How Numerical ALM Models Value Future Discretionary Benefits in Life Insurance

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Numerical ALM modeling

To calculate the present value, F DB, of future discretionary benefits in traditional life insurance, it is necessary to model the statutory balance sheet and to carry out a Monte-Carlo simulation along economic scenarios. Consider yearly time steps t = 1, . . . , T where T corresponds to the projection horizon. The cash-flows, at time t, relevant for market consistent valuation (best estimate calculation according to Solvency II) are premiums, prt, expenses, expt, guaranteed policyholder benefits, gbft, and discretionary policyholder benefits, pht, cf. Section 2. The best estimate, BE, is given as the sum of guaranteed benefits and future discretionary benefits defined in (2.28) and (2.29), that is (3.34) BE = GB + F DB.

This section contains a description of the Asset Liability Management (ALM) model which we have implemented in the programming language R in order to numerically calculate GB and F DB. The algorithm for F DB is quite involved due to the interaction of asset module (Section 3.3), liability module (Section 3.4) and management rules (Section 3.5). The general mechanism behind profit participation in traditional life insurance is described in Section 2 and sketched at the level of model points in Section 3.1.

3.1. Contracts and model points. Our ALM model is concerned with classical life insurance contracts. Each contract (or, more generally, model point), x, has a specified maturity payment, Mx , which is guaranteed and may participate in the company’s earnings. More precisely, each contract x

• has a specific maturity time T x ;

• has associated best estimate mortality and surrender tables;

• has a constant technical interest rate ρ x ; • pays a constant premium prx up to T x − 1;

• has an associated mathematical reserve V x t calculated according to classical actuarial assumptions involving, e.g., ρ x ;

• has an associated bonus account of total declared benefits T DBx t depending on the company’s profits;

• receives either a maturity benefit Mx +T DBx T x−1 at Tx; or a death benefit Mx +T DBx t−1 at t < Tx; or, in case of surrender at t < Tx, a surrender benefit κt(V x t−1 + T DBx t−1 ) where κt is a penalty term which is linear in t such that κ0 = 0.9 and κT x = 1.

When we model the projection of a model point x which has started before valuation time t = 0, it is necessary to distinguish between past and (uncertain) future. Thus we denote by (DBx ) 0 0 those declared benefits associated to x which have been declared in the past, i.e. prior to valuation time t = 0. The modelled projection at t of this account is denoted by (DBx ) 0 t . Similarly, future (a priori, uncertain) declared benefits are denoted by DBx t with DBx 0 = 0. This leads to the decomposition T DBx t = (DBx ) 0 t + DBx t of the total account.

3.2. Economic Scenario Generator (ESG). The proposed ALM model uses the mean-field Libor market model of Section 1 to generate market and book values for four different asset classes: cash, bonds (without default), equity and property. Cash is modelled as a bond with maturity of one year. This corresponds to the roll-over definition of the implied money market account. Market values of property and equity are assumed to follow a geometric Brownian motion where the drift depends on the prevailing one-year forward interest rate, Ft. Further, the drift is modelled as a fixed rate, d, representing rental income or dividend yield. Finally, each property or equity asset may have its own fixed volatility, σ. Hence the market value, MV a t of a given property or equity asset is assumed to be projected according to

(3.35) MV a t = MV a 0 exp (Ft − d − σ 2 /2)t + σWa t

where MV a 0 is the initial market value and Wa t is the asset’s Brownian motion whose correlation with stochastic drivers of other assets remains to be specified. Equity, property and bond portfolio are assumed to be independent in our model. We emphasize that d and σ are assumed to be fixed numbers while Ft is realized as a numerical implementation of the MF-LMM. The numerical implementation we have chosen for this task is a superposition of the mean-field taming [7, Section 4.3] and the anti-correlation approach [7, Section 4.5]; this approach is chosen since it allows flexibility in reproducing cap and swaption prices while at the same time effectively reduces the probability of blow-up. Let Ψm be defined by (1.15). Let v0 > 0 denote a predefined variance threshold and f ≥ 1 a stretching of this threshold.

