How Profit Participation Shapes Risk and Returns in Traditional Life Insurance

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Life insurance with profit participation

Traditional life insurance with profit participation is characterized by a strong interdependence between assets and liabilities. On the liability side, the policyholder contracts are equipped with a minimum guarantee rate and participate in the company’s profit. This profit depends on the book value return of the assets under management. The company’s management may control -to a certain extent- the book value return and the crediting strategy, and this control typically depends on information from liabilities (e.g., guaranteed interest rate) and assets (e.g., amount of unrealized gains). The purpose of this section is to describe these allocation processes and the corresponding control parameters in a mathematically concise manner.

2.1. The portfolio view. The liabilities of a traditional life insurance consist of two items, namely the life provisions, LP, and the surplus fund, SF. Thus the book value of liabilities is BV = LP + SF. In this picture we have assumed that there is no free capital. The life provisions are attributed to individual contracts while the surplus fund belongs to the collective of policyholders as a whole. The former can be further split into a mathematical reserve, V , and declared bonuses (or benefits), DB, such that LP = V + DB. Thus V denotes the sum of all contracts’ mathematical reserves calculated according to actuarial principles and DB is likewise the sum of all declared benefits.

Let A be the set of all assets which are dedicated to the profit participating business. We denote book and market values associated to a particular asset a by BV a and MV a, respectively. The unrealized gains are by definition (2.17) UGa = MV a − BV a and these can be positive or negative. In the latter case we will also refer to UG as unrealized losses. Balance sheet parity dictates that we must have (2.18) BV = X a∈A BV a. In reality, the right hand side of this equation would be expected to be slightly larger than the left hand side. However, since profit participation is calculated only with respect to those assets balancing liabilities, exact balance sheet parity should be assumed.

Let from now on t = 0, . . . , T denote yearly time steps where T is the projection horizon, and T will correspond to 60 years or even more since life insurance is a long term business. For a time dependent quantity ft we denote the increment by ∆ft = ft − ft−1. The return on assets generated by the portfolio at time t is (2.19) ROAt = X a∈At−1 ∆BV a t = X a∈At−1 (BV a t − BV a t−1 ) where At−1 are all assets under management at the previous time step. Notice that ROA is defined in terms of book values. This terminology requires some explanation: Consider an asset a ∈ At−1 with book value BV a t−1 and unrealized gains UGa t−1 = MV a t−1 − BV a t−1 . Assume these values change due to some mechanism (e.g., market movements or depreciation rules) at t to (BV a t ) ′ and (UGa t ) ′ .

Then the uncontrolled return due to a is (BV a t ) ′ − BV a t−1 . However, management may decide to sell a at t and thereby realize the positive or negative value (UGa t ) ′ . In fact, we will allow partial selling such that management may choose u a t ∈ [0, 1] at t, whereby the final book value becomes BV a t = (BV a t ) ′ + u a t (UGa t ) ′ . Similarly, the unrealized gains at t can now be expressed as UGa t = (1 − u a t )(UGa t ) ′ . The corresponding return is thus ROAa t = ∆BV a t = (BV a t ) ′ + u a t (UGa t ) ′ − BV a t−1 , and this depends on the changes in the book value and in the unrealized gains, and on the applied management rule. The life provisions, LPt, can be decomposed as the sum LPt = Vt + DB0 t + DBt: here Vt denotes the sum of mathematical reserves at t, DB0 t are declared bonuses at t which were declared before and up to valuation time t = 0, and DBt are declared bonuses at t which were declared at times 1 ≤ s ≤ t − 1.

The mathematical reserve of a contract is the reserve that is calculated according to classical actuarial principles including safety loadings ([15]). Since DB0 t is guaranteed at time 0 we denote the guaranteed life provisions by LP Gt = Vt + DB0 t . The declared bonuses, DBt, are increased by declarations, ph∗ t , and decreased due to benefits, pht, and surrender gains, sg∗ t . That is, (2.20) ∆DBt = ph∗ t − pht − sg∗ t . The company experiences the following cash flows on the liability side: premiums, benefits and expenses. Premiums paid at t are denoted by prt. Benefits are split into guaranteed benefits, gbft, stemming from LP Gt−1 and discretionary benefits, pht, stemming from DBt−1. That is, the total benefit cash flow at t is bft = gbft+pht. Moreover, the guaranteed benefits can be further decomposed as gbft = abft − ∆DB0 t where abft denotes the benefit calculated according to actuarial principles ([15]) and ∆DB0 t is the reduction of DB0 t .

