Optimal investment for participating insurance contracts under VaR-RegulationInstitute of Insurance Science and Institute of Financial Mathematics, University of Ulm, Germany. Email: thai.nguyen@uni-ulm.de, mitja.stadje@uni-ulm.de

# Optimal investment for participating insurance contracts under VaR-Regulation††thanks: Institute of Insurance Science and Institute of Financial Mathematics, University of Ulm, Germany. Email: thai.nguyen@uni-ulm.de, mitja.stadje@uni-ulm.de

###### Abstract

This paper studies a VaR-regulated optimal portfolio problem of the equity holder of a participating life insurance contract. In a complete market setting the optimal solution is given explicitly for contracts with mortality risk using a martingale approach for constrained non-concave optimization problems. We show that regulatory VaR constraints for participating insurance contracts lead to more prudent investment than in the case of no regulation. This result is contrary to the situation where the insurer maximizes the utility of the total wealth of the company (without distinguishing between contributions of equity holders and policyholders), in which case a VaR constraint may induce the insurer to take excessive risks leading to higher losses than in the case of no regulation, see [3]. Furthermore, importantly for regulators we observe that for participating insurance contracts both relatively small or relatively large policyholder contributions yield rather risky and volatile strategies. Finally, we also discuss the regulatory effect of a portfolio insurance (PI), and analyze different choices for the parameters of the participating contract numerically.

JEL classification: C61, G11, G18, G31

Key words: Non-concave utility maximization, Value at Risk, optimal portfolio, portfolio insurance, risk management, Solvency II regulation

## 1 Introduction

This paper investigates the equity holder’s optimal investment problem of a participating contract under financial regulation. This problem is particularly relevant for insurance companies that operate under Solvency II with a Value-at-Risk (VaR) constraint. Participating contracts are life insurance products which provide the policyholder at least a guaranteed amount in downside market situations and a shared profit in good market scenarios. To participate in the contract, the policyholder pays a premium (participation fee) which is collected together with the (equity holder) insurer’s participation amount in an investment pool. At maturity, the policyholder receives a payoff which is linked to the investment performance. The equity holder’s payoff is determined as the residual amount.

Earlier studies on equity-linked life insurance contracts usually analyze pricing or optimal design problems. Some focus on the policyholder’s perspective assuming specific investment strategies like constant proportion portfolio insurance (CPPI) or generalized constant-mix (see, e.g., [9, 30, 31, 34]). Recently, [14] has incorporated taxes and a so-called fair pricing constraint in the policyholder’s problem.

In this paper, we consider two common contract designs. In the first design we assume that the policyholder is fully protected against an insolvency of the insurance company (i.e., the final payoff to the policyholder always exceeds or equals the guaranteed amount). In the second design, we assume that the equity holder has only limited liability, i.e., the policyholder’s payoff is less than the guaranteed amount in case of a default of the insurance company. Note that in the case of full protection, the equity holder may suffer a negative payoff in case of insolvency. To describe the behavior of the equity holder in the loss domain, we use an -shaped utility function adopted from prospect theory [37, 25, 29].

Risk management and regulations based on a terminal VaR constraint are well-known in banking and insurance regulations. Recall that VaR, defined as an estimate of the maximum portfolio loss given a pre-set significance level, is a quantile measure that controls the tail risk of the terminal portfolio. The problem of utility maximization/optimal asset allocation under VaR-type constraints has been studied extensively in the literature, see e.g. [3, 8, 16, 18, 15].

This paper solves the equity holder’s problem of utility maximization under a regulatory constraint imposed at maturity. We obtain closed-form solutions for various kinds of constraints. We first explicitly solve the problem for the two kinds of contracts mentioned above under a VaR regulation using the martingale approach for non-concave utility maximization problems with constraints. Second, motivated by the fact that regulators usually affect insurance contract designs by imposing a minimum capital requirement which is used to control adverse events, we also introduce a floor on the minimum guarantee rate. This portfolio insurance (PI) constraint enhances the protection for the policyholder especially when a defaultable put is included in the policyholder’s payoff.

