Retrospective Reserves and Bonus with Policyholder Behavior

Legislation imposes insurance companies to project their assets and liabilities. Within the setup of with-profit life insurance, we consider retrospective reserves and bonus, and we study projection of balances with and without policyholder behavior. The projection resides in a system of ordinary differential equations of the savings account and the surplus, and we include the policyholder behavior options surrender and conversion to free-policy. The inclusion results in a structure where a system of ordinary differential equations of the savings account and the surplus is non-trivial. We consider a case, where we are able to find accurate ordinary differential equation and suggest an approximation method to project the savings account and the surplus including policyholder behavior in general.


Introduction
In with-profit insurance, prudent assumptions about the interest rate and biometric risks at initialization of an insurance contract result in a surplus emerging over time. This surplus belongs to the policyholders and must be paid back in terms of bonus. The redistribution of bonus contains certain degrees of freedom, which is a part of the Management Actions. Furthermore, bonus must be taken into account when insurance companies determine their assets and liabilities. Legislation imposes insurance companies to project their balance sheet, and companies must be able to perform projections of assets and liabilities in a number of scenarios of the financial market. This requires a specification of the future dividend strategy and, in general a specification of the Future Management Actions. Management actions may depend on the financial scenario, the present as well as the past entries of the balance sheet and their relations, and other aspects of the financial situation of the insurance company. Therefore, future management actions have a complex nature and are difficult to predict and formalize mathematically. In this paper, we model the projection of the savings account and the surplus of an insurance contract, where we assume the future dividend strategy has a simple structure. We do not incorporate future management actions to their full extent in this model, but the model establishes a foundation for projecting balances in life insurance.
The modeling of surplus and bonus in life insurance is not new. Norberg (1999) introduced the individual surplus of a life insurance contract, and Steffensen (2006) derived differential equations for prospective reserves in the case, where dividends are linked to the surplus. In our model, we also consider dividends linked to the surplus, but distinct from Steffensen (2006), we derive differential equations for the projected savings account and surplus. Jensen and Schomacker (2015) studied the valuation of an insurance contract with the bonus scheme spoken of as additional benefits, where dividends are used to buy more insurance in a scenario-based model for the financial market. Our paper has some similarities with Jensen and Schomacker (2015) in the sense that we also study a scenario-based model with additional benefits. In Jensen and Schomacker (2015) the bonus allocation is discretized, while we allocate bonus continuously, resulting in difference equations in Jensen and Schomacker (2015) and ordinary differential equations in our model. Steffensen (2006) studied prospective reserves, while we focus on the savings account, which is a retrospective reserve including past bonus, and the surplus of an insurance contract. The retrospective approach without bonus is studied in Norberg (1991) and studied with bonus in Asmussen and Steffensen (2020). Bruhn and Lollike (2020) also reflected on the retrospective perspective, and studied retrospective reserves with and without bonus. They model the savings account and the surplus of an insurance contract, and derive ordinary differential equations for the state-wise projections.
This paper serves as an extension to Bruhn and Lollike (2020). The extension resides in the incorporation of the policyholder behavior options surrender and conversion to freepolicy. Upon surrender, the policyholder receives a single payment and all future payments cancel, and with the free-policy option, all future premiums cancel and benefits are reduced by a free-policy factor. We model policyholder behavior as random transitions in the Markov model from the classical life insurance setup extended with surrender and free-policy states as studied in for instance Henriksen et al. (2014). This is in contrast to modeling rational policyholder behavior as in Steffensen (2002). Buchardt and Møller (2015) studied the calculation of prospective reserves without bonus including policyholder behavior using a cash flow approach. We include policyholder behavior options in our model of the retrospective savings account and surplus including bonus, and our approach is based on differential equations of the state-wise projections. Buchardt and Møller (2015) introduced the notion of modified probabilities to calculate prospective reserves including conversion to free-policy. The same modified probabilities appear in our system of differential equations for the state-wise projections of the savings account and the surplus.
