CHAPTER EIGHT
CONTINGENT ANNUITY MODELS
(LIFE ANNUITIES)
In this chapter we consider annuity models under which the making of each scheduled payment
is contingent upon some random event. The reader will note a degree of similarity in notation
and terminology with our discussion of interest-only annuities in Chapter 1. In fact, interestonly annuities can be viewed as special cases of contingent annuities where the scheduled
payments are made with probability 1. In other word, the making of the payments is not
contingent on any probabilistic event.
Generally, the random event upon which each payment is contingent is the continued survival
of a defined entity of interest, such as the continued survival of an identified person. Other
examples of contingent annuities could be a sequence of costs incurred as long as a labor strike
continues, or a sequence of coupon payment made as long as particular corporate bond remains
in good standing.
Contingent annuities are closely related to the contingent payment models of chapter 7. In the
prior chapter, a single payment is made at the time of the failure of the entity of interest; in this
chapter, a sequence of payment is made during the continued survival of the entity of interest,
up until the failure occurs.
8.1 WHOLE LIFE ANNUITY MODELS
In nearly all cases, discrete contingent annuity models are evaluated from a discrete survival
model (life table), such as that presented in chapter 6 of this text. Here we assume that time is
measured in years in the life table, so a contingent annuity with annual payment can be directly
evaluated from the table. In this section we will present the whole life model, in each of the
immediate, due, and continuous cases, and will analyze the model from several perspectives.
8.1.1 THE IMMEDIATE CASE
We begin by recalling the the n-year pure endowment model with present value random
variable 𝑍𝑥:𝑛1 defined by equation (7.20), and expected present value denoted by 𝐴𝑥:𝑛1 or
𝑛 𝐸𝑥 .
Suppose we have a status with identifying characteristic (x) as of time 0, such as a person
alive at time 0 at age x, and a sequence of unit payments scheduled to be made at the end of
each year as long as the status continues to survive. Such a model is called a whole life
contingent annuity-immediate model, since the payment sequence will continue for the whole
of life the status of interest. It Is easy to see that this arrangement is merely a series of t-year
pure endowments for t =1, 2, …. If we let 𝑌𝑥 denote the present value random variable for this
model, then we have
∞
𝑌𝑥 = ∑ 𝑍𝑥:𝑡1
(8.1)
𝑡=1
The expected value of the present value random variable 𝑌𝑥 is denoted 𝑎𝑥 , so we have
∞
𝑎𝑥 = 𝐸[𝑌𝑥 ] =
∞
∑ 𝐸[𝑍𝑥:𝑡1 ]
𝑡=1
=
∑ 𝐴𝑥:𝑡1
𝑡=1
∞
= ∑ 𝑡 𝐸𝑥
(8.2a)
𝑡=1
by using results and notation developed in Chapter 7. Furthermore, since 𝑃𝑟(𝐾𝑥 ≥ 𝑡) =
𝑡 𝑝𝑥
then from Equation (7.21) weh have 𝐴𝑥:𝑡1 = 𝑣 𝑡 . 𝑡 𝑝𝑥 so that Equation (8.2a) can also be written
as
∞
𝑎𝑥 = ∑ 𝑣 𝑡 . 𝑡 𝑝𝑥
(8.2b)
𝑡=1
Suppose 𝑙𝑥 persons each purchase a whole life annuity with unit payments at the end of each
year, with each person paying amount X to purchase the annuity. Then we have an initial fund
of 𝑋 · 𝑙𝑥 dollars at time 0. According to the life table model, 𝑙𝑥+1 persons survive to the end
of the first year, 𝑙𝑥+2 survive to the end of the second year, and so on, until there are no more
survivors. Then the sequence 𝑙𝑥+1, 𝑙𝑥+2, … represents the sequence of payments made under
the annuities in total. This is illustrated in the following figure.
FIGURE 8.1
The total present value at time 0 (age x) of all annuity payment made is
𝑣 · 𝑙𝑥+1 + 𝑣 2 · 𝑙𝑥+2 + 𝑣 3 · 𝑙𝑥+3 + ⋯,
Which is set equal to the initial fund. That is,
𝑋 · 𝑙𝑥 = 𝑣 · 𝑙𝑥+1 + 𝑣 2 · 𝑙𝑥+2 + 𝑣 3 · 𝑙𝑥+3 + ⋯,
(8.3)
If we divide both sides of equation (8.3) by 𝑙𝑥 we obtain
𝑋=𝑣 ·
𝑙𝑥+1
𝑙𝑥
+ 𝑣2 ·
𝑙𝑥+2
𝑙𝑥
+ 𝑣3 ·
𝑙𝑥+3
𝑙𝑥
+ ⋯,
(8.4a)
Where X represents each person’s share of the total present value of all annuity payment made.