Thus when the variance exceeds the threshold, Ψm > v0, the dynamic enters a new regime (anti-correlation), and when Ψm > fv0 the dynamic is supposed to be further affected by a volatility dampening factor. We consider the evolution equation: λ m t, Ψ m t = S(v0 − Ψ m t ) · rm(t) + (1 − S(v0 − Ψ m t )) · exp − max(Ψm t /(fv0) − 1, 0) rm(t) (3.36) en, if m = 2n − 1(3.37) λ m t, Ψ m t = S(v0 − Ψ m t ) · rm(t) − (1 − S(v0 − Ψ m t )) · exp − max(Ψm t /(fv0) − 1, 0) rm(t) (3.38) en, (3.39) if m = 2n rm(t) = a(tm−1 − t) + d e −b(tm−1−t) + c cos θm sin θm (3.40) , t ≤ tm−1 , where S is a sigmoid function with inflection point 0 such that limx→∞ S(x) = 1 and limx→−∞ S(x) = 0. Numerically, this is approximated by a switch at 0. Further, n = 1, . . . , N/2 (we assume that N is even) and en is the n-th standard basis vector. In this case the dimension of the Brownian increment in (1.16), i.e. in (1.2), is d = N.

The parameters which are used to calibrate the model are the so-called angles θm, the Rebonato ([25]) parameters a, b, c, d, and the mean-field parameters f ≥ 1 and v0 > 0. The interest rate scenarios are generated by applying an Euler-Maruyama scheme to the approximating interacting particle system (IPS), and our ALM model uses 5000 such scenarios. Notice that the use of (3.36) and the associated Monte-Carlo simulation of the IPS are justified by Theorem 1.4 since the volatility structure is of the form (1.3) and the map ψ 7→ exp(− max(ψ/(fv0) − 1, 0)) clearly satisfies the point-wise condition (1.11) for all choices f ≥ 1 and v0 > 0; it suffices that the derivative of this map exists and is asymptotically o(1/ψ) as ψ → ∞ but it does not need to be continuous.

3.3. Asset module. The purpose of the asset module is to model book and market values of assets under management and to provide re- and deinvestment strategies. The latter are part of the management rules and described in Section 3.5. The ALM model provides four different asset classes: cash, bonds, equity, and property. Bonds are assumed to be default-free, thus there is no distinction between corporate and government bonds. Cash is equivalent to a bond with a maturity of 1, and correspondingly the interest earned by cash is the prevailing one-year forward rate. Further, book and market values, BV c and MV c , for cash coincide.

That is, BV c t = MV c t = (1 + Ft−1)MV c t−1 . A bond, b, consists of a maturity T b , a nominal payment Nb , a coupon factor Kb , such that the coupon payment is KbNb , a market value MV b and a book value BV b . At each time step t < Tb the market value is determined by the prevailing yield curve, i.e. MV b t = T Xb s=t+1 P(t, s)KbN b + P(t, Tb)N b where P(t, s) is the value of the zero-coupon bond from the mean-field Libor market ESG. The book value is determined by the strict lower-of-cost-or-market (LCM) principle, that is BV b t = min BV b t−1 , MV b t . When a bond b with nominal Nb and maturity T b is bought at time t, the initial book value is BV b t = Nb , and the coupon factor Kb follows from the requirement MV b t = Nb and the prevailing yield curve at t up to T b . I.e., bonds are bought at par. An equity position, e, consists of a market value, a book value, a (constant) volatility factor and a (constant) dividend factor.

The latter is relevant for the company’s surplus which is calculated according to local generally accepted accounting principles (local GAAP), since the dividend affects the book value return. The market value development is given by the geometric Brownian motion (3.35). The book value is given by the strict LCM principle, that is BV e t = min(BV e t−1 , MV e t ). Properties are modelled similar to equities with two distinctions: The dividend factor is interpreted not as a dividend but as rental income.

