis, we also needed m rtality data for the Slovenian populati n before 1971. The 
Human Mortality Database (HMD) contains average central mortality rates for 1930–1933, 
1948–1952, 1952–1954, and 1960–1962. We used this information to interpolate ( )
x
m t
)
 for 
the years 1945–1970. We calculated a log regression line between 1932 and 1985 and made 
an interpolation between 1945 and 1970 using a 95% confidence interval. We chose 1945 as a 
starting year for our analysis because the generation born before this year is highly likely to 
have already retired.   
 
Slovenian mort lity data have some irregul rities that had to be adjust d before we could use 
them for forecasting. For example, at very old ages, the data have very low risk exposures, 
which leads to large sampling errors and highly volatile crude death rates. For example, risk 
exposures for men varied between 567 in 1971 and 1300 in 2007. For the 1971–1980 period, 
no data are available for age groups older than 85. Therefore, we needed a method to 
extrapolate a surviv l function at very old ages, without requiring  accurate mortality data for 
that part of the population. Recent mortality studies suggest that the force of mortality is 
slowly increasing at very old ages and approaching a relatively flat shape (e.g., Pitacco et al., 
2009). In other words, the exponential rate of the mortality increase at very old ages is not 
constant but declines. 
 
We apply the method that Denuit and Goderniaux (2005) proposed to extrapolate death rates
at very old ages. Following this approach, the death rates for very old ages were estimated 
according to the logistic formula proposed here. Parameters were chosen in such a way as to 
maximise the fit.  
 
The log-quadratic regression model is defined as follows: 
 
2
,
ˆln ( )
x t t t x t
q t a b x c x ε= + + +  (9) 
where the one-y a  death probability at time t  ith 
,x t
ε  is independent and normally 
distributed, with a mean of 0 and variance of 
2
σ . If ω  is an age limit, then we have a 
constraint: ( ) 1q t
ω
= . To ensure the concave behaviour of ˆln ( )
x
q t , we implemented a second 
constraint: 
 ( ) 0
x x
q t
x
ω=
∂
=
∂
 (10) 
We obtained the optimal fit (highest
2
) with 130ω =  and a starting smoothing age 75x = . 
However, we used only the results that we obtained from age 85 onward.  
 
Furthermore, some deaths rates were equal to 0 (i.e., there were no deaths in the observed 
period), which happens quite often at younger ages because of the small population. Because 
we used logarithms of death rates for forecasting, we implemented adjustment techniques to 
obtain positive values. In particular, we used interpolation techniques with neighbour central 
death rates to obtain the best estimate for such cases.  
We apply the method that Denuit and Goderniaux (2005) proposed to extrapolat  death rates 
at very old ages. Following this approach, the death rates for very old ages were estimated 
according to the logistic formula proposed here. Parameters were chosen in such a way as to 
maximise the fit.  
 
The log-quadratic regression model is defined as follows: 
 
2
,
ˆln ( )
x t t t x t
q t a b x c x ε= + + +  (9) 
where the one-year death probability at time t  with 
,x t
ε  is independent and normally 
distributed, with a mean of 0 and variance of 
2
σ . If ω  is an age limit, then we have a 
constraint: ( ) 1q t
ω
= . To ensure the concave behaviour of ˆln ( )
x
q t , we implemented a second 
constraint: 
 ( ) 0
x x
q t
x
ω=
∂
=
∂
 (10) 
We obtained the optimal fit (highest
2
) with 130ω =  and a starting smoothing age 75x = . 
However, we used only the results that we obtained from age 85 onward.  
 
Furthermore, some deaths rates were equal to 0 (i.e., there were no deaths in the observed 
period), which happens quite often at younger ages because of the small population. Because 
we used logarithms of death rates for forecasting, we implemented adjustment techniques to 
obtain positive values. In particular, we used interpolation techniques with neighbour central 
death rates to obtain the best estimate for such cases.  
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 257
We obtained the optimal fit (highest R2) with w = 130 and a starting smoothing age x = 
130. However, we used only the results that we obtained from age 85 onward. 
Furthermore, some deaths rates were equal to 0 (i.e., there were no deaths in the ob-
served period), which happens quite often at younger ages because of the small pop-
ulation. Because we used logarithms of death rates for forecasting, we implemented 
adjustment techniques to obtain positive values. In particular, we used interpolation 
techniques with neighbour central death rates to obtain the best estimate for such 
cases. 
4 forecasting Mortality using PoPulation Mortality 
statistics for sloVenia
In this section, we present the results using the stochastic methods introduced in Sec-
tion 2. On the basis of back-testing, we decided on two methods for projecting future 
mortality: the Lee-Carter model and the Poisson log-bilinear model based on the data 
from 1971 to 2008 described in Section 3. The code for this section was programmed 
using Matlab software. 
4.1 The original Lee-Carter model vs. the Poisson log-bilinear model
First, we present the results of the model introduced in Brouhns et al. (2002) using 
the Poisson log-bilinear regression approach. The results of that model are compared 
with the results of the original Lee-Carter (1992) model. In the analysis we used data 
from the Slovenian Statistical Office, adjusted as described in Section 3. The results 
obtained from the Poisson log-bilinear model are represented in Figure 1 by a green 
(light) line, and the results obtained from the original Lee-Carter model are repre-
sented by a blue (dark) line. 
As Figure 1 shows, the betas from both methods exhibit highly erratic behaviour, re-
gardless of the method used. This is mainly a consequence of the small population and 
relatively low number of both exposures and deaths in the Slovenian population as 
compared with larger countries. Figure 1 shows that over the past 40 years, the biggest 
improvements in mortality were in the age group of minors, especially newborns and 
children between the ages of 10 and 14 years. In this age group, the discrepancy be-
tween the LC and Poisson log-bilinear models is also greatest. In the case of the Poisson 
log-bilinear model, beta is somewhat larger for this age group than in the LC method, 
and it is slightly lower for most of the other age groups. The betas indicate trends similar 
to those in other countries, with mortality improvements being greatest in the lower age 
groups. 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011258
Figure 1: Beta(x) as a function of age (male): the Poisson vs. the LC model
Figure 2: Alpha as a function of age (male): the Poisson vs. the LC model
The alphas in both models indicate that there are hardly any differences in the calculated 
values. The only difference is a small discrepancy in alphas for children between the 
ages of 2 and 10 years. Overall, the results of both models are similar to those for other 
Figure 1: Beta(x) as a function of age (male): the Poisson vs. the LC model 
0 20 40 60 80 100 120
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
age
b
e
t
a
(
x
)
male
 