In Section 2, we present the general life insurance setup and the model of the savings account, the surplus, and the dividends. We define the projection of the savings account and the surplus without policyholder behavior and state the results from Bruhn and Lollike (2020) in Section 3. In Section 4, we include policyholder behavior in the classic life insurance setup, and consider the ideal free-policy factor in our retrospective setup including bonus. This free-policy factor does not satisfy the simple structure of the model in Section 3, and therefore we are unable to use the results concerning the projection of the savings account and the surplus. We consider the case with all benefits regulated by bonus. In this case, we actually can project the savings account and the surplus with the ideal free-policy factor. Furthermore, we suggest an approximation of the free-policy factor, for which the state-wise projections of the savings account and the surplus coincide with the state-wise projections using the ideal free-policy factor. We present a method to project the savings account and the surplus with the approximated free-policy factor in a general case.

Life insurance setup
We take the classic multi-state setup in life insurance as a starting point, and extend this with policyholder behavior in Section 4. A Markov process, Z = Z(t) t≥0 , in a finite state space J • = {0, 1, ..., J − 1} describes the state of the holder of a life insurance contract, and payments in the contract link with sojourns in states and transitions between states. We define the transition probabilities of Z as p ij (s, t) = P Z(t) = j Z(s) = i , for i, j ∈ J • and s ≤ t. We assume that the transition intensities Proposition 1. The transition probabilities satisfy the Kolmogorov's forward differential equations Proof. See Buchardt and Møller (2015), Proposition 4.
The processes N k (t) for k ∈ J • count the number of jumps of Z into state k up to time t.
We consider with-profit insurance products, where payments specified in the contract are based on prudent assumptions about interest rate and transition intensities. We call these assumptions the first order (technical) basis, and denote it by (r * , µ * ij ) for i, j ∈ J • , i = j. The third order (market) basis describes the actual development of the interest rate and transition intensities of the insurance portfolio. We denote the third order basis by (r, µ ij ) for i, j ∈ J • , i = j, and we consider it as externally given. Due to the prudent first order basis, a surplus arises, which by legislation is to be paid back to the policyholders as bonus. We use the bonus scheme spoken of as additional benefits, where bonus is used to buy more insurance. We denote this defined contributions since premiums are fixed and benefits are increased by bonus in contrast to defined benefits, where bonus is used to lower premiums and benefits are fixed.
We decompose the accumulated payments of an insurance contract at time t into two payment streams; one that contains the payments not regulated by bonus, B 1 , and one that contains the profile of payments regulated by bonus, B 2 , as presented in Asmussen and Steffensen (2020). We can imagine an insurance contract consisting of a life annuity and a term insurance. Often only the life annuity is scaled by bonus and the term insurance as well as the premiums are fixed. Then the payment stream B 1 consists of the term insurance and the premiums, and the payment stream B 2 consists of the life annuity.
The dynamics of the payment streams are in the following form for i = 1, 2 where b j i (t) denotes the payment rate during sojourn in state j and b jk i (t) the single payment upon transition from state j to state k at time t. We assume that the payment functions b j i (t) and b jk i (t) are deterministic and sufficiently regular. For notational convenience, we disregard lump sum payments at fixed time points during sojourn of states, even though it does not impose mathematical difficulties.
Definition 1. The prospective first order (technical) reserve at time t ≤ n for payment stream dB i (t), i = 1, 2 is given by where n denotes termination of the contract and E * means that we use the first order transition intensities, µ * jk , j, k ∈ J • , j = k, in the distribution of Z.
The principle of equivalence states that V * Z(0) 1 Proposition 2. The prospective first order reserve from Definition 1 has dynamics given by Proof. See Asmussen and Steffensen (2020), Chapter 6.7.

