Testing Actuarial Assumptions With Realistic Life Insurance Data

Table of Links

1. Introduction
2. Mean-field Libor market model
3. Life insurance with profit participation
4. Numerical ALM modeling
5. Phenomenological assumptions and numerical evidence
6. Estimation of future discretionary benefits
7. Application to publicly available reporting data
8. Conclusions
9. Declarations
10. References

Phenomenological assumptions and numerical evidence

This section is concerned with phenomenological assumptions based on numerical experience. The underlying numerical experiments are carried out with respect to realistic life insurance data (cf. Section 3). This implies that the assumptions which are formulated below may hold only in an approximate sense. We consider a given statement or assumption as fulfilled if it holds up 0.5 % of the initial market value, MV0, of the given asset portfolio.

In the following the assumptions are compared to the numerical model over a range of nine parameters. We consider the base case parameterization from Section 3, and then vary the initial amount of unrealized gains and the amount of guaranteed premium payments. In the base case we have UG0/BV0 = 0.05 and premium payments are scaled by the unit factor π0 = 1. In order to test the assumptions over a wide range of parameters we consider all combinations of (4.43) UG0/BV0 : −0.10, 0.05, 0.20, and (4.44) π0 : 0.95, 1, 1.05 which lead to very different relationships between asset and liability portfolios. Indeed, the factor UG0/BV0 controls the market value of assets available to cover the liabilities. Owing to equation (2.22), varying the premium income is equivalent to varying the average technical interest rate. Thus the variation given by π0 can be interpreted as a variation of the relationship between the prevailing yield curve and the guaranteed minimum rate, i.e. as a variation of the relative conditions between current economy and liabilities.

4.1. Assumptions, evidence and discussion.

Assumption 4.1. The liabilities are in run-off such that:

(1) The projection horizon T corresponds to the run-off time of the liability portfolio, that is SFT = LPT = UGT = 0.

(2) The expected life assurance provisions E[LPt] decrease geometrically: there is a fixed 1 ≤ h < T such that E[LPt] can be approximated by LPdt := l h t LP0 where l h t := 2−t/h for t < T and l h T := 0.

(3) Moreover, h can be approximated as the duration of liability cash-flows: h0 = PT t=1 t P(0, t) (gbft + expt − prt) PT t=1 P(0, t) (gbft + expt − prt) cf BS t = ϕ(t, 2h0, h0/2) max(F DB0, 0) h = PT t=1 t P(0, t) (gbft + expt − prt + cf BS t ) PT t=1 P(0, t) (gbft + expt − prt + cf BS t ) (4.45)

Here ϕ(·, 2h0, h0/2) denotes the normal density with mean 2h0 and standard deviation h0/2. The idea of cf BS t is to estimate the effect of future discretionary benefits on the duration of the liabilities. The simplest, albeit quite crude, estimator for F DB is F DB0. Hence these cash-flows are expected to be approximately proportional to F DB0, and we consider only the positive part since future discretionary benefits are positive. The density function serves to distribute the effect of F DB0 over time. The duration, h0, of guaranteed cash-flows is now taken as an estimator for the peak of these cash-flows: the reasoning is that h0 is already an estimator for the time it takes for the declared profits to accumulate, and 2h0 is the point where the policyholder payments from these declared benefits peak.

The first part of assumption 4.1 is trivial once T is chosen sufficiently large. However, that the run-off is geometric and is, moreover, determined (approximately) by formula (4.45) requires some evidence. This is provided by Figure 2. Clearly, the above arguments for h0, cf BS t and h are very heuristic, nevertheless the empirical evidence shows that the overall effects are as desired and the factor (4.45) is a reasonable parameter controlling the geometric run-off.

Assumption 4.2. Let h be given by (4.45). Let σ0 = DB0 0 /LP0, σ1 = max(F DB0/GB, 0.75 · σ0), σt = σ0 max(h − t, 0)/h + σ1 min(t/h, 1) and σ DB t = σ1 min(t/h, 1). Then E[DB0 t ] and E[DBt] can be approximated by DBd0 t and DBdt, respectively, where DBdt = σ DB t LPdt. and DBd0 t + DBdt = σt LPdt for all 0 ≤ t ≤ T . This assumption states that the initial fraction of declared benefits, σ0, remains approximately constant in expectation, but may change over time linearly to σ1 with a speed that is determined by h.

