A. The statewise probability distributions.
The problem is to determine the conditional probability distribution of the liability in (8.1), given the information available at time t. Since the Markov assumption implies conditional independence between past and future for fixed present state of the policy, the relevant functions are the statewise probability distributions defined by
B. A system of integral equations.
A simple heuristic argument will establish that the probabilities in (8.1) satisfy the integral equations
In the first terms on the right here the factor e−Rtsµj·µjk(s)ds is the probability that the policy stays in state until time s (< n) and then makes a transfer to state k (6= j) in the small time interval [s, s + ds). In this case the annuity Bj
is in force during the time interval (t, s], the lump sum bjk(s) falls due at time s, and the interest rate during this time interval is δj, and so the event
Z n
Thus, the corresponding conditional probability is Pk
Summing over all times s and states k, we obtain the first terms on the right of (8.2), which thus is the part of the total probability that pertains to exit from state j before time n.
Likewise it is realized that the last term on the right of (8.2) is the remaining part of the probability, pertaining to the case of no transition out of state j before time n.
This heuristic argument is made rigorous by applying Doob’s optional sam-pling theorem to the martingale generated by the indicator of the event in (8.3) and the stopping time defined as the the minimum of n and the time of the first transition after t from the current state j.
C. A system of differential equations.
Already (8.2) might serve as a basis for computation of the statewise probability functions, but is not convenient since the integrand on the right depends on t.
Introduce the auxiliary functions Qj defined by
Qj(t, u) = Pj
Pj(t, u) = Qj and rearrange a bit to obtain
e−R0tµj·Qj(t, u) =
Now the integrand on the right does not contain t, and we are allowed to differ-entiate along t on the right hand side by simply substituting t for s in minus the integrand. Performing this and cancelling the common factor e−R0tµj·, we arrive at the following main result, where the side condition (7.76) comes directly out of (8.6) by letting t ↑ n:
Theorem. The functions Qjin (8.4) are the unique solutions to the differential equations
Having determined the auxiliary Qj, we obtain the Pj from (8.5).
Remark: The differentiation is in the Stieltje’s sense for functions of bounded variation and does not require differentiability or any other smoothness proper-ties of the functions involved.
D. Computational scheme.
A simple numerical procedure consists in approximating the functions Qj by the functions Q∗j obtained from the finite difference version of (8.7):
Q∗j(t − h, u) = (1 − µj·(t)h)Q∗j(t, u) + h X
k;k6=j
µjk(t)
·Q∗k
t, e(δj−δk)tu + Z t
0
e−δkτdBk(τ )
−e(δj−δk)t Z t
0
e−δjτdBj(τ ) − e−δktbjk(t)
. (8.9) Starting from (8.8) (with Q∗j in the place of Qj), one calculates first the func-tions Q∗j(n − h, ·) by (8.9) and continues recursively until the Q∗j(0, ·) have been calculated in the final step.
The Q∗j(t, u) are defined for t ∈ {0, h, 2h, . . . , n} and u ∈ {a, a + h0, a + 2h0, . . . , b}, say, where the steplengths h and h0 must be sufficiently small and a and b must be chosen such that the supports of the Qj(t, ·) are sufficiently well covered by [a, b]. What is “sufficient” must be decided on in each individual case by judgement and by trial and error. If two states j and k are intercom-municating, then Qj(t, ·) and Qk(t, ·) have the same support. The supports are finite and usually easy to determine in situations where the number of assurance payments has a non-random upper bound. The u-arguments in the functions Q∗k on the right of (8.9) must, of course, be rounded to the nearest point in {a, a + h0, a + 2h0, . . . , b}.
E. Comments on the method.
Before turning to applications, we pause in this paragraph to offer some moti-vation and discussion of our approach. It is not needed in the sequel and may be skipped on the first reading.
The equation (8.7) is just the differential form of the integral equation (8.6).
It does not require that the derivatives ∂t∂Qj(t, u) exist, which they do not in general for the obvious reason that the statewise annuity functions on the right of (8.7) may have jumps.
