The effect of incorporating parameter and process
uncertainty on the claims reserve of Dutch basic health
insurers
Lammert Bakker, s2372622
University of Groningen
master thesis actuarial studies
The effect of incorporating parameter and process
uncertainty on the claims reserve of Dutch basic health
insurers
Lammert Bakker, s2372622
Abstract
Contents
1 Introduction 3
2 Regulatory framework 5
3 Literature review 8
4 Methodology 10
4.1 Generalized linear models . . . 12
4.1.1 Multiplicative model . . . 13
4.1.2 Parameter specifications . . . 15
4.1.3 Distributional assumptions . . . 18
4.1.4 Negative claim amounts . . . 21
4.1.5 In-sample evaluation approach . . . 23
4.2 The IBNS claims reserve . . . 25
4.2.1 Bootstrap methodology for GLMs . . . 25
4.2.2 Solvency II standard formula . . . 28
4.3 Out-of-sample evaluation method . . . 29
4.3.1 First stage: Assessing statistical properties . . . 30
4.3.2 Second stage: Assessing economic properties . . . 34
5 Data 36 6 Results 38 6.1 In-sample evaluation results . . . 38
6.2 Out-of-sample evaluation results . . . 40
6.2.1 First stage: Statistical properties . . . 41
6.2.2 Second stage: Economic properties . . . 44
6.3 Capital requirements . . . 46
7 Conclusion 50
A List of symbols 56
B Iteratively (re-)Weighted Least Squares 58
C Results First stage 63
1
Introduction
Insurance is a form of social good as it protects individuals against financial risks based on risk pooling. As a result, insurers face uncertain cash flows. This is also the case with Dutch basic health insurance companies. Health insurance con-tracts always have a duration of one year which coincides with a calender year. However, the claims originating from a contract are often not settled within the same calender year due to reporting and settlement delays (Kaas et al., 2008). These claims are incurred, but not settled (IBNS). It may take up to 8 years to finalize claims originating from a single accident or calendar year. Hence, an important task of a basic health insurer is to manage its reserves to cover for outstanding loss liabilities (W¨uthrich and Merz, 2015). The importance of proper IBNS claims reserving is also acknowledged by Tee et al. (2017) who state that insurance companies may either get solvency problems resulting from low IBNS re-serves, or lose a significant amount of money because of unnecessary high reserves.
Despite the fact that the financial crisis of 2008 was mainly a banking crisis, Schich (2010) found evidence for adverse effects on insurance companies. Al-though no literature exists linking the 2008 crisis to unexpected losses on the IBNS claims reserves, the crisis clearly showed the importance to safeguard the health of financial institutions. The Solvency II European regulatory framework acknowledges the importance of claims reserving and regulation is in place to en-sure that the buffers of inen-surers are high enough to cover future claims (Solvency directive 2009/138/EC, 2009). Therefore this thesis is embedded within a regula-tory framework implied by Solvency II to make it of practical use.
Furthermore, because of our aging society, citizens (i.e. policy holders) will re-quire more healthcare in the future (Veldman, 2018), increasing the volume of, and reliance on the health insurance companies. In this light, robust health insurers are even more important. Reliable claims reserving techniques are thus essential for both insurers, prudential supervisors and policy holders.
confi-dence level of solvency without incorporating parameter and process uncertainty (Fr¨ohlich and Weng, 2018). It remains unclear whether this also holds for Dutch basic health insurers. What is known is that Solvency II regulation does not take parameter uncertainty into account (Solvency directive 2009/138/EC, 2009; Com-mission Delegated Regulation (EU) 2015/35, 2015; EIOPA, 2011). This raises the question whether the solvency capital requirements attain the required 99.5% probability of solvency.
This study analyzes the effect of incorporating process and parameter uncer-tainty on the estimation of claims reserves for Dutch basic health insurers. Using models with and without parameter and process uncertainty buffers are calcu-lated. The models with parameter and process uncertainty are based on the 99.5% Value-at-Risk (VaR) of the future claim payments. The research question that is answered in this thesis is:
What is the effect of incorporating parameter and process uncertainty on the claims reserve of Dutch basic health insurers?
The approach taken to answer the research question is by assuming the insurer estimates the IBNS claims reserves using a Generalized Linear Model (GLM). These models are widely accepted for claims reserving, see for example (amongst many others) Kaas et al. (2008), England and Verrall (1999, 2002) and Taylor and Mcguire (2016). An explanation of GLMs can be found in Kaas et al. (2008, Chapter 9). This paper uses a simulation approach based on the work of England and Verrall (1999, 2002) to establish a buffer for future claim payments by taking both parameter and process uncertainty into account. Following Solvency II reg-ulation, this buffer should be as such that it equals the 99.5% VaR of the IBNS claims losses. Subsequently, these buffers are compared to the Solvency Capital Requirements (SCR) imposed by Solvency II regulation.
Based on the data of a single health insurer, it is difficult to state that a claims reserving method should be preferred over another for a particular insurance sec-tor. England and Verrall (1999) state that “The effectiveness of a particular re-serving method and modeling can be completely tested only with an extensive case study with data from various lines of business and companies”. Thus, apart from statistical tests and loss functions it is important to consider data from multiple insurers. Therefore, in this thesis, data from all 22 Dutch basic health insurance companies that existed in 2018 are used, covering the period 2006 − 2017. As this covers all available data since the introduction of the new Dutch healthcare law in 2006 this thesis makes it possible to evaluate the effectiveness of the re-serving methods for Dutch basic health insurers. This is a major advantage over other scientific contributions, which often only consider one or two run-off trian-gles whereas this thesis considers 136 run-off triantrian-gles.
The results of this thesis emphasize the importance of incorporating parameter and process uncertainty for reliable claims estimates. Not taking parameter and process uncertainty into account results in claims reserves which do not acquire the required probability of solvency whereas reserves with parameter and process uncertainty do. Furthermore, it turns out that the SCR is economically preferred over the other claims reserving models, both from the insurer’s and prudential supervisor’s perspective.
This thesis is structured as follows: Section 2 provides a brief description of the regulatory framework in which this thesis is embedded followed by the impli-cations for this thesis. Subsequently, Section 3 contains a review of the relevant literature. Next, a description of the methodology can be found in Section 4. This section first discusses the GLM framework, followed by the approach to deal with negative claim amounts and how to obtain the claims reserves (using GLMs and the SCR). The methodology section is concluded by the evaluation method used to compare claims reserving methods. Subsequently, Section 5 describes the Dutch basic health insurance data used in this thesis. The results are presented in Section 6 and the concluding remarks in Section 7.
2
Regulatory framework
the implications for this thesis.
In 2006, a new Dutch health law (”Zorgverzekeringswet”) came into effect. This law made it compulsory for every citizen to have basic health care insurance and, for insurers, to accept all applying citizens. It has three main objectives (Ministerie van Volksgezondheid, 2017). First, it aims to keep healthcare accessi-ble for all citizens. This is achieved by making health insurance compulsory for all citizens. At the same time, health insurers are not allowed to decline anyone ap-plying for basic health insurance or price discriminate in any way. Second, it aims to provide an equal playing field for all insurers. Therefore, a health risk equalisa-tion system (HRES) has been instituted which compensates health insurers based on the risk profile of their portfolio. The latter is required because insurers are obliged to accept all applicants, which may result in a very bad portfolio. Third, healthcare should be efficient by having a good balance between affordability and quality. This is achieved by setting the HRES benefits equal for each risk cat-egory, making more efficient insurers better off than inefficient insurers. At the same time, people are able to switch between insurers if the quality of care pro-vided is too low. The resulting basic health care market may be seen as a strongly regulated competitive market.