As Figure 1 shows, the betas from both methods exhibit highly erratic behaviour, regardless 
of the method used. This is mainly a consequence of the small population and relatively low 
number of both exposures and deaths in the Slovenian population as compared with larger 
countries. Figure 1 shows that over the past 40 years, the biggest improvements in mortality 
were in the age group of minors, especially newborns and children between the ages of 10 and 
14 years. In this age group, the discrepancy between the LC and Poisson log-bilinear models 
is also greatest. In the case of the Poisson log-bilinear model, beta is somewhat larger for this 
age group than in the LC method, and it is slightly lower for most of the other age groups. The 
betas indicate trends similar to those in other countries, with mortality improvements being 
greatest in the lower age groups.  
 
Figure 2: Alpha as a function of age (male): the Poisson vs. the LC model 
0 20 40 60 80 100 120
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
age
a
l
p
h
a
(
x
)
male
 
The alphas in both models indicate that there are hardly any differences in the calculated 
values. The only difference is a small discrepancy in alphas for children between the ages of 2 
and 10 years. Overall, the results of both models are similar to those for other countries. 
Mortality is relatively high for newborns and drops considerably for minors. For male 
teenagers and young adults, mortality increases with a noticeable hump (i.e., the testosterone 
hump) around age 20. After that age, mortality is constant or slightly decreases until age 30, 
when it starts to increase again almost linearly into the oldest ages, as observed by Cairns et 
al. (2006). 
 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 259
countries. Mortality is relatively high for newborns and drops considerably for minors. 
For male teenagers and young adults, mortality increases with a noticeable hump (i.e., 
the testosterone hump) around age 20. After that age, mortality is constant or slightly 
decreases until age 30, when it starts to increase again almost linearly into the oldest 
ages, as observed by Cairns et al. (2006).
Figure 3: Kappa as a function of year (male): the Poisson vs. the LC model
Figure 3 shows that with both methods, kappa decreases substantially over the observed 
period (1971–2008). During that time, there is a continuous improvement in mortality 
for all age groups (see also Figure 1). Given the average value of beta of 0.01, mortality 
has, on average, more than halved in the observed period. Of course, some age groups 
(e.g., minors) experienced a much greater mortality decline than the average, whereas 
older age groups had improvements below or well below the average. Looking at the 
trend for kappa, we can see that with the LC method, kappa decreases almost linearly, 
whereas for the Poisson log-bilinear model, it seems to increase at an even higher rate 
and exhibits a mild curvature. 
Turning to the results for females, the trend in kappa is similar to that for males. Ka-
ppa decreases almost linearly over time under both methods. Again, both methods 
yield similar results, with only slight differences in calculated values for some years. 
Overall, we can conclude that the results of the two methods are relatively robust for 
the values of kappa. The results for females’ alpha, beta, and kappa are presented in 
Appendix 1. 
Figure 3: Kappa as a function of year (male): the Poisson vs. the LC model 
1970 1975 1980 1985 1990 1995 2000 2005 2010
-60
-50
-40
-30
-20
-10
0
10
20
30
40
year
k
a
p
p
a
(
x
)
male
 
Figure 3 shows that with both methods, kappa decreases substantially over the observed 
period (1971–2008). During that time, there is a continuous improvement in mortality for all 
age groups (see also Figure 1). Given the average value of beta of 0.01, mortality has, on 
average, more than halved in the observed period. Of course, some age groups (e.g., minors) 
experienced a much greater mortality decline than the average, whereas older age groups had 
improveme ts below or well below the average. Look ng at the trend for kappa, we can see 
that with the LC method, kappa decreases almost linearly, whereas for the Poisson log-
bilinear model, it seems to increase at an even higher rate and exhibits a mild curvature.  
 
Turning to the results for females, the trend in kappa is similar to that for males. Kappa 
decreases almost linearly over time under both methods. Again, both methods yield similar 
results, with only slight differences in calculated values for some years. Overall, we can 
conclude that the results of the two methods are relatively robust for the values of kappa. The 
results for females’ alpha, beta, and kappa are presented in Appendix 1.  
 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011260
4.2 Projecting kappa using the Poisson log-bilinear model
In this section, we present the results of projecting kappa using the Poisson log-bilinear 
model (Brouhns et al., 2002). In this case, the value of c is equal to –2.43 (for a definition 
of c, see Section 2.3). To check the validity of the model, we examined the statistical prop-
erties of the residuals. Table 5 (in Appendix 2) shows that we cannot reject the hypothesis 
of normally distributed residuals. Both the Jarque-Berra test and the values of kurtosis 
and skewness indicate that the normal distribution is a good approximation for the dis-
tribution of the residuals. Furthermore, in looking at Q-statistics for autocorrelation, we 
observed that there is no statistically significant autocorrelation. 
The previous analysis suggested that for Slovenian mortality statistics, a random walk with 
drift model is suitable for modelling the estimated kt . This is not surprising, as a similar 
observation can be found in many other countries (see, e.g., Tuljapurkar et al., 2000).
Figure 4: Kappa(t) as a function of time (males): the Poisson model
Figure 4 reveals a fairly strong downward trend in mortality. For the low-mortality sce-
nario, the trend is somewhat higher as a result of the greater-than-expected decrease 
in kappa. Likewise, for the high-mortality scenario, the decrease in kappa is somewhat 
lower. We can see that in both cases the trend is negative, which means that the reduction 
in kappa is highly statistically significant. 
Let us now consider the result for females. Again, the ARIMA(0,1,1) proves adequate for 
modelling the dynamics of kappa.
Figure 4: Kappa(t) as a function of time (males): the Poisson model 
1960 1980 2000 2020 2040 2060 2080 2100
-350
-300
-250
-200
-150
-100
-50
0
50
year
k
a
p
p
a
(
t
)
scenarios male
 
 
Figure 4 reveals a fairly strong downward trend in mortality. For the low-mortality scenario, 
the trend is somewhat higher as a result of the greater-than-expected decrease in kappa. 
Likewise, for the high-mortality scenario, the decrease in kappa is somewhat lower. We can 
see that in both cases the trend is negative, which means that the reduction in kappa is highly 
statistically significant.  
 
