Forecasting Lapse, Renewal, and Commencement of Policies in Non-Life Insurance

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Forecasting Lapse, Renewal, and Commencement of Policies in

Non-Life Insurance

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Track: Actuarial Studies

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Forecasting Lapse, Renewal, and Commencement of Policies in

Non-Life Insurance

Suzanne Dechesne February 1, 2021

Abstract

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CONTENTS

1 Introduction

Insurance companies construct financial reports concerning the valuation of the non-life technical provisions (property and casualty insurance) and the related financial position. Furthermore, they conduct analyses on these metrics. One of the financial reports is an outlook, which is a forecast for the balance sheet at the end of the quarter. It contains the expected technical provisions and the direct consequences for the own funds, all according to Solvency II principles. Drawing up the outlook is a continuous process and it is therefore regularly reported during the quarter. Currently, basic analytical models are implemented to obtain forecasts. One of the aspects included in this model is projecting premiums into the future.

This prediction of incoming premiums can be significantly improved. The outlook requires an estimation of the written premiums, both earned and unearned, and of the premiums that are going to be written in the future. The latter are simply termed ‘future premiums’. They play an important role in determining the movement of the best estimate liability. At the end of every quarter, we distinguish between three types of policies. The policies that already existed at the beginning of the quarter and that have not yet expired are labeled existing (EXI). If the contract of a policy has expired but the policyholder has chosen to renew his contract during the quarter, this policy is given the label new business renewed (NBR). Lastly, new policies will be contracted during the quarter that did not exist before. These policies are termed new business added (NBA).

Forecasts of the initiation or termination of policies during the upcoming quarter have to be created. This can be divided in three parts: prediction of the number of lapsed policies (to determine EXI), prediction of the number of renewed policies (NBR), and prediction of fully new policies (NBA). All predictions are made at the same frequency. Standard errors of the forecasts have to be stated by the model as well. Due to limited data availability, the forecast cannot be revised during the quarter. The aim is creating a prediction of the number of policies in force at the end of the quarter. This prediction will remain consistent throughout the quarter, whereas other forecasts incorporated in the outlook, such as reported claims, may be adjusted based on new financial information.

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2 Literature Review

After determining which model provides the best forecasts, this model is used to associate a probability of termination to every policy that is in force at the beginning of the quarter. This facilitates computation of the expected premium income from policies that are already contracted at the insurance company. For the policies that are initiated during the quarter, i.e. the policies labeled NBA, the aggregated premium income is estimated. This is due to the fact that no information is available on individuals that have not yet contracted a policy.

The remainder of this paper is organised as follows. In Section 2, we describe existing literature on the topics of lapse behaviour, cross-buying behaviour, and total demand for non-life insurance. Section 3 provides a characterization of and descriptive statistics on the data. In Section 4, we describe the method that is used to compare the models. Next, Section 5 presents the fundamentals of survival analysis. It also shows the outcomes of some tests and estimations to gain a first insight into the data. Section 6 explains the Cox proportional hazards model. The logistic regression model is discussed in Section 7. Section 8 presents an analysis and comparison of both models. A conclusion is drawn with respect to the optimal model. The method to forecast premium income using the results of this study is given in Section 9. Finally, Section 10 concludes, presents some remarks, and discusses potential developments to be made.

2 Literature Review

The decision to initiate, terminate, or renew a policy is based on multiple factors, differing per individual. Past research has attempted to capture the components that influence this decision in non-life insurance. First, we outline the results of previous studies concerned with the probability of cancellation or non-renewal of policies. Second, the methods and characteristics related to newly acquired customers are discussed. Lastly, a summary of factors influencing non-life insurance demand of the market is presented. This section presents literature regarding the covariates of the models analysed in this paper. The theory regarding the models that are used to create forecasts in this paper is described in later sections.

2.1 Lapse Behaviour

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2.1 Lapse Behaviour

case for many other papers discussed in this section. Such results may not directly apply to other forms of insurance. Nonetheless, it is interesting to examine whether a premium increase incentivizes policyholders to explore other options and eventually terminate their contract.

This idea has been executed by Guelman and Guill´en (2014), who proposed a causal inference framework to measure price elasticity. They applied the framework to car insurance data and found evidence that higher premiums lead to a higher fraction of policies being terminated. Another paper on this issue is Dutang (2012). This paper investigated the effect of price changes on the renewal of non-life insurance contracts in the case of private motor insurance and identified an inverse relationship. The other two main contributions of this paper are: the importance of including a market proxy in the prediction model, and the impact of adverse selection on lapse decisions. The former suggests the presence of a competitive market. The latter implies high-risk individuals are less likely to end their contract than low high-risk-individuals after a premium increase.

The number of policies purchased has also been found to be a relevant factor. Guillen et al. (2008) used both logistic regression and survival analysis techniques to show that the cancellation of one policy is a strong indicator of cancellation of other household policies. They argue that individuals belonging to the same household are likely to behave in the same way. Furthermore, the number of policies purchased has another effect. Staudt and Wagner (2018) showed that this number positively influences customer retention. Hence, a customer having multiple policies contracted at one insurance company is less likely to terminate a policy. Yet if he does, it is presumed his other policies are more likely to be terminated as well.

Staudt and Wagner (2018) demonstrated how the probability of cancellation or non-renewal may differ in the type of the policy. The authors provided evidence that car insurance holders are more likely to end their contract than holders of liability insurance. Furthermore, the authors investigated the influence of the number of damages on the lapse probability. Their numbers are statistically significant but show opposite signs in different years and are therefore assumed not trustworthy. Conducting further analysis including the size and the type of the damages is suggested.

Another aspect that may influence the probability of termination of the policy is the age of the contract. The findings of Reinartz and Kumar (2000) suggest customer satisfaction increases with the length of the relationship to the company. If we associate a higher customer satisfaction with a lower probability of contract termination, this result is consistent with the finding of Verhoef et al. (2001): the lapse rate is inversely related to the contract age. We should, however, keep in mind the conclusion of Oliver (1999). He pointed out that in highly competitive markets, even customers with a high satisfaction level occasionally terminate their contract.

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2.2 Cross-buying Behaviour

they find a negative effect on the lapse rate. Furthermore, G¨unther et al. (2011) succesfully use, among other things, premium, age, gender, whether a partner also has a policy within the company, and having car and/or home insurance as explanatory variables to predict the probability of customer’s risk of leaving the insurance company.

An additional personal determinant of lapse probability is the residence area. Staudt and Wagner (2018) included a dummy variable indicating residence in a rural region in their logistic regression model. Llave et al. (2018) signified the importance of the customer’s proximity to the office of the insurance company and to the offices of competitors. This phenomenon is evidence for the so-called home bias described in behavioural economics. Moreover, they show that a customer is more likely to terminate their policy if nearby customers have done so as well, which corresponds to herding behaviour.

2.2 Cross-buying Behaviour

Newly initiated policies are the result of customer acquisition or follow from cross-buying of policyholders. Since personal characteristics of all potential customers are impossible to gather, we describe the literature on cross-buying behaviour of policyholders in non-life insurance. Re-search has been done to assess the relationship between characteristics of the policy/policyholder and the likelihood of initiating additional policies at the same insurance company.

