Evaluating the impact of non-rational surrender models on liabilities : |a least-squares Monte Carlo Approach

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Evaluating the Impact of Non-Rational

Surrender Models on Liabilities

—A Least-Squares Monte Carlo Approach

Yang Ruan

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Yang Ruan Student nr: 10546367

Email: ruanyoung2013@gmail.com Date: May 10, 2015

Supervisor: Dr. Leendert van Gastel Second reader: . . .

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Impact of Surrender Models on Liabilities — Yang Ruan iii

Abstract

Insurers are always exposed to lapse risks from embedded surrender options in their products. Measuring this kind of risk is a formidable task because surrender probabilities are strongly connected with indi-vidual behavioural preferences, and thus are difficult to be modeled. In an effort to get closer to the policyholders’ actual behaviour, we introduce two recent models that focus on the non-rational side in making surrender decisions. This paper explains how insurers’ liabil-ities, as well as solvency requirements are influenced when these two models are applied in a term contract with surrender option. Thanks to its high-efficiency and accurary, we will use the least squares Monte Carlo technique in our simualtions.

Keywords Life insurance, Asset and liability management, Surrender option, Surrender rate, Non-rational surrender model, Least-squares Monte Carlo simulation, Solvency capital require-ment

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1 Introduction 1 2 Model Summary 3 2.1 Model 0 . . . 3 2.2 Model 1 . . . 3 2.3 Model 2 . . . 4 2.4 Comparison . . . 5 3 Policy Design 7 4 Data 9 4.1 Model Calibration . . . 9 4.2 Short Rates . . . 9 4.3 Surrender Probabilities. . . 10 4.3.1 Model 1 . . . 10 4.3.2 Model 2 . . . 11 4.3.3 Comparison . . . 11

4.4 Cashflows and Best Estimate . . . 11

5 Least Squares Regression 16 5.1 Basis Function . . . 16

5.2 Value at Risk . . . 18

6 Solvency Capital Requirement 22 7 Sensitivity Analysis 24 7.1 Model 1 . . . 24 7.2 Model 2 . . . 24 8 Conclusion 26 Appendix A 27 Appendix B 33 References 35 iv

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Chapter 1

Introduction

As a key feature of most life insurance products, implicit options that includes various embedded rights for the policyholders to modify the contract terms have been studied by a growing number of literatures in recent years. From the perspective of insurers, risk evaluations on implicit options imposed by new industrial regualtions could have great impact on their capital reserve requirement (Gatzert, 2009). One of the most common implicit options is the surrender option. With such an option embedded in the contract, the insured have the right to quit the policy before its maturity and claim an immediate pre-specified cash value after paying a surrender fee. The Exercise of surrender options change the cashflow structure and thus brings uncertainties to insurers’ liability. They may even face liquidity problems if a large number of policyholders surrender their contracts at the same time.

In managing surrender risks, the difficulty comes from understanding the reasons to surrenders and further estimating the surrender probabilities. Bacinello (2005) at-tributed surrenders solely to the economic reason that is driven by the optimal surrender time of the contract. He treated the surrender option as an analogy to American options and computed the contract value in a contingent framework, and assumed policyholders all act according to rigorous valuations. However, reasons such as alternative investment opportunities, personal financial distresses and some unpredictable factors could also affect policyholders’ decisions. Anzilli and De Cesare (2007) proposed a model that in-cludes the influences of personal reasons and insurance company campaigns in addition to fair valuation of the contract. Besides, this model assumes the economic optimality criterion that carried out by an external agency is not strictly followed by policyhold-ers, reflecting irrationality of individual behavior. De Giovanni (2008) also realised that surrender decisions are far from being optimal. He developed a rational expectation surrender model that divides the policyholders’ behaviour into a rational part and an irrational part under two contingent contract value conditions. Moreover, inspired by Kuo et al. (2003), he linked the surrender rates in the rational part with the short-term interest rates to reveal the their dependency. Furthermore, Kim (2005) and Cox and Lin (2006) presented surrender rates by econometric models, selecting macro-economic indicators such as the difference between treasury bond rate and policy credited rate, economic growth rate and unemployment rate as explanatory variables.

Motivation

Solvency II requires rigorous valuations on surrender options as a part of capital reserves. This makes choosing surrender models decisive on insurers’ balance sheet and hedging strategy. In this paper we assume that insurers are presented with two models from Anzilli and Cesare (2007) and De Giovanni (2008) in particular. These two models are similar in that they both take the irrationality of individual behaviour into account. It is interesting to see how they differ in affecting surrender decisions of policyholders

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and liabilities of insurers. The insurance framework would be a term policy embedded with a surrender option.

Least Squares Monte Carlo Simulation

The simulation technique employed in this paper is the least squares Monte Carlo simulation (LSMC), which combines Monte Carlo simulation and least squares regres-sions to approximate a function for certain dependent variable of interest. This technique has been proved to be statistically sound in convergence with the true function and pow-erful in financial application (Stentoft, 2004). Specifically, Longstaff and Schwartz (2001) showed its use in pricing various American options, and Koursaris (2011) demonstrated how it could be applied to derive the liability function for insurers. In this paper we mainly follow the method of Koursaris (2011).

The goal is to predict the best estimate of liability at time 1 for the policy by a function of certain risk drivers. Basically, with LSMC, we first implement simulations on the risk drivers implied by the chosen surrender models, complete the full dataset for the policy and calculate the corresponding best estimate of liability for fitting. Then choose a proper basis function for the least squares regression, and fit the best estimate as a function of the risk drivers. This fitted function can be used to compute approximate best estimate of liabilities by plugging in real-world scenarios of risk drivers.

Compared with traditional Monte Carlo simulation where the number of scenarios expands quickly to millions as the simulation period grows, LSMC approach demands much less scenarios. It specifies a broad range of values that includes the most extreme ones for each risk driver, which will be used as the starting points of further simula-tions. The points, chosen randomly from the range, are called the outer scenarios of the simulation. The number of outer scenarios is normally very large (50, 000 in this paper). Then, on each outer scenario, a small number of paths (2 in this paper) that last consistently to the end are developed. The total number of scenarios of simulation thus equals to only 50, 000 × 2. This makes the simulation fast in computation.

This paper is organised as follows. Chapter 2 explains the chosen surrender models in detail, Chapter 3 introduces the structure of the insurance policy. In Chapter 4 simulations on data of the market rates, surrender probabilities and best estimates are set up for fitting. Then the basis functions are to be found in Chapter 5 by least sqaures regressions. Chapter 6 performs a solvency capital calculation in the light of Solvency II standards. Chapter 7 studies the sensitivity of best estimate on model parameters. The last Chapter concludes the paper.

Reseach Question Measured by LSMC simulations, what is the impact of the two different surrender models have on the liabilities for the policy?

Sub-questions

• How do the chosen surrender probability models differ?