Let us now consider the result for females. Again, the ARIMA(0,1,1) proves adequate for 
modelling the dynamics of kappa. 
 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 261
Figure 5: Kappa(t) as a function of time (females): the Poisson model
As one can see from the corresponding table in Appendix 2, the residuals obtained us-
ing the ARIMA(0,1,1) model are uncorrelated and approximately normally distributed. 
Figure 5 shows that the difference between scenarios increases in time, but the estimates 
remain fairly close.
4.3 Projecting kappa using the LC model
We now consider the case of modelling kappa under the original Lee-Carter model. We 
can see that the simple ARIMA(0,1,0) process seems to be appropriate for the estimated 
tk  for males. The results, presented in Table 5 (in Appendix 2), indicate that the residu-
als are not autocorrelated, whereas the skewness, kurtosis, and Jarque-Berra tests indi-
cate that the hypothesis of normally distributed residuals cannot be rejected. Another 
important consequence of the residual test is that, in the LC model, the standard devia-
tion of residuals is greater than in the Poisson log-bilinear model. The difference between 
the forecasts is thus much smaller in the case of the Poisson log-bilinear model.
The results reveal that the trend is somewhat lower than for the Poisson model; in con-
trast, though, the standard deviation of the residuals is greater than in the Poisson mod-
el. The values for kappa under the high-mortality scenario are thus greater in the original 
LC model than in the Poisson model. The same goes for the central tendency scenario. 
The values for the low-mortality scenario in the LC model are comparable with the val-
ues in the Poisson model. 
Figure 5: Kappa(t) as a function of time (females): the Poisson model 
1960 1980 2000 2020 2040 2060 2080 2100
-300
-250
-200
-150
-100
-50
0
50
year
k
a
p
p
a
(
t
)
scenarios female
 
As one can see from the corresponding table in Appendix 2, the residuals obtained using the 
ARIMA(0,1,1) model are uncorrelated and approximately normally distributed. Figure 5 
shows that the difference between scenarios increases in time, but the estimates remain fairly 
close. 
4.3 Projecting kappa using the LC model 
We now consider the case of modelling kappa under the original Lee-Carter model. We can 
see that the simple ARIMA(0,1,0) process seems to be appropriate for the estimated 
t
κ  for 
males. The results, presented in Table 5 (in Appendix 2), indicate that the residuals are not 
autocorrelated, whereas the skewness, kurtosis, and Jarque-Berra tests indicate that the 
hypothesis of normally distributed residuals cannot be rejected. Another important 
consequence of the residual test is that, in the LC model, the standard deviation of residuals is 
greater than in the Poisson log-bilinear model. The difference between the forecasts is thus 
much smaller in the case of the Poisson log-bilinear model. 
 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011262
In contrast to males, the dynamics for females cannot be modelled as an ARIMA(0,1,0) 
model but can be fitted best by an ARMA(2,2) model. In this case, as presented in Table 
6 (in Appendix 2), the residuals are not autocorrelated (at least up to the relevant number 
of lags) and can be assumed to be normally distributed. 
Once again, we observe that the standard deviation of kappa estimates is significantly 
higher than in the Poisson log-bilinear model. This could be an argument in favour of 
using Brouhns et al. to model future mortality on Slovenian mortality statistics. Namely, 
with the Poisson log-bilinear model, the confidence interval of estimates is much nar-
rower than in the LC model.
4.4 Back-testing
For future projections, we needed a quantitative assessment of both models before decid-
ing which model would better capture the mortality trend. Of course, we could not do 
this graphically, so we used the back-testing technique to determine which model would 
be most appropriate for estimating future mortality. More precisely, we tested the models 
against real data (in our case, number of deaths) for the period 2001–2008. In the first 
step, we fit the model parameters to the data for the period 1971–2000. In the second step, 
we used the values of the parameters obtained in step 1 to predict the number of deaths 
Dx,t in the period 2001–2008. We then used several standard indicators of fit to compare 
the methods. 
Table 1: Comparison of methods using back-testing for the 2001–2008 period (males)
LC Poisson log-bilinear
MSE 21 20.8
MPE 11.7 11.6
R2 0.955 0.96
Source: SORS
First, we examine the results for males for the period 2001–2008. A comparison of the 
methods using back-testing for that period (males) in Table 1 reveals the supremacy of 
the Poisson log-bilinear method, which is even more convincing for females. Of all vari-
ation in the number of deaths for the period 2001–2008, 99% can be explained by this 
method (see Table 2). 
Table 2: Comparison of methods using back-testing for the 2001–2008 period (females)
LC Poisson log-bilinear
MSE 18.5 11.5
MPE 11.2 7
R∧2 0.975 0.990
Source: SORS
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 263
On the basis of the back-testing analysis, we concluded that the Poisson log-bilinear 
model fit the actual central death rates better than the original LC model, and therefore 
we used that model to forecast Slovenian population mortality.
Thus, when projecting the values of kappa under different scenarios for obtaining the 
estimates of future mortality, we can use the following relationship:
(11)
where mx (2008 + t) is the central death rate for year (2008 + t) and age x. Formula  is 
essentially an extrapolation of the classical Lee-Carter model using the projections of 
kappa obtained from ARIMA models. In determining future mortality, we must take 
into account the uncertainty of our estimates. We therefore constructed three scenarios 
that differ with respect to kappa values used when making projections. Under the best-
estimate scenario, we determined future values of kappa by taking kappa to be equal to 
the expected value derived from the ARIMA model. Using equation , we yielded future 
values of kappa through the following relationship: k2008+t = k2008 + ct.
In the high-mortality scenario, we obtained future values of kappa by using the follow-
ing relationship: k2008+t = k2008 + ct + 2σe √t . In the low-mortality scenario, we obtained 
future values of kappa by assuming lower-than-expected values of kappa. In this case, we 
obtained the future values of kappa with k2008+t = k2008 + ct – 2σe √t. 
5 testing the Best-estiMate Valuation of a life annuity
5.1 Cohort vs. period life tables
To calculate the present value of a future obligation arising from life annuity payments, 
actuaries must develop a life table (also called a mortality table or actuarial table). A life 
table shows for each age x the probability that a person of that age will die before his or 
her next birthday (denoted as qx). Life tables are derived from observed and projected 
mortality rates, which can be presented in the following matrix: 
(12)
where 
{qx(t)}, t ∈ (t0,...,tn) represents observed mortality rates, and {qx(t)}, t ∈ (tn + 1,...,tmax) 
represents projected mortality rates. By tn we denote the base year from which projec-
tions are made. The sequence qx(t), qx+1(t + 1),...  is a cohort table. The sequence qx(t), qx+1(t), qx+2(t)... is a period table. This leads to the construction of two types of life tables. 
 