Thus X is the net single premium that each of the 𝑙𝑥 persons should pay to purchase the annuity.
But the general term in equation (8.4a), namely
𝑙𝑥+𝑡
𝑙𝑥
, is simply 𝑡 𝑝𝑥 , the probability of survival
to time t. Thus we have
∞
𝑋 = ∑ 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.4b)
𝑡=1
There is another, very important, random variable approach to understanding the whole life
contingent annuity. If the status of interest fails in the 𝑘 𝑡ℎ time interval, denoted by the event
𝐾𝑥 = 𝑘 − 1, then exactly 𝑘 − 1 annuity payments are made, since no payment would be made
at time k (age x+k). This is represented in the following diagram.
FIGURE 8.2
thus if 𝐾𝑥 = 𝑘 − 1, the present value of the annuity is 𝑎𝑘−1|
̅̅̅̅̅̅̅ , so, in general, the present value
is a random variable denoted by 𝑌𝑥 = 𝑎̅̅̅̅̅
𝐾𝑥 | . The expected value of this present value random
variable is then
∞
∞
𝐸[𝑌𝑥 ] = 𝐸[𝑎̅̅̅̅
𝐾𝑥 | ] = ∑ 𝑎̅̅̅
𝑘| · 𝑃𝑟(𝐾𝑥 = 𝑘) = ∑
𝑘=0
𝑘=0
1 − 𝑣𝑘
· 𝑘 |𝑞𝑥
𝑖
(8.5a)
Using Equation (1.13) for 𝑎̅̅̅
𝑘| and the actuarial notation 𝑘 |𝑞𝑥 to represent 𝑃𝑟(𝐾𝑥 = 𝑘). Then
by using 𝑘 |𝑞𝑥 = 𝑘 𝑝𝑥 −
𝑘+1 𝑝𝑥
, we have
∞
1
𝐸[𝑌𝑥 ] = ∑[ 𝑘 |𝑞𝑥 − 𝑣 𝑘 ( 𝑘 𝑝𝑥 −
𝑖
𝑘+1 𝑝𝑥 )]
𝑘=0
∞
∞
∞
𝑘=0
𝑘=0
𝑘=0
1
𝐸[𝑌𝑥 ] = [∑ 𝑘 |𝑞𝑥 − ∑ 𝑣 𝑘 · 𝑘 𝑝𝑥 + (1 + 𝑖) ∑ 𝑣 𝑘+1 ·
𝑖
𝑘+1 𝑝𝑥 ]
The first summation clearly evaluates to 1 (since (x) must fail sometime), the second
summation evaluates to 1 + 𝑎𝑥 (from Equation (8.2b) and the fact that 0 𝑝𝑥 = 1), and the third
summation evaluates to (1 + i)𝑎𝑥 , again from Equation (8.2b). Thus we have
𝐸[𝑌𝑥 ] =
1
[1 − (1 + 𝑎𝑥 ) + (1 + 𝑖)𝑎𝑥 ] = 𝑎𝑥
𝑖
As already established by Equation (8.2a)
(8.5b)
This random variable approach to the whole life contingent annuity is particularly useful in
finding higher moment of the present value random variable 𝑌𝑥 . First we note that
𝑌𝑥 = 𝑎̅̅̅̅̅
𝐾𝑥 | =
1
1
1
[1 − 𝑣 𝐾𝑥 ] = [1 − (1 + 𝑖) · 𝑣 𝐾𝑥 +1 ] = [1 − (1 + 𝑖) · 𝑍𝑥 ]
𝑖
𝑖
𝑖
8.6
Where 𝑍𝑥 = 𝑣𝐾𝑥 +1 is defined by Equation (7.1a). We can now use this relationship between
the random 𝑌𝑥 and 𝑍𝑥 to find the variance of 𝑌𝑥 . We have
(1 + 𝑖)2 · 𝑉𝑎𝑟(𝑍𝑥 ) 𝑉𝑎𝑟(𝑍𝑥 )
1 − (1 + 𝑖) · 𝑍𝑥
𝑉𝑎𝑟(𝑌𝑥 ) = 𝑉𝑎𝑟 [
]=
=
𝑖
𝑖2
𝑑2
8.7a
𝑖
Since 𝑑 = 1+𝑖. Then using Equation (7.5) for Var(𝑍𝑥 ) we have
2
𝐴𝑥 − 𝐴𝑥 2
𝑉𝑎𝑟(𝑌𝑥 ) =
𝑑2
(8.7b)
Where 𝐴𝑥 and 2 𝐴𝑥 are defined in section 7.1.2
8.1.2.THE DUE CASE
In the annuity-due case, the whole life contingent annuity model is the same as in the annuity