Just as the lapse probability, the likelihood of cross-buying may be influenced by the type of policy. Kumar et al. (2008) showed that cross-buying depends on the type of the policy that is purchased first by the policyholder. Customers buying specific types of policies are more likely to buy additional insurance. Adding to this, Verhoef and Donkers (2005) provided evidence that individuals with automobile insurance are more likely to purchase other policies at the insurance company.

Another factor that should be considered when analyzing cross-buying behaviour is the number of filed claims with respect to the policy under consideration. Kamakura (2008) reported that a larger number of damages leads to more contact with the insurance company, which in turn may lead to a better customer relationship. In Verhoef et al. (2001) it is shown that the customer relationship has a positive effect on the cross-buying behaviour. Combining these findings, we infer that the number of damages reported may positively influence the number of policies that are bought by policyholders.

2.3 Demand for Non-Life Insurance

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3 Data

the period 2000-2011 was performed by Trinh et al. (2016). They showed that among other fac-tors, economic freedom, income, and urbanization are drivers of non-life insurance expenditure. Their results also indicated large differences across countries. Petkovski and Kjosevski (2014) examined 16 countries in Central and South-Eastern Europe and provided evidence for the posi-tive influence of GDP per capita on non-life insurance consumption in the long-run. One should take into account, however, the systemic heterogeneity due to country-specific characteristics.

This is also emphasized by Chang et al. (2013). Using bootstrap panel Granger causality tests, they showed that in many countries, including the Netherlands, there is Granger causality between insurance market development and economic growth, but not vice versa. This implies that the developments of the insurance market are useful in forecasting economic growth, yet economic growth is not likely to yield a proper prediction of the developments of the insurance market. In contrast, the direction of this relationship is reversed for Italy and Canada. Due to the cross-country differences, there exists a lot of diversity in the literature answering these kinds of questions. Simona (2014) provides an overview of the contrasting papers. In addition, the author shows that income is a significant factor for non-life insurance demand in Europe.

3 Data

The forecast models that are compared in this thesis are based on numbers extracted from quarterly policy data from the fourth quarter of 2016 up to and including the third quarter of 2020. Thus, there are sixteen data sets at our disposal. The data provided consists of characteristics of every policy in force, where individual policyholders can be recognized from an identity number. This is important, because there are policyholders who have purchased multiple policies that may even share the same characteristics. A policy will automatically be renewed if no action is undertaken.

Consistent with Solvency II guidelines, the policies are grouped into homogeneous risk groups (HRG’s). They represent an insurance risk class. The ‘type’ of policy mentioned before is the HRG it belongs to. Our data consists of the combined policy data sets from two distribution channels. By assembling the relevant information from the different policy data files over time, a time series of the number of policies over time per HRG and per payment term is created. We also differentiate between EXI, NBR, and NBA.

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4 Methodology

Attribute Variable Description

Identity policyholder IDP Identity number of policyholder

Commencement NEW Contract was initiated during the quarter

Termination END Contract was terminated during the quarter

Duration DUR Number of quarters that have passed since the starting

date of the first contract of the policy

Policy expires EXP Dummy indicating whether the contract term ends

dur-ing the quarter

Contract premium PRE Annual premium to be paid by the policyholder

Payment term PAY The time interval at which the premium is paid

Customer Level CUL Dummy indicating whether the customer is corporate

(1) or private (0)

Covered COV Number of items covered in the policy

High number covered CHI Dummy indicating whether the number of items covered in the policy is larger than one

Number of policies NUM Number of policies purchased by the policyholder High number of policies NHI Dummy indicating whether the number of policies

pur-chased by the policyholder is larger than one

HRG HRG Homogeneous risk group the policy is associated with

Table 1: Variables Used in Data Analysis.

4 Methodology

The goal is to find the optimal model for predicting the incoming premium amount during the upcoming quarter. In order to predict the premium income resulting from policies that were already in force at the beginning of the quarter, we analyze the Cox proportional hazards model and the logistic regression model. The details of these models are explained in Sections 6 and 7. We compare their forecast performances by making use of the rolling-origin evaluations as described in Tashman (2000).

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5 Survival Analysis Basics

These are averaged over all splits to determine the final estimate of the prediction error. This procedure is called fixed-size rolling window and is presented in Swanson and White (1997).

In our models, we do not take calendar time into account. As can be observed from Section 3, only variables regarding the duration of the policy, i.e. the time until the event occurs, are included. The time that has elapsed since the initiation of the policy is termed process time. The exclusion of parameters regarding the calendar time is a convenience matter, because there are only fifteen periods available and therefore the results would not be trustworthy. To take any temporal dependencies into account that might be present, we must withhold all data that occured in the period after the time span of the training set. In the real world, all data about events that occur in the future is unknown. Every test set should follow chronologically after the training set.

5 Survival Analysis Basics

We first approach the problem of predicting the probability that a policy will be terminated from a survival point of view by focussing on the duration of the policy. Survival analysis corresponds to a set of statistical methods that focuses on the expected duration until a specified event occurs. We define an event as the moment the policyholder terminates the contract, which may occur both during and at the end of the contract term. Subsequently, every single policy constitutes one observation, even though a customer might hold multiple policies. The survival time is the time from commencement of the contract until the specified event. In our sample, not all policies have terminated within the studied time period. As a result, we are unable to observe their survival time, leading us to the concept of censoring. All that is known is the lower limit of the survival time, so right-censoring is in order.

In this research, the survival function S represents the probability that a policy is terminated after time t > 0 and hence “survives” from the time of initiation until after the specified time t. The time of initiation differs per policy and is specified by the year and quarter in which the policy commenced. Note that the survival function should be decreasing in t, with S(0) = 1 and limt→∞S(t) = 0. Let T denote the number of time periods between initiation and termination of

the policy. It is a random variable. The hazard function h is a limit describing the instantaneous risk that a policy is terminated at time t:

h(t) = lim ∆t↓0 Pr{t ≤ T < t + ∆t|T ≥ t} ∆t = − S0(t) S(t). (2)

It is the conditional failure rate. The corresponding cumulative hazard function is given by

H(t) = Z t

0

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5.1 Log-Rank Test

The relation between the survival function and the hazard function is given by

S(t) = exp {−H(t)} . (4)

In order to use the survival analysis techniques in Sections 5.1, 5.2, and 5.3, the data is altered to obtain one large dataset including all policies that had been in force during the period starting at the fourth quarter of 2016 up to and including the third quarter of 2020. Let the number of policies in this aggregated data set be denoted by n ∈ N. A survival analysis model that includes multiple covariates is the Cox proportional hazards model. This model is presented in Section 6.

5.1 Log-Rank Test

To obtain a first insight in the potential benefits of adding an additional characteristic to the prediction model, we investigate whether the distributions of the durations of policies grouped according to the characteristic are distinguishable. The most commonly used test to compare survival functions of two different groups is the log-rank test. One of its main advantages is that no knowledge of the shape of the survival curve or distribution of the survival times is required. By performing pairwise log-rank tests, we are also able to establish the potential benefit of adding a categorical covariate that can take on multiple values.