Second, properties, p, are split into building and land value, BV p = BV bu+BV lv , and the building value has a depreciation time, T p , such that BV bu T p = 0. This depreciation time (which is usually not more than 30 years) has to be provided as part of the initial data. The depreciation is linear, i.e. according to (1−(s−t+ 1)(T p −t+ 1))BV bu t−1 for t ≤ s ≤ T p , and the strict LCM applies. Hence the book value development is given by BV p t = BV bu t + BV lv t BV bu t = min (1 − 1 T p − t + 1 )BV bu t−1 , MV bu t BV lv t = min BV lv t−1 , MV lv t where MV bu t and MV lv t follow (3.35) along the same Brownian path. Consequently properties often carry comparatively large amounts of unrealized gains UGp t = MV p t − BV p t .

3.4. Liability module. Let Lt denote the book value of liabilities at time t and assume that Lt = LPt + SFt is a sum of two items: Firstly, the life assurance provision, LPt = Vt + DB0 t + DBt; here Vt is the mathematical reserve at time t which depends only on the survival rates of policyholders but not on future surplus declarations; the term DB0 t contains discretionary benefits that have been credited before valuation time t = 0 and depends only on the survival rates of policyholders but not on future surplus declarations; and the term DBt contains discretionary benefits that have been credited at times 1 ≤ s < t and have not yet been paid out.

Secondly, the surplus fund, SFt, which consists of those profits that have not yet been declared to policyholders, cf. diagram (1) for a schematic overview. This set-up follows the same logic as [17] where Vt, DB0 t + DBt, and SFt are referred to as the actuarial reserve, allocated bonus, and free reserve (buffer account), respectively. Notice that we have split the allocated bonuses according to DB0 t and DBt. Since all cash-flows lead to a corresponding increase or decrease of the cash position in the asset portfolio it follows that Lt has to coincide at all times with the total book value, BVt, of assets under management (3.41) X a∈At BV a t = BVt = Lt = LPt + SFt and the verification of this equality is implemented as an automated test along all scenarios and for all time steps (cf. [19, A. 2.2]). Notice that the equity position in the balance sheet model of [17] is a hybrid of free capital, given in the present notation as the difference BVt − Lt, and hidden reserves, UGt = MVt − BVt. We do retain UGt in the projection since this is indispensable for the formulation appropriate management rules for best estimate calculation.

The ALM model assumes that the following quantities are given as deterministic functions of time:

• premium payments: prt

• guaranteed benefits, including surrender and paid-up payments, due to Vt and DB0 t : gbft

• expense payments: expt

• mathematical reserve: Vt

• previously allocated bonuses: DB0 t

• technical interest rate: ρt.

In fact, all these quantities are given on the level of model points such that the quoted values are the aggregate sums. Moreover, these quantities have been constructed according to classical actuarial assumptions such that benefits, premiums and technical reserves are consistent. The liability module gives rise to two management rules concerning the quantities νt and ηt in (2.24). These rules are specified in Section 3.5.

3.5. Management rules and profit declaration. The company’s management has a certain freedom to act based on market information and the state of assets and liabilities. This freedom concerns decisions about strategic asset allocation and profit declaration. A general understanding of the mechanics of profit declaration may be gained from Figure 1 where the connection between gross surplus, management actions, profit declarations and cash-flows is summarized in a diagram

We recall the definition of the gross surplus (2.23), and note that ROAt = X a∈At−1 cf a t + ∆BV a t + Ft−1Ct−1 where At−1 denotes assets under management at t − 1 other than cash, cf a t is the asset’s cash flow at t, i.e. coupon, nominal, dividend or rental income, ∆BV a t = BV a t − BV a t−1 is the change in book value according to the strict LCM principle, and Ct−1 denotes the amount of cash at t − 1.

Strategic asset allocation.

Rule 3.1. Bonds are valued according to the augmented lower of cost or market principle, i.e. the initial book value is carried forward until the bond is sold or reaches its maturity. Equity and property positions are valued according to the strict lower of cost or market principle, BV · t = min(BV · t−1 , MV · t ). Notice that this rule implies that only bonds can have unrealized losses, all other assets’ unrealized gains must be non-negative.

