Estimating Future Discretionary Benefits Without Monte Carlo Simulation

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

Estimation of future discretionary benefits

Based on the assumptions which are stated and discussed together with numerical evidence in the previous Section 4 we shall now derive estimates, LBd and UBd, for lower and upper bounds of the F DB. The merit of these bounds lies in the fact that they can be computed a priori without the need for Monte Carlo methods. Concretely, we shall estimate the terms I, II, III and COG from the representation (2.32), and the corresponding estimators will be denoted by (c·).

5.1. Estimating I. The portfolio is in run-off, and assumption 4.1 implies correspondingly the trivial estimate (5.50) Ib= 0 for I.

5.2. Estimating II. Assumptions 4.2, 4.4 and 4.5 imply that an upper bound for II can be estimated as IIc := (1 − gph) X T t=2 P(0, t) γ + t σ DB t l h (5.51) t−1 LP0.

5.3. Estimating III. Invoking assumptions 4.2, 4.3 and 4.5 we obtain the estimate III ≤ 2(1 − gph) X T t=1  P(0, t − 1) − P(0, t)  E h DBt−1 + SFt−1 i ≤ 2(1 − gph) X T t=1  P(0, t − 1) − P(0, t) σ DB t + θ  l h t−1LP0 (5.52) =: III d as an upper bound of III.

5.4. Estimating COG. Making use of assumptions 4.1, 4.7, 4.6, 4.2 and 4.3, we model gst by gsb t where the latter is defined in (4.47). Correspondingly the cost of guarantee, COG, defined in equation (2.30) is estimated as (5.53) COG \= E hX T t=1 B −1 t gsb − t i = X T t=1 O − t (1 + ϑ)l h t−1LP0.

and (5.54) O − t := E h B −1 t  Ft−1 + P(0, t) −1 l d t−1 − l d t l h t−1 UG0 (1 + ϑ)LP0 − (1 − σt)(ρt − γt) 1 + ϑ −i is the value at 0 of the floorlet (i.e., negative caplet) with maturity t − 1 and payment min(Ft−1 − kt, 0) at settlement date t, where the strike is given by (5.55) kt := −P(0, t) −1 l d t−1 − l d t l h t−1 UG0 (1 + ϑ)LP0 + (1 − σt)(ρt − γt) 1 + ϑ . In the normal model this value is given by the Black formula ([4, 5]) (5.56) O − t = P(0, t) ·  ± (F 0 t−1 − kt)Φ(−κt) + IVt √ tϕ(−κt)  where Φ and ϕ are the normal cumulative distribution and density functions, respectively. Further, κt := F 0 t−1 − kt IVt √ t where F 0 t−1 = P(0, t−1)−P(0, t) is the forward rate prevailing at time 0, and IVt is the caplet implied volatility known from market data.

5.5. Estimating F DB. These estimates yield a lower bound, LBd, and an upper bound, UBd, for F DB: (5.57) LBd ≤ F DB ≤ UBd where (5.58) LBd := F DB0 − IIc − III d (5.59) UBd := F DB0 + gph · COG. \ If the difference ε = (UBd − LBd)/2 is sufficiently small (e.g., in comparison to MV0), then (5.60) F DB \= LBd + UBd 2 = SF0 + gph LP0 + UG0 − GB + gph · COG \− IIc − III d 2 may be used as a reasonable estimator for F DB with maximal error given by ε.

5.6. Comparison to numerically calculated values. The purpose of this section is to apply the model of Section 3 to a realistic life insurance portfolio, and to compare the results with the estimates involved in calculating (5.60). The life insurance portfolio that we utilize to this end consists of anonymous, aggregated and randomized asset and liability data from multiple life insurance companies. Moreover, the portfolio is perturbed initially according to (4.43) and (4.44) such that we obtain nine different cases. These correspond to quite different relations between and assets and liabilities, and are therefore considered to be representative for a wide variety of possible company data and economic environments. Nevertheless, the conclusions to be drawn from this sections cannot be definite but only those that follow from a specific numerical study. The results from this study are summarized in Table 2.