Our theoretical and numerical results show that already in the case of no regulation there is a moral hazard problem since the insurer does not have an incentive to ensure that there is any capital in the loss states where the terminal wealth falls below the minimal guarantee. The reason is that any terminal wealth in those loss states only benefits the policyholder and comes at the expense of a lower terminal wealth in the more prosperous states where the equity holder receives a positive residual. On the other hand, introducing a VaR constraint as in Solvency 2 forces the equity holders to enlarge the proportion of hedged loss states, leading to a genuine improvement for the policyholders. This result is contrary to the situation where the insurer maximizes the utility of the total wealth of the company without distinguishing between equity holders and policyholders, in which case a VaR constraint may induce the insurer to take excessive risk leading to higher losses than in the case of no regulation, see [3]. This more prudent investment behavior described above is more pronounced if a VaR-based regulation is replaced by a PI-based regulation. Furthermore, the introduction of a full protection makes the equity holder’s investment also generally more prudent in bad market scenarios.

Our derivation of the optimal solutions relies on the combination of a martingale approach for non-concave and non-differentiable objective functions and a point-wise optimization technique with constraints. Note that the classical martingale method can not be directly applied for the equity’s problem as the derived utility function is neither concave nor differentiable. The problem of non-concave utility maximization without constraints has been considered by many authors e.g., [13, 36, 20, 12, 35, 7, 28, 4], using a concavification technique. In a more general framework without constraints, [35] proves the existence of an optimal terminal wealth. However, no specific payoff is provided.

When finishing this paper we noticed that [29] has independently investigated a similar problem for power utility function without regulations. Note that for a piecewise payoff structure, it is very challenging to deal with a VaR constraint because the bindingness should be included in the choice of the corresponding multiplier. Hence, the VaR-constrained problem is more complicated to solve. In this paper, we consider a more general utility function with (independent) mortality risk. To deal with the constrained non concave problems, we extend the technique developed in [15].

The paper is organized as follows: In Section 2, we introduce the asset model and the parametric family of contract payoffs. We then solve the unregulated problems without mortality in Section 3 in connection with the concavification technique. Section 4 discusses the case with independent mortality. In Section 5, we investigate the constrained problems. The results are numerically illustrated in Section 6. All technical proofs are reported in the Appendix.

## 2 The financial market and participating contracts

### 2.1 The financial market

Consider a complete financial market in continuous time without transaction costs that contains one traded risky asset and one risk free asset (the bank account). Let the asset price dynamics for the risky asset and the bank account be given by

 dSt=μtStdt+σtStdWt,S0>0;dBt=rtBtdt,B0=1, (2.1)

where is a standard Brownian motion on the probability space . We assume that and that and are deterministic and bounded processes. This implies that the stochastic differential equations for and have unique solution on . Denote by the filtration generated by the Brownian motion. For the moment, ignoring the insurance risk, we are in a complete market setting which implies the existence of a unique state price density

 dξt=−ξt(rtdt+θtdWt),ξ0=1, (2.2)

where is the market price of risk process. Here is the Arrow-Debreu value per probability (or likelihood) unit of a security which pays out $1 at time if the scenario happens, and else. As this value is high in a recession and low in prosperous times, has the nice property of directly reflecting the overall state of the economy. Therefore, the functional relationship between the optimal wealth and may be used as an interpretation of some of our results. This approached was also used in [33, 3, 8, 18, 15, 2, 23, 24, 21]. See also [17] for an elaborate explanation. We remark that in a consumption based pricing model in equilibrium corresponds to a constant times the marginal utility of consumption and is also called pricing kernel or stochastic discount factor. The insurance company chooses an investment strategy that we describe in terms of the amount (in$) invested in the risky asset at time . We assume that the remaining fraction of wealth is invested in the risk free asset to guarantee that the strategy is self-financing. The wealth process related to the strategy when starting with an initial wealth is then easily seen to satisfy

 dXt=(rtXt+πt(μt−rt))dt+πtσtdWt,X0>0. (2.3)
###### Definition 2.1.

A strategy is said admissible if it is adapted with respect to the natural filtration and . Furthermore, (the solution of SDE (2.3)) exists and a.s. for all . The set of all admissible strategies is denoted by .