The most well known and widely used log-rank test is called the Mantel-Haenszel log-rank test. Let {1, ..., I} denote the indicators of the distinct times that an event occurs in either of the groups. For both groups j, j = 1, 2, and each time ti an event has occurred, i = 1, ..., I, the

observed number of events oij is calculated together with the number eij that would be expected

if there were no differences between the groups. Let nij denote the total number of policies of

group j in force just before time ti. Let ni= ni1+ ni2 and oi = oi1+ oi2.

The Mantel-Haenszel log-rank test includes a test statistic equal to the sum of the squared differences between observed and expected number events for both groups, divided by the hy-pergeometric variance: χ2_MH,j = h PI i=1(oij − eij) i2 PI i=1Vij → χ2(1) for j = 1, 2, (5)

where the hypergeometric variance is given by

Vij = eij ni− oi ni ni− nij ni− 1 for i = 1, ..., I, j = 1, 2. (6)

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5.2 Kaplan-Meier Estimation Method

a score test on the Cox proportional hazards model when comparing two groups.

A pairwise log-rank test shows significant differences between the majority of HRGs. Further log-rank tests show the presence of significant differences between policies with corporate and private policyholders and between policies terminated either at their expiration date or during their contract term. Also, log-rank tests provide evidence of significant differences between policies with different payment terms, except between quarterly and semi-annual payments.

The log-rank test uses equal weighting, thereby assuming that the survival probabilities are equal for policies with different moments of initiation. The Kaplan-Meier estimation method, presented in Section 5.2, makes the same assumption. The log-rank test will presumably not detect any differences between separate groups if their survival curves cross. In addition, the log-rank test does not estimate the size of the difference between the types of policies. For these reasons, it is important to plot the estimated survival curves.

5.2 Kaplan-Meier Estimation Method

One technique to quickly estimate the survival function of different types of policies is the Kaplan-Meier method. It is a non-parametric method that provides estimates of the survival probabilities considering ordered discrete times ti, i = 0, . . . , T , with t0 = 0 and corresponding

S(t0) = 1. If we let ni be as above and let di denote the number of policies terminated at time

ti, i = 1, . . . , I, then the Kaplan-Meier estimate of the survival probability is given by

SKM

(ti) = SKM(ti−1)

ni− di

ni

, i = 1, ..., I. (7)

No assumptions regarding the distribution are required. Hence, the estimator is often used to graphically check the fit of parametric models. The Kaplan-Meier method is useful only when the predictor variable is categorical. If one wants to take other non-categorical factors into account, the Cox proportional-hazards model provides the right tools.

The log-rank test shows significant differences between the survival curves of policies grouped by HRG. However, a plot of the survival curves shows that many of them cross at some point(s) in time. This holds for partitioning on payment terms as well. When partitioning on customer level, the survival curve of policies being held by private customers decreases more rapidly than the survival curve of the policies classified as corporate (Figure 1). Hence, the duration of policies held by private customers is generally shorter. In addition, the survival curve of policies held by corporate policyholders resembles an exponential survival function, which is indicated by the dashed red line. We can therefore cautiously deduce from this graph that corporate customers have a constant hazard rate with respect to time.

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5.3 Nelson-Aalen Estimation Method

Figure 1: Kaplan-Meier Survival Curves for Separate Customer Levels

5.3 Nelson-Aalen Estimation Method

An alternative estimator of the survival distribution that does not require any assumptions on the distribution, is the Nelson-Aalen estimator. It provides an estimate of the cumulative hazard function, HNA_{(t) =} X i:ti≤t di ni , (8)

that in turn can be used to find an estimate of the survival function:

SNA_{(t) = exp {−H}NA_{(t)} .} ₍₉₎

An advantage of this estimation method is that Pn

i=1HNA(ti), i.e. the expected number of

events, is equal to the observed number of events. The relationship between Kaplan-Meier survival curve estimates and the Nelson-Aalen cumulative hazard function estimates is given by

SKM(t) = Y j:tj≤t 1 −dH NA_(t j) dtj , (10)

which can be confirmed by inferring from Equation (8) that dHNA(tj)/dtj = dj/nj. The

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6 Cox Proportional Hazards Model

Recall that the hazard function h describes the conditional failure rate, being a limit describing the instantaneous risk that the event occurs at the specified time t given that it has not been terminated before t. To study the dependence of this hazard function for observation i on p covariates, given by Xi = (Xi1, . . . , Xip)0, Cox (1972) developed the Cox proportional hazards

model:

h(t, Xi) = h0(t) · expβ0Xi , i = 1, ..., n. (11)

The baseline hazard function h0 corresponds to the value of the hazard function if all

covari-ates were equal to their baseline value. We call β0Xi the index of the hazard function of policy

i, i = 1, ..., n. The values β = (β1, ..., βp)0 describe the influence of the explanatory variables on

the hazard. Hence, the hazard function depends parametrically on the covariates and it depends nonparametrically on time. The model is known as the proportional hazards model, because the hazard ratio h(t, Xi)/h(t, Xj) of observations i and j is constant over time. If no assumptions

are made about the form of the baseline hazard function, this model is called semi-parametric. Note that it can be rewritten into a multiple linear regression of the logarithm of the hazard function with the baseline hazard being analogous to the intercept term of the regression.

The relative hazard ratio φi = exp(βi), i = 1, . . . , p, describes the effect of covariate i on

the hazard function. If the relative hazard ratio is equal to one the covariate does not influene the baseline hazard, if it is larger than one the covariate increases the baseline hazard, and if it is smaller than one the covariate leads to a reduction in the baseline hazard. An increase in this conditional failure rate implies that the event is more likely to occur and a decrease would lead to the event being less likely to occur, given time t. Note that φi should be independent of

process time. Hence, one of the main assumptions of the Cox proportional hazards model is that the hazard curves for the different groups are independent of time. They should be proportional and thus may not cross.

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6.1 Maximum Partial Likelihood Estimation

Suppose there are no ties in the data, i.e. ti 6= tj for i 6= j, i, j = 1, ..., n. Cox (1972) suggested

maximizing the following partial likelihood to estimate β:

PL(β) = Y i:Ci=1 exp{β0Xi} P j:tj≥tiexp{β 0_X j} , (12)

where Xi = (Xi1, ..., Xip)0. Oakes (1981) points out that every i-th term in the product of

Equation (12) is the conditional probability that the event occuring at time ti is the event of

observation i, given that an event occurs at ti and given the set of observations that is at risk

at time ti.

The corresponding log partial likelihood

`(β) = X i:Ci=1  β0Xi− log   X j:tj≥ti exp{β0Xj}     (13)

does not depend on the baseline hazard. The partial score function is derived by differentiating the log partial likelihood function over β,

s(β) = X i:Ci=1 " Xi− P j:tj≥tiexp{β 0 Xj}Xj P j:tj≥tiexp{β 0_X j} # , (14)

and the Fisher information matrix is derived by taking the negative of the second derivative of the log partial likelihood. It is a p × p matrix given by

I(β) = X

i:Ci=1

V (β, ti) . (15)

Here V (β, ti) denotes the variance matrix of the covariates at time ti, written as

V (β, ti) = P j:tj≥tiexp{β 0_X j}XjX0j P j:tj≥tiexp{β 0 Xj} − P j:tj≥tiexp{β 0_X j}Xj exp{β0Xj}Xj 0 P j:tj≥tiexp{β 0_X j} 2 for i ∈ {j : j = 1, ..., n, Cj = 1} . (16)

Setting the score function equal to zero yields the maximum partial likelihood estimator b

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6.2 Adjusting for Ties

Nonetheless, our data contains tied event times. Multiple policies may be terminated at the same time. Suppose there are I ordered distinct survival times ti, i = 1, ..., I, as in Section 5.2.