Moreover, E[DBt] increases linearly from 0 to σ1 in h time steps, whence E[DB0 t ] cannot decrease too quickly. Numerical evidence for this assumption is provided in Figure 3. This figure shows that the assumption appears to be satisfied only up to time h and then holds rather poorly at later projection times. Nevertheless, we view this assumption as justified, since the overall behavior is represented correctly, and errors at later points in time are discounted. Moreover, given that this assumption relies on the calculation of h in Assumption 4.1 and the expectation future profits as estimated by the fraction F DB0/GB in the definition of σ1 the results in Figure 3 show a remarkable stability.

Assumption 4.3. The relation SF0/LP0 =: ϑ remains constant in expectation: E[SFt] = ϑE[LPt] for all 0 ≤ t ≤ T. (Cf. [19, A. 3.10]) Assumptions 4.2 and 4.3 are statements about time-homogeneity. Management rules concerning bonus declarations should remain reasonably constant in the long run such that σ and ϑ do not vary too strongly. The relevant point in this context is that these quantities should not vary arbitrarily but follow from target rates set by management rules. Assumption 4.3 is comparable to the assumption concerning the ‘annual interest rate’ in [17, Section 4.2]. Figure 4 contains empirical observation for Assumption 4.3. Clearly, this figure only shows evidence for E[SF]t/E[LPt] ≤ ϑ. Nevertheless, Assumption 4.3 is kept in its present form since a statement such as 0.75 · ϑ ≤ E[SF]t/E[LPt] ≤ ϑ, while being more accurate, introduces unnecessary complexity.

Assumption 4.4. The surrender gains, sg∗ t = χtDBt−1, can be estimated on average with the same factor, γt, as the technical gains in (2.23): E[sg∗ t ] ≤ max(γt, 0)E[DBt−1] ≤ max(γt, 0)DBdt. The factor γt contains mortality, cost and surrender margins as a fraction of the full life assurance provision, LPt−1. It is therefore reasonable to expect that the same factor can be used as an upper bound on the surrender margin arising from declared bonuses, DBt−1, alone. Since the technical gains may also be negative only the positive part, max(γt, 0), is considered.

Assumption 4.5. (1) The correlation between B −1 t and DBt−1 is negative: Cor[B −1 t , DBt−1] ≤ 0

(2) The relative covariance of B −1 t Ft−1 and DBt−1 + SFt−1 satisfies Cov h B −1 t Ft−1 E  B −1 t Ft−1  , DBt−1 + SFt−1 E [DBt−1 + SFt−1] i = Cor h B −1 t Ft−1, DBt−1 + SFt−1 i · CV h B −1 t Ft−1 i · CV h DBt−1 + SFt−1 i ≤ 1 for 1 ≤ t ≤ T where CV denotes the coefficient of variation. The first part of this assumption is straightforward since a low discount factor is related to high nominal rates and this yields high nominal returns and consequently high reserves DBt−1. In the long run this argument applies also to the second part of the assumption, i.e. to the correlation of B −1 t Ft−1 and DBt−1 + SFt−1 since, after some time, the behaviour of B −1 t Ft−1 should be dominated by B −1 t . In the short run when the correlation might be positive, however, we expect the product, CV [B −1 t Ft−1] · CV [DBt−1 + SFt−1] of the coefficients of variation to be small. Numerical evidence for these heuristic ideas is provided in Figure 5.

According to (2.23) the gross surplus is given by gst = Ft−1(LPt−1 + SFt−1) + Ft−1UGt−1 − E[∆UGt|Ft−1] + ROAt − E[ROAt|Ft−1] − (ρt − γt)Vt−1 where ρt and γt may, in general, also depend on the stochastic interest rate curve via dynamic surrender. For the purpose of estimating terms III and COG in (2.32), we make the following simplifying assumptions. The principle idea behind these assumptions is that the main source of stochasticity in gst is the forward rate Ft−1 whence all other quantities are replaced by their expected values. The simplified model of gst will be denoted by gsb t .