In a complete market, it is known from the martingale method that choosing a portfolio is equivalent to choosing a terminal wealth which can be financed by . The set of admissible terminal wealth values is defined by

 X:={X∈L1(Ω,FT,P):X≥0andE[ξTX]≤X0}.

Hence, the dynamic problem is equivalent to the static one

 supXT∈XE[U(XT)],

where is some strictly increasing and differentiable concave utility function. This is the classical Merton problem considered by e.g. [32, 33]. The solution to this problem is also called the Merton solution. Let us first emphasize that the classical martingale method can not be directly applied when the utility function is neither concave nor differentiable. This issue will be addressed further in the sequel.

### 2.2 Participating contracts and payoffs

We assume that the representative policyholder invests in a single-premium equity-linked life insurance contract with a maturity of years with . At the initiation of the contract, the policyholder invests a lump sum ; the shareholder provides an initial equity . Consequently, the initial portfolio value is given by the sum of both contributions, i.e., . We denote by the share of the policyholder’s contribution (or equivalently the debt ratio of our insurance company). Below is also called the policyholder’s participation rate. At maturity and in case of solvency, the policyholder receives some guaranteed amount (we can, for example, choose , where is the guaranteed rate). If the terminal portfolio value exceeds the guaranteed rate, i.e., if , the surplus is shared between insurance company and policyholder. In this paper, we assume that the policyholder receives a surplus equal to 111 denotes the maximum ., where the surplus (bonus) rate is the percentage of surpluses that is credited to the policyholder. If the terminal portfolio value is less than the guarantee (default event), the policyholder receives the remaining amount. To exclude unrealistic cases, we assume throught the paper that and . To summarize, the policyholder receives the following payoff at time :

 VpL(XT):= {XTif XT≤LT,LT+δ[αXT−LT]+else\,, = LT+δ[αXT−LT]+−[LT−XT]+.

Hence, the policyholder takes a long position in the bonus option and a short position in a defaultable put and benefits from the potential upsides over the final maturity guarantee. This type of defaultable contracts is frequently used in the literature on insurance contracts, see, for example [10, 22, 5]. The equity holder always obtains the residual asset value

 VpE(XT):=XT−VL(XT)= ⎧⎪⎨⎪⎩0if XT≤LT,XT−LTif LT

where we introduced which is the threshold where the participation bonus kicks in. Agreeing on the contract, the insurer takes a long position of the call and short positions of the bonus call with strike price . For our analysis, we introduce

 ~δ:=αδ;ˆLT:=˜LT−LTandf(x)=(1−˜δ)x−(1−δ)LT. (2.5)

Hence, is the actual (achieved) bonus rate of the policyholder, is the difference between the bonus threshold and the guarantee, and is the payoff that the insurer receives in case that the wealth is greater than the bonus threshold . Like any profit-seeking company, the life insurance company sets up its investment mix to primarily maximize the benefits of its shareholder. Hereby, we assume that the shareholder values her benefits through a utility function defined on the positive real line, which is twice differentiable and satisfies the usual Inada and the asymptotic elasticity (AE) (see [27]) conditions

 limx↓0U′(x)=∞;limx↑∞U′(x)=0;limx↑∞xU′(x)U(x)<1. (2.6)

We assume furthermore that .222The case where is easier and can be treated without concavification procedure, see more in Remark 3. As usual, we denote by the inverse of the first derivative of the utility function . For optimal terminal wealth representations we introduce

 h(x):=I(x/(1−~δ))+(1−δ)LT1−~δ. (2.7)

Note that is a decreasing mapping from to .

### 2.3 Full protection and S-shaped utility function

In this subsection, we consider the case where the contract at maturity gives the policyholder at least some guaranteed amount (see, among many others, [10, 22]). More precisely, at time the policyholder receives the following payoff

 VnpL(XT):=LT+δ[αXT−LT]+,

and the equity holder takes the residual portfolio value

 VnpE(XT):=XT−VL(XT)={XT−LTif XT≤˜LT,XT−LT−δ[αXT−LT]+if XT>˜LT.