Given that the assumptions which go into the derivation of the estimation formula (5.60) are verified numerically in Section 4 it is to be expected that the empirical and estimated values in Table 2 are consistent. This is indeed the case, and it can be observed that the values agree quite well in most cases, particularly when viewed relative to the initial market value MV0 = LP0 + SF0 + UG0. Table 2 is compartmentalized as follows. The top two lines consist of the parameters π0 and UG0/BV0 which serve to vary the relationship between assets and liabilities. The quantities LP0, SF0, MV0 and UG0 are starting values, and F DB0 and GB are calculated deterministically.

The value E[B −1 T MVT ] corresponds to the expected discounted final market value of the asset portfolio. Since the company is modeled with respect to run-off assumptions, this quantity is expected to be small, as is verified here. In fact, smallness of this term is a necessary condition: if it is not verified then the projection horizon has to be increased. The quantity T AX represents the value of corporate tax payments. The term V IF corresponds to the value of in-force business. This is the value of the life insurance portfolio from the shareholder’s perspective. The market consistent embedded value that is often disclosed in financial reports is defined accordingly as MCEV = MV0 +V IF. It is the shareholder’s objective to maximize V IF, cf. the control problem in Section 2.4. It can be observed in Table 2 that this value is subject to quite a significant variation depending on the initial conditions.

The value of in-force business is defined in Section 2 as the difference of shareholder gains and cost of guarantee, V IF = SHG − COG. In Table 2, SHG and COG are calculated numerically by means of the model set-up in Section 3. The values COG \MC and COG \correspond to the Monte-Carlo valuation of (5.53) and the valuation of (5.53) by means of the Black formula (5.56), respectively. If the model were calibrated perfectly we would have COG \MC = COG \. For our purpose of estimating the upper bound it is relevant that COG \≥ COG. This holds for all cases. In fact, we observe an over-estimation. This over-estimation is related to Figure 10 where it can be seen graphically that COG \MC , and therefore also COG \, does not take the COG-reducing effect of management rules into account. When COG and COG \MC are very low, which occurs in every third column corresponding to a high fraction UG0/BV0, we observe that COG \MC is much smaller than COG \indicating imperfect model calibration. In these cases it may also happen that COG \MC < COG, however since this occurs only for negligible values of COG this observation does not impede the applicability of the upper bound formula (5.59).

The terms I, II and III in Table 2 are calculated numerically according to their defining formulas in Theorem 2.1. It can be observed that the estimates Ib = 0 and IIc are quite sharp, and that IIc might, at least according to this study, equally well have been set to 0. Indeed, IIc corresponds to shareholder gains on surrender penalties stemming from future profit declarations. This involves surrender probabilities and penalty factors which typically decline to 0 towards the maturity of a contract. From these considerations it can already be deduced that IIc should be expected to be quite small in comparison to F DB. The third estimate, III d, shows an over-estimation. This can be related to Assumption 5 where the relative covariance is estimated by 1 while Figure 5 shows that this covariance might have been estimated by 0. If this were the employed estimate then the factor 2 in (5.52) would be replaced by 1, and III d and III in Table 2 would agree quite well. However, we do not strive to have the sharpest estimates LBd and UBd, and therefore prefer these more generous -but also more stable- assumptions.

The leakage and Monte Carlo error results in Table 2 show the usefulness of representation (2.32) in numerical calculations. Both are much smaller when the representation formula is used to compute the future discretionary benefits. This is not surprising since F DB0 depends only on the guaranteed benefits but is independent from stochastic scenarios, and this already determines F DBrep to a large extent – as can be verified by reading off the other terms from Table 2.

The boxed results containing F DBCF , F DBrep and F DB \ contain the core of Table 2. First of all, notice that F DBCF and F DBrep coincide up to their respective Monte Carlo errors, as it should be according to Theorem 2.1. Moreover, the difference δ = F DB \−F DBrep is quite small compared to MV0, and satisfies |δ| < ε in all cases. Notice, finally, that F DB0 while always in the interval from LBd to UBd is not generally a good estimator for F DB because it would neglect the effect of the guarantee cost COG.