Note that the latter can only decrease since new profits are declared to DBt by construction. Finally, the sum of expense cash flows at t is denoted by expt. The gross surplus at t, which corresponds to the company’s profit under locally generally accepted accounting principles (local GAAP), is now defined as (2.21) gst = ROAt − ∆LP Gt + prt − gbft − expt + sg∗ t = ROAt − ∆Vt + prt − abft − expt + sg∗ t . Let ρt be the weighted average technical interest rate at t. That is, ρt = P c∈Lt−1 ρcV c t−1/Vt−1 where Lt−1 denotes the liability portfolio at t − 1 and c ∈ Lt−1 is a contract with guarantee rate ρc and mathematical reserve V c t−1.

If the premium were calculated without any safety loadings, we would have ∆V c t − prc t + abf c t + expc t = ρ cV c t−1 . Thus the difference ρtVt−1 − ∆Vt + prt − abft − expt + sg∗ t =: γtVt−1 can be viewed as a technical gain due to factor loadings, and we take this equation as the definition for the technical gains rate (2.22) γt = ρtVt−1 − ∆Vt + prt − abft − expt + sg∗ t /Vt−1 The gross surplus equation can now be expressed succinctly as (2.23) gst = ROAt − (ρt − γt)Vt−1. If the gross surplus is negative, the shareholder has to cover this and incurs a cost of guarantee. Thus we define cogt = gs− t where k − = − min(k, 0) is the negative part of a real number. If the gross surplus is positive, this profit is shared between policyholder, shareholder and tax office according to positive factors gph, gsh and gtax satisfying gph + gsh + gtax = 1. Let k + = max(k, 0) be the positive part of a real number.

Thus, at t, the shareholder receives a cash flow shgt = gsh · gs+ t representing shareholder gains, and the tax office receives taxt = gtax · gs+ t . However, the policyholder collective need not be credited the full amount gph · gs+ t at t. Rather, management may choose a control parameter νt ∈ [0, 1] such that νt · gph · gs+ t is credited at time t and then paid out at later times s > t when the corresponding policies which were credited at t reach their maturities. The remaining fraction (1 − νt) · gph · gs+ t is allocated to the surplus fund SF. Conversely, the declaration may be, again at the management’s discretion, augmented by choosing ηt ∈ [0, 1] and declaring the amount ηt ·SFt−1. The total declaration at time t is therefore (2.24) ph∗ t = νt · gph · gs+ t + ηt · SFt−1. We define µ t s as the fraction of the declaration ph∗ t that is paid out at s > t. Notice that µ t s is a model dependent quantity. It follows that ph∗ t gives rise to delayed cash flows µ t t+1ph∗ t , . . . , µt T ph∗ t and that the policyholder cash flow can in turn be expressed as (2.25) pht = Xt−1 s=1 µ s tph∗ s . A schematic overview of the declaration procedure, including the management controls which are denoted by u, ν and η, is contained in the diagram in Figure 1.

2.2. Evolution of policyholder profits. The evolution equation (2.20) can be adjoined by one for the surplus fund, namely (2.26) ∆SFt = (1 − νt)gph · gs+ t − ηtSFt−1 = gph · gs+ t − ph∗ t . The sum DBt + SFt represents the pot of shared profits incurred up to time t. Its evolution can be expressed as (2.27) ∆(DBt + SFt) = gph · gs+ t − pht − sg∗ t which, crucially, holds independently of declaration management rules regarding ν and η. Let MVt = P a∈At MV a t denote the market value of assets at t. Further, let B −1 t denote the stochastic discount factor associated to a risk neutral interest rate model, and let E[ ] be the expectation associated to the risk neutral measure. In Section 3 this measure is fixed as the spot measure associated to the mean-field Libor market model of Section 1 but for the present purpose the specific choice is not relevant. The value of guaranteed benefits is defined as (2.28) GB = E hX T t=1 B −1 t (gbft + expt − prt) i and the value of future discretionary benefits is (2.29) F DB = E hX T t=1 B −1 t pht i . Further, we consider the value of shareholder gains, SHG = E[ PT t=1 B −1 t shgt], the shareholder’s cost of guarantee (2.30) COG = E[ X T t=1 B −1 t cogt], and the value of tax payments, T AX = E[ PT t=1 B −1 t taxt]. The no-leakage principle ([19]) states that (2.31) MV0 = GB + F DB + SHG − COG + T AX + E[B −1 T MVT ].

