A. Payment streams and their discounted values. First some basic definitions and results are quoted from Norberg (1990).
Consider a stream of payments commencing at time 0. It is defined by a finite-valued payment function A, which for each time t ≥ 0 specifies the total amount A(t) paid in [0, t]. Negative payments are allowed for; it is only required that A be of bounded variation in finite intervals and, by convention, right-continuous. This means that A = B − C, where B and C are non-negative, nondecreasing, finite-valued, and right-continuous functions representing the outgoes and incomes, respectively, of some business. In the context of insurance B represents benefits and C represents contributed premiums on an insurance policy (or a portfolio of policies). The payment function extends in a unique way to a payment measure on the Borel sets, which is also denoted by A.
Assuming that payments are valuated by a piecewise monotone and contin-uous discount function v, the present value of A at time t is
V (t, A) = 1 v(t)
Z
[0,∞)
v(τ ) dA(τ ) (9.1)
(the sum of all payments in small intervals discounted at time 0, multiplied by the interest factor 1/v(t) for the interval from 0 to t). Just to obtain transparent formulas, it will be assumed throughout that v is of the form
v(t) = e−R0tδ, (9.2)
with piecewise continuous interest intensity δ. (The shorthand exemplified by Rt
0 δ =Rt
0 δ(τ ) dτ will be in frequent use throughout.)
B. Definitions of retrospective and prospective reserves. The restriction of A to a (measurable) time set T is the measure AT counting only those A-payments that fall due in T ; AT{S} = A{S ∩ T }.
At any time t ≥ 0 the payment stream splits into payments after time t and payments up to and including time t; A = A(t,∞)+ A[0,t]. The present value in (9.1) splits correspondingly into
V (t, A) = V+(t, A) − V−(t, A), (9.3) where
V+(t, A) = V (t, A(t,∞)) = 1 v(t)
Z
(t,∞)
v(τ ) dA(τ ) (9.4)
is the discounted value of future net outgoes (in the insurance context benefits less premiums), and
V−(t, A) = V (t, −A[0,t]) = 1 v(t)
Z
[0,t]
v(τ ) d(−A)(τ ) (9.5)
is the value, with accumulation of interest, of past net incomes (in the insurance context premiums less benefits). The quantity V−(t, A) is observable by time t and can suitable be called the individual retrospective reserve of the policy at time t. If the future development of (v, A) were known, then V+(t, A) would be the appropriate amount to set aside to cover future excess of benefits over premiums on the individual policy. However, if the future course of (v, A) is uncertain, it is not possible to provide V+(t, A) as a prospective reserve on an individual basis.
Assume now that A and, possibly, also v are stochastic processes on some probability space (Ω, F, P ). An operational definition of the prospective reserve must depend solely on information that is at hand at the moment when the reserve is to be provided. Let F = {Ft}t≥0be a family of sub-sigmaalgebras of F, Ft representing some piece of information available at time t. The family F may be increasing, that is, Fs⊂ Ft, s < t, but this is not required in gen-eral. Reserves are defined as conditional expected values, given the information provided by F. At time t the prospective F-reserve is
VF+(t, A) = EFtV+(t, A), (9.6) and the retrospective F-reserve is
VF−(t, A) = EFtV−(t, A), (9.7) where the subscript on the expectation sign signifies conditioning. The prospec-tive F-reserve meets the operationality requirement formulated above as it is determined by the current information. Even though the retrospective individ-ual reserve in (9.5) is observable by time t, it may be judged relevant to calculate retrospective reserves with respect to some more summary information F. For a given realization of {v(τ )}0≤τ ≤t, Ft may be thought of as a classification of
the policies, whereby all policies with the same characteristics as specified by Ft
are grouped together. Forming the mean, conditional on Ft, means averaging over all policies in the same group, roughly speaking.
The reserves are conditional means of the present values V±(t, A). Other features of the conditional distributions of these random variables may be of interest. In particular, as measures of variability, introduce the variances
VF±(2)(t, A) = V arFtV±(t, A). (9.8) C. Relationships between reserves. When only one payment stream is consid-ered, notation can be saved by dropping the symbol A from V (t, A). Thus, abbreviations like V (t) and VF±(t) will be frequently used in the sequel.
By (9.1) – (9.3), the value of A at time 0 is related to its value at any time t ≥ 0 by
V (0) = v(t)V (t)
= v(t){V+(t) − V−(t)}.
Taking expectation gives
E V (0) = E {v(t)V (t)} (9.9)
= E {v(t)(V+(t) − V−(t))}. (9.10) The equivalence principle of insurance states that premiums and benefits should balance on the average as seen at the outset, that is,
E V (0) = 0. (9.11)
It does not imply EV (t) = 0 for t > 0 unless v is a deterministic function, confer (9.9). Taking iterated expectations in (9.10), the equivalence requirement can be cast as
E {v(t)VF−(t)} = E {v(t)VF+(t)} (9.12) if v(t) is determined by Ft, and
E VF−(t) = E VF+(t) (9.13)
if v is deterministic. Relation (9.9) is a special case of (9.13).
Let F0 = {Ft0}t≥0 be some sub-sigmaalgebra representing more summary information than F = {Ft}t≥0 in the sense that Ft0 ⊂ Ft, t ≥ 0. The rule of iterated expextations yields the following relationship between reserves on different levels of information:
VF±0(t) = EFt0VF±(t). (9.14) By the general rule V arX = E V arYX + V ar EYX, variances denoted as in (9.8) are related by
VF±(2)0 (t) = EFt0{VF±(2)(t) + (VF±(t))2} − (VF±0(t))2. (9.15) There exist also useful relationships between reserves at different times for a fixed source of information, F. The discounted values of future net outgoes at two different points of time, t < u, are related by
v(t)V+(t) = Z
(t,u]
v(τ ) dA(τ ) + v(u)V+(u).
Similarly, for s < t,
v(t)V−(t) = v(s)V−(s) + Z
(s,t]
v(τ ) d(−A)(τ ).
Taking expectations in these two identities, yields a pair of basic relationships under the assumption that v is deterministic. If {A(τ )}τ≥u depends stochasti-cally only on Fu for given Ft and Fu, then
v(t)VF+(t) = Z
(t,u]
v(τ ) d EFtA(τ ) + v(u)EFtVF+(u). (9.16)
Likewise, if {A(τ )}τ≤s depends stochastically only on Fsfor given Fs and Ft, then
v(t)VF−(t) = v(s)EFtVF−(s) + Z
(s,t]
v(τ ) d(−EFtA)(τ ). (9.17)
D. Right-continuity of the reserve processes. As defined by (9.6) and (9.7) the reserves are right-continuous stochastic processes. They could alternatively be made left-continuous by letting the integrals in (9.4) and (9.5) extend over [t, ∞) and [0, t), respectively. This would be in keeping with tradition, but the right-continuous versions are chosen here since they fit into the general apparatus of stochastic integrals and differential equations and thus are the more convenient quantities to deal with in anticipated applications of the theory to complex models. Anyway, the right-continuous and the left-continuous versions differ only at points of time where non-null amounts fall due with positive probability.