Scand. Actuarial J . 1993; 2: 100-106
ORIGINAL ARTICLE
Identities for Present Values of Life Insurance Benefits
: RAGNAR NORBERG
Norberg R. Identities for present values of life insurance benefits. Scand. Actuarial J. 1993; 2: 100- 106.
The rule of integration by parts produces useful formulas for the present value of a payment stream. Applied to life insurance, utilizing the counting process nature of the development of the policy, the rule induces three classes of identities. some generalizing certain classical relationships between life annuities and assurances and some not hitherto encountered in the literature. Ke), ~i'ord.~: Life znsurunce, payment streams, discounting, counting processes.
1. INTRODUCTION A. Background
Consider a single life aged .I- whose remaining life time T has survival function of the form P ( T > t ) = ,p, = exp( -Sbp,, ,;d r ) , where ,LL, +, is the mortality intensity at age x
+
T . Assume that interest is earned at constant intensity 6 so that the annual discount rate is v = exp( -6). A classic in life insurance mathematics is the identitywhere a,,l =
S;1
cr,p, ds is the expected present value of a life annuity payable, contingent on survival, continuously with level intensity 1 up to time n, ,E, = c " ,p,is the expected present value of a pure endowment of 1 payable, contingent on survival, at time n, and
A!;;I
= ?;1 v T , p , p , + , d ~ is the expected present value of a term insurance wilh sum 1 payable immediately upon death within n years.Roberts (1984) extended ( 1.1) to more geh'eral non-level benefits and interest and pointed oul that it rests on the corresponding relationship between the present values. Hoem (1969) generalized (1.1) in another direction, to a multi-state life policy described by a lime continuous Markov chain. In a more recent paper by the author (Norberg. 1990) these two modes of generalization were merged and carried further.
B. Outline of tlte puper
Section 2 lists the basic concepts and notation needed in the sequel. The key to generalizations of (1.1) is the rule of integration by parts. Some useful variations of it are presented, and in Section 3 these are turned into lucid equivalent formulas for the present value of a general payment stream, which induce an untraditional class of formulas for present values of life insurance benefits. In Section 4 the counting process nature of the developmenl of a life insurance policy is utilized in an efficient
proof of the relationship expressing a general life annuity by endowments and assurances. Section 5 presents a class of relationships not hitherto encountered in the literature. The final Section 6 renders some comments on the results.
2. PREREQUISITES AND PRELIMINARIES A . Present vulues in liji insurance
The framework of our discussions is the theory of payment streams and discounting and the description of standard forms of payments in life insurance presented in Norberg ( 1990, 1991). For a brief outline see Norberg ( 1992), Paragraphs 2A and 3A. Let it suffice here to just introduce those elements that are needed in the present context.
A stream of payments commencing at time 0 is represented by a payment function A , which to each z ( 2 0 ) specifies the total amount A, paid during the time interval [0, t ] . It is right-continuous and of bounded variation. When payments are evaluated in accordance with the discount function 11, the present value at time 0 of the payments provided by A in the time interval ( t , u ] is
Throughout
j':
signifies integration over ( t , u] for u finite (andj,"
for u = co). We shall assume that o, = exp(-J&
6, ds), 6 , being the interest intensity at time z, hencedv, = - u,6, d r . (2.2)
A life insurance treaty specifies a set
2
= {0, 1, . . . , J ) of possible states of the policy, such that at any time after issue the policy is in one and only one state, commencing in state 0 at time 0, say. Let N{k count the number of transitions from state j to state k ( # j ) in the time interval (0, z ] . The counting processes (NJk),, + kare taken to be right-continuous and increasing with jumps of size 1, and it is assumed that only a finite number of jumps can occur in any finite time interval and that no two processes can jump.at the same time. Let I{ denote the indicator of the event that the policy stays in state j at time z. The processes (IJ)j,a are related to the counting processes by the fact that I' increases/decreases (by 1) upon a transition intolout of state j. Thus
where a dot in the place of a subscript signifies summation over that subscript, e.g. N" = C k , k + , Nlk,
We shall consider two forms of payments that are standard in insurance. First, the general life annuity provides payments during sojourns in a given state j. It has payment function of the form
102 R. Norberg Scand. Actuarial J . 2 where A' is a payment function specified in the contract. Second, the life assurance
provides payments upon transitions from a given state j to another given state k. It has payment function of the form
where a/ik is the sum payable upon a transition from j to k at time z.