• What is the difference on solvency capital requirement for the two models? • What is the sensitivity of the best estimate against parameter changes in

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Chapter 2

Model Summary

This chapter describes several models on surrender probability in the context of an insurance contract with a surrender option. Model 1 is referred to Anzilli and Cesare (2007) and Model 2 is given by De Giovanni (2008). Both models are derived from the original Model 0. Define a time variable t that takes integer values starting from 0. Also define the term irrationality solely as the violation of the surrender standard established from economic analysis on surrender values and continuation values in Model 1 and Model 2.

2.1 Model 0

As the basis model of surrender, Model 0 has a form of pt= 1 − e−Θt,

where Θtis the annual surrender rate between t and t + 1, and ptis the annual surrender probability. The value of surrender probability depends solely on the surrender rate, which is specified as given by insurers. It is assumed that the surrender rate Θt only allows for surrenders that stems from economic contract valuation, and therefore no irrationality is implied. On the contrary, the next two models are more complicated and take irrationality into account.

2.2 Model 1

Established by Anzilli and Cesare (2007), Model 1 allows for both personal reasons and economic convenience reasons to surrender decisions.

Let Stbe the value one gets by surrendering the contract in the time period [t, t + 1), and let Ctbe the remaining value of the contract if he stays with the contract in [t, t+1). The ratio

θt= St Ct ,

is defined as the economic decision variable in the period [t, t + 1). It has a value larger than zero by nature. Assume that this decision variable is calculated by an external agency and is not observed by policyholders. When the surrender value Stis larger than the continuation value Ct and the decision variable is larger than one, it is optimal to exercise the surrender option. On the opposite, when St is smaller than Ct and the decision variable is smaller than one, it is unwise to exercise the option. Then the annual probability to surrender by economic reasons, Kt, can be denoted by a function

Kt= f (θt),

meaning that rational policyholders refer to the decision variable θt only to carry out economic surrender decisions. As long as the decision variable is larger than one, they

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will exercise the surrender option. If the decision variable is smaller than one, they will stay with the contract for sure. Therefore the ideal probability function is

Kt=

1 if θt> 1 0 if θt≤ 1

.

However, since the decision variable is not observable, and not all of the policyholders have rational thinking, the true function must deviate from the ideal pattern. Incorrect decisions are often made, especially when surrender value and continuation value are very close to each other, making it hard to judge for policyholders. Also, the higher the value of surrender is relative to the value of continuation, the more likely policyholders will surrender. To add these irrationality, choose an interval [θmin, θmax] that includes 1, such that when θt falls within this interval, Kt is an increasing function of θt. For simplicity, a linear increasing relation is defined in this interval. The other parts of the domain remain the same as the ideal function. By doing so one gets a continuous piecewise function Kt= f (θt, θmin, θmax) =    1 if θmax< θt θt−θmin

θmax_−θmin if θmax≥ θt> θmin 0 if θmin≥ θ_t

,

with 0 ≤ θmin ≤ 1 ≤ θmax.

It can be seen that apart from θt, parameters θmin and θmax that represent the irra-tionality also affect the decision-making. As the interval [θmin, θmax] grows in length, policyholders are more likely to make wrong decisions.

Now we turn to personal reasons. Define Pt as the annual probability to surrender by personal reasons. It accounts for anything that leads to surrenders except for the economic analysis on the contract represented by Kt. In practice policyholders may sur-render the contract out of all kinds undetectable personal reasons, making it difficult to find out the true values of Pt. Anzilli (2007) took Ptfixed at either 0 or 0.01. Compara-ibly surrender probabilities from economic reasons could reach as high as 1. Therefore the surrender probability by personal reasons is much smaller than economic reasons.

Combining surrenders from both personal reasons and economic reasons gives the annual proabability

pt= 1 − (1 − Pt)(1 − Kt(θt, θmin, θmax)) that a policyholder will surrender between time [t, t + 1).

The surrender rate is not present in the formula, but it can be derived with a linkage to Model 0. Since Model 1 is a variant of Model 0, the term (1 − Pt)(1 − Kt) is equivalent to e−Θt _{in Model 0. Therefore, the implied surrender rate for Model 1 is}

Θt= −log[(1 − Pt)(1 − Kt)].

2.3 Model 2

Based on contigent claims of the contract value, De Giovanni (2008) built a rational expectation model that classifies surrenders as rational and irrational. He expressed the general formula of surrender probability pt for a population of policyholders between time t and t + dt as

pt= 1 − e− Rt+dt

t h(u) du,

where h(t) is the hazard rate that represents the intensity of surrenders in the interval (t, t+dt). Two different situations in terms of the contract value are discussed seperately. First, when the continuation value of the contract is larger than the surrender value of the contract, it is not sensible to exercise the surrender option. In this situation, all of

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Impact of Surrender Models on Liabilities — Yang Ruan 5

the surrenders are identified as irrational. Their hazard rate is assumed to be constant at a rate of

h(t) = θI,

with I stands for ”irrational”. Therefore, the irrational surrender probability for the contract in this situation is given by

pI_t = 1 − e−θIdt= 1 − e−θI.

dt = 1 because the time variable t has a scale of one year. De Giovanni (2008) chose a range of [0, 3] for the level of θI, which corresponds to [0, 0.95] for pI_t.

Next, in case the surrender value of the contract is larger than the continuation value, it is optimal to exercise the surrender option. Rational surrenders would only occur in this situation. However, not all of the surrenders in this situation are rational, there are also some irrational surrenders among them. The reason for that is those who make irrational surrenders in the first situation do not refer to the contigent contract value to make surrender decisions. They surrender no matter if it is optimal to do so. Therefore, in the second situation the hazard rate

h(t) = θR(rt) + θI,

with R stands for ”rational”, is contributed by both rational and irrational surrenders. θR_(r

t) measures the rational surrenders component and θI still indicates the irrational surrenders component. Specifically, θR(rt) is described by a function

θR(rt) = A2r2t,

where rt is the expected one-year short rate at time t. This function says that the intensity for rational surrenders is dependent on the one-year market interest rate. As the interest rate gets higher, opportunity costs of staying with the contract rises and thus surrender intensity also rises. Parameter A reflects policyholders’ sensitivity to the change of interest rate. It is taken to the second power because a non-decreasing function is needed to describe the relationship between θR(rt) and rt. De Giovanni (2008) also chose the second order term for rt because he wanted to avoid negative θRin case rt is negative. However, formulated in this way, the surrender rate would go up unreasonably even if the market rate keeps falling below zero. Rational policyholders would certainly not surrender the term contract when the market rate is negative, and thus the surrender rate should be zero in this case. Therefore in our study, negative market rates are set to zero before plugging them into this function.

As a whole, the probability to surrender in the second situation is pR_t = 1 − e−(θR(rt)+θI)_.