An approximation proposed by Breslow (1974) for the partial likelihood, taking into account the tied event times, is given by

PLBreslow(β) = I Y i=1 exp{β0Si} P j∈Dk:tk≥tiexp{β 0 Xj} di, (17)

where we set Di = {j : policy j is terminated at time ti} ⊂ {1, ..., n} with ∪Ii=1Di = n and

Di∩ Dj = ∅ to define

Si =

X

j∈Di

Xi. (18)

As above, di denotes the number of terminated policies at time ti, i = 1, ..., I. An alternative

approximation proposed by Efron (1977) provides estimates that are even closer to the true values. Efron noticed that the denominator of the Breslow approximation is always larger than it should be. He suggested the following approximation:

PLEfron(β) = I Y i=1 exp{β0Si} Qdi−1 k=0 P j∈Dk:tk≥tiexp{β 0_X j} −_dk_iPj∈Diexp{β 0_X j} . (19)

Both approximations provide estimates of β close to zero if the number of ties is relatively large compared to the number of policies in force just before the time of termination. The Efron approximation provides more accurate estimations than the Breslow approximation and it is therefore is used in this paper.

6.3 Testing Linearity and Additivity

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6.4 Testing Proportionality of Hazards

6.3.1 Transformations

Let us first introduce the concept of martingale residuals. In this context, a martingale residual is the difference between the observed value and the expected value of the variable indicating whether the policy was terminated. Positive values indicate that the policy was terminated earlier than expected, negative values indicate that the policy was in force for a longer period than expected or that it was censored.

Therneau et al. (1990) suggested plotting the martingale residuals of a model where all coefficients equal zero against continuous covariates to assess how these covariates should be transformed. They show that if the true impact of covariate j on the hazard function is of the form exp {βj· f (X·j)} for some differentiable function f , then the expectation of the martingale

residual for policyholder i given covariate Xij is equal to c · f (Xij), i = 1, ..., n, j = 1, ..., p.

Here, c ∈ R is a constant that is affected by the amount of censoring. As a result, the plot of a covariate against the corresponding martingale residuals will provide an insight intp how that particular covariate should be transformed.

6.3.2 Interactions

The residual plots above may provide useful insights when the model is additive in its covariates, but fails when interaction terms are present in the correct model. In an adjusted variable plot, residuals of a regression that does not include the examined covariate are plotted against the residuals of a regression of the examined covariate on all other covariates. The adjusted variable plot will be linear if the model is also linear in all (possibly transformed) covariates. However, unlike the martingale residual plot, a non-linear adjusted variable plot does not inform us about the true form of the model. It only verifies the additivity assumption.

6.4 Testing Proportionality of Hazards

As mentioned in the introduction of this section, the name of this model refers to the ratio of hazards h(t, Xi)/h(t, Xj) of observations i and j being constant over time. This proportional

hazards assumption should be tested. It comes down to testing whether all regression effects βi are independent over time. Winnett and Sasieni (2001) describe a method that makes use

of scaled Schoenfeld residuals. The Schoenfeld residuals si ∈ Rp for observation i are given

by the differences between the observed and expected covariates for each event, which can be formulated as si = Xi− P j:tj≥tiexpβ 0_X j Xj P j:tj≥tiexpβ 0_X j for i ∈ {j : j = 1, ..., n, Cj = 1} . (20)

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6.5 Dependent Observations

If the values of β depend on time, the model may be written as

h (t, Xi) = h0(t) exp(β + g(t))0Xi

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for some vector g that is a function of time. If the function g is different from zero, the hazard ratios are not constant over time. Grambsch and Therneau (1994) proved that

V−1(β, ti) E [si] ≈ g(ti). (22)

Hence, plotting the coefficient of the linear regression of the scaled Schoenfeld residuals on time is equivalent to testing the proportional hazards assumption. If the coefficients of the linear regression are changing over time, the assumption does not hold.

Multiple policies may be held by one individual policyholder. We assume that the decisions of one policyholder do not depend on the decisions of other policyholders. We do, however, expect one of his decisions to depend on the others. If this is the case, we anticipate there to be correlation in the errors. To incorporate this expectation into the Cox proportional hazards function, random effects are added to the survival model. These random effects describe the shared additional risk (frailty) for policies to be terminated if they are held by the same policyholder. The “population”, being the total set of policies in this situation, is not homogeneous. We are dealing with clustered data where the policies are divided into groups based on their shared policyholder. Incorporating random effects on the policyholder level captures the unobserved covariates that cause this heterogeneity.

We assume the policies from the same policyholder to share the same frailty. This random effect is therefore assumed to represent the common risk for all policies held by one policyholder. Suppose there are m policyholders that together hold n policies. Formally, to include frailties, the traditional model specified in Equation (11) is extended to

h (t, Xi, Qi) = h0(t) · expβ0Xi+ ω0Qi , i = 1, ..., n, (23)

where the values ωj in ω ∈ Rm represent the frailty of a policy being held by policyholder j,

which is indicated by the binary values Qij in Qi ∈ Rm, j = 1, ..., m. Equation (23) takes on

the form of a penalized Cox model with β and ω denoting the unconstrained and constrained coefficients, respectively. The corresponding penalized partial loglikelihood becomes

PPL (β, ω) = PL (β, ω) − p(ω; θ), (24)

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The general form of a shared frailty model is

h t, Xi, Ωj(i) = h0(t)Ωj(i)expβ0Xi , i = 1, ..., n, j = 1, ..., m, (25)

where Ωj(i)is the common frailty of group j that observation i belongs to. The random variables

Ωj are independently and identically distributed with mean one and variance θ ∈ R. Many

distributions may be chosen for these frailties, but the gamma distribution with parameters α = β = θ−1 is frequently used. It is therefore also employed in this paper. Therneau et al. (2003) show using the Laplace transformation of the gamma distribution that if the penalty function is chosen as p(ω; θ) = 1 θ m X j=1 [ωj− exp {ωj}] , (26)

the solution to the gamma frailty model is equivalent to solution of the model specified in Equation (23) with Ωj = exp {ωj}. Larger heterogeneity between groups is reflected in larger

values of θ. In fact, association between observations in a group is given by Kendall’s tau, which is θ/(2 + θ). If there is no association effect present in the data and thus θ = 0, this indeed corresponds to the values ωj = 0 for all j = 1, ..., m indicating no dependence of occurence of

an event on other outcomes.