Savings account, surplus and dividends
Similar to Asmussen and Steffensen (2020), the insurer returns the surplus to the insured through a dividend payment stream, D, and let Q(t) be the number of payment processes B 2 (t) bought over time [0, t]. As we use dividends to buy B 2 (t) at the price of V * Z(t) 2 (t), we must have The policyholder experiences the total payment process with dynamics which is the payment process guaranteed at time t.
We denote by X(t) the savings account of an insurance contract, which is the technical value of future payments guaranteed at time t ≥ 0, i.e. we have the following relation between X(t) and Q(t) .
We desire that the savings account is equal to zero at the beginning of the insurance contract, X(0−) = 0. Then by the principle of equivalence, we have the initial condition Q(0−) = 1.
Due to the relationship between X and Q, the payment process, dB(t), is a linear function in X, which we now denote by Proposition 3. The savings account, X, has dynamics where the sum-at-risk is given by is the technical value of guaranteed payments after the transition from state j to state k.
The surplus is the difference between past premiums less benefits over time [0, t] accumulated with the market interest rate and the savings account at time t. We denote the surplus by Y (t) Throughout this paper, we consider the market interest rate as externally given. In a setup with a stochastic market interest rate, the market interest rate over time [0, t] is known at time t such that Y (t) is adapted to the interest rate filtration and therefore known at time t. We have that Y (0−) = 0.
Proposition 4. The surplus, Y , has dynamics where the surplus contribution is given by Proof. See Asmussen and Steffensen (2020), Chapter 6.7.
The dividend payments stream, dD Z(t) (t), describes how the surplus is returned to the insured. We assume that the dividend process is continuous and depends on the savings account and the surplus, such that the dynamics are The dynamics of the savings account and the surplus are affine if and only if the dividend process is. The main result of this paper relies on affinity in the dynamics of the savings account and the surplus, and therefore we make the assumption that the dividend process is affine in X(t) and Y (t) for sufficiently regular and deterministic function δ j 0 , δ j 1 and δ j 2 , j ∈ J • . This is a restriction in the degree of freedom in the dividend allocation strategy of the insurance companies, and therefore of the future management actions in the model.

State-wise projections without policyholder behavior
In order to satisfy legislation, insurance companies and present research focus on the projection of balances in life insurance using simulation methods. Both the savings account, X, and the surplus, Y , are entries of the balance sheet, and the projection of these requires simulation of the market basis i.e. the interest rate (investment returns) and the biometric risks. It is computational heavy to simulate the biometric history of an entire insurance portfolio, and therefore we eliminate the biometric part of the simulation study by studying state-wise projections.
Definition 2. We define the state-wise projections of the savings account, X, and the surplus, Y , byX for j ∈ J • . The expectation is taken under the market basis. The subscript Z(0) denotes that the expectation is the conditional expectation given Z(0).
Bruhn and Lollike (2020) derived differential equations for the state-wise projection of the savings account and the surplus from Definition 2 to use for projection of the savings account and the surplus in a given interest rate scenario. The theorem below states the main result of Bruhn and Lollike (2020), and the purpose of this paper is to extend these differential equations to a setup including policyholder behavior.
Lemma 1. The dynamics of the savings account, X, from Proposition 3 and the dynamics of the surplus, Y , from Proposition 4 are in the form for deterministic functions α j i,H and λ jk i,H for i = 0, 1, 2, H = X, Y and j, k ∈ J • , j = k. See Appendix A for the expressions of α and λ for the savings account and the surplus.
Theorem 1. Let X and Y have dynamics in the form of Lemma 1. Then the statewise projections of X and Y from Definition 2 satisfy the following system of ordinary differential equations for j ∈ J • , and X(0−) = Y (0−) = 0.
Kolmogorov's forward differential equations from Proposition 1 can be used to calculate the transition probabilities in Theorem 1.