The health insurance law will not be discussed in detail in this study. However, an important implication from this law for basic health insurers is that the cov-erage of the basic health insurance policy is dictated by the Dutch government. Therefore, the insurance coverage is not stable over time. For this reason, the resulting claim data is also not stable over time and is not expected to become so in the future, making it difficult to accurately estimate the IBNS claims reserve solely on historic data. However, although this problem is acknowledged and might be partially solved by introducing expert judgement into the models, this thesis assumes that claims reserves are estimated solely on the basis of historic data.
sol-vency capital requirements (SCR) and provides a regulatory framework in which this thesis is nested. From hereon, the first pillar will simply be referred to as Solvency II.
The buffers determined by the Solvency II framework are based on a one-year horizon. This implies that the buffers should be sufficiently high for the insurer to survive the coming year. Obviously, it is not unimaginable for an insurance company to go bankrupt and it is unrealistic to obtain sufficiently high buffers for the probability of bankruptcy to be 0%. However, the SCR are designed to be high enough for the probability of bankruptcy to be 0.5%. This implies the buffers should be high enough to survive a 1-in-200 year loss event on a one year horizon. We follow Solvency II on both points. As such, the constructed reserves used to answer the research question are based on a one year horizon and a 99.5% confidence level.
Obviously, Solvency II encompasses more risks than the risk in the IBNS claims reserve. However, for this thesis, the risk in the IBNS claims reserve is the only risk of interest and the remaining risks are disregarded. The reserve needed for IBNS claims for Dutch basic health insurers is determined by the premium and reserve risk submodule within the health underwriting risk module. The underwriting risk is defined as “the risk of loss or of adverse change in the value of insurance liabilities, due to inadequate pricing and provisioning assumptions”. For reserve risk, this implies that the capital requirements should be high enough to cover un-expected losses with respect to the initial estimations of the IBNS claims reserve that occur once every two-hundred years.
In the calculation of these capital requirements parameter uncertainty is not taken into account. However, Fr¨ohlich and Weng (2018) stress the importance of incorporating parameter uncertainty in the calculation of reserve risk under Sol-vency II. This was concluded after assessing the probability of solSol-vency introduced by Gerrard and Tsanakas (2011) and Fr¨ohlich and Weng (2015) with and without parameter uncertainty. The type of insurance considered by Fr¨ohlich and Weng (2018) is not specified, however, we expect that the results of Fr¨ohlich and Weng (2018) also apply to Dutch basic health insurers.
to 0 in order to calculate the capital requirements solely for the reserve risk. The standard formula will be outlined in Section 4.
However, it is relevant to explain the definitions for the claim amounts used by the standard formula. These claims do not simply equal the claim amounts paid by the insurer to the health practitioners. Instead, they are aimed to represent the true risk borne to the insurer. This implies that the obligatory own contributions of policy holders are subtracted from the claims filed, as well as claims recovered from third parties. As a result, the net claim amounts might be negative which complicates the analysis. However, we follow the Solvency II definition for the claim amounts.
Summarizing this section, the main implications of the regulatory framework on this thesis are threefold. First, the reserves are determined for a one-year horizon. Second, the constructed reserves to answer the research question aim to have a 99.5% confidence level. Last, the net claim amounts are used to reflect the true risk borne by the insurer.
3
Literature review
A vast literature exists on IBNS claims reserving. The most commonly used claims reserving methods are the deterministic chain ladder and Bornhuetter-Ferguson approach (Kaas et al., 2008). The chain ladder method only uses observed data to predict future claims, while the Bornhuetter-Ferguson method also incorporates prior knowledge regarding the ultimate claim amount for a particular claim year. The advantage of these methods is that they are easy in use. However, a disadvan-tage is that these deterministic methods only provide the expected claims reserve, which is a point estimate.
Hoedemakers et al. (2005) and Tee et al. (2017). Another reason for the popularity of these models is that the used distributions have a sub-exponential tail, which is regarded to be empirically relevant for claim size data (Hoedemakers et al., 2005).
A comprehensive overview of various stochastic claims reserving methods can be found in W¨uthrich and Merz (2008). However, little literature exists on claims reserving methods specifically for health insurers. One of the exceptions is Wiehe (2005), who describes deterministic claims reserving methods used by health in-surers which are similar to claims reserving methods used by general inin-surers. This makes us believe that standard claims reserving methods are used by health insurers to estimate the outstanding claims reserve.
According to England and Verrall (2002), IBNS claims reserves are subject to process and parameter uncertainty. Traditional reserving methods often do not take parameter and process uncertainty into account. Spierdijk and Koning (2014) do incorporate parameter and process uncertainty for determining appro-priate buffers for disability insurers. Similarly, Kaas et al. (2008) and Pinheiro et al. (2003) apply it on claim data from Taylor and Ashe (1983). Unfortunately Taylor and Ashe (1983) do not specify the type of insurance. Tee et al. (2017) also use GLMs incorporating process and parameter uncertainty to estimate the claims reserve of an Estonian insurer of which the type of insurance is not specified. This makes us conclude that there are only few scientific contributions which evaluate claims reserves with parameter and process uncertainty on an empirical data set of known origin. To that end, this study will be a major contribution to the literature.
Buffers for IBNS claims are often equal to a high quantile of the IBNS claims distribution. This quantile is frequently referred to as the Value-at-Risk (VaR), see for example Kaas et al. (2008, Chapter 5.6). Obviously, different VaR models result in different claims reserves. To determine a preferred model, the VaR mod-els have to be evaluated.
VaR models which have been found to pass the accuracy tests, the deviation of the forecasted claim amounts from their realisations is measured using loss functions. These loss functions can be seen as a function to evaluate the economic properties of a reserve. The model with the lowest loss function will be the preferred model in this study.
As most scientific contributions only consider one or two run-off triangles to empirically evaluate claims reserving models, the approach described by Abad et al. (2015) is not common in claims reserving. The amount of available data is simply too small. Abad et al. (2015)practice is however common practice in VaR evaluations. As the data set used in this thesis is not limiting, we are able to follow Abad et al. (2015).
4
Methodology
This section describes the methodology to investigate the effect of incorporating parameter and process uncertainty on the claims reserve of Dutch basic health insurers. A lot of notation is necessary to mathematically formulate the method-ology. Therefore, a list with the interpretation of the most important symbols is provided in Appendix A as a reference.
The claims reserve should be high enough to cover the unknown future claim payments. As previously explained, this study considers a one year horizon instead of the full run-off period implying that we are only concerned with the claims re-serves for claims paid in the next year. It is assumed that historical claim payment data and patterns can be extrapolated to estimate these future claim amounts.
We let Yi,j denote the claim amount paid for year of origin i, i = 1, . . . , T , in
development year j, j = 1, . . . , d. Here T is defined as the latest observed year of origin for a particular insurer and d the maximum development year. It is common in the scientific literature to refer to these claims as “incremental claims”. How-ever, unless specified otherwise, all claim amounts in this thesis are incremental claims. The claims Yi,j are paid in financial year i + j − 1. Thus, the claims Yi,j
have already been observed when i + j − 1 ≤ T and are, hereafter, denoted by yi,j.
The notation Yi,j remains for the unknown (future) claims.
are sometimes referred to as “the lower triangle”. Based on the observed claim amounts yi,j with i + j − 1 ≤ T , claims reserving methods estimate the lower
triangle, or in our case, the claims to be paid in the next calender year, that is, Yi,j with i + j − 1 = T + 1.