When forecasting the survival probability, two kinds of predictive probabilities can be cal-culated. The first is related to policyholders that are present in both the training and the test set. A conditional survival probability, given the specific policyholder, can be calculated by ap-plying the frailty estimated using the training set. Second, one can compute a marginal survival probability where the frailty is averaged over the population. This is done as follows:

EΩ[S(t, Xi, Ω)] = EΩ[exp {−ΩH(s, Xi)}] = 1 θ 1/θ Γ 1 θ −1Z ∞ 0 exp −z 1 θ + H(s, Xi) z1θ−1 dz = 1 θ 1/θ Γ 1 θ −1 1 θ+ H(s, Xi) −1/θZ ∞ 0 exp {−u} u1θ−1 du = [1 + θH(s, Xi)]−1/θ, (27)

where we have used the fact that Ω ∼ Γ 1_θ,1_θ. This expectation is applied when forecasting the survival probabilities of policies held by policyholders that were not present in the training set.

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7 Logistic Regression survival probability as Pr (ti ≤ t + 1|ti> t) = 1 − Pr (ti > t + 1|ti> t) = 1 − S(t + 1) S(t) . (28)

The predicted value of the outcome variable, i.e. the variable indicating whether policy i held by policyholder j is terminated during the quarter, i = 1, ..., n, j = 1, ..., m, is given by

b Y_iCox=      1 −hexp{−H0(t+1)} exp{−H0(t)} iΩj(i)exp{β0Xi}

if a frailty is known for policyholder j, 1 − h 1+θH0(t) exp{β0Xi} 1+θH0(t+1) exp{β0Xi} i1/θ otherwise. (29)

The predicted values are inserted into Equation (1) to determine the forecast error.

7 Logistic Regression

We now approach the problem of predicting the probability that a policy will be terminated as the prediction of a binary event. Contrary to survival analysis, the main variable under investigation is not the time unit the event takes place, but the binary variable END that indicates whether or not the event has occurred. The logistic regression model enables us to calculate the probability of an event happening based on a set of explanatory variables that are not necessarily categorical. It is a form of a generalized linear model with the link function being a logit link. Relevant in this paper is the binary logistic regression model. The logistic regression model is fully parametric and does not necessarily include a time unit.

7.1 Independent Observations

The logistic regression model does not require any assumptions on the distributions of either END or the explanatory variables. The model requires independence of the observations, linear-ity of the model and other assumptions that are implicitly made when setting up the regression. Let pi= Pr (ENDi= 1|Xi) denote the probability that the event occurs given explanatory

vari-ables Xi ∈ Rp, i = 1, .., n. The value of the explanatory variables is observed at the start of the

quarter. A logistic regression model is given by

log pi 1 − pi = β0Xi for i = 1, ..., n, (30)

where β ∈ Rp are the parameters of the model. Hence, if we make the assumption that no interaction effects are present, then βj denotes the change in the log odds per unit change of the

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the above expression yields

pi = 1 + exp{−β0Xi}

−1

. (31)

The main advantage of this model is that is simple to understand and the estimated param-eters can easily be interpreted. Just as for the Cox proportional hazards model, the assumption of the linearity and additivity in the covariates should be tested. A linear relationship between the log odds of the dependent variable and the explanatory variables is assumed. To test the validity of this assumption graphically, one might look at plots of the log odds values against continious covariates. A linear relation confirms the assumption is correct.

An mentioned above, one essential assumption of the logistic regression model is independence of the observations. In our sample, an observation is given by a single policy. Note, however, that a policyholder might hold multiple policies at the insurance company. The assumption of independence might not hold. Bonney (1987) describes a regressive logistic model, which is a model that allows for dependence between the observations. We will modify this model to suit our purposes.

Consider a sample of n policies that are held by m ≤ n policyholders. The sample consists of {X_i, Yi}, where Yi = ENDi is the binary variable indicating whether the policy was terminated

and Xi= (1, Xi1, Xi2, ..., Xip)0 is the vector of explanatory variables. Let Nj = {N1j, N j 2, ..., N

j nj}

denote the set of nj policies that are purchased by policyholder j, j = 1, ..., m. We assume the

policyholders to make decisions independent of each other, but the decisions made by a single policyholder are allowed to be correlated. Let us illustrate this matter by means of an example. Suppose the sample consists of seven policies that are purchased by three different policy-holders, with N1 _{= {1, 2, 3, 4}, N}2_{= {5}, and N}3 _{= {6, 7}. The assumptions above imply that}

for Y = (Y1, ..., Yn)0 ∈ Rn and X = (X1, ..., Xn)0 ∈ Rp×n, we have

Pr (Y|X) = Pr (Y1, Y2, Y3, Y4|X1, X2, X3, X4) · Pr (Y5|X5) · Pr (Y6, Y7|X6, X7) . (32)

The m = 3 conditional probabilities can each be decomposed into a product of nj probabilities,

1 ≤ j ≤ m. Also, we make the additional assumption that a policy does not depend on the explanatory variables of other policies. As an illustration, the decomposition of the first conditional probability on the right hand side of Equation (32) is given by

Pr (Y1, Y2, Y3, Y4|X1, X2, X3, X4) =Pr (Y1|X1) · Pr (Y2|Y1, X2) ·

Pr (Y3|Y1, Y2, X3) · Pr (Y4|Y1, Y2, Y3, X4) .

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rise to the following general formulation of the joint probability of a set of events:

Pr (Y1, Y2, ..., Yn|X1, X2, ..., Xn) = m Y j=1 nj Y i=1 Pr Yi|{Yk : k ∈ Nj, k ≤ i}, Xi . (34)

The standard logistic regression model can be extended to incorporate this dependence. Building on the method proposed by Bonney (1987), we define the (i, k)-th element of Z ∈ Rn×(n−1) as

Zik=    ˜ Zk = 2Yk− 1 if {i, k} ⊂ Nj, i < k and 1 ≤ j ≤ m, 0 otherwise. (35)

Hence, in our example the matrix Z is given by

Z =              0 0 0 0 0 0 0 ˜ Z1 0 0 0 0 0 0 ˜ Z1 Z˜2 0 0 0 0 0 ˜ Z1 Z˜2 Z˜3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Z˜6              . (36)

Let Zi denote the i-th row of Z. We assume the log odds ratio to be of the form

ψi = logit [Pr (Yi = 1|{Yk: i, k ∈ nj, k 6= i}, Xi)]

= β0Xi+ γι0n−1Zi

= β0+ β1Xi1+ β2Xi2+ ... + βpXip+ γ (Zi1+ Zi2+ ... + Zi,n−1) ,

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where β = (β0, β1, ..., βp)0∈ Rp+1is the coefficient vector specifying the relationship between the

covariates and the response variable, γ ∈ R is a parameter denoting the strength of interactions, and ιn−1 is a (n − 1)-vector of ones. We only estimate an interaction effect on a general basis

and not per individual policyholder, because the policy data on which the estimated model is implemented will contain policies from new policyholders as well. With this new characterization of the log odds ratio, Equation (32) becomes

Pr (Y|X) = n Y i=1 exp{ψiYi} 1 + exp{ψi} . (38)

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8 Results

from one policyholder:

odds [Pr (Yi = 1|Yk= 1, Xi)]

odds [Pr (Yi = 1|Yk= 0, Xi)]

= expβ

0_X

i+ γ(Zi1+ ... + Zi,k−1+ 1 + Zi,k+1+ ... + Zi,n−1)

expβ0_X

i+ γ(Zi1+ ... + Zi,k−1− 1 + Zi,k+1+ ... + Zi,n−1)

= exp{2γ} for i, k ∈ nj, k 6= i, j = 1, ..., m.