Remark

De Giovanni (2008) actually formulated the rational surrender rate as θR(rt) = A2rt2+B, with A, B ≥ 0. The parameter B is deleted because in calibrating the model we found out that the value of B is always negative. This would easily drag the value of θR to negative if rtis close to zero. Besides, in the formula of pRt , it is difficult to differentiate the effect of B and θI because as scale parameters they are added up together.

2.4 Comparison

A major difference between the two models is that in terms of making surrender deci-sions, Model 1 assumes every policyholder has both irrationality and rationality in him, while Model 2 allows some policyholders with pure irrationality. To see this, in Model 1,

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by setting θminand θmax at values other than 1, policyholders would inevitably deviate from fully rational behaviour. But they still refer to θt, the variable that sets the stan-dard of rationality, to make decisions. While in Model 2, irrational surrenders simply ignore the economic analysis of surrender and continuation values, and surrender in any case.

Irrationality is composed of two elements: the act to surrender in unfavourable sit-uations, and the failure to surrender in favourable situations. Both models manage to consider it in entirety but in different ways. In Model 1 policyholders have the possibil-ities to make irrational surrenders when θt falls between [θmin, 1], and to forgo rational surrenders when θt is between [1, θmax]. In Model 2, the first element is reflected by θI for constant irrational surrenders. And the second element is reflected by the fact that those who only surrender in favourable situation may give up the right decision, because apart from the decision variable θt, they hinge their decisions also directly on the movement of market rates.

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Chapter 3

Policy Design

Consider a policy that starts at time t = 0 and comes to maturity at time t = T . It charges an one-off premium at the outset and pays back a lump sum benefit VT at maturity. The premiums are invested in a zero coupon bond that has the same maturity with the policy to cover future payouts. Assume there is no mortality risk, and the technical rate set by the insurer for this policy is RT, then the book value of the policy at time t is

Vt=

VT (1 + RT)T −t

for t = 0, 1, 2, . . . , T.

A surrender option that gives the policyholders the right to quit before maturity and receive an instant surrender value is embedded in the contract. Assume policyholders do not have to pay a surrender fee to get the surrender value. Also, surrenders can occur only at the beginning of each year, and one cannot surrender at the beginnig of the conctract (time 0), neither can he surrender at the maturity (time T ). The instant surrender value St to be paid at time t equals to the contemporary book value Vt, or

St= Vt for t = 1, 2, . . . , T − 1,

so if one exercises the surrender option at the beginning of the (t + 1)th year (time t) he gets a cash amount of Vt.

Suppose there are certain number of policyholders in the contract and the expected final benefit combined equals to VT. The current one year spot rate is r0. Denote the future one year short rate at time t as rtfor t = 1, 2, . . . , T − 1. It will be modeled by a Hull-White interest rate simulation. The expected yearly surrender probability for the contract, obtained from the models decribed in chapter 2, is ρt for t = 1, 2, . . . , T − 1. Then the expected cashflows Lt is given by

L1 = V1ρ1, Lt= Vtρt t−1 Y j=1 (1 − ρj) for t = 2, 3, . . . , T − 1, and LT = VT T −1 Y j=1 (1 − ρj).

We are interested in the best estimate of liability in one year later at time 1, and it can be computed as BE = E " L1+ T X t=2 Lt Qt−1 j=1(1 + rj) |F1 # ,

where Ft represents the information available up to time t. 7

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To reach a surrender decision, a rational policyholder would compare the surrender value with the continuation value of the contract. The continuation value Ct of this policy is equal to its market value, which the present value discounted from the final benefit using the expected annual market short rates, given by

Ct= E " VT QT −1 j=t (1 + rj) |Ft # for t = 1, 2, . . . , T − 1.

The market value CT at maturity equals to the final benefit VT by definition.

If the surrender value St is larger than then continuation value Ct, it is optimal to exercise the surrender option immediately. Otherwise, it is not sensible to surrender. As the insurer of this policy there are two major risks. One is the uncertainty of future short rate that influences discount factors and the market value, and the other one is the surrender possibility that changes the payment structure of cashflows.

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Chapter 4

Data

This chapter explains how data of the two major risk drivers, the one-year short rate and the surrender probability, as well as the best estimate of liability at time 1 are prepared for the LSMC simulation. Assume the policy’s starting time t = 0 is the first day of 2014 with a maturity time T of 30 years.

4.1 Model Calibration

To make the two models comparable, the parameters are needed to be calibrated on the same set of data. Due to a lack of existing data on surrender probabilities, we will generate some with Model 1 and 2 themselves. First assign 0.85 and 1.5 to the parameters θmin and θmax in Model 1, and 20 to A in Model 2. The parameter Pt in Model 1 and θI are both fixed to 0.01. With these parameter values the two models produce similar results on surrender probabilities over 30 years. Then generate 50, 000 paths of thirty-year surrender probabilities from each model and combine them into one data set. Calibrating Model 1 and Model 2 respectively on this set of data using least squares technique to find out the optimised parameters, we get 0.9147 and 1.3155 for θmin and θmax, and 9.7311 for A.

4.2 Short Rates

The expected future one-year short rate is used in the surrender rate models as an input variable. It is also used as the discount rate for the calculation of policy’s best estimate of liability. We are interested in simulating the future annual short rates rtat time t for the coming 30 years, and this is done with a Hull-White interest rate model.

The money market interest rate (spot rate) r0 was at 0.241% on the last day of 20131. Specify an interval [−4%, 8%] for the possible interest rate r1 to develop the LSMC’s outer scenarios (stressed points) at time 1. As an example, a total of 100 values are selected in a quasi-random manner within the interval in this section to generate 100 outer scenarios. For each outer scenario, two inner scenarios are further developed. Next, apply the Hull-White model to each of these initial outer stressed points to create 200 paths of interest rate simulations for 30 years. The term structure chosen is the one starts at 31-12-2013, with 30 years of maturities2.

According to Hull (2011), the Hull-White model is givin by r(t + δt) = r(t) + (γ(t) − ar(t))δt + σZ√δt. The variable γ(t) is a function of

γ(t) = dF (0, t) dt + aF (0, t) + σ2 2a(1 − e −2at_), 1 De Nederlandsche Bank (2014).http://www.dnb.nl 2_{De Nederlandsche Bank (2014).} http://www.dnb.nl 9

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where the variable F (0, t) is the instantaneous forward curve that is derived from the term structure at time 0 (Hull, 2011). The parameter δt is the increment of time step, a is the mean reversion rate, σ is the expected interest rate volatility and Z is a randomly distributed standard normal variable. The time step δt is equal to 1 year. The interest rate volatility σ, calculated by taking the standard deviation of the historical dutch annual market interest rates from 1999 to 20133, is 1.496%. And the mean reversion rate is not calibrated but taken as given.