immediate case, except that payments are scheduled at the beginning of each year instead of
the end. Since the status (x) is known to exist at time 0, then the payment scheduled at time 0
(age x) is certain to be made. Thus we have the same series of contingent payments as in the
immediate case, plus the certain payment of 1 at time 0. If we let 𝑌𝑥̈ denoted the present value
random variable in the annuity-due case, then we have
∞
𝑌𝑥̈ = 1 + 𝑌𝑥 = 1 + ∑ Z𝑥:𝑡1
(8.8)
𝑡=1
Then the expected value of 𝑌𝑥̈ , denoted 𝑎̈ 𝑥 , is easily found as
𝑎̈ 𝑥 = 𝐸[𝑌𝑥̈ ] = 1 + 𝐸[𝑌𝑥 ]
∞
= 1 + ∑ 𝑡 𝐸𝑥
𝑡=1
∞
= ∑ 𝑡 𝐸𝑥
𝑡=0
∞
= ∑ 𝑣 𝑡 . 𝑡 𝑝𝑥
𝑡=0
(8.9)
Where we note that 0 𝐸𝑥 = 𝑣 0 . 0 𝑝𝑥 = 1. Furthermore, since 𝐸[𝑌𝑥 ] = 𝑎𝑥 we have the important
relationship
𝑎̈ 𝑥 = 𝑎𝑥 + 1
(8.10)
Again we note 𝑎̈ 𝑥 = 𝐸[𝑌𝑥̈ ] is called the EPV or APV or NSP for the whole life contingent
annuity-due.
The group deterministic interpretation is parallel to that in the immediate case, except that there
will be a payment of 𝑙𝑥 dollars at time 0 (age x) itself. Thus we would modify Equation (8.3)
to read
𝑋 · 𝑙𝑥 = 𝑙𝑥 + 𝑣 · 𝑙𝑥+1 + 𝑣 2 · 𝑙𝑥+2 + ⋯
(8.11)
and Equation (8.4a) to read
𝑋=
Substituting 𝑡 𝑝𝑥 for
𝑙𝑥+𝑡
𝑙𝑥
𝑙𝑥
𝑙𝑥+1
𝑙𝑥+2
+𝑣·
+ 𝑣2 ·
+⋯
𝑙𝑥
𝑙𝑥
𝑙𝑥
(8.12a)
𝑙
, for 𝑡 = 1,2, …, and 𝑣 0 · 0 𝑝𝑥 = 1 for 𝑙𝑥 , we reach
𝑥
∞
𝑋 = ∑ 𝑣 𝑡 . 𝑡 𝑝𝑥
(12.b)
𝑡=0
showing again that the NSP and the EPV are the same.
With respect to the random variable approach in the case of the annuity-due, we note that if
failure occurs in the 𝑘 𝑡ℎ time interval, as indicated by the event 𝐾𝑥 = 𝑘 − 1, then exactly k
Annuity payments are made since a payment is made at the beginning of the 𝑘 𝑡ℎ year itself (at
age x+k-1).
FIGURE 8.3
In general, the present value is a random variable denoted by
𝑌𝑥̈ = 𝑎̈ ̅̅̅̅̅̅̅̅̅
𝐾𝑥 +1| =
1 − 𝑣 𝐾𝑥 +1
𝑑
(8.13)
With expected value given by
∞
𝐸[𝑌𝑥̈ ] = 𝐸[𝑎̈ ̅̅̅̅̅̅̅̅
𝐾𝑥 +1| ] = ∑ 𝑎̈ ̅̅̅
𝑘| · 𝑃𝑟 (𝐾𝑥 = 𝑘 − 1)
𝑘=1
8.14
Since 𝑌𝑥̈ = 1 + 𝑌𝑥 , it follows immediately that
𝑉𝑎𝑟(𝑌𝑥̈ ) = 𝑉𝑎𝑟(𝑌𝑥 ) =
2
𝑉𝑎𝑟(𝑍𝑥 )
𝐴𝑥 − 𝐴𝑥 2
=
𝑑2
𝑑2
(8.15)
as established by Equation (8.7b)
8.1.3 THE CONTINUOUS CASE
In this section we return to the abstract notion of continuous payment, introduced in section
1.2.3 in case of interest-only annuities. Although continuous payment annuities cannot exist in
practice, there is some theoretical value in studying them. Furthermore, continuous annuities
might be considered good approximations to annuities payable very frequently, such as weekly
or event monthly.