Thus, when projecting the values of kappa under different scenarios for obtaining the 
estimates of future mortality, we can use the following relationship: 
 
2008 2008
(2008 ) (2008) exp( ( ))
x x x t
m t m β κ κ
+
+ = −  (11) 
where (2008 )
x
m t+ is the central death rate for year (2008 + t) and age x. Formula (11) is 
essentially an extrapolation of the classical Lee-Carter model using the proj ctio s of kappa 
obt ined from ARIMA models. In determining future mortality, we must take into account the 
uncertainty of our estimates. We therefore constructed three scenarios that differ with respect 
to kappa values used when making projections. Under the best-estimate scenario, we 
determined future values of kappa by taking kappa to be equal to the expected value derived 
from the ARIMA model. Using equation (11), we yielded future values of kappa through the 
following relationship: 
2008 2008t
ctκ κ
+
= + .  
 
In the high-mortality scenario, we obtained future values of kappa by using the following 
relationship: 
2008 2008
2
t
ct t
ε
κ κ σ
+
= + + . In the low-mort lity scenario, we obtained future 
values of kappa by assuming lower-than-expected values of kappa. In this case, we obtained 
the future values of kappa with 
2008 2008
2
t
ct t
ε
κ κ σ
+
= + − .  
5 TESTING THE BEST-ESTIMATE VALUATION OF A LIFE ANNUITY 
5.1 Cohort vs. period life tables 
To calculate the present value of a future obligation arising from life annuity payments, 
actuaries must develop a life table (also called a mortality table or actuarial table). A life table 
shows for each age x the probability that a person of that age will die before his or her next 
birthday (denoted as q ). Life table  are d rived from observ d and pr ject d mortality rates, 
which can be resented in the f llowing matrix:  
 
min min min
max max max
0 max
0 max
( ) ( ) ( )
( ) ( ) ( )
x x n x
x x n x
q t q t q t
q t q t q t
 
 
 
 
 
 
 
L L
M
M
 (12) 
where  
}
0
( ) , ( , , )
x n
q t t t t∈ K represents observed mortality rates, and 
max
( ) , ( 1, , )
x n
q t t t t∈ + K represents projected mortality rates. By 
n
t  we denote the base year 
from which projections are made. The sequence 
1
( ), ( 1),
x x
q t q t
+
+ K
 
is a cohort table. The 
sequence 
1 2
( ), ( ), ( )
x x x
q t q t q t
+ +
K  is a period table. This leads to the construction of two types 
of life tables. In a period life table, the present value of a life annuity takes into account age-
specific mortality rates at age x and older ages observed in a given calendar year. So a period 
life table includes different generations in a single table. In a cohort life table, a life annuity is 
calculated on the basis of observed birth cohort, which means that for each generation, we can 
construct one life table. Cohort life tables better explain improvements in mortality for each 
generation separately, so for best-estimate calculations, cohort life tables are used (see Pitacco 
et al., 2009). 
To calculate the age cohort life table, we first chose the base cohort birth year τ . We then 
calculated the life table by taking diagonal probabilities from birth year τ as follows: 
 
1
( ) (1 ( )) ( )
x x x
l q x lτ τ τ
+
= − + ⋅  (13) 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011264
In a period life table, the present value of a life annuity takes into account age-specific 
mortality rates at age x and older ages observed in a given calendar year. So a period life 
table includes different generations in a single table. In a cohort life table, a life annuity is 
calculated on the basis of observed birth cohort, which means that for each generation, 
we can construct one life table. Cohort life tables better explain improvements in mortal-
ity for each generation separately, so for best-estimate calculations, cohort life tables are 
used (see Pitacco et al., 2009).
To calculate the age cohort life table, we first chose the base cohort birth year τ . We then 
calculated the life table by taking diagonal probabilities from birth year τ as follows:
(13)
We can calculate an annuity of size 1 that is payable yearly at the beginning of each year 
while an insured is alive from the following:
(14)
5.2 Selection effect
The standardised mortality ratio (SMR) is used as an index for comparing mortality 
experiences between two groups: actual deaths in a particular population (e.g., life an-
nuity owners) with expected deaths, if “standard” age-specific mortality rates were to be 
applied. The SMR is defined as follows:
(15)
A life annuity purchaser is, most likely, a healthy person with particularly low mortal-
ity in the first years of the life annuity payment and, in general, a longer-than-average 
expected lifetime. Therefore, to calculate the best estimate of an insurance annuity, we 
had to adjust mortality projections to include this effect in the projections. Pitacco et al. 
(2009) suggest the following model for age 60 and older:
(16)
Here we denote by mxHMD (t) the population central death rates and by mxLIM (t) the annu-
ity purchaser central death rates. This leads us to produce SMR in the form e f̂(x), which 
can be used to adopt mortality projections for the insurance market. Slovenia mortality 
experience for annuity purchasers is not directly available, and because pension reform 
started only 10 years ago, there are no adequate statistical data for the conclusion regard-
birthday (denoted as q ). Life tables are derived from observed and projected mortality rates, 
which can be presented in the following matrix:  
 
min min min
max max max
0 max
0 max
( ) ( ) ( )
( ) ( ) ( )
x x n x
x x n x
q t q t q t
q t q t q t
 
 
 
 
 
 
 