Table 1: Observed and unobserved claim payments in a run-off triangle
Year of origin Development year
1 2 3 4 5 1 y1,1 y1,2 y1,3 y1,4 y1,5 2 y2,1 y2,2 y2,3 y2,4 Y2,5 3 y3,1 y3,2 y3,3 Y3,4 Y3,5 4 y4,1 y4,2 Y4,3 Y4,4 Y4,5 5 y5,1 Y5,2 Y5,3 Y5,4 Y5,5
To comply with the one year horizon imposed by Solvency II, the one year reserve imposed by the GLMs is calculated as
Ryr = X
i+j−1=T +1
Yi,j, with i = 1, . . . , T and j = 1, . . . , d (1)
where the superscript yr (year) indicates that we consider the one year reserve. If we take the run-off triangle in Table 1 as an example, the one year reserve equals Ryr = Y
5,2+ Y4,3+ Y3,4+ Y2,5.
Additionally, we define the reserve Rf ull as the sum of all future claims in the
lower triangle, that is, Rf ull =P
i+j−1>TYi,j, where the superscript f ull indicates
this reserve considers the full run-off, i = 1, . . . , T and j = 1, . . . , d. This notation is necessary because although Solvency II regulation requires insurers to have suf-ficient capital to survive at least one year, the standard formula uses the reserve Rf ull to calculate the SCR.
4.1
Generalized linear models
In this study GLMs are used to obtain the claims reserves with parameter and process uncertainty. GLMs have two main benefits over the standard linear model (Hoedemakers et al., 2005; Kaas et al., 2008). Firstly, the error term does not need to be normally distributed and homoscedastic. Secondly, the GLMs do not require the random variable to be a linear function of the explanatory variables as is the case in the standard linear model. GLMs offer the possibility for the random variable to be linear to the explanatory variables on another scale, for example the logarithmic scale.
The general framework for GLMs may be described by three components: the link function, a stochastic component and the variance function. The first component is the link function g(·), linking the expected value µi,j = E(Yi,j) to
the linear predictor as
ηi,j = g(µi,j). (2)
Examples of link functions are the identity and logarithmic link functions. The linear predictor ηi,j is defined as
ηi,j = Xi,jξ, (3)
where Xi,j is a 1 × p vector of covariates, ξ is a p × 1 vector of parameters and p
denotes the number of parameters excluding the dispersion parameter. The linear predictor is a linear function in the parameters ξ with regressors Xi,j.
The second component of the general GLM framework is a stochastic compo-nent stating that the claims Yi,j conditional on the covariates Xi,j, i = 1, . . . , T ,
j = 1, . . . , d are independent random variables with a probability distribution from the exponential family. The exponential family contains distributions such as the normal, Poisson, binomial, gamma and inverse Gaussian distribution. The probability density function of the exponential family equals
f (yi,j; θi,j, φ|Xi,j) = exp
yi,jθi,j− b(θi,j)
a(φ) + c(yi,j, φ)
, (4)
where a(·), b(·) and c(·) are known functions with dispersion parameter φ, location parameter θi,j. Often, a(·) is written as a(φ) = φ or ai,j(φ) = φ/zi,j with known
weight zi,j corresponding to observation yi,j. The latter is especially useful for
the IWLS algorithm used to calculate the maximum likelihood estimates of ξ (see Appendix B). Subsequently, the mean of Yi,j is obtained as the first derivative of
b(θi,j) with respect to θi,j, that is
The variance equals the second derivative of b(θi,j) multiplied by a(φ)
(Hoedemak-ers et al., 2005), that is
V ar(Yi,j|Xi,j) = b00(θi,j)a(φ). (6)
Finally, the third and last component of the general GLM framework is the variance function. Assuming a(φ) = φ, (6) may be rewritten as
V ar(Yi,j|Xi,j) = φV (µi,j), (7)
where V (µi,j) is called the variance function. The variance function V (µi,j)
de-pends on the distribution of Yi,j|Xi,j. For example, if
Yi,j|Xi,j ∼ Poisson(µi,j), (8)
it follows that
V (µi,j) = µi,j (9)
and φ = 1 such that
V ar(Yi,j|Xi,j) = µi,j. (10)
4.1.1 Multiplicative model
The next methodological step is to specify the covariates for the GLMs. Similar to Kaas et al. (2008) and Tee et al. (2017) this thesis considers the multiplicative model for the claims Yi,j as this has been proven empirically relevant. The
multi-plicative model contains, besides a constant c, parameters αi for the year of origin
i, βj for development year j and γk for calender year k = i + j − 1. That is, we
assume
µi,j = E(Yi,j) = cαiβjγk. (11)
Due to the multiplicative structure of (11), the link function equals the log link function. That is,
ηi,j = g(µi,j) = log(µi,j), (12)
implying the linear predictor is defined as
ηi,j = g(µi,j) = c + αi+ βj + γk. (13)
Although it seems that the covariates Xi,j are missing in these last three equations,
“dummyfication” may be used to write µi,j in the conventional GLM form. That
is, write ηi,j = log(µi,j) = Xi,jξ where Xi,j contains the dummyfied risk factors
in this thesis.
To avoid the dummy trap, some coefficients should be left out. Because the logarithm of these coefficients is set to 0, the coefficient themselves are set to 1, i.e. α1 = β1 = γ1 = 1. Similarly, because k = i + j − 1, multicollinearity might
arise in some parameterizations which can be prevented by assuming additional structure on the parameters.
The interpretation of the αi, βj, and γk coefficients needs further explanation.
The parameter αi corresponds to the marginal effect of the year of origin i on
the random claim amount Yi,j. If all other covariates are kept fixed, the marginal
effect of the claim amount originating from year i instead of year 1 is the factor αi. For example, when a severe flue epidemic materializes in claim year i it may
be expected that the claim amounts Yi,j with j = 1, . . . , 5 are larger than in claim
year 1 without an epidemic, ceteris paribus. This would result in αi > 1.
Similarly, βj corresponds to the marginal effect of development year j on the
claim size Yi,j. It represents the claims payment pattern for each accident year.
This parameterization assumes that the payment pattern of all accident years is the same. Most claims are paid out in the first two development years, thus the parameters β1 = 1 and β2 are substantially larger than those of subsequent
de-velopment years. However, it is observed that the introduction of new regulation changes the payment pattern. Health providers need to get acquainted to the new regulation and subsequently file the claims later than they used to. Therefore, a lower observation for the first development year might actually imply an increase in the claims in de subsequent development years. This is not captured by the βj
parameters. Additional dummy variables could be introduced to account for the policy change effect.
Finally, γk is a parameter corresponding to the marginal effect of the calendar
year k = i + j − 1, which can also be explained as a measure of price inflation over the years. For example, the handling costs of individual claims might increase over the years, which has an effect on all claim amounts in a specific calender year k.
Difficulties arise if we try to extrapolate the calendar estimates ˆγk as there
is no information about the future calendar years k > T . This may be resolved by assuming that γk follows a geometric pattern, with γk ∝ γk. By using this
extrapolation problem, implying that they assume price inflation does not exist. Although the latter might seem a bit too simplistic, we let the data determine which parameter specification is preferred.
The maximum likelihood estimates of c, αi, βj and γk are calculated using
the iterative weighted least squares (IWLS) algorithm (Nelder and Wedderburn, 1972). As it is often difficult to obtain an exact solution for the maximum likeli-hood estimates, algorithms like the IWLS algorithm are used (Hoedemakers et al., 2005). A detailed explanation of the IWLS algorithm may be found in Appendix B.
Once the parameters are estimated the future claim amounts are predicted by their conditional expectation, which may be approximated by
ˆ
µi,j = g−1(ˆηi,j) (14)
= ˆc ˆαiβˆjˆγk. (15)
Although we should theoretically correct (14) for Jensen’s error because of the logarithmic link function, common practice is to ignore it. This has been done by all previously mentioned authors who consider claims reserving using GLMs, except for the log-normal model.