In this full protection case, the insurer takes more risk because she may have a negative payoff in the insolvency case where . Taking potential losses for the insolvency case into account we assume that the insurer evaluates the investment optimality using an -shaped utility (convex on the loss domain and concave on the gain domain) suggested by prospect theory [37] . In particular, the insurance company’s utility function takes the following form

 US(x):={−Ulo(−x)if% x<0,U(x)if x≥0,

where and are two utility functions. We assume furthermore that satisfies the Inada and (AE) conditions. For example, we can take , for some loss aversion degree , see [37]. Note that for such fully protected contracts, the guarantee level is considered as the reference point which is naturally used to distinguish gains and losses.

### 2.4 Contracts with mortality risk

Mortality is one of the most important risk factors in life insurance as it strongly affects the pricing and premium principles. Note that when non-tradable stochastic mortality is considered, the market becomes incomplete. For detailed discussion on general stochastic mortality models and their applications we refer for instance to [19, 11]. In this section we incorporate mortality risk into the optimal investment problem of the equity holder. For simplicity, we assume that the premature death of the policyholder is modelled by the event , where is a binomial random variable which is independent of the financial risk [1, 6] and the reference portfolio value . This assumption allows to employ a separating technique in our optimization problem.

First, consider the case where the policyholder receives full protection. We suppose that the policyholder receives the amount if she is alive at maturity. If she dies before maturity the guarantee will be paid at maturity to the policyholder’s relative (inheritress). So, the policyholder’s payoff is given by

Hence, the equity holder’s payoff will be given by the difference between the terminal portfolio value and the amount paid to the policyholder, i.e.,

 Vnp,dE(XT):=XT−Vnp,dL(XT)=VnpE(XT)1d=0+(XT−LT)1d=1.
###### Remark 1.

For the sake of simplicity, we have assumed that is a Bernoulli variable. This can be generalized to the case where for some and .

Recall that and are the payoffs defined in the absence of mortality risk in Section 2.3.

For the case of a defaultable contract, we still assume that the policyholder receives the amount if she is alive at maturity and in case of death the guarantee will be paid.333For a defaultable contract, an alternative assumption is that there is no guaranteed payment in case of premature death. In this situation there is no need to use an -shaped utility function as the insurer’s terminal wealth is always non-negative. Therefore, the policyholder’s payoff is given by

and

 Vp,dE(XT):=XT−Vp,dL(XT)=VpE(XT)1d=0+(XT−LT)1d=1.

### 2.5 Insurer’s optimization objective

The insurer wants to solve the following optimization problem

 supXT∈XE[˜US,j,d(XT)],where˜US,j,d:=US∘Vj,dE,j∈{p,np}. (2.8)

We observe that the payoffs and , admit a path-independent structure which allows to apply the martingale approach. When mortality is ignored we drop in the superscript and still denote the equity holder’s derived utility function by

 ˜US,j:=˜US∘VjE,j∈{p,np}.

We remark that for the defaultable contract without mortality the use of an -shaped utility is not needed, i.e., , and hence .

We furthermore assume the following integrability condition:

Assumption (U): For any , we have

 E[U((1−˜δ)−1I(λξT))]<∞andE[ξTI(λξT)]<∞.

It is straightforward to check that Assumption (U) holds for commonly used utility functions like power, logarithmic, exponential. This condition is needed to guarantee the existence of the Lagrangian multiplier .

Now, the independence of and reference portfolio allows us to write

 E[˜US,j,d(XT)]=(1−P(d=1))E[˜US,j(XT)]+P(d=1)E[US(XT−LT)].

Introducing , the optimization problem (2.8) can be rewritten as

 supXT∈XE[˜US,j,ϵ(XT)], (2.9)

where

 ˜US,j,ϵ(x):=(1−ϵ)˜US,j(x)+ϵUS(x−LT).

Note that is the survival probability 444If the policyholder’s age is then using the standard actuarial notations. and is an -shaped utility function for all . In particular,

 ˜US,j,ϵ(XT):=⎧⎪⎨⎪⎩−ϵjUlo(LT−XT)if XT≤LT,U(XT−LT)if LT˜LT, (2.10)

where and and is defined by

 Uϵ(x):=(1−ϵ)U(f(x))+ϵU(x−LT). (2.11)

Hence, is the utility of the equity holder in case of mortality if is greater than the bonus threshold. Furthermore, is a linear combination of and , weighted by and respectively. Clearly, is a strictly increasing and concave function on for all . Its first derivative is given by

 U′ϵ(x)=(1−ϵ)(1−˜δ)U′(f(x))+ϵU′(x−LT). (2.12)

Moreover, for any we have , which implies that

 U(f(x))≤Uϵ(x)≤U(x−LT),∀x≥˜LT.