Figure 4.1 and figure 4.2 show the results for the example of 200 times simulation. In both figures the colored thin lines are simulated short rate paths, and the black thick line denotes the yield curve at time 0 with maturites up to 30 years. Figure 4.1 uses a mean reversion rate a of 0.1 while figure 4.2 sets a at 0.5. Clearly a higher mean reversion rate leads to more convergent simulation results for short rates. And the simulation paths in both figures center on the yield curve. Since the money market interest rate r0 was at 0.241% on the last day of 2013 and the fitting range chosen is [−4%, 8%], it is reasonable for some simulation paths to fall below zero in some years.

In the rest of the paper the Hull-White mean reversion rate a is set to 0.5 because it gives more realistic future rates within a tight range. Once the short rates are simulated, the continuation value of the policy Ct, or the market value, can be calculated with the formula of Hull-White zero coupon bond price (Thetaris, n.d.). Choose a technical rate RT of 3.8% for the policy, which is denoted by the yellow level dashed line in figure 4.2.

4.3 Surrender Probabilities

Models in Chapter 2 typically assume policyholders can surrender at any time of a year, and produce surrender probabilities that describe the likelihood of surrenders in one year of time. However in the designed contract the policyholders can only surrender at the beginning of each year. Clearly the setting of the models is what prevails in practice. To reconcile this contradiction, assume all of the surrenders that are scattered within a whole year in reality are now brought forward to the beginning of the year, so that we can still use the surrender probabilities generated by these models to obtain number of surrenders for the policy.

For each model, altogether 100, 000 simulations (50, 000 outer scenarios, each one with 2 inner scenarios) of short rates and continuation values are conducted in the way described in the previous section. As the direct input of the surrender models, continuation values from each outer scenario are averaged over the two inner scenarios, making a reduction to 50, 000 paths. This generates a dataset of surrender probability with 50, 000 paths also. And each path has 29 values, from time t = 1 to t = 29, that make up a complete contract up to maturity.

4.3.1 Model 1

As has been calibrated, for economic surrender probability Kt(θt, θmin, θmax), θmin = 0.9147 and θmax = 1.3155. The annual probability to surrender by personal reasons is fixed at Pt= 0.01 for all time periods. That is to say policyholders reveal irrationality when the surrender value is between 91.47% and 131.55% of the continuation value. The decision variable θtvaries with the future short rates. From the resulting simulated values for Kt, it has a mean of 0.1245. Therefore the economic reasons are dominant in surrender decisions compared with personal reasons.

The simulation results for surrender probabilities can be found in figure 4.3. For each year from t = 1 to t = 29 there are 50, 000 simulated surrender probabilities, displayed by the black dots that span vertically. The black curve in the lower half shows the trend of the mean of surrender probabilities each year. The pink, green, yellow, red,

3_{De Nederlandsche Bank (2014).}

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blue and light blue stars that mark the data each year from top to bottom in sequence represent the 95th, 90th, 75th, 50th, 25th and 10th quantiles respectively. The lowest possible surrender probability is 1% because Pt is fixed at 0.01.

It can be seen that the data are mostly concentrated within a band of 1% to 30% each year, and there is a general increasing trend over time. The mean of surrender proability starts at about 5% at time 1 and rises to about 20% in the end. The minimum surrender probability also rises in the last year. These all suggest that it is more favourable to surrender as the contract approaches maturity.

Moreover the distribution of the data is increasingly concentrated from the begin-ning to the end period. A clear convergent tendency is observable as the span of black dots shrinks gradually over time. The reason of convergency is that in later years the continuation value is discounted through less future short rates and thus less volatilities are involved, stablising the the continuation values and the surrender probabilities. 4.3.2 Model 2

In the first situation of Model 2, the hazard rate of pure irrationality θI is still fixed at 0.01, making a constant irrational surrender probability pI_t of about 1%. In another word, we assume that one in a hundred policyholders will always surrender regardless of the market moves. This value is equal to the surrender probability by personal reasons in Model 1. In the second situation, following the calibration, parameter A is set to 9.7311 for θR(rt) = A2r2t. After taking a power of second order, the high value of A2 reflects the high sensitivity of policyholders’ surrenders on interest rate movements. To mention it again, due to the quadratic form of θR(rt), negative short rates rt are converted to zero before plugging into θR(rt), otherwise the hazard rate would go up unreasonably even if the interest rate keeps falling below zero.

Figure 4.4 shows the simulation results. Again, pink, green, yellow and red markers represent the 95th, 90th, 75th, 50thquantiles respectively, and the black line denotes the mean. The minimum surrender probability is about 1% in each period, derived from pure irrational surrenders in the first situation. Surrender probabilities higher than the minimum value are obtained only in situation two when rational surrenders take part. The quantiles show that a majority of the surrender probabilities each year are equal to the minimum level, indicating in most cases it is not rational to surrender.

There is a general increasing trend for the probabilities, and the distribution of them gets wider over time. The mean value starts at about 3% and rises to about 9% in the end. The 95th quantile at time 1 is very large because the short rates r1 are the outer stressed points that consist of many high rates randomly drawn from the interval [−4%, 8%].

4.3.3 Comparison

Some similarities and differences between surrender probabilities generated by two mod-els are summarised as follows. Probabilities from both modmod-els have increasing trends during the contract life. They both have a minimum surrender probability of at least 1% in each period. However, Model 1 on average generates surrender probabilities about twice as high as Model 2. It could even reach 100% surrenders in the early fifteen years. Besides, probabilities of Model 1 converge to more small ranges year by year, while probabilities of Model 2 diverge. Therefore in final years Model 1 gives more predictable results than Model 2.

4.4 Cashflows and Best Estimate

Suppose the total final benefit VT to be paid by the insurer is 10000. Then the yearly surrender values Stcan be specified. The cashflows are composed of the surrender values

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payable from time t = 1 to t = 29 and the benefit payable at time t = 30. Figure 4.5 and 4.6 display the cashflow structures generated from Model 1 and 2 respectively. Again, the black dots are cashflow values each year for different simulation paths, the black curve denotes the mean, and the pink, green, yellow and red stars denote 95th, 90th, 75th and 50th quantiles.

For Model 1 the cashflows have a decreasing trend in general. The highest cashflows on average are obtained at year 7. Cashflows at the final year only increase for a small amount compared with year 29. On contrary, Model 2 has a cashflow structure that would be normally expected for a term insurance product. The cashflows before maturity each year are low and grow at a mild pace, while the cashflow at maturity is much higher. What caused the unusual cashflow structure of Model 1 is its relatively high surrender probabilities, which increase the early cash payables before maturity.

Figure 4.7 shows the distributions of the best estimate of liability at time 1 for both models. The means of best estimate for Model 1 and 2 are 3743.8 and 3907.8, and the standard deviations are 305.62 and 392.16 respectively. Both distributions have a long right tail, but comparably Model 1’s distribution is more negatively skewed, while Model 2’s distribution is more balanced. With smaller standard deviation and mean, Model 1 is likely to produce smaller values for the best estimate.