4.1.2 Parameter specifications
In the explanation of GLMs, so far the multiplicative model was introduced in the previous section. Within this model, parameter restrictions may be imposed resulting in different parameter specifications. This thesis will investigate various parameter specifications nested in the multiplicative model. Two of these param-eter specifications have, to the best of our knowledge, not been specified before and extend the multiplicative model. The parameter specifications considered in this thesis are listed in Table 2 and are mainly based on Kaas et al. (2008). All of these 7 parameter specifications will now be briefly explained.
The first specification has a parameter for each accident and development year and assumes a geometric pattern for the calender year effect. It has the same economic interpretation as has been explained in the previous section (4.1.1). To avoid the dummy trap, we set α1 = β1 = 1. Furthermore, to avoid
multicollinear-ity, γ is calibrated on calender years k > 2.
Kremer (1982)and is often referred to as the chain-ladder specification because the same parameters are used as with the deterministic Chain-ladder method. This specification assumes the price inflation per calender year to equal 0%. Further-more, payment patterns are assumed to be constant over time and each accident year has a specific parameter. To avoid the dummy trap, we again set α1 = β1 = 1.
Model specification 3 assumes the payment pattern of each accident year to be the same, together with a constant percentual price inflation over the years. It no longer assumes an accident year specific parameter αi. This has the main benefit
that if the observation YT ,1 of the latest observed accident year T is substantially
lower than the first development year of earlier accident years, the claims for sub-sequent development years will be less underestimated. To avoid the dummy trap, we set β1 = 1. To avoid multicollinearity, γ is calibrated on calender years k > 1.
Similar to parameter specification 3, specification 4 does not consider the col-umn variables βj, but only assumes an accident year effect αi together with a
constant calender year effect γ. For the same reasons as before, we set α1 = 1 and
calibrate γ on calender years k > 1. Although this parameterization seems very nonintuitive, we will let the data speak about the method’s appropriateness.
Table 2: Parameter specifications of various methods
Parameter specification Parameters used Number of parameters (p)
1 c, αi, βj, γk T + d + 2 2 c, αi, βj T + d + 1 3 c, βj, γk d + 2 4 c, αi, γk T + 2 5 c, αi, βj, γk 4 6 c, αi, βj, γk, δf yi,j T + d + 3 7 c, αi, βj, γk, δ pc i,j T + d + 3
seem meaningless to estimate these parameterizations. However, exploring pa-rameterizations without development year specific parameters enables us to test the assumption that development year specific parameters are important.
The following parameter specification (6) is similar to specification 1 but addi-tionally includes the parameter δi,jf y, where the superscript f y indicates “first year” and where the subscript is needed to indicate the corresponding random variable Yi,j. This parameter corresponds to the marginal effect on the development years
j > 1 given that the claim amount in the first development year of accident year i is lower than that of accident year i − 1. For example, a political policy change might decrease the percentage of claims filed in the first development year due to adaptation time of health practitioners, but increase the claim amount filed in the subsequent development years. Similar as before, dummyfication is used to obtain the required GLM form. Therefore, let Ii,jf y equal 1 for development years j > 1 of accident year i = 2, . . . , T if the claims Yi,1 are lower than Y(i−1),1 and
zero otherwise. More specifically,
Ii,jf y=
1, if Yi,1 < Y(i−1),1, 1 < i < T and j > 1,
1, if YT ,1 < YT −1,1, j > 1, and PT −1 i=1 Pd j=1I f y i,j > 0, 0, otherwise. (16)
Note that the second case in (16) is needed because δf yi,j can not be estimated from the data if PT −1 i=1 Pd j=1I f y i,j = 0. If I f y
i,j = 0 for all i = 1, . . . , T and j = 1, . . . , d for
some observed run-off triangle, δi,jf y can not be estimated for that run-off triangle. If this is the case, it has been implemented that parameter specification 6 equals specification 1.
The last parameter specification (7) is similar to specification 6, only now parameter δi,jpc corresponds to the marginal effect on development years j > 1 if a major policy change occurs. The superscript pc indicates “policy change” and the subscript is needed to indicate the corresponding random variable Yi,j.
change for development years j > 1. More specifically,
Ii,jpc = (
1, if Major policy change in years i or i − 1 but not in T and j > 1 0, otherwise.
(17) Note that if the policy change occurs in year T the effect can not be measured from the historic data. Again, if Ii,jpc = 0 for all i = 1, . . . , T and j = 1, . . . , d for some observed run-off triangle (i.e. no major policy change has been observed), δpci,j can not be estimated for that run-off triangle. If this is the case, it has been implemented that parameter specification 7 equals specification 1.
Both method 6 and 7 have, to the best of our knowledge, never before been specified. Furthermore, these models have a different expected mean than the general multiplicative model in (11). Under these new specifications, the mean is given by
µi,j = E(Yi,j) = cαiβjγkδi,j, (18)
where δi,j refers to either δf yi,j or δ pc
i,j. Future claim payments are again estimated
using the conditional expectation given by ˆ
µi,j = g−1(ˆηi,j) (19)
= ˆc ˆαiβˆjγˆkδˆi,j. (20)
Now that the seven different parameter specifications have been established, the distributional assumptions need to be specified in order to fully specify the GLMs. These will be discussed in the upcoming section.
4.1.3 Distributional assumptions
After having explained the general GLM framework and various parameter spec-ifications, this section explains the distributional assumptions for the claims Yi,j.
The random variables Yi,j with i = 1, . . . , T, j = 1, . . . , d are often assumed to be
this thesis because they have proven to be empirically relevant.
Pinheiro et al. (2003) compare the Poisson and gamma model with parameter specification 2 on the insurance data from Taylor and Ashe (1983) of unknown type. They found both models yielding approximately the same claims reserves and were unable to appoint a preferred model. Using the same data, England and Verrall (1999) compare the Poisson, gamma and log-normal model. They found the prediction errors of the three models to be of equal size and do not appoint a preferred model. However, both authors did not consider out-of-sample testing.
Similarly, Tee et al. (2017) compare the Poisson, gamma and log-normal but do consider out-of-sample testing. However, they use a single data set originat-ing from an Estonian insurance company of unknown type. They found that the gamma and over-dispersed Poisson model result in approximately the same point estimates, which are somewhat higher than those of the log-normal model. Fur-thermore, the over-dispersed Poisson model results in lower prediction errors as a percentage of the reserve estimates than the gamma and log-normal model and was therefore the preferred model. The log-normal model produced irrationally high prediction errors as a percentage of the reserve estimates and was deemed inappropriate.
We continue by describing the Poisson, gamma and log-normal model. Obvi-ously, the Poisson model assumes the claim amounts Yi,j to follow a Poisson(λi,j)
distributed with expected value µi,j as in (11). The Poisson(λ) probability
den-sity function can be written in the form of (4), with ai,j(φ) = φ, b(θi,j) =
exp(θi,j) and c(yi,j, φ) = − log(yi,j!), where θi,j = log(λi,j). The Poisson model
has a log link function such that ηi,j = log(µi,j) and the linear predictor equals
ηi,j = c + αi+ βj + γk. The variance function of the Poisson model is defined as
var(Yi,j) = φV (µi,j) = φµi,j with φ = 1.
ter Berg (1980) already acknowledged that the gamma distribution with log-linear link could be used for modelling total claim amounts. Mack (1991) followed ter Berg (1980) and used the gamma model to estimate the reserves of two sets of automobile insurance data. The gamma model was found to be an appropriate model for the first data set but the fit for the second was less satisfactory. Fur-thermore, the gamma and chain-ladder model resulted in almost the same point estimates making the authors conclude that the parametric gamma method should first be considered before using the chain-ladder heuristic.