2.3. Balance sheet representation of future discretionary benefits. Putting the above together yields a representation which can be stated without any cash flows: Theorem 2.1 ([14]). (2.32) F DB = F DB0 + gph · COG − I − II − III where F DB0 = SF0 + gph LP0 + UG0 − GB I := E h B −1 T DBT + SFT + gph(UGT + VT + DB0 T ) i II := (1 − gph)E hX T t=2 B −1 t sg∗ t i III := (1 − gph)E hX T t=1 Ft−1B −1 t (DBt−1 + SFt−1) i

2.4. Optimal control. The management’s objective is, according to corporate practice, to maximize the shareholder value of in-force business, V IF = SHG − COG. In view of the no-leakage principle (2.31) this means that management rules should be designed so as to minimize F DB + T AX + E[B −1 T MVT ] −→ MIN since MV0 and GB cannot be altered by management rules (assuming independence of guaranteed benefits and profit sharing, i.e. static policyholder behaviour).

Focusing on the F DB-part in the minimization problem, if we choose νt = ηt = 0 for all t = 1, . . . , T it follows that ph∗ t = 0 and the future discretionary benefits are minimized as F DB = 0. Clearly, this strategy is unrealistic. Thus we have to reformulate the control problem F DB + T AX + E[B −1 T MVT ] −→ MIN subject to certain constraints which reflect the management’s goal of providing competitive declared bonuses to the policyholders. Typical constraints to this effect are:

(1) There is a fixed θ > 0 such that SFt ≤ θLPt. This is a run-off assumption ensuring that relation between different balance sheet items remains realistic throughout the projection and it automatically reduces the terminal value E[B −1 T MVT ].

(2) The profit share ph∗ t aims to follow a target reflecting the policyholders’ expectations. This target could be defined, e.g., in terms of a combination of the prevailing yield curve at t to reflect the expectation of a financially rational policyholder, and of the previous profit sharing rate to avoid large jumps in the profit sharing rate.

A concrete implementation of such a set of rules is presented in Section 3.5. Another way to view this control problem goes as follows. The numbers µ s t defined in (2.25) allow us to express the future discretionary benefits as F DB = E hX T t=2 B −1 t Xt−1 s=1 µ s t ph∗ s i = E h T X−1 t=1 ph∗ t qt i where qt := E[ PT s=t+1 B−1 s µ t s |Ft]. Notice that qt ≥ 0 and ph∗ t ≥ 0 by construction. This formulation allows to directly discern the impact of ph∗ t on the F DB-minimization. If we assume that the strategic asset allocation (SAA) is not a control parameter but a constraint, namely that the SAA should be kept approximately constant, the remaining actuated variables are ν and η in (2.24) and the realizations of unrealized gains in the calculation of ROA.

The latter can be expressed as ut = (u a t ) where a ∈ At runs over all assets held at t and u a t ∈ [0, 1] controls the fraction of a to be sold at t. Let tat denote an appropriate target at t, defined e.g. as in item (2) above. Thus we may rephrase the control problem as the search for a strategy (νt, ηt, ut) such that (2.33) E h T X−1 t=1 ph∗ t qt + ω(ph∗ t − tat) 2 i −→ MIN where ω is a weighting factor representing the target’s importance and ph∗ t = ph∗ t (νt, ηt, ut) = νt gph ROA′ t + X a∈At u a t UGa t − (ρt−1 − γt−1)Vt−1 + + ηt SFt−1, and the minimization is subject to the constraints SFt ≤ θLPt with a fixed number θ > 0 and that the SAA is kept (approximately) constant. Here, ROA′ t = BV ′ t − BVt−1 denotes the change in the portfolio’s book value before application of management rules (i.e., realizations of unrealized gains).

In practice, the admissibility of controls u a is often restricted further: while it is advantageous from the shareholder’s perspective to realize unrealized losses in cases where ph∗ t > tat, realizations of positive unrealized gains in cases where ph∗ t < tat might be prohibited to avoid future cost of guarantee payments. Indeed, when the gross surplus is negative it is certainly desirable, again from the shareholder’s point of view, to realize positive unrealized gains in order to avoid capital injections.

Authors:

FLORIAN GACH

SIMON HOCHGERNER

EVA KIENBACHER

GABRIEL SCHACHINGER

This paper is available on arviv under CC by 4.0 Deed (Attribution 4.0 International) license