To present the optimal terminal wealth we introduce the inverse marginal utility function . From (2.11) we observe that for all we have

 I(y/˜δϵ)+LT≥Iϵ(y)≥(1−~δ)−1[I(y/˜δϵ)−(1−δ)LT], (2.13)

where is defined by

 ˜δϵ:=(1−˜δ)(1−ϵ)+ϵ. (2.14)

Note that for the case without full protection coincides with on the gain domain . However, when the portfolio value falls below the liability level , the insurer now partially suffers a smaller loss measured by due to the presence of the defaultable put. The upper bound of loss is then given by

 qp:=ϵUlolo(LT)

## 3 Unconstrained problem and concavification

To discuss the solution of the general problem (2.8), we first study the simple case for the payoff (given in (2.2)) without mortality, namely

 supXT∈XE[U(VpE(XT))], (3.1)

where is a utility function satisfying condition (AE) and the integrability assumption (U). Note that the insurer’s payoff is positive almost surely and hence, the use of an -shaped utility function is not needed. Moreover, as mentioned above, the classical martingale method can not be directly applied for (3.1) because the derived utility function is neither concave nor differentiable. Motivated by non-linear compensation schemes for a fund manager, the maximizing the utility of terminal wealth for non-concave utility functions has been considered by many authors e.g., [13, 20, 12, 35, 7]. Similar characterisations can be found in [28, 31, 4] where non-linear contract payoffs or changing preferences are considered. Let us briefly summarize the basic ideas of the concavification method which relies on convex analysis.

Convex conjugate: Intuitively, the convex conjugate of can be related to a family of upper-half hyperplanes whose intersection equals to the region below the graph of . In particular, each member of this family is the smallest affine function of the form which dominate , i.e.,

 ˜Up(x)≤xy+c,∀x∈[0,∞).

Then, for a given slope , the corresponding conjugate of is the smallest constant being determined by

 (˜Up)∗(y):=supx≥0(˜Up(x)−xy).

Note that is not differentiable at the utility changing points and . Therefore, it is important to first compute the above supremum on each interval , and . The convex conjugate follows from a comparison among these three local maximums. Since is concave on each of these intervals the corresponding supremum defines a convex and decreasing function in . Therefore, can be seen as a decreasing, convex and differentiable function except on a finite number of points (which are explicitly determined below as the tangency point depending on our parameters). Classical results from convex analysis ensure that the double conjugate is the smallest concave function which dominates , and on an interval where the concavification is needed is linear [12, 35]. The optimal terminal wealth is then given555 stands for right derivative of by for some Lagrangian multiplier determined via the budget constraint. For more details, see [13, 20, 12, 35, 7].

###### Remark 2.

We will see later in Section 5 that when a VaR constraint is additionally imposed the optimal solution can not be directly derived from the right derivative of the convex conjugate . As mentioned above, may be linear in some interval which means that the concave hull of the utility function is neither strictly concave nor smooth, and the results in [3] for concave optimization under a VaR constraint cannot be applied.

To obtain explicit solutions, below we use the Lagrangian approach to determine the optimal solution and point out the links to the concavification points of the derived utility function which is determined below.

### 3.1 Concavification

In this section, we determine the concavification of the derived utility function defined in (2.11). Recall that is a strictly increasing and concave function on for all with first derivative given by (2.12). Furthermore, let and define

 Υϵ,q(x):=Uϵ(x)−xU′ϵ(x)+q. (3.2)

The parameter represents the upper bound of the losses in case of an -shaped utility function and is used to deal with negative wealth. In particular, we set in the case of full protection and in the case where default put and death probability are considered simultaneously, see Section 4 below.