L L
M
M
 (12) 
where  
}
0
( ) , ( , , )
x n
q t t t t∈ K represents observed mortality rates, and 
max
( ) , ( 1, , )
x n
q t t t t∈ + K represents projected mortality rates. By 
n
t  we denote the base year 
from which projections are made. The sequence 
1
( ), ( 1),
x x
q t q t
+
+ K
 
is a cohort table. The 
sequence 
1 2
( ), ( ), ( )
x x x
q t q t q t
+ +
K  is a period table. This leads to the construction of two types 
of life tables. In a period life table, the present value of a life annuity takes into account age-
specific mortality rates at age x and older ages observed in a given calendar year. So a period 
life table includes different generations in a single table. In a cohort life table, a life annuity is 
calculated on the basis of observed birth cohort, which means that for each generation, we can 
construct one life table. Cohort life tables better explain improvements in mortality for each 
generation separately, so for best-estimate calculations, cohort life tables are used (see Pitacco 
et al., 2009). 
To calculate the ag  c hort life table, w  fir t chose the base cohort birth year τ . We then 
calculated the life table by taking diagonal probabilities from birth year τ as follows: 
 
1
( ) (1 ( )) ( )
x x x
l q x lτ τ τ
+
= − + ⋅  (13) 
We can calculate an annuity of size 1 that is payable yearly at the beginning of each year 
while an insured is alive from the following: 
 
1
0
0
1, 0
( ) (1 )
(1 ( ), 0
x
kk
x
x jk
j
k
a i
q x j k
ω
τ
τ
−
−
−
+=
=
= 
 
= ⋅ + 
− + + >
 
 
∑
∏
&&  (14) 
5.2 Selection effect 
The standardised mortality ratio (SMR) is used as an index for comparing mortality 
experiences between two groups: actual deaths in a particular population (e.g., life annuity 
owners) with expected deaths, if “standard” age-specific mortality rates were to be applied. 
The SMR is defined as follows: 
 
tan
ˆ ( )
ˆ ( )
xt x
s d
xt x
TR m t
SMR
TR m t
=
∑
∑
 (15) 
A life annuity purchaser is, most likely, a healthy person with particularly low mortality in the 
first years of the life annuity payment and, in general, a longer-than-average expected 
lifetime. Therefore, to calculate the best estimate of an insurance annuity, we had to adjust 
mortality projections to include this effect in the projections. Pitacco et al. (2009) suggest the 
following model for age 60 and older: 
 ln ( ) ( ) ln ( )
LIM HMD
x x xt
m t f x m t ε= + +
) )
 (16) 
Here we denote by ( )
HMD
x
m t
)
 the population central death rates and by ( )
LIM
x
m t
)
 the annuity 
purchaser central death rates. This leads us to produce SMR in the form 
ˆ
( )x
e , which can be 
used to adopt mortality projections for the insurance market. Slovenia mortality experience 
for annuity purchasers is not directly available, and because pension reform started only 10 
We can calculate an annuity of size 1 that is payable yearly at the beginning of each year 
while an insur d is alive from the following: 
 
1
0
0
1, 0
( ) (1 )
(1 ( ), 0
x
kk
x
x jk
j
k
a i
q x j k
ω
τ
τ
−
−
−
+=
=
= 
 
= ⋅ + 
− + + >
 
 
∑
∏
&&  (14) 
5.2 Selection effect 
The standardised mortality ratio (SMR) is used as an index for comparing mortality 
experiences between two groups: actual deaths in a particular population (e.g., life annuity 
owners) with expected deaths, if “standard” age-specific mortality rates were to be applied. 
The SMR is defined as follows: 
 
tan
ˆ ( )
ˆ ( )
xt x
s d
xt x
TR m t
SMR
TR m t
=
∑
∑
 (15) 
A life annuity purchaser is, most likely, a healthy person with particularly low mortality in the 
first years of the life annuity payment and, in general, a longer-than-average expected 
lifetime. Therefore, to calculate the best estimate of an insurance annuity, we had to adjust 
mortality projections to include this effect in the projections. Pitacco et al. (2009) suggest the 
following model for age 60 and older: 
 l ( ) ( ) ln ( )
LIM HMD
x x xt
m t f x m t ε= + +
) )
 (16) 
Here we denote by ( )
HMD
x
m t
)
 the population central death rates and by ( )
LIM
x
m t
)
 the annuity 
purchaser central death rates. This leads us to produce SMR in the form 
ˆ
( )x
e , which can be 
used to adopt mortality projections for the insurance market. Slovenia mortality experience 
for annuity purchasers is not directly available, and because pension reform started only 10 
Seznam potrebnih popravkov članka: »ADOPTION OF PROJECTED MORTALITY TABLE 
FOR THE SLOVENIAN MARKET USING THE P ISSON LOG-BILINEAR MODEL TO TEST 
THE MINIMUM STANDARD FOR VALUING LIFE ANNUITIES« 
 
1. stran 252, odstavek (5): 
Beseda »standard« se pravilno deli »stan-dard » in ne kot »stand-ard« (Merriam-
Webster Encyclopedia) 
 
2. V zadnj m stavku pred naslovom 2.2 je oklepaj, medtem ko zaklepaj manjka. 
Pravilno je: »...the future with a time-series approach (e.g., as a random walk with a 
drift or an autoregressive integrated moving average - ARIMA model). 
 
3. V enačbi (3) se med D
x,t
 in nadaljevanjem izpiše kvadratek namesto znaka, ki bi 
označeval porazdeljevanje (» «).  
 
4. V glavi na vsaki drugi strani (kjer je sez m avtorjev) je za začetnico imen  SAMBT 
vezaj namesto pike (biti torej mora »J. SAMBT«, ne pa »J- SAMBT«). 
 
5. Prvi stavek, ki sledi naslovu 2.3 ima v referenci, torej v oklepaju odvečno podpičje 
»(Lee-Carter, 1992;). V stavku, ki sledi: je priimek »Cart-er« napačno deljen. Pravilno 
je »Car-ter«. Ali ga je sploh potrebno deliti?  
 