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Figure 4.1: Hull-White simulations (200 times) on one-year short rate with a = 0.1.

Figure 4.2: Hull-White simulations (200 times) on one-year short rate with a = 0.5.

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Figure 4.4: Surrender probabilities generated by Model 2.

Figure 4.5: Cashflows generated by Model 1.

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Least Squares Regression

We are interested in studying the relations between the best estimate of liability at time 1 and the risk drivers, that is, the one-year short rate r1 at time 1 and the surrender probability ρ1at time 1. To do so, a basis function of best estimate needs to be estimated using the OLS fitting technique. Follow the example of Koursaris (2011), we will select the basis function from a group of polynomial regression functions with an order up to 3 according to the criteria that the function has the best fit as well as significance in coefficients.

5.1 Basis Function

First, include only r1, the one-year short rates at time 1 generated from outer stressed points, as the independent variable, the set of polynomial functions are:

BE = β0+ β1r1+ , (1)

BE = β0+ β1r1+ β2r12+ , (2) and BE = β0+ β1r1+ β2r12+ β3r13+ . (3) Same formulas are used when the risk driver ρ1 is the only independent variable:

BE = β0+ β1ρ1+ ; (4)

BE = β0+ β1ρ1+ β2ρ12+ ; (5) BE = β0+ β1ρ1+ β2ρ12+ β3ρ13+ . (6) Next, when both risk drivers r1 and ρ1 are included, the preliminary function is:

BE = β0+ β1r1+ β2ρ1+ . (7) Theoretically the surrender probability is negatively affected by the contemporaneous short rate. Therefore a cross term r1ρ1, which measures the change of either risk driver’s influence on the best estimate when the other risk driver changes, is added to reflect the interaction effects:

BE = β0+ β1r1+ β2ρ1+ β3r1ρ1+ . (8) The functions grow further in length as the polynomial order raises to 2 and 3:

BE = β0+ β1r1+ β2r12+ β3ρ1+ β4ρ12+ β5r1ρ1+ , (9) BE = β0+β1r1+β2r12+β3r13+β4ρ1+β5ρ12+β6ρ13+β7r1ρ1+β8r12ρ1+β9r1ρ12+. (10) Detailed regression results on the group of polynomial functions of both models can be found in Appendix A.

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For Model 1, all of the coefficients in function (1) to (6) are highly significant. The plots of these functions with only one risk driver are shown in figure 5.1 and 5.2, where the red, yellow and green lines represent the first, second and third polynomials respectively. Judging from the Akaike Information Criterion (AIC), with only one risk driver allowed, the short rate r1 fits much better than the surrender probability ρ1, and function (3) has the best fit. Now observe the case with both risk drivers. Moving from function (7) to function (8) it can be seen that the interaction term r1ρ1 is significant and the fit of the model is improved. In function (9) the coefficient β4 of ρ12 is not significant. By dropping ρ12 and forming a refined function:

BE = β0+ β1r1+ β2r12+ β3ρ1+ β4r1ρ1+ , (11) all the other coefficients remain significant and the AIC value is lowered. Therefore, function (11) is preferred. Function (10) is the third order polynomial that includes the most terms. However, coefficients β3, β5, β6, β8 and β9 are insignificant, and hence the function is invalid. Besides its AIC is higher than function (11), which has the lowest AIC within the group. Therefore the basis function of Model 1 is function (11):

BE = 3892.6 − 77.25r1+ 1.08r12− 4.79ρ1+ 0.66r1ρ1.

The short rate r1, with a coefficient of −77.25, has a huge negative influence on the best estimate. This reflects the fact that higher r1 generates lower discount factors used to compute the best estimate of liability at time 1. The second order term r12 has a positive coefficient, but with a value of only 1.08 its impact is negligible. Note the scale of the coefficient for surrender probability ρ1 is much smaller than short rate r1, but the changes of ρ1 is presumably larger than that of r1, so it is not negligible. Keeping the short rate constant, the marginal effect of ρ1 on BE is −4.79 + 0.66r1. That is to say only when r1 is larger than 7.26% does an increase in ρ1 affects BE positively.

For Model 2, figure 5.3 and 5.4 display the plots for functions with only one risk driver. Short rate r1 has similar decreasing distribution as Model 1. While for ρ1, the majority of them cluster at the minimum level of about 1% in Model 2, higher values are more scarcely distributed than Model 1. Also three fitting curves for ρ1 are somewhat more linear alike than those curly lines for Model 1.

In the case when r1 is the only risk driver, function (1) and (2) are valid, but function (3) is invalid due to the insignificance of β2 and β3. As for the case with only ρ1, again function (4) and (5) are valid but (6) is not. Compare function (7) and (8) one can see that the cross term r1ρ1 is significant and thus is reasonable to be included. However, the second order function (9) is not significant in most coefficients. So we try a refined function

BE = β0+ β1r1+ β2r12+ β3ρ1+ , (12) which is valid. Then we add the cross term r1ρ1 to function (12) and reach function (11) again, because we want to take the interaction effect between short rate and surrender probability into account. It turns out that function (11) is significant in all coefficients and has the lowest AIC value. The third order function (10) is not valid because none of the coefficient but β0 and β1 are significant. As a result the basis function of Model 2 is still function (11). The specific function is:

BE = 4064.2 − 81.31r1+ 1.00r12− 4.85ρ1+ 0.45r1ρ1.

Marginal effect of a one percent increase in surrender probability to BE is −4.85+0.45r1. It is positive only when short rate r1 is as high as 10.8%, a very unlikely occurrence. Coefficients in this basis function are similar to those in Model 1. The most notable difference is the scale parameter β0. It is about 170 higher for Model 2. The AIC value of Model 1’s basis function (657229.9) is smaller than that of Model 2’s basis function (706619.5). Therefore Model 1 fits better than Model 2.

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The 3D surface plots of both basis functions are displayed in figure 5.5 and 5.6. In general, best estimates increase as r1 and ρ1 decrease for both models. Also, the slope of either one of the risk drivers is larger when the other risk driver’s value increases. Comparably, Model 2 has a steeper surface than Model 1. In figure 5.7 we see the difference between the best estimates generated by two basis functions (Model 1 minus Model 2). Clearly the best estimate of Model 1 is larger than that of Model 2 only when ρ1is higher than about 80% and r1 is higher than about 6%. In most other cases Model 1 gives lower best estimates.

5.2 Value at Risk

The best estimate of liability at time 1 can be calculated with the basis functions obtained in the previous section for both models by plugging in observed short rate r1 and surrender probability ρ1 at time 1. For the purpose of risk management, it is necessary to calculate the 99.5% VaR of the best estimate according to Solvency II. Therefore, a large number of real world values for both risk drivers are run through the basis functions to get a distribution of the best estimates.