Concavification of the derived utility function crucially depends on the sign of at the utility changing point . In particular, the concavification area depends on and , see the Lemma below. From (3.2) we easily observe that

 Υϵ,q(˜LT)=U(ˆLT)−˜δϵU′(ˆLT)˜LT+q≥U(ˆLT)−U′(ˆLT)˜LT+q=Υ1,q(˜LT),

where is defined by (2.14). The concave hull of the derived utility function is then characterized in the following lemma.

###### Lemma 3.1.

Let , and defined as in (3.2). If then there exists a positive number satisfying , i.e.,

 U(ˆy1,q−LT)−U′(ˆy1,q−LT)ˆy1,q+q\lx@stackrel!=0. (3.3)

If then there exists a positive number satisfying , i.e.,

 Uϵ(ˆyϵ,q)−ˆyϵ,qU′ϵ(ˆyϵ,q)+q\lx@stackrel!=0. (3.4)

In fact, is the tangency point of the straight line starting from the point to the curve in the first case and is the tangency point of the straight line starting from to the curve in the second case.

Proof. We prove the first property. To this end, note that the left hand side of (3.3) is an increasing, continuous function in due to the concavity of . By Inada’s condition, it takes values in and the conclusion follows from the intermediate value theorem. The second statement can be proved in the same way using the asymptotic elasticity condition of .

Below we solve the optimization problem (3.1) using a Lagrangian approach. The optimal terminal wealth will be expressed as a function of the price density price and a Lagrangian multiplier defined via the budget equality.

### 3.2 Unregulated optimal wealth for VpL without mortality

In this section, we ignore mortality risk and assume that the equity holder’s payoff is defined by (2.2). In this context, we take in (3.2) and . Then, the concave hull of the derived utility function intrinsically depends on how to determine the tangent line to the curve starting from the origin. In particular, from Lemma 3.1, if the concave hull is linear in and coincides with the curves and in and in respectively. In this case, the utility changing points and play an essential role in the optimal terminal wealth. For the case that , the concave hull is linear in and coincides with the curve in . In this case, only the tangency point matters for the optimal terminal wealth.

###### Theorem 3.1.

The optimal solution to the unconstrained problem (3.1) is given by

where

 ξ0˜L:=(1−˜δ)U′(ˆLT)λ,ξ0U:=U(ˆLT)˜LTλ,ˆξ1,0:=U′(ˆy1,0−LT)λ,ˆξ0,0:=(1−˜δ)U′(f(ˆy0,0))λ.

The Lagrangian multiplier is defined via the budget constraint .

Proof. See Section A.1.

Let us give some comments on Theorem 3.1. First, when or equivalently, the concavification point of lies on the interval , the optimal terminal wealth takes a four-region form determined by the utility changing points and . For or equivalently, for the concavification point being in the interval , the optimal terminal wealth takes a two-region form and is determined by . When , concavification is needed on the interval . The concave hull is then equal to the linear segment connecting zero with and coincides with on . In this case the optimal terminal wealth takes a three-region form. In all cases, the optimal terminal wealth ends up with zero from a certain value of the price density on, i.e., in the worst economic states. This reflects the moral hazard problem that the insurer does not have an incentive to ensure that there is any capital in case the terminal wealth falls below the minimal guarantee. The reason is that any terminal wealth in those loss states only benefits the policyholder and comes at the expense of a lower terminal wealth in the more prosperous states. We will later see that introducing a VaR constraint ameliorates this situation.

Optimal policy of wealth distribution: Like the classical concave utility maximization problem, the optimal terminal wealth is given as a function of the state price density at maturity and the Lagrangian multiplier which is determined via the budget equation. In particular, we observe from Theorem 3.1 that for , the four-region form solution is characterised by wealth levels defined by corresponding to the partition of the terminal market states with boundary points . In other words, the wealth level at time is respectively assigned to on the sub-interval , to on , to on and finally to zero on . Therefore, it is convenient to represent the terminal wealth in dependence of the state price as

For simplicity, below we drop the dependence of the terminal wealth on the Lagrangian multiplier and the price density. In the same spirit as above, the optimal wealth for the case and the case can be respectively represented by the three-region and the two-region wealth distributions as

These representations of wealth distribution will be used in the rest of the paper for notational convenience.