B. Integration by parts
Let A and B be real-valued functions of bounded variation defined on the interval [ t , u]. In all that follows a key tool is the rule of integration by parts, which states that
where A , - = lim,,, A,. Subtracting and adding B,A, on the right and putting B, - B, =
j:l
dB,, gives the equivalent formLikewise, subtracting and adding B , A , on the right of ( 2 . 6 ) gives
In differential form ( 2 . 6 ) reads
The minus sign in A , is effective only at common points of discontinuity of A and B and can be left out if one of them is continuous.
3. GENERAL IDENTITIES FOR PRESENT VALUES
A . Three expressions for the present value of a payment stream
The class of identities we shall present is valid for any payment function, whether of the forms ( 2 . 4 ) - ( 2 . 5 ) or not. Putting v in the role of B in ( 2 . 6 ) - ( 2 . 8 ) and applying ( 2 . 2 ) , yields
= " ( A ,
-
A , )+
lu
v , ( A , - A , ) 6 , d7All these expressions are easy to interpret as they represent four different ways of disposing of the payments, which must all produce the same present value. For
instance, (3.2) corresponds to postponing the encashment of the total amount to time u and meanwhile cashing currently the interest earned on the savings, and (3.3) corresponds to cashing the total amount in advance as a loan at time t and thereafter paying interest currently on the outstanding principal until the loan is fully repaid at time u.
If the payment stream A is stochastic, the relationships are valid almost surely and, of course, remain valid if we take the expected value. For an insurance policy they generate a class of identities expressing expected present values as integrals of expected amounts. They seem not to be noted in previous literature.
B. Examples
Consider the term insurance described in Paragraph 1A. The amount paid during the time interval (O,T] is 1 , :,
,,
( I , denotes the indicator function of the event A), and its expected value is the probability of death in that interval, 1 - , p x . Using (3.2) (or either of the two other equivalent forms) with t = 0 and u = n, we obtain (1.1). This proof is different from the one usually seen.As another example, take the life annuity described in Paragraph lA, for which the amount paid during (0, r ] is
Jh
l{,,,, ds and the expected amount is e,;l =l'l,
, p , ds, the expected lifetime in (O,T]. From (3.2) and (3.3) we obtain4. LIFE ANNUITIES EXPRESSED BY ENDOWMENTS AND
LIFE ASSURANCES
A . The generul cluss of relationships
Consider the present value (2.1) of the general life annuity given by (2.4). Letting A' and u,I{ take the roles of A and B in (2.6), gives
u, 1: dA< = V , I', A i - v , 1'; Aj -
Iu
A/ - d(ur I:).Reshaping the differential in the last term as (use (2.9) and then (2.2) and (2.3))
104
R.
Norberg Scand. Actuarial J . 2 Other appearances of the identity are obtained if we start from (2.7) or (2.8). Forinstance, the latter gives
Taking expectation in (4.1) yields
The notation speaks for itself on the background of (4.1). It is explicated in Norberg ( 1990), where (4.3) was proved by application of (3.1) to the periods between transitions. The present proof is more efficient and, unlike the former, exhibits the underlying identity (4.1) between the present values.