Assumption 4.6. In gsb t the technical interest rate ρt and the technical gains γt are deterministic functions of t. More precisely, ρt is calculated as the LP Gt-weighted mean guaranteed rate, and γt is calculated according to (4.46) γt(1 − (σt−1 − σ DB t−1 ))LP Gt−1 = ρt  1 − (σt−1 − σ DB t−1 )  LP Gt−1 − ∆LP Gt + prt − gbft − expt Notice that (4.46) coincides with (2.22) up to the substitution Vt−1 = (1 − (σt−1 − σ DB t−1 ))LP Gt−1, which is consistent with Assumption 4.2, and we omitted the surrender gains sg∗ t . This definition allows us to compute γt a priori from the given (deterministic) data. Figure 6 contains plots of γt and also of γt + E[sg∗ t ], and confirms that the effect of sg∗ t should be expected to be quite negligible.

Assumption 4.7. In gsb t the return ROAt is predictable, i.e. Ft−1-measurable, and realizations of unrealized gains are determined by a fixed number 1 < d < T: (1) ROAt − E[ROAt|Ft−1] = 0; (2) Ft−1UGt−1 − E[∆UGt|Ft−1] = P(0, t) −1 (l d t−1 − l d t )UG0 where l d t := 2−t/d for t < T and l d T := 0. Moreover, d is given as the duration of the initial bond portfolio: d = (PT t=1 P b t P(0, t) cf b t )/( PT t=1 P b P(0, t) cf b t ) where b runs over all initially held bonds. To compare the first part of this assumption against numerical evidence, we consider the quantity (ROAt − E[ROAt|Ft−1])/BVt−1 since ROAt is the total return on the book value BVt−1. Figure 7 shows observations of expected values and standard deviations, that is E[(ROAt − E[ROAt|Ft−1])/BVt−1] and SD[(ROAt − E[ROAt|Ft−1])/BVt−1]. While the expected values vanish due to the tower property of the conditional expectation, the fact that also the standard deviations are (relatively) small is significant for Assumption 4.7.

The second part of Assumption 4.7 is studied numerically in Figure 8. Again it can be observed that the assumption is not satisfied accurately, but nevertheless the overall behaviour is reflected quite well. Invoking the above assumptions 4.1, 4.7, 4.6, 4.2 and 4.3, we define gsb t : = Ft−1E[BVt−1] + P(0, t) −1 (l d t−1 − l d t (4.47) )UG0 − ρtVt−1 + γtLPt−1

 Ft−1 + P(0, t) −1 l d t−1 − l d t l h t−1 UG0 (1 + ϑ)LP0 − (1 − σ)(ρt − γt) 1 + ϑ  (1 + ϑ)l h t−1LP0 to be used as a model for gst in the estimation of COG. Due to its relevance we state this conclusion as an equation: (4.48) gst = gsb t with the understanding that this equality is to be understood in the approximative sense. Since we aim to use gsb t in order to calculate an upper bound, COG , for the cost of guarantee, COG, equation (4.48) is justified if (4.49) E hX T t=1 B −1 t gs− t i ≤ E hX T t=1 B −1 t gsb − t i . Empirical evidence for assertions (4.48) and (4.49) is contained in Figures 9 and 10.

Remark 4.8. The above assumptions are a streamlined version of those in [14], enhanced with numerical tests. Concretely, Assumption 4.1 corresponds to [14, Ass. 4.1 & 4.2]; Assumption 4.2 corresponds to [14, Ass. 4.3]; Assumption 4.3 corresponds to [14, Ass. 4.4]; Assumption 4.4 corresponds to [14, Ass. 4.5]; Assumption 4.5 has similar elements as [14, Ass. 4.10], but the assumption is not as strong ([14, Ass. 4.10] is not supported by our numerical results as the dotted line in Figure 5 shows); Assumption 4.6 corresponds to [14, Ass. 4.8]; Assumption 4.7 corresponds to [14, Ass. 4.9]. The necessity of [14, Ass. 4.6 & 4.7] is no longer given since we have simplified the calculation of the lower bound estimate.