A distribution of 3000 real world values for short rate at time 1 is created by a one-period Hull-White simulation. For surrender probability, also 3000 values, with a boundary from 0% to 100%, are drawn from a normal distribution with a mean equal to 4.13% and a standard deviation equal to 9.17%. The values of the two statistics are determined by taking the average of the mean and the standard deviation of ρ1 generated from Model 1 and Model 2 in the Data section. As a result, 9 million different real world values of the best estimate of liability are generated, and the distributions are shown in figure 5.8 for both models.

The two distribution functions have very similar shape, and Model 1’s function is to the top left side of Model 2. The means of best estimate are 3810.8 and 3978.2 for Model 1 and Model 2 respectively. This is 4.2% lower for Model 1. The means have both increased for about 70 compared with the those generated from the simulated distributions in figure 4.7. On the other hand, the standard deviations have both gone down for about two thirds to 108.29 and 117.15 for Model 1 and Model 2 respectively. Therefore the real world distributions of best estimate of liabilities are much concentrated in higher values for both models than the simulated distributions.

The 99.5% VaR for Model 1 and Model 2 are 4106.2 and 4292.6 respectively. The difference between them is only 186.4, or about 4.3% lower for Model 1 than Model 2. Most part of the difference can be explained by the gap in the intercept β0 between two basis functions.

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Figure 5.1: Fitting functions with r1 as the only risk driver for Model 1.

Figure 5.2: Fitting functions with ρ1 as the only risk driver for Model 1.

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Figure 5.4: Fitting functions with ρ1 as the only risk driver for Model 2.

Figure 5.5: Basis function Model 1.

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Figure 5.7: Differences of BE generated from basis functions (Model 1 minus Model 2).

34000 3600 3800 4000 4200 4400 4600 0.5 1 1.5 2 2.5 3 3.5 4x 10 −3 Best estimate Probability Model 1 Model 2

Figure 5.8: Real-world distribution of best estimates of liability at time 1 for Model 1 and Model 2.

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Solvency Capital Requirement

The policy faces two specific risks according to the Technical Specifications of Quanti-tative Impact Study 5 of Solvency II (European Commission, 2010). The first one is the market interest rate risk in the general Market Risk Module, and the second one is the lapse risk in the Life Underwriting Risk Module (Assume there is no operational risk). A solvency capital requirement (SCR) that meant to absorb the two types of risks is needed to be reserved on the basis of the best estimate of liability as a safety buffer. SCRs of the two risk modules are calculated separately by introducing shocks to risk drivers and detecting the changes on the net asset value (NAV), defined as the present value of assets minus liabilities.

In this regard we will need to assume some assets to balance the liabilities. To hedge the market risk of the policy, the assumed assets should have similar cash inflow structures as the outflow structures of the policy. Therefore, we use the average cashflows for both models in figure 4.5 and 4.6 as references to create our assets.

As a result, for Model 1, we assume that the policy has an asset that generates fixed cash inflows each year. Specifically, the asset earns linearly increasing payments from 180 at time 1 to 280 at time 7, and then gets linearly decreasing payments from 280 at time 7 to 80 at time 30. For Model 2, assign the policy a 30-year bond with a face value of 2800. The yearly interest payment of this bond increases linearly from 50 at time 1 to 250 at time 29. These two assets are comparable because they have the same present value given the current term structure.

Combining the assets and the liabilities of the policy, we can derive the basis func-tions of the NAV with the same LSMC technique used in the previous chapter. Appendix B shows the results. Basis functions in no shock section use the same old data as Chapter 4, while basis functions in market interest rate shock and surrender rate shock sections make use of the data that have applied respective shocks. The same sets of real world interest rates and surrender probabilities used in Chapter 5 are passed through each function to obtain real world distributions of the NAV. We are interested in the 0.5% VaR value of the NAV, which is equivalent to the 99.5% VaR of liabilities minus assets that measures the level of the 0.5% most insolvent case.

For the SCR of market interest rate risk, shocks on the term structure are applied to test the policy’s sensitivity to interest rate movements. A given set of change factors (European Commission, 2010)1are multiplied to the rates in the current term structure, which is lifted up in the upward shock and brought down in the downward shock. Similarly, the SCR of lapse risk is derived by shocking the surrender rates up and down according to some given factors each year (European Commission, 2010)2. In Model 1, the change factors are multiplied before the surrender rate Θt, and in Model 2 they are multiplied before θI and θR(rt) + θI in the two situations respectively.

Full results of SCR calculations are displayed in figure 6.1. In each risk module,

1

SCR.5.21., QIS5, Solvency II

2_{SCR.7.50. and SCR.7.51., QIS5, Solvency II}

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Figure 6.1: Solvency capital requirements for both models.

shocked NAVs are deducted from the NAV without shock to obtain ∆ NAVs. The higher ∆ NAV is the SCR for that risk module.

The SCRs of market risk module are 6.03 for Model 1 and 27.22 for Model 2. These numbers are very small for both models compared with their best estimate of liabilities of about 4000. This is because we chose assets in such a way that their cashflows mimic the average of policy payouts, and therefore they are able to eliminate the risk of yield curve movements to a great extent. However the market risk cannot be completely ruled out. Because unlike the determinstic assets cashflows, the policy payouts are dependent on the stochastic short rates. Model 2’s NAV changes more significantly in both market up and down shocks than that of Model 1, indicating that Model 2 is more volatile to market risks.

In the life underwriting risk module, since the assets are immune to lapse risks, they remain unchanged. So the effects of surrender shocks on the liability side can be screened out accurately. For both models, the surrender up shock leads to higher NAVs, and surrender down shock leads to lower NAVs. This implys a negative relation between surrender probabilities and the best estimate of liabilities for both models. The ∆ NAV is similar for two models in surrender up shock, but it is much larger for Model 1 than Model 2 in surrender down shock. Clearly Model 1’s liabilities increase more dramatically than Model 2 under the down shock, rendering larger risks for Model 1 in this case.

The final SCR of the policy is determined by a formula that takes SCR of both two risk modules and their correlation into account3. Risk diversification between different risk modules reduces the capital needs to be reserved. Model 1’s SCR is about half of Model 2, and this is mainly due to its lower SCR in the market risk module.

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Sensitivity Analysis

In this chapter we will study the sensitivity of the 99.5% VaR of best estimate of liabilities to parameter changes for both models.

7.1 Model 1

We first see the sensitivity brought by parameter Pt, which is the surrender probability due to personal reasons. This parameter is assumed to be given and to remain the same over the whole contract periods because it is impossible to be calibrated. In Chapter 4 we used 0.01 for Pt. Now change its value in the interval from 0 to 0.1. The results are shown in figure 7.1. There is a clear negative relation between Pt and the 99.5% VaR value. In fact Pt acts as a base of the surrender probability. When Pt is raised up, surrender probabilities each year are also lifted up. Therefore, we can deduct a negative relation between overall surrender probability and the 99.5% VaR value as well.