The gamma model is similar to the Poisson model but assumes the claims Yi,j
to be gamma response variables. The gamma distribution has a shape parameter α > 0 and a rate parameter β > 0. The gamma(α, β) probability density func-tion can be written in the form of (4), with ai,j(φ) = −φ, b(θi,j) = log(θi,j) and
c(yi,j, φ) = log(φ)φ
1 φ− 1
log(yi,j) − log
Γ(φ1)
, where φ = α1 and θi,j = βα.
The gamma model uses a log link function and linear predictor ηi,j = c + αi+
βj + γk. The gamma model has mean µi,j = cαiβjγk and variance V ar(Yi,j) =
φV (µi,j) = φµ2i,j with φ > 0. Future claims are again predicted using (14) and the
parameters are estimated using the IWLS algorithm.
Although the results from Tee et al. (2017) might indicate that the log-normal model is unable to produce reliable claims estimates, several authors successfully used it to model claims reserves. Examples are, amongst many others, Kremer (1982), Taylor and Ashe (1983), Verrall (1991). The latter two used the data from Taylor and Ashe (1983) of unknown insurance type and concluded that the log-normal model was appropriate for estimating the claims reserve.
The log-normal model assumes the log of the claims to be normally distributed, that is,
log(Yi,j) = N (µi,j, σ2). (21)
This results in a standard linear regression model but, as the linear regression model is a particular case of a GLM, it will be formulated in the GLM context. The probability density function of log(Yi,j) can be written in the form of (4),
with a(φ) = φ, b(θi,j) = 12θi,j2 and c(yi,j, φ) = − y2 i,j 2φ − 1 2log(2πφ), where φ = σ 2 and θi,j = µi,j.
the same as for the Poisson and gamma models, but we do no longer assume the multiplicative model. Instead, the model has become additive. Furthermore, the error terms are assumed to be normally distributed with mean 0 and variance σ2.
This implies that the variance function is V (µi,j) = 1. The parameters are again
estimated using the IWLS algorithm, which corresponds to a standard linear re-gression for the normal distribution.
To get the estimated claims ˆµi,j we can not simply exponentiate the linear
predictor as this gives the median for the log-normal distribution (England and Verrall, 2002; Tee et al., 2017). Instead, the expectations of the log-normally distributed claims amounts are given by
ˆ
µi,j = exp(ˆηi,j+
1 2σˆ
2). (22)
The reason provided by England and Verrall (2002) to add the correction term
1 2σˆ
2 in (22) and not in (14) is that in the latter we use the claims themselves as a
response variable while in the log-normal model we use the logarithm of the claims as a response variable.
4.1.4 Negative claim amounts
When the GLMs are estimated, difficulties arise because of negative claim amounts in the observed run-off triangle. These negative claims arise from recovering claims from third parties. To give an example, consider a car accident between two drivers, driver A and B. Driver A is insured at the health insurer in question and is badly injured. This implies that immediate medical treatment is needed which will, at first, be paid by the health insurer of driver A. For now, we assume these costs are paid in the first development year. However, driver B is found to be fully responsible for the accident. Therefore, the health insurer will start a law-suit against driver B (or his/her insurance company) to recover the costs made for driver A. Lawsuits considering very large claim amounts tend to take a long time. In this example, we assume the insurer of driver A receives the recovered claim amount in development year 5. The recovered claim amount in development year 5 might be higher than the sum of paid claims, resulting in a negative claim amount in development year 5 for the insurance company of driver A.
capital requirements are based on claim triangles containing negative values. To make a fair comparison between the GLMs and Solvency II based reserves, we need a way to cope with the negative claim amounts. Fortunately, this has been dealt with in the past by several authors.
Mack (1994) suggests to ignore negative claims by simply removing them from the data, causing missing values. Kunkler (2006) mentioned a completely differ-ent approach to modelling negative claims. He proposes to use a mixture model suitable for all stochastic claims reserving models using GLMs. It is originally developed by Kunkler (2004) to model zero’s in claims triangles. A mixture data triangle is defined including whether a specific increment is positive or negative. Subsequently, a positive claims triangle is defined by multiplying the original tri-angle with the mixture tritri-angle and separate parameters are defined to model these negative claims. A Bayesian approach is used to obtain estimates of the future reserves. However, this model adds extra complexity and parameters to be estimated.
A more frequently used approach is the constant method. All claims are in-creased by a sufficiently high constant making all claims positive. After the analy-sis, the same constant is subtracted from the estimated future claims. For example, Verrall and Li (1993) formalized the constant solution for the log-normal model by using a three parameter log-normal distribution. However, it turns out that the estimated future claims depend on the chosen constant. De Alba (2002) recog-nizes the instability of the three parameter log-normal distribution and takes the threshold parameter as a known constant. The constant solution has also been proposed by Verrall (1996) for an over-dispersed Poisson model.
In this thesis the constant method is used to solve the problem of negative claims since it is easy to implement and relatively frequently used. In our opinion, ignoring negative claims altogether such as suggested by Mack (1994) seems to be unsuitable because these negative claims are realistic and contain information on the claim payment process. The constant method is also preferred over the mixture model because the latter adds an extra layer of complexity and requires the estimation of more parameters.
The constant C is taken to be deterministic and run-off triangle specific. The following procedure is used to find the constant C. First, consider the smallest observed claim amount yi,j with i+j −1 ≤ T , i = 1, . . . , T and j = 1, . . . , d. If this
constant is set to equal the absolute value of the smallest observed claim amount multiplied by a predetermined constant. That is,
C = M |min(mini+j−1≤T(yi,j), 0)|, (23)
with constant M > 1, i = 1, . . . , T and j = 1, . . . , d. This results in positive claim amounts yi,j+ = yi,j+ C.
To give an example, consider the arbitrary run-off triangle in Table 3 and further assume M = 1.1. The minimal claim amount is −10. From (23), it follows that C = 11. Thus, all claim amounts in the run-off triangle are increased by 11 resulting in the positive claim values y+i,j between brackets.
Table 3: Example of a run-off triangle before and after applying the constant method
Year of origin i Development year j
1 2 3
1 1001 (111)2 60 (71) -10 (1)
2 80 (91) 73 (84)
3 105 (116)
1 Original claim values y i,j
2 The values between brackets denote the claim values y+i,j = yi,j+C after applying the constant method
4.1.5 In-sample evaluation approach
In the previous explanation of the GLMs, we have specified 7 different parame-ter specifications to estimate the IBNS claims reserves. Only the best parameparame-ter specifications are included in the analysis of the effect of including parameter and process uncertainty on the claims reserves of Dutch basic health insurers. Even-tually, the aim is to have a good out-of-sample fit (i.e. the most accurate future claim estimates as possible). Increasing the number of parameters increases the in-sample fit, resulting in zero deviance when each observation gets its own param-eter. However, adding too many parameters in the model results in over-fitting, leading to bad out-of-sample estimates (Taylor and Mcguire, 2016).
context by, among many others, Hoedemakers et al. (2005), Kaas et al. (2008) and Taylor and Mcguire (2016) and is found to be an acceptable criterion for model selection. The AIC makes a trade-off between the simplicity of the model (i.e. number of parameters used) and the goodness of fit of the model. The AIC equals −2L + 2p, where L equals the value of the log likelihood function and p equals the number of parameters, excluding the dispersion parameter.
As long as different models have the same dependent variable, the model re-sulting in the lower AIC value is preferred over the other. The models do not need to be nested models. However, to avoid over-parameterization or mining a model without a theoretical basis, the procedure in model comparison is to go from general to more specific models. That is, a general model is determined contain-ing parameters which are theoretically expected to have a significant explanatory power. Subsequently, parameter restrictions are added making the model more specific.