6. V stavku takoj za formulo (6) je pri ϕ  namesto ničle, črka »O«. Pravilno je »... where 
0
p
ϕ ≠ , 0
q
ψ ≠ , and 
t
ξ is a Gaussian ...«  
 
7. Zadnji stavek prvega odstavka razdelka 4.1. naj se spremeni iz »The results obtained 
from the Poi son logbilinear model are represented in Figure 1 by a green line, and 
the results obtained from the original Lee-Carter model are represented by a blue 
line.« v »The results obtained from the Poisson log-bilinear model are represented in 
Figure 1 by a green (light) line, and the results obtained from the original Lee-Carter 
mod l are repre nted by a blue (dark) line.« 
 
8. Stran 263, zadnji stavek prvega odstavk . Pri besedi »observedv izpisu ista vidni 
prvi dve črki. 
 
9. Forumla (15) se pravilno glasi: 
stand
ˆ ( )
ˆ ( )
xt x
xt x
ETR m t
SMR
ETR m t
=
∑
∑
. V enačbi (15) je v 
imenovalcu mišljen »stand«, ne pa »s tan d«. 
 
10. Na strani 265 je v drugem odst vku, drugi stavek, za besedo »from« en presledek 
preveč – torej : »As we see from, in gen ral, SMR depends n age…” 
 
 
11. V tabeli 5 za ženske namesto »Female« napišemo »Females« saj imamo tudi za 
moške »Males«, hkrati pa imamo tudi v tabelah 6 in 7 za ženske »Females«.  
 
12. V tabelah 5, 6 in 7 napišemo »Males« in »Females« z velikimi začetnicami. 
 
13. Pri sliki 6 je oznaka za prekinjeno skalo zamaknjena. Predlagamo, da črtico 
odstranimo.  Vstavi naj se sledeča slika: 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 265
ing the selection effect. As a result, we chose an alternative solution introduced by the 
Associazione Nazionale fra la Imprese Assicuratrici (2005), which has been used to build 
mortality statistics for the Italian insurance annuity industry. 
The idea is to use SMR from another population with similar characteristics as the popu-
lation for which we want to introduce the selection effect. As we see from, in general, 
SMR depends on age. Let us denote by SMRx
RC (t) the reference country’s standardised 
mortality ratio between the insured and general population for the particular year t. We 
can then calculate life insurance market central death rates as follows:
(17)
5.3 Applying the selection effect to Slovenian mortality projections
To obtain SMRx
RC (t), we included an element of selection that emerged from data from the 
United Kingdom, where annuity and pension market income is well developed. We used 
UK data for the period 1999–2002 collected by the Continuous Mortality Investigation 
Bureau and published in number 23 of the Continuous Mortality Investigation Reports 
(2009), which pertains to the experience of portfolios of immediate and deferred life 
annuities. The 1999–2002 mortality investigation presents the so-called 00 series base 
mortality tables adopted by UK actuaries. The statistical base is extensive: it involves 
more than 20 million lives exposed to risk.
In particular, we used the mortality investigation of life office pensioners (insured to de-
ferred annuities) - PNM00 tables for men and PNF00 tables for women, which show the 
mortality rates for each age from 20 to 120 years, distinguished between lives (i.e., heads 
insured) and amounts (i.e., weighted by the benefit).
By comparing the mortality of UK insured lives with those of the total UK population 
(taken from English Life Table No. 16, 2000–2002), we were able to quantify the increased 
survival of the insured population. To take into account the impact of economic wealth 
of the insured on selection, we weighted mortality rates by the size of annuity, as has 
been statistically proved in other markets (e.g., Germany; see DAV, 2005).
Such a selection factor is structured to represent the mortality of the insured’s deferred 
annuity. In the case of an immediate annuity, further selectivity should be added. As a 
result, we added an extra selection factor, calculated as the ratio between the mortality 
of deferred annuity owners and immediate annuitants in the United Kingdom. Figure 6 
shows the combined selection factors for immediate annuitants used for the population 
mortality tables for Slovenia. 
Starting from the particular generation cohort life table derived from historical data and 
stochastic projections described in the previous section, and then in applying the selec-
tion factors, we obtained the projected and selected mortality table for this particular 
years ago, there are no adequate statistical data for the conclusion regarding the selection 
effect. As a result, we chose an alternative solution introduced by the Associazione Nazionale 
fra la Imprese Assicuratrici (2005), which has been used to build mortality statistics for the 
Italian insurance annuity industry.  
 
The idea is to use SMR
 
from another population with similar characteristics as the population 
f r which we wa t to introduce the selection effect. As we see from (16), in general, SMR  
depends on age. Let us denote by ( )
RC
x
SMR t the reference country’s standardised mortality 
ratio between the insured and general population for the particular year t. We can then 
calculate life insurance market central death r tes as follows: 
 ( ) ( ) ( )
LIM RC HMD
x x x
m t SMR t m t= ⋅
) )
 (17) 
5.3 Applying th  selection effect t  Slovenian mor ality projections 
To obtain ( )
RC
x
SMR t , we included an element of selection that emerged from data from the 
United Kingdom, where annuity and pension market income is well developed. We used UK 
data for the period 1999–2002 collected by the Continuous Mortality Investigation Bureau 
and publish d in number 23 of the Continuous Mortality Investigation Reports (2009), which 
pertains to the experience of portfolios of immediate and deferred life annuities. The 1999–
2002 mortality investigation presents the so-called 00 series base mortality tables adopted by 
UK actuaries. The statistical base is extensive: it involves more than 20 million lives exposed 
to risk. 
 
In particular, we us d the mortality investigation of lif  office pensioners (i sured to deferred 
annuities) - PNM00 tables for men and PNF00 tables for women, which show the mortality 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011266
generation. Such derived life tables are considered the best estimate of an annuity pur-
chaser in Slovenia. These rates can be compared with the current minimum standard: 
DAV 1994 R.
Figure 6: Selection factor for Slovenia annuity owners
5.4 Testing the minimum standard
In this section, we compare the Slovenian annuity life table with DAV 2004 and DAV 
1994. Looking at the results presented in Table 3, the net single premium for an annuity 
based on the Poisson model is up to 12% less than the DAV 2004 single premium annui-
ties. This gap is a consequence of the 15% margin incorporated in the new German tables 
(DAV, 2005) and the fact that mortality rates derived from the Poisson model represent 
the best estimate of future annuitant mortality. Those tables cannot be directly compared 
in this respect. Under the best-estimate scenario, we determined future kappa values by 
taking kappa to be equal to the expected value. By comparing the low-mortality scenario 
with DAV 2004, we observed only minor differences in rates (a 1–7% higher single pre-
mium in the case of DAV 2004). 
The comparisons also show that for current generations who have not yet retired, the 
DAV 1994 tables underestimate the best-estimate annuity in Slovenia by 2%. For males 
who will retire in the future (deferred annuitants), the difference is almost 4%. As a con-
sequence, we believe that the DAV 1994 tables should not be used for the best-estimate 
valuation of annuity liabilities in the Solvency II framework. 
0 20 40 60 80 100 120
0
.
5
0
.
6
0
.
7
0
.
8
0
.
9
1
.
0
Selection factors
Age
s
e
l
e
c
t
i
o
n
 