B. Exumpfes
Again ( 1.1 ) is a special case, this time by considering a unit paid at time 0. Another special case of (4.1) is
where (TL?),~ =
l;;
tlZt,p, dt is the expected present value of an increasing life annuity payable, contingent on survival, with intensity T at time T up to time n,(TA)Lnl
=S;;U'Z
,p,pX+d~
is the expected present value of an increasing term insurance with sum T payable immediately upon death at time T < n . Startinginstead from (4.2), we find the equivalent
where (Da),,l and (DA)ld are the expected present values of the decreasing life annuity and assurance defined by replacing s with n - s in the definitions of the
corresponding increasing benefits. ,
.
5. LIFE ANNUITIES EXPRESSED BY DISCOUNTED CONTRACTUAL
PAYMENTS
A . The general class of relationships
Finally, we establish yet another type of expression for the present value of a general life annuity. Introduce
the present value at time 0 of payments provided in (0, T] by the contractual function A J (as if the policy were staying uninterruptedly in state j from time 0 to,
and including, time t). Then dA; = v, d A J . Letting
AJ
and It take the roles of A and B in (2.6), givesNow, use (2.3) again to obtain
The identity can be recast by starting from (2.7) or (2.8), the latter yielding
The relationship (5.2) says e.g. that a disability annuity is equivalent to placing the current cash value of the outstanding part of the contractual annuity to the creditldebit of the insured's account upon each onset/cessation of disability. The identities (4.1), (4.2), and (5.1) can be expounded in a similar way.
B. Example
Pursuing the second example in Paragraph 4B, we obtain from (5.1) and (5.2) the transparent formulas
where
4
=j;,
u.' d . ~ is the present value of a t-year annuity-certain.6. COMMENTS ON THE R E S ~ L T S A. Independence of model assumptions
All relationships encountered above are valid almost surely for the present values themselves, hence they are inherited by the expected present values irrespective of specific model assumptions. The only assumptions that are really needed are those made in Paragraph 2A about the sample path properties of the counting processes and the interest process. Thus, the counting processes need not be Markov, and the discount function may well be a stochastic process. In fact, the interest intensity need not exist; it suffices that the discount function is of bounded variation (see Norberg (1990)). It should also be noted that the contractual payments A' and aJk need not be deterministic, but can be allowed to depend on the past in a more or less complex manner.
106
R.
Norberg Scand. Actuarial J . 2B. Signrficance of the results
The results derived here are mainly of theoretical importance. They exhaust (presumably) a topic that has attracted considerable attention in the actuarial literature and occurs in countless theorems, examples, and exercises in textbooks. It is worthwhile identifying the general relationships underlying all the apparently dissociated special cases.
The classical relationship ( 1.1) is held to be of practical value as it can be turned to express an assurance by the simpler survival benefits. As we have seen in Sections 4 and 5, this is not the aspect that accommodates generalizations. On the contrary, the general life annuities are expressed by assurances. Clearly, relation (4.3) could never be helpful in evaluation of the present value on the left hand side. Hopefully, this piece of knowledge is useful to those who might have wondered about the practical potential of the kind of relationships studied here.
REFERENCES
Hoem. J. M. (1969). Markov chain models in life insurance. Bliitter der Deutschen Gesellschafi fur
F'rr\ictterungsmnlhematik 9, 91 - 107.
Norberg. R. (1990). Payment measures, interest, and d i s c o u n t i n g - an axiomatic approach with applica- tions to insurance. Scund. Actuurial J . 1990, 14--33.
Norberg, R. (1991). Reserves in life and pension insurance. Scund. Acttrurial J. 1991, 3-24.
Norbcrg, R . (1992). Hattendorff's theorem and Thiele's differential equation generalized. Scand.
Actuariul J. 1992, 2 - 14.
Roberts, L. (1984). Generalized annuities and assurances, and their inter-relationships. J. ln~cicure of
Actuurrrs 111, 555-564.
Received July 1993
Address for correspondence: Ragnar Norberg
Laboratory of Actuarial Mathematics Universitetsparken 5
DK-2100 Copenhagen U Denmark