Alternatively to the AIC, the F-test may also be used to compare the in-sample fit of the parameter specifications (Ter¨asvirta and Mellin, 1986). The F-test is especially convenient for the quasi-Poisson model, as the AIC is not defined for this model as it is estimated by quasi-maximum likelihood. For a detailed explanation of the quasi-maximum likelihood approach for GLMs we refer to Wedderburn (1974). However, the F-test requires the models to be nested. The null hypothesis of the F-test states that the full model (model 2) does not provide a better fit than the reduced model (model 1). The F-test statistic equals
F = SSR1−SSR2 p2−p1 RSS2 n−p2 ,
where SSRi and pi respectively equal the sum of squared residuals and the number
of parameters of model i, and n equals the number of observations, for i = 1, 2. Under the null hypothesis, the F-test statistic F follows an F-distribution F(·) with (p2−P1, n−p2) degrees of freedom. The p-value then equals p = 1−Fp2−P1,n−p2(F ),
4.2
The IBNS claims reserve
Now that the distributional assumptions and various parameter specifications for the GLMs have been set out, this section describes the procedure to estimate the reserves with parameter and process uncertainty. First, the bootstrap methodol-ogy for the GLM model is described to obtain IBNS claims reserves with parameter and process uncertainty. Subsequently, we briefly explain the standard formula for the reserve risk SCR.
4.2.1 Bootstrap methodology for GLMs
The expected value of the IBNS reserves equals the sum of the estimated future claims ˆµi,j with i + j − 1 = T + 1, i = 1, . . . , T and j = 1, . . . , d. The expected
value of the reserve is thus given by ˆ
Ryr = X
i+j−1=T +1
ˆ
µi,j, for i = 1, . . . , T and j = 1, . . . , d. (24)
However, this technical provision does not take parameter and process uncertainty into account. Parameter uncertainty arises because ˆµi,j depends on the estimated
GLM parameters. This results in uncertainty in ˆµi,j. Similarly, process
uncer-tainty arises because realisations of Yi,j with i + j − 1 > t deviate from E(Yi,j).
This difference is referred to as process uncertainty. It is important to take both parameter and process uncertainty into account to obtain the true probability of solvency (Fr¨ohlich and Weng, 2018). The bootstrap method can be used to assess the reserve risk including these two types of uncertainty in an easy way (England and Verrall, 1999).
This section provides the bootstrap method based on England and Verrall (1999), Pinheiro et al. (2003) and Tee et al. (2017). Measures of accuracy for pa-rameter and process uncertainty can be derived from the bootstrap method. The bootstrap method creates pseudo data by drawing many samples with replacement from the original data, making it possible to do statistical inference. Bootstrap methods for claims reserving always consider the resampling of the residuals rather than bootstrap the data themselves. For standard linear regression models, the residuals are simply the observed values minus the fitted values. However, this is not generally the case for GLMs.
may be used for the bootstrap method in claims reserving are the Anscombe and Deviance residuals. However, although possible, Deviance residuals are less suit-able for bootstrapping as they can not be found analytically (England and Verrall, 1999). Furthermore, because the use of Pearson residuals is more common than Amscombe residuals, only the former are considered in this bootstrap procedure.
The Pearson residuals are defined as ri,j =
yi,j − ˆµi,j
pV (ˆµi,j)
, for i + j − 1 ≤ T. (25)
Thus, the observed claims yi,j can be written as
yi,j =
q
V (ˆµi,j)ri,j + ˆµi,j, (26)
which turns out to be convenient for the bootstrap methodology. These Pearson residuals are then used to calculate the deviance or Pearson scale parameter
θp =
P(ri,j)2
n − p , (27)
where n equals the number of observations and p the number of parameters. Us-ing (26) and the sampled Pearson residuals bootstrap pseudo data are created. However, England and Verrall (1999) state that the Pearson residuals need a bias correction resulting in the adjusted Pearson residuals. These adjusted Pearson residuals are calculated by
ri,ja = ri,j
r n
n − p, (28)
where the superscript a indicates that the Pearson residuals are adjusted.
After calculating the estimated reserves corresponding to each bootstrap sam-ple, confidence intervals are provided for the claims reserve including process and parameter uncertainty.
The bootstrap methodology to obtain a (1 − α) confidence level for the claims reserve including parameter and process uncertainty, with α ∈ (0, 1), is as follows: 1. Add a constant C = M min(mini+j−1≤T(yi,j), 0), where M > 1 to yi,j to make
sure all y+i,j = yi,j + C are positive, for i + j − 1 ≤ T with i = 1, . . . , T and
j = 1, . . . , d.
3. From the estimated GLM, the mean ˆµi,j is obtained for all i = 1, . . . , T and
j = 1, . . . , d. It is used to calculate the Pearson residuals ri,j by (25) and the
Pearson scale parameter θp by (27), that is for all i, j such that i + j − 1 ≤ T
with i = 1, . . . , T and j = 1, . . . , d. Hereafter, the adjusted Pearson residuals rai,j are calculated using (28).
4. Next, the original one year reserve ˆRyr is estimated as ˆRyr =P
i+j−1=T +1(ˆµi,j−
C) with i = 1, . . . , T and j = 1, . . . , d.
5. Now the bootstrap procedure starts. First, obtain a bootstrapped sample of Pearson residualsera
i,j by drawing with replacement from the Pearson residuals
ra i,j.
6. Continue by calculating the bootstrapped claim datayei,j =pV (ˆµi,j)er
a i,j+ ˆµi,j
for the upper triangle.
7. Reestimate the GLM with IWLS, but now using the bootstrapped claim data yei,j to obtain the bootstrapped future claim values eYi,j. The value
P
i+j−1=T +1( eYi,j − C) equals the claims reserve including parameter
uncer-tainty. However, we want to include both parameter and process unceruncer-tainty. 8. Thus, for each eYi,j with i + j − 1 = T + 1 a random variable eYi,jb is drawn such
that process uncertainty is also included. The superscript b indicates the index of the bootstrap iteration with b = 1, . . . , B and B a large integer. For the (quasi-) Poisson model, Yb
i,j is calculated by a drawing from the Poisson
distribution with mean ˆµi,j/θp multiplied by θp. Similarly, for the log-normal
model, draw from a log-normal distribution with mean ˆµi,j and variance σ2.
Finally, for the gamma distribution eYi,jb is drawn from the gamma distribution with shape parameter ˆa = θ1
p and rate parameter ˆb = 1
θpµˆi,j. After obtaining
e Yb
i,j the bootstrap reserve including parameter and process uncertainty is
calculated as eRbpp =P
i+j−1=T +1( eY b
i,j− C) with i = 1, . . . , T and j = 1, . . . , d.
9. Repeat steps 5 to 8 B times, with B being a large number. This results in the bootstrapped reserves eR1
pp, . . . , eRBpp. In this thesis, B = 10, 000 is taken
because additional analysis revealed that the estimated reserves did not vary much when B was further increased.
10. Finally, calculate the (1 − α)% quantile of eR1pp, . . . , eRBpp denoted by ˆRpp which
equals the claims reserve including both parameter and process uncertainty with confidence level (1 − α). Thus, we have now calculated the expected IBNS claims reserve ˆRyr without parameter and process uncertainty as well
as the claims reserve ˆRpp with parameter and process uncertainty. This
con-cludes the bootstrap method.
α = 0.025 and α = 0.005 in the described bootstrap methodology, respectively. Because we are ultimately interested in the 99.5% confidence level for the claims reserve, the tail is more important than the center of the distribution. By choosing these values of α it is observed whether the claims reserving models have a cor-rect behavior in the tail of the distribution. Only calculating the claims reserves with a 99.5% confidence level would not suffice as there is not a sufficiently large amount of data available to reliably evaluate such a high quantile of the future claims distribution.