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An advantage of this model is that it can easily be estimated using ordinary statistical analysis software. To create forecasts, the logistic regression model without a Z-variable is estimated by using the test set. The predicted outcome variables from this estimation are used to create the Z-variables for the test set. The predicted value of the outcome variable resulting from the regressive logistic regression model is given by

b

Y_iRLR= cPr (Yi = 1|Xi, Zi) =pbi, for i = 1, ..., n. (40)

8 Results

Because this a prediction study, the goal is to create the best predictions for the outcome variable. This is different from identifying the true causes of a policy being terminated. Hence, we are less interested in the value and/or significance of the coefficients. Nevertheless, a correct specification of the model enhances the predictive power. Since all variables included are believed to be relevant to the occurrence of the event, they are all included in the model that predicts the outcome variables. When scatter plots are presented, the relationship between the presented variables is identified by applying a locally weighted smoothing method.

8.1 Results Cox Proportional Hazards with Shared Frailty Model

Martingale residual plots can only be depicted for continuous variables. Figure 3 (Appendix A.2) depicts the martingale residual plots of the annual contract premium including some trans-formations. From these figures, we conclude that in the Cox proportional hazards model with shared frailty, the logarithm of the annual contract premium should be used. As expected, an increase in low level premiums leads to a larger change in the probability of termination than a similar change in high level premiums. The logarithm consistently shows an approximately linear relation for all data sets, where the other transformations do not.

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8.1 Results Cox Proportional Hazards with Shared Frailty Model

or more policies. We replace this variable by a dummy NHI , indicating whether the number of policies held by the policyholder is larger than one, and an interaction term NHI∗NUM .

Figure 6 (Appendix A.2) shows the development of the coefficient of the linear regression of the scaled Schoenfeld residuals of EXP , i.e. the variable indicating whether the contract term of the policy ends during the quarter, over time. If the proportional hazards assumption regarding this variable hold, this plot would show a horizontal line. Clearly, this is not case. The variable EXP has therefore been removed from further analysis in this survival model. No other variables lead to graphs that are not linear.

To test the validity of the additivity assumption in this model, adjusted variable plots are used. Two models are estimated. The first is a Cox proportional hazards with shared frailty regression on all variables except the logarithm of annual contract premium. The second is a regression of this logarithm on all other variables. Their residuals are plotted in Figure 7 (Appendix A.2). It can be observed that the relationship between these residuals is linear, which confirms the appropriate use of the additivity assumption.

For illustration purposes, the results of the model estimation regarding the first quarter of 2019 are presented in Table 2. Note that these results stem from estimation of the models on random subsets of the data. They are therefore not applicable in reality. We find a higher premium to significantly increase the probability of termination. To find the effect of the payment term, a payment term of 6 months is set as base category. Relative to this category, policies for which premiums are paid on a monthly basis have a significantly higher probability of being terminated. The opposite holds for policies with yearly paid premiums. The effects of having monthly or yearly paid premiums compared to quarterly payments is statistically significant. Compared to policies that only cover one item, policies covering multiple items are found to generally have a probability of termination that is significantly higher. Every additional item from two items onwards that is covered by the policy decreases the probability of cancellation. Corporate policyholders are less likely to terminate their policy than private policyholders. This effect is also statistically significant.

The above results make sense intuitively. Less easy to explain are the effects resulting from the policy being associated with a particular homogeneous risk group. These effects result from characteristics specific to the HRG that we are unable to capture. From Table 2, we also infer that there is a large positive effect on the probability of termination from the policyholder having purchased multiple policies. One can think of many explanations for this positive influence. For example, having purchased more than one policy leads to a larger cumulative premium that in turn may result in a larger willingness to cancel one or multiple policies. The effect of having purchased an additional policy is smaller when the number of policies purchased by the policyholder is larger than one.

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8.1 Results Cox Proportional Hazards with Shared Frailty Model P olicy termination Cox PH with shar ed fr ailty Cox PH without shar ed fr ailty R egr essive lo git L o git b β se( b β ) b φ = exp( b β ) p-v alue b β se( b β ) b φ = exp( b β ) p-v alue Estimate Std. Error p-v alue Estimate Std. Error p-v alue In tercept

EXP DUR log

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8.2 Results Regressive Logistic Regression Model

found for the other time periods. The rolling-origin evaluations lead to an average root mean square error of 0.043298. It should be noted that estimating such models may take the R programming language up to several hours. This is due to the involvement of the shared frailty effects. A similar model that does not include the shared frailty effects is alo presented in Table 2. It can be observed that the coefficients do not differ much from the original model. In addition, this models leads to an average root mean squared error of 0.043301. Hence, one should make the trade-off between time and model fit when deciding on which model to apply.

8.2 Results Regressive Logistic Regression Model

To test the linearity assumption in the regressive logistic regression model, Figure 8 (Appendix A.2) presents plots of the log odds ratio against the continuous variables under consideration. The annual contract premium, the number of quarters that have passed since the starting date of the first contract of the policy, and the number of items covered in the policy are all linearly related to the logit value. The number of policies held by the policyholder is not. As was the case for the Cox proportional hazards model, the effect of the number of policies held changes from having purchased one policy to having purchased two or more policies. Therefore, we again replace this variable by the dummy NHI and the interaction term NHI∗NUM .

Table 2 presents the results of the estimation of the regressive logistic regression model regarding the first quarter of 2019. When the contract term ends during the upcoming quarter, the probability that the policy will be terminated during this quarter increases. This probability increases with the time that this policy has been in force. What is remarkable, is that some of the coefficients are of opposite sign from the corresponding coefficients in the Cox proportional hazards model. This is for example the case for the coefficients of the sixth HRG. We conclude that the effects of this variable is ambiguous.

The resulting average RMSE of the regressive logistic regression model is 0.042055. Hence, we may conclude that this model provides better forecasts than the Cox proportional hazards with shared frailty model, with the additional advantage of a much lower estimation time. Removing the dependency measure (the Z-variable) leads to a RMSE of 0.042254, slightly above the model that includes this measure. The latter RMSE is still below the RMSE of the Cox proportional hazards model that includes shared frailty.

9 Forecasting Premium Income

9.1 Unearned Premium Reserve and Future Premiums

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9.2 Creating Confidence Intervals

already existed at the beginning of the quarter is estimated by the product of the current annual premium, the fraction of this amount that would potentially be earned during the forecasted quarter and the probability that it will not be terminated during the quarter. At this point, it is important to elaborate on the concepts of unearned premium reserve (UPR) and future premiums (FP). The total of the incoming premiums is termed total future premiums (TFP) and can be distinguished into UPR and FP. The UPR is the amount of premium that has been received but is not yet earned. These premiums are paid in advance, as is for example the case with payments on a yearly basis. Insurance companies are required to keep the unearned premium income as a reserve and consider it as a liability in their accounting books, since it has to be returned to the policyholder upon cancellation of the policy.