###### Remark 3.

Concavification is not needed for the case where . In this situation, it can be deduced directly from the proof of Theorem 3.1 in Appendix A.1 that the optimal terminal wealth is given by

### 3.3 Full protection and S-shaped utility functions without mortality

We now turn our attention to the case where the policyholder receives a full protection as discussed in Section 2.3 without mortality (i.e., ). Noting that the insurer’s payoff may take negative values, we consider the following unconstrained optimization problem with an -shaped utility function

 supXT∈XE[˜US,np(XT)],where˜US,np:=US∘VnpE. (3.5)

We assume that is convex on and piecewise concave on each interval and . Losses are bounded by . The concave hull of the derived utility function is characterized by Lemma 3.1 which determines the tangent line to the curve starting from the with in this context.

###### Remark 4.

We observe that for any , , which implies that whenever they both exist. Similarly, whenever they both exist. This means that the concavification area for the full protection case using an -shaped utility function will be reduced in comparison to the case of a defaultable contract, assuming that the insurer uses the same concave utility function for the gain part.

Having discussed the concave hull of the -shaped utility function we are in a position to present the optimal solution to problem (3.5).

###### Theorem 3.2.

The optimal solution to the -shaped utility unconstrained problem (3.5) is given by

where

The Lagrangian multiplier is defined via the budget constraint .

Proof. See Appendix A.1.

We observe from Theorem 3.2 that the optimal terminal wealth takes an all-or-nothing form. Hence, giving a full protection to the policyholder, the insurance company needs to take the possibility of having negative terminal wealth into account. In line with descriptive decision theory, we assume a convex utility function for the loss area . As numerically shown in Figure 6, the insurer is induced to take a more prudent investment strategy. Mathematically, this follows from and provided they exist, see Remark 4. Now, using the budget constraint we conclude that the Lagrangian multiplier of the -shaped utility problem is greater than the one in the case with a default put. This means that an -shaped utility function leads to a shift-to-the-right effect on the optimal terminal wealth.

###### Remark 5.

We observe that for (i.e., the policyholder has no initial contribution) with a defaultable contract, the problem becomes simpler with a call-option-form payoff . When (i.e., the policyholder fully contributes to the investment pool), the insurer’s payoff also takes a call-option-form payoff . As numerically illustrated in Figure 4, these cases lead to a riskier investment, see more in [13]. For a fully protected contract it can also be observed that the insurer’s payoff is given by if and for and for when . Note that these situations are not interesting in practice as the contract would in each case not be acceptable to one of the two respective sides.

### 3.4 The optimal strategy

We have determined the optimal terminal wealth as a function of and a multiplier that satisfies the budget constraint. For the reader’s convenience, we briefly discuss how to deduce the optimal strategy by applying the martingale representation theorem. Let be the optimal terminal wealth and define . Then, the process is a martingale under . Therefore, by the martingale representation theorem, there exists a square integrable adapted process such that

 ξtZt=X0+∫t0ςsdWs,t∈[0,T],

or equivalently, . On the other hand, applying Itô’s lemma we get

 dZt=d(ξtZtξ−1t) =ξ−1td(ξtZt)+ξtZtdξ−1t+d(ξtZt)dξ−1t

Identifying the last equation with (2.3) we deduce that the optimal strategy is given by

 π∗t=σ−1tξ−1tςt+σ−1tZtθt,X∗t=Zt.

Note that is square integrable and is hence admissible.

The same argument can be applied for the constrained problems below. More explicit representations can be provided for power and logarithmic utility functions.

## 4 Optimal wealth with mortality risk

We now consider Problem (2.8) with mortality, i.e., . Recall that for both payoffs, the derived utility function defined in (2.10) takes an -shaped form. The concave hull of the derived utility function , related to the existence of a tangency line starting from , is now characterized by Lemma 3.1 with , being defined in (2.15). The optimal terminal wealth of problem (2.9) is characterised by the following theorem.

###### Theorem 4.1.

For any , the optimal terminal wealth of Problem (2.9) is given by

where

 (4.1)

The Lagrangian multiplier is defined via the budget constraint.

Proof. See Appendix A.1.