If one should attempt to construct differential equations along the lines of Norberg (1994), the starting point would be the martingale M defined by
M (t) = P
Z n 0
e−R0τδdB(τ ) ≤ u | Ft
,
where Ft = σ{X(τ ); τ ≤ t} is the information generated by the state process
up to time t. Using the Markov property of conditional independence between past and future, given the present, we find
M (t) =X
j
Ij(t) Pj
t, eR0tδ
u −
Z t 0
e−R0τδdB(τ )
.
Now, the recipe would be to apply the change of variable formula to the ex-pression on the right and then to identify the martingale component that is predictable (and of bounded variation) and hence constant. Accomplishing this without caring about justification, would lead to the first order partial differen-tial equations
∂
∂tPj(t, u) + ∂
∂uPj(t, u)(δju − bj(t))
+ X
k;k6=j
µjk(t) (Pk(t, u − bjk(t)) − Pj(t, u)) = 0,
valid between jumps of the contractual annuity functions Bj, and
Pj(t−, u) = Pj(t, u − ∆Bj(t)),
valid at jumps of the Bj, and subject to the condition that the Pj(n, u) are 0 and 1 according as u < 0 or u ≥ 0.
The approach requires that the functions Pj possess first order derivatives in both directions. As we have seen already in the introductory example of Section 2 they generally do not. This difficulty might possibly be circumvented by conditioning on the ultimate state at time n, but proving differentiability of the conditional probabilities would require additional assumptions and would not be straightforward.
These remarks serve to show that the method developed in Section 4 is not just one among several candidate approaches to the problem; it is the only mathematically sound solution we are able to offer. One general conclusion we can extract is that there is no single general technique for solving the bulk of problems of the kind considered here; the method will have to be designed for each individual problem at hand and will depend on the model assumptions and the functional of interest.
8.5 Applications
A. The Poisson distribution.
In continuance of the example in Paragraph 6B of Norberg (1994), consider the special case with two states, J = {1, 2}, no interest, δ1 = δ2= 0, and the only payments being an assurance of 1 payable upon each transition, b12= b21= 1.
Then, taking n = 1, the present value in (8.1) is just the number of transition in the time interval (t, 1], N12(1) + N21(1) − N12(t) − N21(t). Furthermore, take µ12= µ21= µ, a constant (> 0). Then it is seen from the defining relation (8.4) that the functions Pj and Qj are all the same. Denoting this function by P , we can work with (8.2), which becomes
P (t, u) = Z 1
t
e−µ(s−t)µP (s, u − 1)ds + e−µ(1−t)1[0 ≤ u]. (8.1) Using that P (t, u) = 0 for u < 0, we readily obtain from (8.1) that P (t, u) = e−µ(1−t) for 0 ≤ u < 1. Then, for 1 ≤ u < 2, it follows from (8.1) that P (t, u) = µ(1 − t)e−µ(1−t)+ e−µ(1−t). Proceeding by induction we obtain, for each u ≥ 0, that
P (t, u) =
[u]
X
i=0
(µ(1 − t))i
i! e−µ(1−t),
which is the Poisson distribution with parameter (1 − t)µ, of course.
As a check on the accuracy of the numerical method, we list the computed values of P (0, u), u = 0, 1, . . . , 8, for t = 0 and µ = 1 together with the exact values of the Poisson probabilities (in parantheses): 0.3670 (0.3679), 0.7358 (0.7358), 0.9202 (0.9197), 0.9813 (0.9810), 0.9965 (0.9864), 0.9994 (0.9994), 0.9999 (0.9999), 0.9999 (0.9999), 1.0000 (1.0000). These results were obtained with h = 1/200, h0 = 1/100, a = −0.5, and (truncating the infinite support) b = 9.5.
B. The term insurance policy.
To analyse the term insurance policy in Paragraph 2A, take J = {1, 2}, n = 30, µ12(t) = 0.0005 + 0.000075858 · 100.038(30+t), δ1 = ln(1.045), b1 = −0.0042608, b12= 1, and all other intensities and payments null.