The premiums of policies currently in force that are not yet received but are expected to in the future are included in the FP. This concerns the premiums to be received from current contracts until the contract boundary and the recognition of contracts that will expire in the next fixed period of time that might be prolonged. The latter is required by a regulation of Solvency II, stating that the insurance company may not unilaterally cancel the contract in a set period of time before the end of the contract term. The contract will therefore be prolonged in the absence of action from the policyholder. The set period of time before the end of the contract term is called the recognition period. Its length is company-specific and therefore not mentioned in this paper. An entire additional contract term of premiums must be included in the FP for the policies that are in their recognition period.

Note that we distinguish between policies from which we expect to receive and earn premiums in the upcoming quarter, policies from which we expect to receive unearned premiums in the upcoming quarter, and policies that have to be included in the future premiums. We take the probability of lapse into account, thereby making realistic predictions of the premium income in the upcoming quarter. Forecasts of both the UPR and FP are based on the probabilities of policy termination that are estimated in this paper.

A prediction can be made regarding the number of policies that will be initiated during the quarter (NBA), but it would merely be based on 15 data points. On top of that, it is challenging to associate a newly initiated policy with an annual premium. The fraction of premium that would be earned during the quarter, taking into account the data at which the policy was initiated, would be difficult to estimate as well. As a consequence, the premium forecast will not be trustworthy. We do expect there to be a seasonal effect on a yearly basis. For these reasons we opt to assume the premium income arising from newly initiated policies to be equal to the equivalent quarter of the year before.

9.2 Creating Confidence Intervals

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illus-9.2 Creating Confidence Intervals

trate how the confidence interval for the second quarter of 2019 is constructed. The regressive logistic regression model is fitted to the data of the first quarter of 2019, as in Section 8.2. Bootstrap samples are generated of the policies in force at the beginning of the second quarter of 2019. For every policy in the bootstrap samples, a probability that it will be terminated during this quarter is estimated based on the results of the fitted model. The estimated value of the outcome variable and thus the estimated probability of termination is denoted by bYi.

Let cPI denote the expected value of the premium income and define n2018Q2 and n2019Q2 as the number of policies in force at the beginning of the second quarter of 2018 and of 2019, respectively. We assume the decision to terminate the policy to be made mid-quarter and therefore termination of a policy to occur during the second month of the quarter. Let πi

denote the fraction of the annual premium income P REi that would be received during the

second quarter if policy i is not terminated and let αi denote the corresponding fraction if it

is terminated. Separating the premium incomes resulting from policies labeled EXI, NBR, or NBA, the expected value of the premium income that is received during the second quarter of 2019 is given by c PI = cPIEXI+ cPINBR+ cPINBA = n2019Q2 X i=1 h πi(1 − bYi) + αiYb_i · PREi i + cPINBA = n2019Q2 X i=1 h πi(1 − bYi) + αiYb_i · PREi i + n2018Q2 X j=1 h πj· 1{Yj=0}+ αj· 1{Yj=1} · PREj i . (41)

Exact definitions of πi and αi are presented in Appendix A.4. The amount of premiums that

is earned out of the received premiums (ERPI) can be estimated as well. Similarly, we can determine the amount of premiums that is earned out of the unearned premium reserve (EUPR). The amount that is reserved at the beginning of the quarter for policies that are terminated during the quarter has to be returned (RUPR). The expected change in the unearned premium reserve, denoted ∆ [UPR, is given by

\

∆UPR = cPI − \ERPI − \EUPR − \RUPR. (42)

The premiums that are included in the future premiums at the beginning of the period as a result of the recognition requirement are known. This “old” recognition amount (OR) is at the end of the period replaced by a “new” recognition amount (NR). At the beginning of the quarter, we therefore make an estimate of the new recognition amount. Not taking the recognition period into account, the amount of premium income that was included in the future premiums with regard to the analyzed quarter is denoted as FPINC. An amount equal to the

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10 Conclusion and Discussion

premiums arising from policies that are renewed during the quarter (FPNBR). Our estimate of

the additional future premiums resulting from policies that are initiated during the quarter is equal to the equivalent amount of the second quarter of 2018. We denote this estimate by

c

FPNBA. Finally, we estimate the change in future premiums as follows:

[

∆FP = dNR − OR − FPINC− cFPEND+ cFPNBR+ cFPNBA. (43)

Algebraic definitions of the values in Equations (42) and (43) are given in Appendix A.4. A histogram of the resulting bootstrap estimates is presented in Figure 2. The vertical red line represents the true value resulting from the subset of policies. Table 3 presents the lower and upper limit of the confidence intervals, the point estimates and the corresponding true values of the premium income, the change in unearned premium reserve, and the change in future premiums. The true values all lie within the generated confidence intervals and are quite close to the point estimates.

10 Conclusion and Discussion

The aim of this paper is finding the optimal model to forecast the incoming premiums during the upcoming quarter. At the beginning of every quarter, a data set containing characteristics of every policy in force is available. With this information, we are able to predict the probabil-ities that these policies will be terminated during the quarter and thereby estimate the related premium income. We divide policies that are in force at the end of the quarter into three groups: policies that already existed at the beginning of the quarter and have not expired during the quarter (EXI), policies that already existed at the beginning of the quarter and expired during the quarter, but of which the contracts have been renewed (NBR), and policies that were initi-ated during the quarter (NBA). The total premium income arising from newly added policies is estimated by the value of the previous year.

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10 Conclusion and Discussion

(a) Histogram of the Bootstrap Expected Premium Income Values

(b) Histogram of the Bootstrap Expected Values of the Change in UPR

(c) Histogram of the Bootstrap Expected Values of the Change in FP

Figure 2: Histograms of the Bootstrap Estimates with True Values Displayed by Red Lines

Lower limit Estimate Upper limit True Value PI

∆UPR ∆FP

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REFERENCES

This research can be improved in a number of ways. Currently, the personal characteristics of policyholders are not available for research purposes. Among these personal characteristics are for example age, gender, and area of residence. The possibility of requesting this information is currently being developed. Personal characteristics might be incorporated in future forecasting models of the outlook. In addition, past literature has studied extensively the relationship between the number and size of claims made and the probability of policy termination and cross-buying. Such data would enhance our forecasts even more. In an optimal scenario, we would make use of the difference between the requested claim amount and the amount that is paid. This would be a proxy for customer satisfaction. Another variable that may improve the forecast performance, is whether a change in premium is going to occur during the quarter.

Moreover, the forecast procedures could provide better estimates of the total premium income if the date of termination were specified in more detail, for example including the month of termination. The conditional probability of a policy being terminated during the upcoming quarter, given that it has not been terminated up until now, has in this paper been computed with time increments of a quarter. If the termination months were known, this conditional probability could for example be replaced by a probability including multiple smaller time increments, which would then be denoted in months. As mentioned in Section 9, a prediction of the number of initiated policies based on a few data points is unreliable. Future forecasts can include a development factor based on expectations from the business and possibly even distinguish in trend or seasonal effects.

In this paper, we make one-period forecasts based solely on the information at the begin-ning of the quarter. Every quarter, all information that is known from previous quarters is disregarded. Future research might track the characteristics of the policy from its initiation onwards and include time-varying covariates into the prediction models. Both the Cox propor-tional hazards model and the logistic regression model can be adapted to include time-varying covariates.