State-wise projections with policyholder behavior
Now, we extend the setup from Section 2 to include policyholder behavior. We include the policyholder behavior options surrender and conversion to free-policy. Upon surrender, the policyholder receives a single payment and all future payments cancel. With the freepolicy option, all future premiums cancel, and benefits are reduced by a free-policy factor, f , that depends on the time at which the policyholder goes from premium paying to freepolicy. We study how the introduction of policyholder behavior affects the dynamics of the savings account, X, from Proposition 3 and the surplus, Y , from Proposition 4. The objective is to be able to perform state-wise projections of the savings account and the surplus including policyholder behavior.
We model policyholder behavior in the classic way by extending the state space of the Markov chain, Z, to include surrender and free policy states as presented in Henriksen et al. (2014). We extent the state space of Z from Section 2 as illustrated in Figure 1.
The state J corresponds to surrender, and we assume that surrender can only happen from state 0. The state space J f denotes the free-policy states, and it is a copy of J in the sense that it holds the same number of states and that state i ∈ J f corresponds to state i − (J + 1) ∈ J. We assume that conversion to free-policy can only occur from state 0 and that the transition intensities in J f equal the transition intensities in J, but we can easily relax this assumption. We assume throughout the rest of this paper that Z(0) ∈ J. The classical 7-state model from for example Buchardt and Møller (2015) is contained in this setup, where state 0 in our model corresponds to the premium-paying active state.
In order to model payments including policyholder behavior, we need to decompose the payment streams from Equation (1) in benefits, dB + i (t) and premiums, dB − i (t) for i = 1, 2. The sojourn payments and payments upon transition are then decomposed into b j+ i and b j− i , and b jk+ i and b jk− i respectively. We consider defined contributions such that the payment stream increased by bonus only contains benefits i.e. b j− 2 (t) = b jk− 2 (t) = 0 for all t ≥ 0 and j, k ∈ J, j = k. The payment stream B 1 contains premiums and benefits not regulated by bonus.
The technical benefit and premium reserves respectively in the non-free-policy states, Z(t) ∈ J, are given by for i = 1, 2.
Defined contributions imply that for Z(t) ∈ J.
We introduce the duration in the free-policy states U , Payments in the free-policy states equal a free-policy factor, f ∈ [0, 1], times the benefits in the corresponding premium-paying state. We allow the free-policy factor to depend on the savings account, i.e. f (t, X(t)), and the benefits are reduced with the free-policy factor evaluated at the time of conversion to free-policy, f (t − U (t), X(t − U (t))). We introduce the mapping of Z that returns the corresponding premium-paying state if Technical reserves in the free-policy states equal the free-policy factor times the technical benefit reserve in the corresponding premium-paying state.
Definition 3. The technical reserve in the free-policy states is The inclusion of policyholder behavior changes the payment process from Equation (3) and the sum-at-risk from Proposition 3. Now, the payment process and the sum-at-risk depend on time, the savings account, and the duration in the free-policy states.
Proposition 5. The total payment process guaranteed at time t including policyholder behavior is where the continuous payment functions during sojourns in states and the payment functions upon transition between states are b j (t, x,u, x f ) for j, k ∈ J∪J f , j = k. We assume that there are no continuous payments in the surrender states, and that there is no payment upon transition between J and J f . Proposition 6. Including policyholder behavior, the sum-at-risk from Proposition 3 is The last line corresponds to the sum-at-risk upon conversion to free-policy, where u = 0.
Remark 1. In the last line of the sum-at-risk from Proposition 6, we have that g(k) = g(J + 1) = 0 = j, and by Equation (5), the sum-at-risk upon conversion to free-policy is The dynamics of the savings account, X, and the surplus, Y , including policyholder behavior are equal to the dynamics given in Proposition 3 and Proposition 4, where the payment process and the sum-at-risk are given by Proposition 5 and Proposition 6. Thus, the dynamics of the savings account are and the dynamics of the surplus are where the surplus contribution is given by The dividend strategy δ is given by Equation (4).
The above dynamics of the savings account and the surplus contain the free-policy factor, f , and duration, U (t), which imply that they are not in the form of Lemma 1. Therefore, we cannot use Theorem 1 to project the savings account and the surplus.