Again, as a check on the accuracy of the numerical method, we list the com-puted values of P (0, u) together with exact values (in parantheses): P (0, u) = 0 for u < −0.0705 (0 for u < −0.0709), P (0, u) = 0.8453 for u ∈ [−0.0705, 0.1965) (0.8452 for u ∈ [−0.0709, 0.1961)), P (0, 0.2) = 0.8491 (0.8490), P (0, 0.4) = 0.9465 (0.9467), P (0, 0.6) = 0.9774 (0.9776), P (0, 0.8) = 0.9918 (0.9919), P (0, 1) = 1.0000 (1.0000). These results are based on h = 1/1000, h0= 1/2000, a = −0.1, and b = 1.1.
C. A combined insurance policy.
In our final numerical example we consider what will be referred to as the combined policy, which is the same as the one in Paragraph B, but with a disability pension added. More specifically, 1 is payable upon death, an annuity with level intensity 0.5 is payable during disability, and premium is payable with level intensity in active state. The relevant model entities are J = {1, 2, 3},
n = 30, b13= b23 = 1, b1= −0.013108 (net premium when the intensities are as specified below), b2= 0.5, and, adoping the standard Danish technical basis except for the recovery intensity, δ = ln(1.045) (independent of state), and
µ13(t) = µ23(t) = 0.0005 + 0.000075858 · 100.038x, µ12(t) = 0.0004 + 0.0000034674 · 100.06(30+t), µ21(t) = 0.005,
all other payments and intensities being null.
Figure 2 about here
Fig. 2: Probability distribution of the present value of the combined insurance policy in (a) state 1 and (b) state 2.
This example is a follow-up of Paragraphs 4B-C in Norberg (1994), where the first three moments of the present value are calculated for the combined policy.
D. Numerical evaluation of multiple integrals.
Numerical integration in higher dimensions is in general complicated, and there exists no technique held to be universally superior. The technique developed here can be used to evaluate integrals that, possibly after a reinterpretation, can be recognized as a probability related to a present value for a suitably specified policy. Just to illustrate the idea, suppose T1 and T2are independent positive random variables with cumulative distribution functions F1and F2with densities f1 and f2, respectively, and that we seek P[(T2∧ 1) − (T1∧ 1) ≤ u].
It is realized that this probability is found as P (0, u) for the policy with J = {1, 2, 3, 4}, µ12(t) = µ34(t) = f1(t)/(1 − F1(t)), µ13(t) = µ24(t) = f2(t)/(1 − F2(t)), b1(t) = −1, b2(t) = 1, n = 1, and no interest. Countless examples of this kind can be constructed.
Reserves
Prospective and retrospective reserves are defined as conditional expected val-ues, given some information available at the time of consideration. Each speci-fication of the information invoked gives rise to a corresponding pair of reserves.
Relationships between reserves are established in the general set-up. For the prospective reserve the present definition conforms with, and generalizes, the traditional one. For the retrospective reserve it appears to be novel. Special attention is given to the continuous time Markov chain model frequently used in the context of life and pension insurance. Thiele’s differential equation for the prospective reserve is shown to have a retrospective counterpart. It is pointed out that the prospective and retrospective differential equations have, respec-tively, the Kolmogorov backward and forward differential equations as special cases. Practical uses of the differential equations are demonstrated by examples.
9.1 Introduction
A. Sketch of the idea. The concept of prospective reserve is no matter of dispute in life insurance mathematics. It is defined as the conditional expected present value of future benefits less premiums on the policy, given its present state. A straightforward generalization is obtained by conditioning on some other piece of information, e.g. on the policy’s staying in some subset of the state space.
It is proposed here to define the retrospective reserve analogously as the condi-tional expected present value of past premiums less benefits.
B. An example: insurance of a single life. A person aged x buys a life insurance policy specifying that the sum assured, b, is payable immediately upon death before age x + n and that premiums are to be contributed continuously with level intensity c throughout the insurance period. Let Tx denote the person’s remaining life length after the policy is issued at time 0, say. Assume that the survival functiontpx = P {Tx > t} is of the form tpx = e−R0tµx+sds, with
119
continuous force of mortality, µ. Finally, assume that interest is earned with a constant, nonrandom intensity δ so that v = e−δ is the annual discount factor and i = eδ− 1 is the annual rate of interest (1 + i = v−1 is the annual interest factor).