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A Appendix

A

Appendix

A.1 Descriptive Statistics

Period 2017Q1 2017Q2 2017Q3 2017Q4 2018Q1 2018Q2 2018Q3 2018Q4 EXI+NBR NBA Period 2019Q1 2019Q2 2019Q3 2019Q4 2020Q1 2020Q2 2020Q3 EXI+NBR NBA

Table 4: Number of Policies in Employed Data Sets

Number 1 2 3 >4

Fraction

Table 5: Number of Policies per Policyholder

HRG 1 2 3 4 5 6 7 8 9 Duration (quarters) Non-renewals1 Discontinued2 Lapse3 New4 (a) Categorized on HRG

Payment term One month Three months Six months Twelve months

Duration (quarters) Non-renewal1 Discontinued2 Lapse3

New4

(b) Categorized on Payment Term

Table 6: Various Statistics

1_{The number of non-renewals as a fraction of the total number of contracts that expired during the}

quarter (in %).

2_{The number of discontinued policies during contract term as a fraction of the total number of contracts}

not ending during the quarter (in %).

3_{Total lapse (non-renewals+discontinued policies) (in %).}

4_{Total number of new policies added during the quarter as a fraction of the existing policies at the}

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A.2 Testing Cox Proportional Hazards with Shared Frailty Model Assumptions

A.2 Testing Cox Proportional Hazards with Shared Frailty Model Assump-tions

Figure 3: Martingale Residuals of the Annual Contract Premium and its Transformations

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A.2 Testing Cox Proportional Hazards with Shared Frailty Model Assumptions

Figure 5: Martingale Residuals of the Number of Policies Held by the Policyholder

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A.3 Testing Regressive Logistic Regression Model Assumptions

Figure 7: Adjusted Variable Plot of the Logarithm of the Annual Contract Premium

A.3 Testing Regressive Logistic Regression Model Assumptions

Figure 8: The Log Odds Ratio Against the Values of Several Variables

A.4 Algebraic Definitions

If the payments of policy i are made every quarter, six months, or year, let PMi denote the

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in the quarter. The length of a contract is one year. Then

πi =          1 4 if PAYi∈ {1M, 3M}, 1 2 · 1[PMi∈{M1,M2,M3}] if PAYi= 6M, 1[PMi∈{M1,M2,M3}] if PAYi= 12M, (44) αi =    1 6 if PAYi= 1M, 1 6 · 1[PMi=M1]+ 1 12 · 1[PMi=M2] if PAYi∈ {3M, 6M, 12M}. (45)

The estimated value of the earned out of received premiums is given by

\

ERPI = \ERPIEXI+ \ERPINBR+ \ERPINBA

= n2019Q2 X i=1 h ξi(1 − bYi) + αiYb_i · PREi i

+ \ERPINBR+ \ERPINBA

= \ERPIEXI+ \ERPINBR+

n2018Q2 X j=1 h ξj· 1{Yj=0}+ αj· 1{Yj=1} · PREj i , (46) where ξi=    1 4 if PAYi = 1M 1 4 · 1[PMi=M1]+ 1 6 · 1[PMi=M2]+ 1 12· 1[PMi=M3] if PAYi ∈ {3M, 6M, 12M}. (47)

The estimated value of the earned premiums out of the unearned premium reserve can be written as \ EUPR = n2019Q2 X i=1 h ζi(1 − bYi) + τiYb_i · P REi i , (48) with ζi=          0 if PAYi = 1M, 1 6 · 1[PMi=M3]+ 1 12· 1[PMi=M2] if PAYi = 3M, 1 4 · 1[PMi∈{M/ 1,M2,M3}]+ 1 6 · 1[PMi=M3]+ 1 12· 1[PMi=M2] if PAYi ∈ {6M, 12M}, (49) τi=          0 if PAYi= 1M, 1 6 · 1[PMi=M3]+ 1 12· 1[PMi=M2] if PAYi= 3M, 1 6 · 1[PMi∈{M/ 1,M2}]+ 1 12· 1[PMi=M2] if PAYi∈ {6M, 12M}. (50)

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A.4 Algebraic Definitions estimated by \ RUPR = n2019Q2 X i=1 κiYb_i· P RE_i, (51) where κi =                0 if PAYi = 1M, 0 if PAYi = 3M, 1 12· 1[PMi=M1−4]+ 1 6 · 1[PMi=M1−5]+ 1 4 · 1[PMi=M1−6] if PAYi = 6M, M3−PMi+6 12 · 1[PMi∈{M/ 1,M2,M3}] if PAYi = 12M. (52)

Note that if M1 = 1, then PMi = M1 − 4 implies that the payment month of policy i is the

tenth month of the previous year.

Let Kt denote the set of policies that expires in the recognition period at the beginning of

quarter t. Then OR = X j∈Kt PREi, (53) and d NR = X j∈Kt+1 1 − bYi · PREi. (54)

Let EMi denote the expiration month of policy i. This implies that the contract of policy i,

in absence of renewal, ends on the last day of month EMi− 1. The estimate of the amount of of

premium income that was included in the future premiums with regard to the analyzed quarter is given by FPINC= n2019Q2 X i=1 πi· 1[EXPi=0]+ π ∗ i · 1[EXPi=1] · PREi, (55) where π_i∗=    1 12· 1[EMi=M2]+ 1 6 · 1[EMi=M3] if PAYi = 1M, 0 if PAYi =∈ {3M, 6M, 12M} (56)

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terminated during the current quarter is estimated by

d F PEND= n2019Q2 X i=1 υiYb_i· PRE_i, (57) with υi=                EMi−M2 12 if PAYi= 1M, j EMi−M2 3 k ·1 4 if PAYi= 3M, j EMi−M2 6 k ·1 2 if PAYi= 6M, 0 if PAYi= 12M, (58)

where b·c is the floor function. EMi − M2 denotes the remaining matury of the contract of

policy i at the moment of (potential) termination.

The additional future premiums resulting from policies that are renewed during the quarter is estimated by c FPNBR= n2019Q2 X i=1 1_[EXP_i_=1]· ηi 1 − bYi · P REi, (59) where ηi=                3 4 · 1[EMi=M1]+ 5 6 · 1[EMi=M2]+ 11 12· 1[EMi=M3] if PAYi = 1M, 3 4 · 1[EMi∈{M1,M2,M3}] if PAYi = 3M, 1 2 · 1[EMi∈{M1,M2,M3}] if PAYi = 6M, 0 if PAYi = 12M, (60)

Forecasting Lapse, Renewal, and Commencement of Policies in Non-Life Insurance

Forecasting Lapse, Renewal, and Commencement of Policies in

Non-Life Insurance

Forecasting Lapse, Renewal, and Commencement of Policies in

Non-Life Insurance

Contents

1

Introduction

2

Literature Review

3

Data

4

Methodology

5

Survival Analysis Basics

6

Cox Proportional Hazards Model

7

Logistic Regression

8

Results

9

Forecasting Premium Income

10

Conclusion and Discussion

References

A

Appendix