i
n
 
%
males
females
 
 
14. Pod sliko 6 naj se ostrani tekst »own works«. Podobno velja pri tabelah – tekst »own 
works« naj se odstrani izpod tabel 1,2 in 5. 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 267
Table 3: Immediate annuity: Age at issue 60/birth year 1950 (annuity starts in 2010)
Net single 
premium
R94 R04
Poisson model, 
central rates
Poisson, 
low-mortality 
scenario
R94/Poisson
Male 18.10488 20.36943 18.50437 18.98097 0.978
Female 20.49378 22.00024 20.98405 21.76160 0.977
Table 4: Deferred annuity: Age at issue 60/birth year 1980 (annuity starts in 2040)
Net single 
premium
R94 R04
Poisson model, 
central rates
Poisson, 
low-mortality 
scenario
R94/Poisson
Male 19.72056 22.94550 20.46949 21.28829 0.963
Female 22.81581 24.42694 22.87369 23.58179 0.997
Notes: R94 – DAV 1994 annuity life table; R04 – DAV 2004 annuity life table; Poisson model – Slovenian an-
nuity life table based on the Poisson log-bilinear model. 
6 conclusion
In this article, we have presented an application of the Lee-Carter methodology to calcu-
late the best-estimate value of an insurance annuity in Slovenia. In particular, we focused 
on forecasting life expectancies on a time-series basis. We tested two different stochastic 
methods for forecasting mortality (basic Lee-Carter and Poisson log-bilinear). On the 
basis of back-testing analysis, we concluded that the Poisson log-bilinear model provides 
a better fit than the original Lee-Carter model for past observed central death rates for 
Slovenia. We therefore used a Poisson log-bilinear model to forecast mortality. Given 
that a life annuity purchaser is, most likely, a healthy person with longer-than-average 
life expectancy, we also incorporated the selection effect into the results. Because Slov-
enian mortality statistics for annuity purchasers are not directly available, we chose se-
lection statistics from the UK experience and compared them with German statistics. By 
multiplying the selection factor, which depends on age, with cohort population mortality 
rates, we derived the best estimate of selected mortality rates for an annuity purchaser. 
We then compared those rates with the current minimum standard in Slovenia: the DAV 
1994 R mortality rates.
The net single premium based on the Poisson model is 2–4% higher than that calculated 
by the current minimum standard in Slovenia. Therefore, the DAV 1994 R annuity tables 
are inappropriate for the best-estimate valuation of annuity liabilities in the Solvency II 
framework. In other words, technical provisions for annuities based on the DAV 1994 R 
tables are underestimated by 2–4%, which is not insignificant.
After 21 December 2012, the use of only unisex tables will be allowed for premium cal-
culation. This is also the case for life annuities. In this respect, further research is needed 
to take into account male and female selection in the Poisson framework. 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011268
references
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uncertainty: theory and calibration. The Journal of Risk and Insurance, 73 (4), 687–718.
Carter, L. R. (1996). Forecasting U.S. mortality: a comparison of Box – Jenkins ARIMA and structural time 
series models. The Sociological Quarterly, 37 (1), 127–144.
Denuit, M. & Goderniaux, A.-C. (2005). Closing and projecting life tables using log-linear models. Bulletin of 
the Swiss Association of Actuaries, (1), 29–48.
European commission, Internal market and services DG (2010). QIS5 Technical Specifications. https://eiopa.
europa.eu/fileadmin/tx_dam/files/consultations/QIS/QIS5/QIS5-technical_specifications_20100706.pdf 
(accesed May 20, 2011).
Goodman, L. A. (1979). Simple models for the analysis of association in cross-classifications having ordered 
categories. Journal of the American Statistical Association, 74, 537–552.
Institute of Actuaries and the Faculty of Actuaries (2009). Continuous Mortality Investigation Reports 
(2009), No. 23.
Insurance Supervision Agency. Decision on detailed rules and minimum standards relating to calculation of 
technical provisions, Uradni list RS, št. 3/01, 69/01, 85/05, 66/2008. http://www.a-zn.si/Documents/acts/en/
azn00022en.pdf (accessed May 20, 2011).
DAV-Unterarbeitsgruppe Rentnersterblichkeit (2005). Herleitung der DAV-Sterbetafel 2004 R für Rentenver-
sicherungen, Blätter der DGVFM, XXVII, 199-313.
Lee, R. D. & Carter, L. R. (1992). Modelling and forecasting U.S. mortality. Journal of the American Statistical 
Association, 87/419, 659-671.
Lee, R. (2000). The Lee-Carter Method for foresting mortality, with various extensions and applications. 
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Olivieri, A. (2001). Uncertainty in mortality projections: an actuarial perspective. Insurance: Mathematics 
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Pitacco. E. et al. (2009). Modelling Longevity Dynamics for Pension and Annuity Business. Oxford University 
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Tuljapurkar, S., Li, N., & Boe, C. (2000). A universal pattern of mortality decline in the G7 countries. Nature, 
405, 789–792.
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 269
aPPendix 1: Comparing the results of both methods for females’ alphas and betas
Figure 7: Beta(x) as a function of age (females): the Poisson vs. the LC model
Figure 8: Alpha as a function of age (females): the Poisson vs. the LC model
Appendix 1: Comparing the results of both methods for females’ alphas and betas 
Figure 7: Beta(x) as a function of age (females): the Poisson vs. the LC model 
0 20 40 60 80 100 120
0
0.005
0.01
0.015
0.02
0.025
0.03
age
b
e
t
a
(
x
)
female
 
Figure 8: Alpha as a function of age (females): the Poisson vs. the LC model 
0 20 40 60 80 100 120
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
age
a
lp
h
a
(
x
)
female
 