Choice of free-policy factor
The extension of the classic life insurance setup without bonus to include policyholder behavior is described in existing literature. We refer to Buchardt and Møller (2015) for a description of this extension. Without bonus, we usually choose the free-policy factor according to the equivalence principle such that there is no jump in the technical reserve upon conversion to free-policy, i.e.
where the superscript • refer to the setup without bonus.
Without bonus, the technical reserve, V * (t), is the technical value of future payments guaranteed at time t, since all payments are guaranteed. In our setup with bonus, this corresponds to the savings account, X(t). Therefore, the ideal free-policy factor in the setup with bonus is the free-policy factor, where the sum-at-risk of the savings account upon conversion to free-policy is equal to zero, resulting in no jump in X upon conversion to free-policy. From Remark 1, we conclude that This free-policy factor is nonlinear in the savings account, which implies that the dynamics of the savings account and the surplus from Equations (6) and (7) do not satisfy the linearity assumption in Lemma 1 with this choice of free-policy factor.
Let X id be the savings account and let Y id be the surplus with the ideal free-policy factor from Equation (8) above. Similar to Definition 2, the state-wise projections of the savings account and the surplus are given bỹ for j ∈ J ∪ J f .

The case with all benefits regulated by bonus
We consider the case, where all benefits are regulated by bonus such that the payment stream not increased by bonus, B 1 , only includes premiums i.e. B + 1 = 0. In this case, we show that the dynamics of the savings account and the surplus with the ideal free-policy factor from Equation (8), are in the form of Lemma 1 such that we can use Theorem 1 to find differential equations for the state-wise projections of the savings account and the surplus.
Again, we consider the example of an insurance contract consisting of a life annuity and a term insurance. In the case B + 1 = 0, both products are regulated by bonus, in contrast to the case where only the life annuity is scaled by bonus.
The assumptions of defined contributions and B + 1 = 0 imply that the total payment process has dynamics where Q(0−) = 1 due to the principle of equivalence.
In the continuous payment functions during sojourns in states and the payment functions upon transition between states from Proposition 5, the terms including the free-policy factor are multiplied by either b j+ 1 , b jk+ 1 or V * j+ 1 for j, k ∈ J. In the case B + 1 = 0, these are all equal to zero and therefore the free-policy factor does not appear in the payment functions.
The continuous payment functions during sojourns in states and the payment functions upon transition between states from Proposition 5 are in this case for j, k ∈ J ∪ J f .
Similar to the payment functions, the terms including the free-policy factor in the sumat-risk from Proposition 6 are multiplied by V * j+ 1 for j ∈ J, except for the sum-at-risk upon conversion to free-policy. Thus, in the case B + 1 = 0, the sum-at-risk is With the free-policy factor from Equation (8), the last line in the sum-at-risk above is equal to zero. Therefore, in the case B + 1 = 0 with the free-policy factor from Equation (8), neither the payment functions (11) and (12) nor the sum-at-risk (13) depend on the duration in the free-policy states, and they are linear in the savings account. This implies that the dynamics of X id (t) and Y id (t) are in the form of Lemma 1, leading to the result in Theorem 1. Hence, in this case, we actually have differential equations for the projected savings account and surplus with the free-policy factor from Equation (8) for j ∈ J ∪ J f andX id (0−) =Ỹ id (0−) = 0. The expressions forα j andλ jk are in Appendix B.
We compare the differential equations of the projected savings account and surplus in the case B + 1 = 0 using the free-policy factor from Equation (8) with the differential equations without policyholder behavior. This comes down to a comparison of the coefficients α j and λ jk from Appendix A andα j andλ jk from Appendix B. The coefficient α j and the correspondingα j consist of the same terms, butα j is decomposed in the cases j ∈ J and j ∈ J f in the same sense as the payment functions and the sum-at-risk from Equations (11), (12) and (13), since in the free-policy states, we only consider benefits. This also goes for λ jk andλ jk .