4.2.2 Solvency II standard formula
As has been explained in Section 2, Solvency II regulation measures the risk in the IBNS claims reserve through the premium and reserve risk submodule of the health underwriting risk. As only the reserve risk is of interest, the premium vol-ume is simply set to zero to obtain the SCR for the reserve risk.
The reserve risk is measured through the run-off ratio. The run-off ratio is de-fined as the claims paid Yi,j for accident years i < t in calender year k = t plus the
expected value at time t + 1 for claims originating from accident years i < t and subsequently dividing this sum by the expectation of the claims reserve at time t for accident years i < t. This run-off ratio is calculated for all insurers for all available historic years. Subsequently, these observed run-off ratios are assumed to be independent and log-normally distributed with a variance offset to the size of the insurer. The standard deviation of the run-off ratio is calculated and used to calculate the SCR, see Solvency directive 2009/138/EC (2009), Commission Delegated Regulation (EU) 2015/35 (2015) and EIOPA (2011) for a detailed ex-planation of the standard formula.
The Solvency capital requirement (SCR) for the reserve risk on a one-year horizon is calculated as
SCRR = 3σrRˆf ullins , (29)
where σr = 0.05 equals the standard deviation of the run-off ratio and where ˆRinsf ull
Officially, the SCR equals the capital needed to cover unexpected losses and does not include the expected value of the claims reserve. By adding the insurers expected value of the claims reserve, ˆRinsf ull, a reserve is obtained which should be sufficiently large to pay out next years claims with a confidence level of 99.5%. Thus, the claims reserve using the Solvency II methodology with the insurers true expected value of future claims ˆRf ullins is given by
ˆ
RSCRins = ˆRf ullins + 3σrRˆf ullins. (30)
Although the estimated reserve ˆRf ullins corresponds to all future claims, future claims for years after the next one only make up about 5−20% of the total claims reserve. Furthermore, the insurer’s expectation for claims paid in the next calender year alone is not available. Therefore it seems reasonable to simply take the expected value of the full run-off.
4.3
Out-of-sample evaluation method
Up to this point, two methods have been described to calculate the IBNS claims reserves, one general method using GLMs and one using the Solvency II standard formula. This section explains the procedures used to evaluate the out-of-sample (or forecasting) performance of the claims reserving methods.
After the claims reserve is calculated for a specific run-off triangle a compari-son is made to the realizations of the future claims in order to assess the claims reserving model. Recall that these reserves can be seen as a VaR, being a high quantile of the future claims distribution. A two step approach is used to compare the various VaR models. The first stage tests whether a reserve has the necessary statistical properties. A detailed explanation of the four statistical tests used for this purpose is provided in Section 4.3.1.
Although it is a necessary requirement for a proper VaR model that the un-conditional coverage and independence properties hold, these statistical tests are not sufficient to rank the reserving models passing all four tests. Therefore, in the second stage of the out-of-sample evaluation method, the economic properties of the claims reserving models are assessed using loss functions. Only models passing all four statistical tests are evaluated. The loss functions are explained in more detail in Section 4.3.2.
re-serve. Firstly, it enables us to assess which reserving methods with parameter and process uncertainty are most suitable to estimate the IBNS claims reserve for Dutch basic health insurers. The size of the claims reserves of these preferred models can be compared to their corresponding claims reserves without parameter and process uncertainty. This will reveal the impact of including these types of uncertainty on the size of the claims reserve. Secondly, the individual evaluation stages also provide valuable insights in the effects of taking parameter and process uncertainty into account. For example, by assessing the probability of solvency of the expected claims reserve without parameter and process uncertainty in the first stage and evaluating the corresponding economic properties in the second stage.
4.3.1 First stage: Assessing statistical properties
The first stage tests whether a reserve has the necessary statistical properties. These statistical properties are required to assure that the estimated quantile be-haves like the intended (theoretical) quantile. According to Christoffersen (1998), two statistical properties should hold for a successful VaR model. The first is the unconditional coverage property, stating that the theoretical confidence level of the claims reserving model should equal the probability of realizing a loss larger than the resulting VaR. If the probability of observing a loss exceeding the VaR is larger than the theoretical confidence level, the VaR model underestimates the risk in the future claims. On the other hand, if the probability of observing a loss exceeding the VaR is lower than the theoretical confidence level, the risk in the claims reserve is over-estimated which leads to excessively high claims reserves. The unconditional coverage property is tested by the Unconditional Coverage (Ku-piec, 1995; Christoffersen, 1998) and Binomial tests (Jorion, 2007).
property hold.
All tests require an exception indicator. That is, if the total amount of ob-served “future” claim payments is larger than the estimated reserve, the value 1 is assigned to the exception indicator and 0 otherwise. Let L be the total amount of observed future claims in year T +1, that is L =P
i+j−1=T +1Yi,j, with i = 1, . . . , T
and j = 1, . . . , d. Let N be the number of estimated reserves and corresponding observed “future” claim amounts. Furthermore, let ˆR denote any of the previously defined reserves (i.e. ˆRyr, ˆRSCR
ins or ˆRpp) which theoretically resembles a 1 − α
con-fidence level for the claims reserves. Then, the indicator variable is defined as
Iq =
(
1, if Lq > ˆRq, for q = 1, . . . , N
0, otherwise, (31)
where ˆRq denotes the estimated reserve of run-off triangle q and Lq to the
corre-sponding realised “future” claim payments.
The first of the four tests is the likelihood ratio test of Unconditional Coverage (Christoffersen, 1998). The unconditional coverage is tested by comparing the null hypothesis E(Iq) = α against the alternative hypothesis E(Iq) 6= α while it is
assumed that the reserve violations Iq are independent. The VaR violations Iq are
assumed to be bernoulli distributed random variables with P(Iq = 1) = α. The
probability mass function of the bernoulli distribution is given by
f (x; α) = (
αx(1 − α)1−x, if x = 0, 1
0, otherwise. (32)
Then, the likelihood under the null hypothesis is given by
L(α; I1, . . . , IN) = (1 − α)n0αn1, (33)
where n1 =
PN
q=1Iq is the number of observed VaR violations and n0 = N − n1.
The likelihood under the alternative hypothesis is given by
L(π; I1, . . . , IN) = (1 − π)n0πn1, (34)
where π is estimated by ˆπ = n1/N . Subsequently, the likelihood ratio test for
Unconditional Coverage is given by
LRuc = −2log(L(α; I1, . . . , IN)/L(ˆπ; I1, . . . , IN)), (35)
degree of freedom. The p-value of the Unconditional Coverage test is given by p = 1 − Fχ2
1(LRuc). The null hypothesis is rejected when p < ζ, where ζ equals
the significance level of the test statistic.
Similar to the Unconditional Coverage test, the Binomial test (Jorion, 2007) is used to assess whether the number of times the future claim payments exceed the reserve is consistent with the theoretical confidence level α. The number of failures PN
q=1Iq is assumed to be binomially distributed B(N, α) conditional on
the failures being independent, where N is the number of independent experiments and α the theoretical confidence level of the estimated reserve. The probability mass function of the binomial distribution is given by
f (x; α) = ( N
xα
x(1 − α)N −x, if x = 0, 1, . . . , N
0, otherwise. (36)
If the distributional assumption holds, the expected number of failures equals N α with a standard deviation of pNα(1 − α). To test the null hypothesis P(Iq =
1) = α, the binomial distribution is approximated by the normal distribution. The test statistic of the Binomial test is then given by
Z = n1− N α
pNα(1 − α), (37)
which asymptotically follows a standard normal distribution. Subsequently, the p-value of the test statistic is given by p = 2(1 − FN (0,1)(|Z|)). The null hypothesis
is rejected when p < ζ.