Rule 3.2. New bonds are bought at par with a time to maturity of 10 years.

Rule 3.3. The strategic asset allocation is kept approximately constant: the market value ratios (that is, cash amount over total market value, bonds over total market value, equities over total market value, and properties over total market value) are kept constant up to a predefined deviation. When an asset class breaches this bound, the portfolio is rebalanced such that the original targets, given by the market value ratios at t = 0, are restored. The rebalancing is such that placement of assets with minimal unrealized gains is prioritized (to avoid unintended book value return).

Negative surplus.

Rule 3.4. When the gross surplus is negative, unrealized gains are realized until the gross surplus equals 0, or no more positive unrealized gains exist. The selling order is: bonds before equity before property, and within those classes positions with large amounts of positive unrealized gains are prioritized. This rule is in place in order to avoid shareholder capital injections as much as possible. The gross surplus is updated after the application of this rule.

Positive surplus. The following rules apply if the gross surplus that is calculated by the model after the portfolio has been aligned according to the strategic asset allocation is positive. The gross surplus is updated after the application of each of the following rules. Let ϑ = SF0/LP0. Let τt denote the declared total participation rate at t. Notice that a participation rate of τt means that the amount of bonus declaration is given by ph∗ t = νt · gph · gs+ t + ηt · SFt−1 = X x∈Xt τt − ρ x + V x t−1 where Xt is the set of model points x which are active at t, and ρ x and V x t−1 are the technical interest rate and previous mathematical reserve. Let v = 1/100. We define the target rate τ ∗ t = min (τt−1 + L 10 t )/2, τt−1 + v and the target amount of participation tat = P x∈Xt (τ ∗ t − ρ x )+V x t−1 . This choice for v means that the target participation rate increases at most 1 percentage point from the previous participation rate thus avoiding large upwards jumps. The target is defined in terms of the previous participation rate, τt−1, and the prevailing 10Yforward rate, L 10 t , it is therefore a combination of previous profit participation and general market expectation.

Rule 3.5. If τt > τ ∗ t and there are bonds b ∈ At with unrealized losses, UGb t < 0, then these losses are incurred until either τt = τ ∗ t or all unrealized gains are non-negative.

Rule 3.6. Assume gst > 0. We distinguish two cases:

(1) If gph · gs+ t ≥ tat: νt = min 1, X x∈Xt (τ ∗ t − ρ x )+V x t−1/(gph · gs+ t ) η (1) t = 0

(2) If gph · gs+ t < tat: νt = 1 η (0) t = X x∈Xt (τ ∗ t − ρ x )+V x t−1 − gph · gs+ t + . SFt−1 η (1) t = min 1 2 , η (0) t unless SFt−1 = 0, in which case we set η (1) t = 0. Given νt according to (1) or (2), we set η (2) t = (1 − νt − θνt) · gph · gs+ t + SFt−1 − θ(Vt + DB0 t + DBt−1 − pht − sg∗ t ) (1 + θ)SFt−1 unless SFt−1 = 0, in which case we set η (2) t = 0. Finally, we define ηt = min max(η (1) t , η (2) t ), 1 . The purpose of this rule is to follow the above defined target rate by managing the surplus fund allocations, however to reach this target not more than half of the existing surplus fund is provided. The exception to this limit on surplus fund injections is the case η (2) t > η(1) t . This last term is to ensure that the inequality (3.42) SFt−1 + gph · gs+ t − ph∗ t = SFt ≤ θLPt = θ Vt + DB0 t + DBt−1 + ph∗ t − pht − sg∗ t holds for all time.

Order of management rules.

Rule 3.7. The rules are applied in the order in which they are stated.

As a consequence it may happen, within one accounting step, that, e.g., Rule 3.4 is carried out after the rebalancing step in Rule 3.3. Thus capital gains may be realized by, e.g., selling an equity position so that the distribution of assets may no longer be in line with the strategic asset allocation. In such a case a misalignment of asset positions is carried forward along one accounting year and then rebalanced at the end of this year. This rule is chosen nevertheless in this form since short term misalignment is acceptable.

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