The case B + 1 = B + 2 corresponds to the case B + 1 = 0, since the total payment process when which has the same form as the payment process in the case B + 1 = 0, but whereQ(0−) = 2 since Q(0−) = 1 due to the principle of equivalence. When the benefits in B 1 are equal to the benefits in B 2 , all benefits are regulated equally by bonus, and therefore we can rewrite the case B + 1 = B + 2 to be in the form of B + 1 = 0. Hence, the results above also apply for B + 1 = B + 2 .
If benefits not regulated by bonus cancel due to conversion to free-policy, we have that B + 1 = 0 after conversion to free-policy, and the result above still applies, and we have differential equations for the projections of the savings account and the surplus. We can imagine an insurance contract consisting of a life annuity and a term insurance, where the life annuity is regulated by bonus, and the term insurance cancels upon conversion to free-policy. Throughout this paper, we assume that payments in the free-policy states equal a free-policy factor times the benefits in the corresponding premium-paying state. The example does not comply with this assumption, but we can easily extend our setup to include this case.

Approximation of the free-policy factor
In the general setup, B + 1 (t) ≥ 0 for t ≥ 0, we cannot project the saving account and the surplus by Theorem 1, since the assumptions are violated. The dynamics of the savings account and the surplus depend on the duration in the free-policy states, U , and the derivation of Theorem 1 relies on linearity of X and Y in the dynamics from Lemma 1. When the free-policy factor depends on the saving account, the linearity of the dynamics breaks. This motivates an approximation of the ideal free-policy factor from Equation (8), which does not depend on X.
Just before conversion to free-policy, the policyholder must be premium paying and active, i.e. Z(t−) = 0. A reasonable approximation of the free-policy factor is thereforê .
We have not developed methods to calculate the projection of a fraction containing the savings account, X(t), in both the nominator and the denominator. Therefore, we cannot continue with the approximation above. Alternatively, we can project the nominator and denominator in the free-policy factor separately and obtain the approximatioñ Corollary 1. Let X id be the savings account and Y id be the surplus modeled with the ideal free-policy factor from Equation (8), and let X ap be the savings account and Y ap be the surplus modeled with the approximated free-policy factor from Equation (16). The state-wise projections are given by Equations (9) and (10), and In the case where all benefits are regulated by bonus, B + 1 = 0, we have that Proof. Assume all benefits are regulated by bonus, B + 1 = 0. The state-wise projections of the savings account and the surplus with the ideal free-policy factor satisfy the differential equations in Equations (14) and (15).
The sum-at-risk with the approximated free-policy factor,f , is In the case B + 1 = 0, Equations (11), (12) and (13) in Section 4.2, state that only the sum-at-risk depends on the free-policy factor.
The dynamics of X ap and Y ap are in the form of Equations (6) and (7) with the payment functions from Equations (11) and (12) and the sum-at-risk from Equation (17). This implies that the dynamics of X ap and Y ap are in the same form as in Lemma 1, since they do not depend on the duration, U , and they are linear in X ap (t) and Y ap (t).
Theorem 1 gives differential equations of the projected savings account and surplus,X j ap andỸ j ap . These differential equations can be expressed in terms ofα andλ from the differential equations (14) and (15) By inserting the expression forf from Equation (16), the differential equations (18) and (19) are equal to the differential equations (14) and (15). We have that for j ∈ J ∪ J f as desired.
Corollary 1 implies that in the case B + 1 = 0, we can project the savings account and the surplus with the approximated free-policy factor and actually obtain the same accurate projections as with the ideal free-policy factor. Based on this result, we considerf to be a reasonable approximation of f , that does not depend on the savings account, but instead on the projected savings account.