The third test is the Independence test by Christoffersen (1998). The Inde-pendence test only considers the indeInde-pendence property and does not consider the number of VaR violations at all. Christoffersen (1998) proposes to test the null hypothesis of independence against the alternative hypothesis that there exists a first order Markov process. That is, for the alternative hypothesis the sequence {Iq} is assumed to be a first order Markov chain with transition probability matrix
Π1 = " 1 − π01 π01 1 − π11 π11 # , (38)
mass function equals f (Iq; π01, π11) = 1 − π01, if Iq = 0|Iq−1 = 0 π01, if Iq = 0|Iq−1 = 1 1 − π11, if Iq = 1|Iq−1 = 0 π11, if Iq = 1|Iq−1 = 1 0, otherwise. (39)
If the independence property holds it is expected that π01 = π11, as this would
imply that the probability of observing a VaR violation is independent from last period and viceversa. The likelihood function for the alternative hypothesis that the independence property does not hold is given by
L(Π1; I1, . . . , IN) = (1 − π01)n00π01n01(1 − π11)n10πn1111, (40)
where nij is the number of observations with value i followed by j. The estimate
of the transition matrix Π1 is given by ˆΠ1 and is computed as
ˆ Π1 = " n00 n00+n01 n01 n00+n01 n10 n10+n11 n11 n10+n11 # . (41)
Thus, the maximum likelihood estimates for π01and π11are given by ˆπ01= n00n+n0101
and ˆπ11 = n10n+n1111, respectively. The transition probability matrix under the null
hypothesis is given by Π2 = " 1 − π2 π2 1 − π2 π2 # , (42)
with corresponding likelihood
L(Π2; I1, . . . , IN) = (1 − π2)n00+n10π2n01+n11. (43)
The maximum likelihood estimate for ˆπ2 is given by ˆπ2 = (n01+ n11)/N . Then,
the likelihood ratio test of Independence is formulated as
LRind = −2log(L( ˆΠ2; I1, . . . , IN)/L( ˆΠ1; I1, . . . , IN)), (44)
where the subscript ind denotes that we are dealing with the Independence test. LRind asymptotically follows a chi-square distribution with 1 degree of freedom.
The p-value of the test statistic is given by p = 1 − Fχ2
1(LRind) and the null
hy-pothesis is again rejected when p < ζ.
Coverage test combines the Unconditional Coverage and Independence test into a single test statistic. The likelihood test of Conditional Coverage is then given by
LRcc= −2log(L(α; I1, . . . , IN)/L( ˆΠ1; I1, . . . , IN)), (45)
where the subscript cc indicates that we are dealing with the Conditional Coverage test. Thus, the test assesses the null hypothesis of the Unconditional Coverage test against the alternative hypothesis of the Independence test. For a formal proof see Christoffersen (1998). The likelihood ratio test of Conditional Coverage asymptotically follows a chi square distribution with 2 degrees of freedom. The p-value of the test statistic is given by p = 1 − Fχ2
1(LRind) and the null hypothesis
is again rejected when p < ζ.
4.3.2 Second stage: Assessing economic properties
The economic properties of the VaR models passing the first stage are further evaluated using loss functions. These loss functions make it possible to rank the remaining models. The loss functions resemble the preferences of the user. This thesis considers two regulator’s loss functions (RLFs) and a firm’s loss function (FLF) which have been proposed by proposed by Abad et al. (2015). It must be noted that the insurer’s and prudential supervisor’s true risk appetite might deviate from the proposed loss functions. However, this is beyond the scope of this study.
The RLF takes the perspective of a prudential supervisor who is mostly con-cerned with the solvency of the health insurer. Therefore, the RLF only penalizes a claims reserving model when the claims paid exceed the estimated claims re-serves, that is, when Lq > ˆRq such that Iq = 1, q = 1, . . . , N .
Sarma et al. (2003) defined the quadratic RLF
RQ = (
( ˆRq− Lq)2, if Lq > ˆRq
0, otherwise. (46)
Thus, if the claim realisations Lq are higher than the estimated reserve ˆRq, the
relative loss function because this study considers run-off triangles from different insurers with varying sizes, making it a natural choice to consider loss functions penalizing VaR exceedances relative to the VaR. One of the RLF’s proposed by Caporin (2008) is
RC = (
(Lq− ˆRq)2/ ˆRq, if Lq > ˆRq
0, otherwise, (47)
which equals the quadratic RLF divided by the estimated reserve. Although these two RLFs are different, Abad et al. (2015) concluded that, in their case, the pre-ferred VaR model was invariant to the RLF used. We let the data speak whether this also holds for Dutch basic health insurers.
Finally, the FLF represents the perspective of the health insurance company. A larger loss than the predicted VaR is obviously not in the interest of the insurer. However, although an excessively high VaR mitigates the risk of exceedances, it results at the same time in the insurer holding more capital than necessary. This may be seen as an opportunity cost to the insurer. Therefore, Abad et al. (2015) suggest the FLF FABL = ( (V aRt− rt)2, if rt < V aRt β(rt− V aRt), otherwise, (48)
where β is the cost of capital. Again, VaR exceedances are penalized according to the quadratic RLF but when no VaR exceedances materialize, the difference between the realised loss and VaR is penalized by the cost of capita β. Abad et al. (2015) suggest to take an interest rate as cost of capital β. However, this thesis assumes a cost of capital of β = 6% which is consistent with the cost of capi-tal assumed by Solvency II regulation (Commission Delegated Regulation (EU) 2015/35, 2015).
5
Data
The data used in this thesis contain information about all 22 Dutch basic health insurance companies that existed in 2018. The data cover the period from 2006 up to and including 2017 and has been adjusted for mergers and acquisitions. As the current healthcare law came into effect in 2006, claims data from earlier years are not applicable. The length of the data period is not equal for all insurers, as some of the current basic health insurers were not in business in the first years after the institution of the new healthcare law.
Two variables from the data are used. The first is a full run-off triangle with claim amounts yi,j for each insurer, with i = 1, . . . , T and j = 1, . . . , d. These
ob-served claim amounts have been adjusted for claims recovered from third parties as well as the own risk contribution from policyholders. By doing so, the historic claim data represent the true risk borne by the insurer. However, it does result in the disadvantage that claim amounts may be negative. These negative claims arise mainly in the later development years and are the result of long lawsuits against third parties to recover part of the claims paid in earlier development years (see Section 4.1.4).
The second considered variable is the insurer’s expected value of future claim payments Rf ullins which is used to calculate the Solvency capital requirements. The insurers use internal models to forecast the future claim amounts. As each insurer might use a slightly different model which might also change over the years, the methods used by the insurers to obtain Rf ullins will not be considered in this thesis.
This thesis assumes that the claims for all accident years are completely settled after d = 5 years. This seems to be reasonable because the average proportion of claims in development years d > 5 compared to the claims in development years d ≤ 5 for a single accident year is −0.03%. We further set T = 5 as the latest observed claim year in each run-off triangle. If negative claims are observed in the run-off triangle we set C = M min(mini+j−1≤5(yi,j, 0)) to make the claims
y+i,j = yi,j+ C positive, with constant M = 1.1, i = 1, . . . , 5 and j = 1, . . . , 5. The
choice to set M = 1.1 has been a pragmatic choice. As such, M is large enough to make all claims positive, but sufficiently low to not change the claim values too much. Furthermore, a sensitivity analysis revealed that the value of M has no effect on the estimated claims reserves as long as a high quantile of the future loss distribution is considered (see Section 6.3).