We remark that the no-mortality case is already given in Theorem 3.2. Let us now consider the case where , i.e., the policyholder dies almost surely before the maturity of the contract. In this case, assuming that the insurer uses an -shaped utility function

 US,0(XT):={−Ulo(LT−XT)if XT≤LT,U(XT−LT)if LT

we obtain the following optimal solution.

###### Corollary 4.1.

Suppose that . The optimal terminal wealth (with and without default put) is given by , where satisfies the budget constraint.

## 5 Optimal investment under regulations

### 5.1 The VaR-constrained problem

As seen in the previous section, under the absence of regulation the equity holder will optimally choose a strategy which may lead to insolvency at maturity. In this case, the policyholder suffers severe losses since the terminal portfolio value may be zero for very bad market scenarios. In practice, an appropriate investment must take some regulatory constraints into account. According to Solvency II, the insurance company needs to ensure that a VaR-type regulation (i.e., a default probability constraint) shall be satisfied. In addition, the insurance company is interested in achieving at least a target payment to serve the promised guaranteed amount to the policyholder. Therefore, the equity holder (the insurance company) has to choose an optimal dynamic portfolio under a VaR constraint, which can be stated as

 supXT∈XE[˜US,j,ϵ(XT)],s.t.P(XT

for some probability default level . It is clear that the introduction of a probabilistic constraint complicates the non-concave problem. In the spirit of the Lagrange method, it is natural to consider the following auxiliary utility function

 ˜US,j,ϵλ2(x):=˜US,j,ϵλ2(x)−λ21x

where is a multiplier. Problem (5.1) boils down to maximize over the set . However, it is not clear at first sight how to link the optimality of the auxiliary utility function with the optimality of the initial utility function under the VaR constraint as depends on , and different will lead to different optimal solutions. However, a good choice for should reflect the bindingness of VaR constraint. Below we show that this goal can be achieved using a special form of which follows from a careful comparison of the local maximizers in the unconstrained problem.

###### Theorem 5.1.

Let and define so that . Suppose that . Then, the VaR-constrained problem (5.1), , admits the following optimal solution:

• If then

• If then

• If then

In each case, the Lagrangian multiplier is defined via the budget constraint.

Proof. See Section A.3.

We observe first that when // in the first/second/third case, the VaR constrained is not binding and the corresponding unconstrained solution given by (4.1) is still optimal for (5.1). On the other hand, when the VaR constraint is binding (active) the terminal wealth will be (partially) shifted to the right of the concavification point and we can observe that for , there exists such that

 XVaR,j,ϵ,∗T(ξT)≥Xj,ϵ,∗T(ξT)% for allξT≥ξ∗j,

meaning that the VaR-terminal wealth dominates the unconstrained terminal wealth for the most negative loss states (due to the fact that and are decreasing in ). Hence, introducing a VaR-constraint, forces the equity holders to enlarge the proportion of hedged loss states, leading to a genuine improvement for the policyholders. This result is contrary to the situation where the insurer maximizes the utility of the total wealth of the company without distinguishing between equity holders and policyholders, in which case a VaR constraint may induce the insurer to take excessive risk leading to higher losses than in the case of no-regulation, see [3]. However, there is still a region of market scenarios in which the optimal terminal wealth equals zero, which means that a VaR regulation does not lead to a full prevention of moral hazard. The intuitive reason is that under a VaR regulation, the equity holder is only required to keep the portfolio value above with a given probability . Once the regulation is probabilistically fulfilled the equity holder can push the remaining risk into the tail to seek a higher potential wealth level in good market states. This is consistent with the classical VaR-constrained asset allocation problem with concave utility functions [3]. However, in our case with participating contracts, the use of a VaR constraint does not lead the insurance company to bigger losses than in the case of no regulations as in the classical VaR problem. On the contrary, our results show that for an insurance company with equity holders, a VaR constraint strictly improves the risk management for the loss states.

### 5.2 The PI-constrained problem

In this section, we try to better protect the policyholder from the equity holder’s gambling investment strategies by, instead of having a VaR constraint, assuming that the equity holder has to keep the portfolio value almost surely above some given level minimum capital requirement . Hence the insurance company needs to solve

 supXT∈XE[˜US,j,ϵ(XT)],