3.7.1 The set-up of payments and the financial market
In the classical actuarial set-up of payments (see Hoem [33]), S = (Z0, X) with r being deterministic. In this set-up, the surplus was studied in Ramlau-Hansen [58].
3.7. A COMPARISON WITH RELATED LITERATURE 71 Mathematically, it is only a small step to let r depend on X, still linking payments and intensities toX only (notZ0). However, conceptually, it is a large step since it is actually the step from working with deterministic financial market modelling to stochastic financial market modelling. This was done in Norberg [51]. A stochastic rate of interest was introduced in the analysis of surplus in Norberg [53].
Generalizing the financial market and linking the payments to this financial mar- ket is a natural step. Conceptually, this is another large step because it opens for an explicit interaction between the insurance contract and the financial market intro- ducing financial mathematics as an integrated part of life insurance mathematics.
One way of approaching this interaction is, simply, to assume that the insurance contract is integrated completely in the financial market and consider it as a dy- namically traded security itself. This was done in Chapter 2 and is the starting point for the reserves and surplus as introduced in this chapter.
3.7.2 Prospective versus retrospective
We have here introduced two notions of surplus, the retrospective surplus and the prospective surplus. This terminology is inherited from the theory of retrospective and prospective reserves as introduced in Norberg [50]. There the prospective reserve at time t is defined as the expected present value of benefits minus premiums, i.e.
−B, over (t, T] conditioned on information formalized by a sigma-algebra Gt ⊆ Ft, where G is not necessarily a filtration. The retrospective reserve is defined as the expected present value of premiums minus benefits, i.e. B, over [0, t] conditioned on Gt. Thus, the two reserves differ in the sign on B and by the time interval over which B is considered.
TakingGt=FtS, we see thatLt is a retrospective reserve and Vt is a prospective reserve. Thus, the retrospective and the prospective surplus equal the retrospective and the prospective reserve, respectively, with subtraction of the technical prospec- tive reserve. We can, in fact, represent the retrospective surplus and the prospective surplus as expected values themselves,
←F−∗t = EQ Lt−Vt∗| FtS
, (3.43)
−
→F∗t = EQ Vt−Vt∗| FtS
. (3.44)
Furthermore, as was shown in (3.11) and (3.19) the retrospective surplus and the prospective surplus are, in fact, reserves themselves. The retrospective and prospec- tive surplus are the retrospective and prospective reserves, respectively, correspond- ing to the payment process C∗+Be.
3.7.3 Surplus
In the traditional approach to emergence of surplus (see Sverdrup [67]), one defines the contribution to the surplus C as the payment stream in the form, dCt = ctdt,
fulfilling
Vt′ =Vt∗+EQ Z T
t
Zt0
Zu0d(−Cu) St
. (3.45)
From the corresponding differential equations one easily gets ct=α∗t. Using that Vt′ =Vt−EQ
Z T t
Zt∗ Zu∗d
−Beu
St
, (3.45) can be written as
Vt−Vt∗ =EQ Z T
t
Zt
Zu
d
−Cu−Beu
St
,
and Sverdrup [67] can be said to be based on prospective reasoning. Note that our prospective surplus does not show up here since we have enforced dCt =ctdt.
In Ramlau-Hansen [58] a realized profit, which corresponds to our retrospective surplus with Be = 0 and (r∗, g∗, h∗) =
b r,bg,bh
, is introduced and studied inten- sively. Ramlau-Hansen [58] remarks that the realized profit only differs from the difference between the ”second-order retrospective premium reserve” and the ”first- order prospective reserve” by a martingale term. Since the second-order retrospec- tive reserve in Ramlau-Hansen [58] is rather a retrospectively calculated prospective reserve Vbt (as pointed out in Norberg [50]) and the first-order prospective equals V∗, this difference corresponds to our prospective surplus, still with Be = 0. Thus, the remark by Ramlau-Hansen [58] can be seen as consequence of the fact that the retrospective surplus and the prospective surplus only differ by a martingale term.