At any time t ∈ [0, n] the policy is either in state 0 = ”alive” or in state 1
= ”dead”. The prospective reserves in the two states, indicated by subscripts 0 and 1, are
V0+(t) = Z n
t
vτ−tτ−tpx+t{bµx+τ− c} dτ (9.1) (by the usual heuristic argument, the sum of expected discounted benefits minus premiums in small time intervals (τ, τ + dτ ), 0 < τ < t, and, of course,
V1+(t) = 0. (9.2)
The statewise retrospective reserves as defined above are
V0−(t) = c Z t
0
(1 + i)t−τdτ (9.3)
(trivial) and
V1−(t) = 1 1 − tpx
Z t 0
(1 + i)t−τ{c (τpx− tpx) − bτpxµx+τ} dτ (9.4) (use the same kind of argument as in (9.1) noting that, conditional on death within time t, the probability of survival to τ is (τpx− tpx)/(1 − tpx) and the probability of death in (τ, τ + dτ ) isτpxµx+τdτ /(1 − tpx), 0 < τ < t) .
The state at time t is X(t) = 1[Tx≤ t], the ”number of deaths” of the person within time t. This is the information on which the reserves in (9.1) – (9.4) are based.
Now, suppose the complete prehistory of the policy is currently recorded, so that it is known at any time if the person is alive or dead and, in the latter case, when he died. The information available at time t is the pair (X(t), min(Tx, t)).
Denote the reserves correspondingly by a double subscript. The reserves in state 0 remain as above, V0,t±(t) = V0±(t), and so does the prospective reserve in state 1, of course, V1,T+x(t) = V1+(t) = 0. Only the retrospective reserve in state 1 is affected by the additional information on the exact time of death. It now becomes simply the value at time t of past premiums less the benefit payment,
V1,T−x(t) = c Z Tx
0
(1 + i)t−τdτ − b(1 + i)t−Tx. (9.5)
The quantity in (9.1) is what traditionally is referred to as the prospec-tive reserve. The notion of retrospecprospec-tive reserve launched here differs from the traditional one, which in the present example is
−V0(t) = 1
tpx
Z t 0
(1 + i)t−ττpx(c − bµx+τ)dτ. (9.6) This quantity emerges from the principle of equivalence, which requires that benefits and premiums should balance in the mean at the outset:
Z n 0
vττpx(bµx+τ− c)dτ = 0. (9.7)
SplittingRn 0 into Rt
0+Rn
t in (9.7) and substituting from (9.1) and (9.6), yields
−V0(t) = V0+(t) . (9.8)
Thus, the traditional concept of ”retrospective reserve” is rather a retrospective formula for the prospective reserve, valid for b and c satisfying the equivalence principle. For general b and c the quantity−V0(t) is not an expected value, and it has no probabilistic interpretation in the present model involving one single policy. In an extended (artificial) model, with m independent replicates of the policy issued at time 0, it may be interpreted as the almost sure limit of the total accumulated surplus per survivor by time t as m tends to infinity.
As compared with (9.8), the retrospective reserve introduced here is related to the prospective reserve under the equivalence principle by the identity
tpxV0−(t) + (1 −tpx)V1−(t) =tpxV0+(t) . (9.9) C. Outline of the paper. In Section 2 the present notions of reserves are defined for quite general stochastic payment streams and discounting rules, and certain relationships between them are established. No particular reference to the in-surance context is made at this stage. In Sections 3 and 4 the framework of the further discussions is presented: payments of the life annuity and life insurance types in a continuous time Markov chain model. A useful auxiliary result is that a Markov process behaves like a composition of mutually independent Markov processes in disjoint intervals when its values at the dividing points between the intervals are fixed. This together with standard results for Markov chains is used in Section 5 to investigate the properties of reserves in the Markov chain case. The prospective reserve, given the state at the time of consideration, is the traditional one, which satisfies the well-known generalized Thiele’s differen-tial equations (see e.g. Hoem, 1969a), here also referred to as the prospective differential equations. The statewise retrospective reserves turn out to satisfy a set of retrospective differential equations, different from the prospective ones. It
is pointed out the differential equations for the reserves have the Kolmogorov differential equations for the transition probabilities as special cases. Surpris-ingly, maybe, it is the retrospective equations that generalize the Kolmogorov forward equations, while the prospective equations generalize the Kolmogorov backward equations. In Section 6 some more examples are supplied.