Next we choose a variation range of [0.5, 1] and [1, 1.5] for parameter θmin and θmax respectively. Figure 7.2 shows that both θmin and θmax have positive relations with 99.5% VaR in general. Based on these sets of range choice, the model produces lowest surrender probabilities when θmin equals to 1 and θmax equals to 1.5. Because in such case nobody surrender when θtis smaller than 1, and 100% surrender comes only when θt is as high as 1.5. With the same logic, highest surrender probabilities are produced when θmin equals to 0.5 and θmax equals to 1. Thus we see again the negative relation between overall surrender probabilities and 99.5% VaR of BE.

7.2 Model 2

Parameter θI is analogous to Pt in Model 1 because it also provides a base value for surrender probabilities. We choose a same range [0, 0.1] for θI. The other parameter A has a value of 9.7311 after calibration. We will study its value from 5 to 15 in this case. From figure 7.3 one sees that θI and A both have a negative relation with 99.5% VaR of BE. This relation is much more significant for either parameter when the other one has smaller values. Highest VaR value is obtained when θI and A are at their minimum. The same is true for the opposite. Since smaller θI and A both contribute to lower surrender probabilities each year, one can derive a negative relation between overall surrender probabilities and the 99.5% VaR value, just as the relation reflected by Model 1.

Therefore, it seems no matter which model is applied for our policy, early surren-ders will be preferred by the insurers because they lower the value of best estimate of liabilities.

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Figure 7.1: Sensitivity of 99.5% VaR value to parameter p in Model 1.

Figure 7.2: Sensitivity of 99.5% VaR value to parameter θmin and θmax in Model 1.

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Conclusion

In this paper we studied the impact of applying two advanced surrender probability models on the liabilities of an insurance product with surrender option. These two models are advanced in the sense that they both take irrationality of policyholders into account. Using LSMC simulation method, we found that the two models share the same basis function for the best estimate of liabilities, providing a firm ground for comparison. The simulation result shows that Model 1 generates larger best estimate of liabilities than Model 2 only in some extreme cases when both market rate and surrender probability in the first year are very high. In most other occasions Model 2 yields higher best estimate of liabilities. This conforms to what we found in real world scenarios.

Given the real world distributions for both models in Chapter 5, we see that the means of best estimate of liabilities are 3810.8 for Model 1 and 3978.2 for Model 2. And the standard deviations of both models are very close. On average Model 1 tends to produce best estimates of liabilities that are 4.2% smaller than Model 2 does, which is large enough to make significant impact on insurers’ liabilities. Also it turns out that the 99.5% VaR of best estimate of liabilities, stands at 4106.2 for Model 1 and 4292.6 for Model 2, is 4.3% lower for Model 1 than Model 2. This means that if insurers use Model 1 instead of Model 2 as their internal model for solvency II, they will be demanded 4.3% less capital for the best estimate.

Therefore, to answer the research question, Model 1 gives much lower best estimate of liabilities of the policy than Model 2 does. With only these two models available, insurers’ balance sheets would look stronger if Model 1 is applied.

Compared with the best estimate of liabilities, the solvency capital requirements of the policy for both models are much smaller. They are only 0.3% of the best estimate for Model 1 and 0.5% for Model 2. And Model 1’s overall SCR is about half of that of Model 2. Their low values are mainly due to the risk diversification between market risk and lapse risk modules. By choosing some fixed income assets that closely replicate the average policy payouts, we managed to hedge the market risks of the liability side greatly. Model 2 is more vulnerable to market risk because its market SCR is four and half times larger than Model 1. On contrary, Model 1 is much sensible to surrender rate shocks than Model 2 since it has about one third more lapse SCR than Model 2.

Moreover we looked at the sensitivity of the best estimate on model parameters, and found out a negative relationship between the overall surrender probabilities and best estimate of liabilities for both models.

We have seen in this paper how a mere switch of surrender model could have caused crucial consequences on liabilities and risks. However, it should be noted that the two models we studied are only similar in their idea of including irrationality, but different in nature. Choosing a convincing model with strong predicting power seems to be the most central issue when it comes to lapse risk mitigation. But still, what can be learned is that insurers must take close look at the impact before employing their internal surrender models.

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Appendix A: Regression results

for Chapter 5

Crucial regression statistics such as p-value and AIC that are supplementary to Chapter 5 are displayed in this section. Notations x1 to x9 represent r1, r12, r13, ρ1, ρ12, ρ13, r1ρ1, r12ρ1 and r1ρ12 respectively. Numbers in the round parentheses before regression functions correspond to the equation number in Chapter 5.

Model 1

r1 as the only risk driver Linear regression model: (1) y ~ 1 + x1

Estimated Coefficients:

Estimate SE tStat pValue

(Intercept) 3890.4 0.906 4294 0

x1 -72.366 0.22582 -320.46 0

> AIC(fit) 658312.7

Linear regression model: (2) y ~ 1 + x1 + x2 Estimated Coefficients:

(Intercept) 3879.3 1.0705 3623.9 0

x1 -78.006 0.36921 -211.28 0

x2 1.3982 0.072576 19.265 2.1203e-82

> AIC(fit) 657945

Linear regression model: (3) y ~ 1 + x1 + x2 + x3 Estimated Coefficients:

(Intercept) 3881.8 1.3574 2859.7 0 x1 -78.677 0.4334 -181.53 0 x2 0.97609 0.16017 6.0941 1.1088e-09 x3 0.070205 0.023751 2.9559 0.0031187 > AIC(fit) 657938.2 27

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ρ1 as the only risk driver Linear regression model: (4) y ~ 1 + x4

(Intercept) 3793.4 1.446 2623.4 0

x4 -9.4402 0.12319 -76.633 0

> AIC(fit) 708581.3

(Intercept) 3811.9 1.5407 2474.1 0

x4 -18.014 0.29151 -61.793 0

x5 0.19284 0.0059559 32.377 1.3041e-227

> AIC(fit) 707545.8

(Intercept) 3825 1.6789 2278.3 0 x4 -27.21 0.55964 -48.62 0 x5 0.63195 0.0236 26.778 7.549e-157 x6 -0.0044901 0.00023356 -19.224 4.5948e-82 > AIC(fit) 707179.5

r1 and ρ1 as risk drivers Linear regression model: (7) y ~ 1 + x1 + x4 Estimated Coefficients:

(Intercept) 3894.1 0.93685 4156.6 0

x1 -71.085 0.24063 -295.41 0

x4 -1.2045 0.079401 -15.17 7.2605e-52

> AIC(fit) 658085.1

(Intercept) 3902.2 0.98215 3973.1 0 x1 -73.489 0.25659 -286.4 0 x4 -5.4066 0.18108 -29.858 3.5218e-194 x7 0.80592 0.031261 25.781 1.3119e-145 > AIC(fit) 657426.8

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Linear regression model:

(9) y ~ 1 + x1 + x2 + x4 + x5 + x7 Estimated Coefficients:

(Intercept) 3892.7 1.2002 3243.4 0 x1 -77.24 0.37158 -207.87 0 x2 1.0836 0.077079 14.059 8.2866e-45 x4 -4.8247 0.22186 -21.747 2.2599e-104 x5 0.00098432 0.0038314 0.25691 0.79725 x7 0.65629 0.034545 18.998 3.4022e-80 > AIC(fit) 657231.8

(11) y ~ 1 + x1 + x2 + x4 + x7 (the final basis function) Estimated Coefficients:

(Intercept) 3892.6 1.192 3265.7 0 x1 -77.25 0.36953 -209.05 0 x2 1.0814 0.076606 14.117 3.6488e-45 x4 -4.7935 0.18586 -25.791 1.0206e-145 x7 0.659 0.032889 20.037 5.86e-89 > AIC(fit) 657229.9

(10) y ~ 1 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 Estimated Coefficients:

(Intercept) 3893.5 1.5329 2539.9 0 x1 -77.712 0.50084 -155.16 0 x2 0.97045 0.16033 6.0528 1.4332e-09 x3 0.026266 0.025288 1.0387 0.29896 x4 -4.9652 0.46122 -10.765 5.3665e-27 x5 0.0034133 0.01618 0.21095 0.83293 x6 -1.8095e-05 0.00015142 -0.1195 0.90488 x7 0.76333 0.097714 7.8119 5.7438e-15 x8 -0.014283 0.011976 -1.1927 0.233 x9 -8.1797e-05 0.0020464 -0.039972 0.96812 > AIC(fit) 657237.6

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Modal 2

r1 as the only risk driver Linear regression model: (1) y ~ 1 + x1

(Intercept) 4066 1.4708 2764.5 0

x1 -78.107 0.3666 -213.06 0

> AIC(fit) 706765.8

(Intercept) 4059.8 1.7435 2328.6 0

x1 -81.238 0.60134 -135.1 0

x2 0.77625 0.11821 6.5669 5.1886e-11

> AIC(fit) 706724.7

(Intercept) 4060 2.211 1836.3 0 x1 -81.275 0.70595 -115.13 0 x2 0.75336 0.2609 2.8876 0.0038836 x3 0.0038068 0.038687 0.098401 0.92161 > AIC(fit) 706726.7

ρ1 as the only risk driver Linear regression model: (4) y ~ 1 + x4

(Intercept) 3945.3 1.8183 2169.8 0

x4 -12.498 0.21667 -57.682 0

> AIC(fit) 735844.7

(Intercept) 3950.4 2.1485 1838.7 0

x4 -17.008 1.0337 -16.454 1.1416e-60

x5 0.11949 0.026777 4.4622 8.1286e-06

> AIC(fit) 735826.8

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(Intercept) 3945.9 3.47 1137.2 0 x4 -12.357 2.9985 -4.1211 3.7765e-05 x5 -0.17616 0.1809 -0.97379 0.33017 x6 0.0044963 0.0027209 1.6525 0.098439 > AIC(fit) 735826

r1 and ρ1 as risk drivers Linear regression model: (7) y ~ 1 + x1 + x4 Estimated Coefficients:

(Intercept) 4068.1 1.4923 2726.1 0

x1 -77.074 0.3875 -198.9 0

x4 -1.4019 0.17124 -8.187 2.7403e-16

> AIC(fit) 706700.8

(Intercept) 4073.4 1.9015 2142.1 0 x1 -77.758 0.41677 -186.57 0 x4 -6.0952 1.0669 -5.7131 1.1158e-08 x7 0.68419 0.15352 4.4568 8.3367e-06 > AIC(fit) 706683

(9) y ~ 1 + x1 + x2 + x4 + x5 + x7 Estimated Coefficients:

(Intercept) 4063.2 2.7686 1467.6 0 x1 -80.825 1.0707 -75.489 0 x2 0.99757 0.12385 8.0547 8.1401e-16 x4 -3.9955 1.8943 -2.1092 0.034932 x5 0.06146 0.11178 0.54981 0.58245 x7 -0.010437 0.85661 -0.012184 0.99028 > AIC(fit) 706621.2

(Intercept) 4060.2 1.7421 2330.6 0

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x2 1.067 0.12158 8.7757 1.7508e-18

x4 -1.7699 0.17617 -10.047 1.0025e-23

> AIC(fit) 706625.9

(11) y ~ 1 + x1 + x2 + x4 + x7 (the final basis function) Estimated Coefficients:

(Intercept) 4064.2 2.2139 1835.7 0 x1 -81.311 0.60496 -134.41 0 x2 1.0012 0.12367 8.0962 5.7969e-16 x4 -4.8522 1.0772 -4.5045 6.6675e-06 x7 0.45265 0.15606 2.9005 0.0037271 > AIC(fit) 706619.5

(10) y ~ 1 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 Estimated Coefficients:

(Intercept) 4051.2 7.7714 521.3 0 x1 -76.457 3.2983 -23.181 3.0048e-118 x2 1.0825 2.0762 0.52139 0.6021 x3 0.0061167 0.040678 0.15037 0.88047 x4 8.6037 8.2351 1.0448 0.29614 x5 -0.39242 1.8603 -0.21094 0.83293 x6 -0.052473 0.049064 -1.0695 0.28486 x7 -4.9264 3.4675 -1.4207 0.1554 x8 -0.11225 2.0645 -0.054372 0.95664 x9 0.44877 0.28643 1.5667 0.11718 > AIC(fit) 706625.9

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Appendix B: Basis functions of

NAV, used for SCR calculations

No shock

Model 1

N AV = 144 + 4.8382r1− 0.20406r12+ 5.0823ρ1− 0.71858r1ρ1.

Model 2

N AV = −87.832 + 4.5195r1+ 6.6803ρ1− 0.74083r1ρ1.

Market interest rate shock

Model 1 up shock

N AV = 144.74 + 1.3022r1− 0.20193r12+ 2.265ρ1− 0.62706r1ρ1.

Model 1 down shock

N AV = 144.27 + 9.6667r1− 0.24263r12+ 8.8307ρ1− 0.79407r1ρ1.

Model 2 up shock

N AV = −70.312 + 2.2258r1− 0.15768r12+ 1.4309ρ1− 0.62139r1ρ1.

Model 2 down shock

N AV = −113.98 + 7.9989r1+ 0.33747r12+ 14.217ρ1− 0.79513r1ρ1.

Surrender rate shock

Model 1 up shock

N AV = 188.9 + 3.1622r1− 0.20965r12+ 4.6237ρ1− 0.70204r1ρ1.

Model 1 down shock

N AV = 53.819 + 7.3543r1− 0.21077r12+ 5.9681ρ1− 0.72944r1ρ1.

Model 2 up shock

N AV = −39.035 + 3.5621r1+ 6.2841ρ1− 0.74703r1ρ1. 33

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Model 2 down shock

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