This follows immediately from the fact that ZF0 is a martingale. This is true even with Be = 0, although ZF0 is no zero-mean martingale then.
In Norberg [53] the surplus is defined as the realized profit in Ramlau-Hansen [58]. Norberg [53] sets out by derivingct in (3.45) in the traditional way and obtains that this equals the systematical contribution to the individual surplus. Again, this is just a consequence of the fact that F is a martingale and the retrospective and prospective surplus therefore only differ by a martingale term, even with Be= 0.
Finally, consider the special case of our surplus appearing from the set-up of payments and the financial market considered in Norberg [53]: Invest in Z0 only, let the payments depend on X only and price, referring to diversifiability, risk to zero (g = 0). Let the technical basis coincide with the first order basis. Then (3.10) coincides with (3.13) and reduces to
d←F−∗t =rt←F−∗tdt+
(rt−brt)Vbt+Rbtbgtàt
dt−RbtdMt+dBet, whereas α∗t coincides with α∗tQ and reduces to
α∗t = (rt−brt)Vbt+Rbtbgtàt.
Letting Be = 0 and defining, appropriately, the first order intensity by (1 +bg)à, these quantities equals the corresponding quantities in Norberg [53].
3.7. A COMPARISON WITH RELATED LITERATURE 73
3.7.4 Information
In (3.43) and (3.44) we have represented the retrospective and the prospective sur- plus as expected values conditioned on FtS. In Norberg [50] and Norberg [53], of such expected values conditioned on the full information are spoken of as individual quantities. It is one of the main ideas in Norberg [50] to relax this information and work with different versions of the reserve and the surplus corresponding to different sub-sigma-algebras Gt ⊆ FtS, not necessarily filtrations. This idea also plays a role in Norberg [53]. We shall not go into a study of the surplus under such relaxed in- formation, but just remark that different sub-sigma-algebrasGt⊆ FtS replacing FtS
in (3.43) and (3.44), correspondingly, define different versions of the retrospective and prospective surplus, respectively. In Norberg [53] the relaxation is restricted to concern information on diversifiable risk. We have no problem with relaxation of any kind of risk since we have, appropriately, adjusted for a possible non-zero price of risk by taking expectation under the measure Q.
3.7.5 The arbitrage condition
Last but not least, we compare the two apparently different requirements on the dividends showing up in Norberg [53] and in our framework. Our requirement is that dividends should be arbitrage free, leading to
EQ Z T
0−
1 Zt0dBt
= 0, (3.46)
whereas in Norberg [53] the requirement is E
Z T 0−
1 Zt0dBt
H′T ∨ GT
= 0, (3.47)
where H′T contains a part of the information over (0, T] on diversifiable risk in the environment and GT contains all information over (0, T] on non-diversifiable risk in the environment. It can easily be shown that (3.46) follows from (3.47) if diversifiable risk is priced to zero,
EQ Z T
0−
1 Zt0dBt
= EQEQ Z T
0−
1 Zt0dBt
H′T ∨ GT
= EQE Z T
0−
1 Zt0dBt
HT′ ∨ GT
= 0.
By (3.47), the company is allowed to carry a part of the diversifiable risk only. This may be an unfulfillable hard requirement. Taking instead expectation under the risk adjusted measure this requirement is not necessary. The company is then properly paid for the risk they are left with when all accounts have been settled at time T. However, in our set-up, the owners of the insurance company are, of course, welcome on the market if they want to get rid of some of this risk. This would be a matter of hedging, to some extent, the contingent claim LT.
We hereby indicate that our set-up imposes a very important separation of the investment strategy underlying the investment of payments of the insurance con- tract, θ, which should be based on the objectives of the policy holder, and the investment strategy of the owners of the insurance company, which should be based on the objectives of these owners and could include an extent of hedging of the claim LT.