Projections with the approximated free-policy factor
In the general setup, B + 1 (t) ≥ 0 for t ≥ 0, with the approximated free-policy factor from Equation (16), we have linearity in the dynamics of the savings account and the surplus, but also dependence of duration in the payment functions from Proposition 5 and the sum-of-risk from Proposition 6. Therefore, we cannot use Theorem 1 to project the saving account and the surplus. This motivates an extension of Theorem 1 including duration dependence, where linearity in the dynamics of the savings account and the surplus are preserved.
Lemma 2. The dynamics of the savings account, X ap , from Equation (6) and the dynamics of the surplus, Y ap , from Equation (7), with the approximated free-policy factor, f , from Equation (16), can be written in the form for all t ≥ 0 and j ∈ J. See Appendix C for the expressions ofᾱ,β,λ andγ for the savings account and the surplus.
We consider the difference between the case with all benefits regulated by bonus with the free-policy factor from Equation (8) from Section 4.2 and the general case with the approximated free-policy factor. This comes down to a comparison of the coefficientsα andλ from Appendix B with the coefficientsᾱ,β,λ andγ from Appendix C. Apart from the sum-at-risk upon conversion to free-policy and the duration dependent terms, the coefficients are equal. In the first case, the sum-at-risk upon conversion to free-policy is equal to zero, while in the second case, it is added toᾱ andλ. The duration dependent terms from Propositions 5 and 6 are equal to zero in the case with all benefits regulated by bonus, while in the general case they appear inβ andγ. Now, we extent the result of Theorem 1 to include the duration dependence from the dynamics of the savings account and the surplus in Lemma 2.
Theorem 2. Let X ap and Y ap have dynamics in the form of Lemma 2 and Z(0) ∈ J.
The state-wise projections of the savings account and the surplus,X j ap andỸ j ap defined in Corollary 1, satisfy the system of differential equations below whereX j ap (0−) =Ỹ j ap (0−) = 0 andf is the approximated free-policy factor from Equation (16) and pf Z(0)j (0, t) are thef -modified probabilities, defined as Proof. See Appendix D. Buchardt and Møller (2015) derived forward differential equations for the samef -modified probabilities in the case where j ∈ J f . In the case where j ∈ J, thef -modified probabilities are in fact the ordinary transition probabilities that satisfy Kolmogorov forward differential equations from Proposition 1. Therefore, for a general j ∈ J ∪ J f we use the following forward differential equations to calculate thef -modified probabilities Theorem 2 enables us to project the savings account and the surplus in a setup with the policyholder behavior options surrender and free-policy with the approximated freepolicy factor from Equation (16). For instance in the example with an insurance contract consisting of a life annuity and a term insurance, where the life annuity is regulated by bonus and the term insurance and the premiums are fixed.
Remark 2. Let the savings account and the surplus have dynamics in the form of Lemma 2, but with a general free-policy factor,f , that does not depend on the savings account. Then Theorem 2 holds withf -modified probabilities.
Based on the results of Bruhn and Lollike (2020), also presented in Section 3, we cannot in general project the savings account and the surplus including policyholder behavior with the ideal free-policy factor from Equation (8), since we have duration dependence and the linearity assumption breaks. In the case, where all benefits are regulated by bonus from Section 4.2, we can actually find accurate differential equations for the projected savings account and surplus with the ideal free-policy factor. In Section 4.3, we suggest an approximation to the ideal free-policy factor, and Corollary 1 states that in the case, where all benefits are regulated by bonus, the projections based on the ideal free-policy factor coincide with the projections based on the approximated free-policy factor. This implies that the approximated free-policy factor is a reasonable approximation for the ideal free-policy factor. Hence, we consider Theorem 2 as a good extension of Theorem 1 to include policyholder behavior outside the case, where all benefits are regulated by bonus.

D Proof of Theorem 2
We only present the proof of the differential equation forX j , since the differential equation forỸ j is obtained using the same calculations.
We consider the integral equation forX j (t) We calculate E Z(0) 1 {Z(t)=j} dX(s) Z(s−) = g for both terms in the dynamics of X(t) from Lemma 2.