Figure 9: Kappa as a function of year (females): the Poisson vs. the LC model 
Appendix 1: Comparing the results of both methods for females’ alphas and betas 
Figure 7: Beta(x) as a f nction of age (females): the Poisson vs. the LC model 
0 20 40 60 80 100 120
0
0.005
0.01
0.015
0.02
0.025
0.03
age
b
e
t
a
(
x
)
female
 
Figure 8: Alpha as a function of age (females): the Poisson vs. the LC model 
0 20 40 60 80 100 120
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
age
a
lp
h
a
(
x
)
female
 
Figure 9: Kappa as a function of year (females): the Poisson vs. the LC model 
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011270
Figure 9: Kappa as a function of year (females): the Poisson vs. the LC model
1970 1975 1980 1985 1990 1995 2000 2005 2010
-60
-50
-40
-30
-20
-10
0
10
20
30
40
year
k
a
p
p
a
(
x
)
female
 
Appendix 2: Residuals  
Table 5: Summary statistics for the ARIMA model 
 Poisson model LC model 
 males female males female 
 Mean  2.88E-16 -3.24E-16 -5.96E-16 -0.052288 
 Median -0.307543  0.199426 -3.33E-15  0.183130 
 Maximum  6.586123  6.081238  10.12461  7.569289 
 Minimum -7.556903 -6.402445 -11.78582 -11.08728 
 Std. Dev.  3.349367  3.093559  4.889802  4.813961 
 Skewness -0.049737 -0.028232 -0.409455 -0.559959 
 Kurtosis  2.642194  2.224110  3.041657  2.718712 
 Jarque-Bera  0.212627  0.933006  1.036536  2.000007 
 Probability  0.899143  0.627192  0.595551  0.367878 
 Sum  1.95E-14 -7.99E-15 -2.13E-14 -1.882381 
D. MEDVED, A. AHČAN, J. SAMBT, E. PITACCO  |  ADOPTION OF PROJECTED MORTALITY FOR THE ... 271
aPPendix 2: Residuals 
Table 5: Summary statistics for the ARIMA model
Poisson model LC model
Males Females Males Females
Mean  2.88E-16 -3.24E-16 -5.96E-16 -0.052288
Median -0.3075432  0.199426 -3.33E-15  0.183130
Maximum  6.586123  6.081238  10.12461  7.569289
Minimum -7.556903 -6.402445 -11.78582 -11.08728
Std. Dev.  3.349367  3.093559  4.889802  4.813961
Skewness -0.049737 -0.028232 -0.409455 -0.559959
Kurtosis  2.642194  2.224110  3.041657  2.718712
Jarque-Bera  0.212627  0.933006  1.036536  2.000007
Probability  0.899143  0.627192  0.595551  0.367878
Sum  1.95E-14 -7.99E-15 -2.13E-14 -1.882381
Sum Sq. Dev.  403.8574  344.5238  860.7658  811.0976
No. of observations  37  37  37  36
Source: SORS
Table 6: LC model residuals for kappa 
Males Females
AC  PAC  Q-Stat  Prob AC  PAC  Q-Stat  Prob
1 -0.210 -0.210 1.7712 0.183 0.105 0.105 0.4340 0.510
2 -0.089 -0.139 2.0984 0.350 0.063 0.053 0.5952 0.743
3 0.052 0.001 2.2112 0.530 -0.331 -0.348 5.1487 0.161
4 -0.075 -0.080 2.4579 0.652 -0.093 -0.027 5.5211 0.238
5 0.304 0.295 6.6169 0.251 0.191 0.295 7.1360 0.211
6 -0.122 -0.009 7.3086 0.293 0.098 -0.073 7.5702 0.271
7 0.015 0.071 7.3194 0.396 0.108 -0.006 8.1237 0.322
8 -0.213 -0.284 9.5842 0.295 -0.359 -0.272 14.420 0.071
9 0.034 -0.025 9.6439 0.380 -0.043 0.094 14.513 0.105
10 0.201 0.054 11.807 0.298 0.108 0.266 15.128 0.127
11 -0.214 -0.094 14.338 0.215 0.408 0.219 24.232 0.012
12 0.179 0.159 16.178 0.183 0.055 -0.234 24.407 0.018
13 -0.193 -0.073 18.426 0.142 -0.038 0.097 24.493 0.027
14 0.001 -0.016 18.426 0.188 -0.227 0.035 27.701 0.016
15 -0.010 -0.211 18.433 0.241 -0.134 -0.122 28.874 0.017
16 0.070 0.151 18.770 0.281
ECONOMIC AND BUSINESS REVIEW  |  VOL. 13  |  No.  4  |  2011272
Table 7: Brouhns et al. residuals for modelling kappa
Males Females
AC  PAC  Q-Stat  Prob AC  PAC  Q-Stat  Prob
1 -0.255 -0.255 2.6156 0.106 -0.072 -0.072 0.2090 0.648
2 -0.118 -0.196 3.1852 0.203 -0.161 -0.167 1.2718 0.529
3 -0.089 -0.195 3.5248 0.318 -0.050 -0.079 1.3799 0.710
4 -0.029 -0.161 3.5618 0.469 -0.013 -0.053 1.3870 0.846
5 0.261 0.181 6.6234 0.250 0.020 -0.008 1.4055 0.924
6 -0.225 -0.153 8.9865 0.174 -0.094 -0.113 1.8171 0.936
7 -0.003 -0.060 8.9868 0.254 0.130 0.114 2.6304 0.917
8 -0.177 -0.257 10.540 0.229 0.035 0.024 2.6912 0.952
9 0.160 -0.003 11.866 0.221 -0.154 -0.127 3.9196 0.917
10 0.048 -0.076 11.990 0.286 0.128 0.134 4.7928 0.905
11 -0.230 -0.238 14.929 0.186 0.155 0.160 6.1273 0.865
12 0.310 0.213 20.473 0.059 0.152 0.214 7.4576 0.826
13 -0.156 -0.042 21.929 0.056 -0.251 -0.155 11.245 0.590
14 0.046 -0.084 22.063 0.077 -0.087 -0.047 11.717 0.629
15 -0.025 -0.028 22.104 0.105 0.133 0.075 12.877 0.612
16 -0.038 -0.016 22.201 0.137 -0.064 -0.038 13.157 0.661