For the continuous case, we return to the future lifetime random variable 𝑇𝑥 , defined in Section
7.3.1, in place of the discrete duration at failure random 𝐾𝑥 , used thus far in this chapter. If
failure occurs at precise time t, which is measured in years and denoted by the event 𝑇𝑥 = 𝑡,
for the status of interest with identifying characteristic (x) at time 0, then continuous annuity
payment (at an annual rate of 1 unit of money) will be made for exactly t year. The present
value of this continuous annuity is 𝑎̅𝑡|̅ , so, in general, the present value is random variable
which we denoted by
𝑌̅𝑥 = 𝑎̅ ̅̅̅̅
𝑇𝑥 | =
1 − 𝑣 𝑇𝑥
𝛿
(8.16)
The expected value of this present value random variable, denoted 𝑎̅𝑥 , is given by
∞
𝑎̅𝑥 = 𝐸[𝑌̅𝑥 ] = 𝐸[𝑎̅ ̅̅̅̅
̅𝑡|̅ · 𝑓𝑥 (𝑡)𝑑𝑡
𝑇𝑥 | ] = ∫ 𝑎
(8.17a)
0
Where 𝑓𝑥 (𝑡) is the probability density function of the random variable 𝑇𝑥 . Recall from section
5.3 that this PDF is given by 𝑓𝑥 (𝑡) = 𝑡 𝑝𝑥 · 𝜇𝑥+𝑡 . Then we can write Equation (8.17a) as
∞
𝑎̅𝑥 = ∫ 𝑎̅𝑡|̅ · 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
0
We evaluate the integral using integration by parts to obtain
(8.17b)
∞
∞
∞
∫0 𝑎̅𝑡̅| 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
∞
𝑡
=
−𝑎
̅
·
𝑝
+
∫
𝑣
·
𝑝
𝑑𝑡
=
∫
𝑣 𝑡 · 𝑡 𝑝𝑥 𝑑𝑡
̅
𝑡 𝑥 0
𝑡 𝑥
𝑡|
𝑣 𝑡 𝑑𝑡
−𝑡 𝑝𝑥
0
0
Since 𝑎̅𝑡|̅ · 𝑡 𝑝𝑥
∞
is 0 at both the upper and lower limits. Thus we have
0
∞
𝑎̅𝑥 = ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝑑𝑡
(8.17c)
0
a convenient form evaluating 𝑎̅𝑥 from some parametric survival models with known
conditional survival function 𝑡 𝑝𝑥 =
𝑆0 (𝑥+𝑡)
𝑆0 (𝑥)
Returning to the random variable 𝑌̅𝑥 , we observe that it is closely related to the random variable
𝑍̅𝑥 = 𝑣 𝑇𝑥 , defined by Equation (7.34) we have
𝑌̅𝑥 = 𝑎̅ ̅̅̅̅
𝑇𝑥 | =
1 − 𝑣 𝑇𝑥 1 − 𝑍̅𝑥
=
𝛿
𝛿
(8.18a)
Which we can also write as
𝑍̅𝑥 + 𝛿. 𝑌̅𝑥 = 1
(8.18b)
Equation (8.18a) enables us to easily find the variance of 𝑌̅𝑥 as
2 ̅
1 − 𝑍̅𝑥
𝑉𝑎𝑟(𝑍𝑥̅ )
𝐴𝑥 − 𝐴̅𝑥 2
𝑉𝑎𝑟(𝑌̅𝑥 ) = 𝑉𝑎𝑟 (
)=
=
𝛿
𝛿2
𝛿2
(8.19)
8.2 TEMPORARY LIFE ANNUITY MODELS
8.2.1 THE IMMEDIATE CASE
The immediate n-year temporary life annuity, payable to a status with identifying characteristic
(x) at time 0, will make a payment at the end of each year for n years at the most, provided the
status continues to survive. If we let 𝑌𝑥:𝑛|
̅̅̅ denote the present value random variable for this
model, then in terms of a series of t-year pure endowments we have
𝑛
1
𝑌𝑥:𝑛|
̅̅̅ = ∑ 𝑍𝑥:𝑡
𝑡=1
(8.20)
The expected value of this present value random variable (EPV) is denoted 𝑎𝑥:𝑛|
̅̅̅ and is given
by
𝑎𝑥:𝑛|
̅̅̅ = 𝐸[𝑌𝑥:𝑛|
̅̅̅ ]
𝑛
= ∑ E[Z𝑥:𝑡1 ]
(8.21)
𝑡=1
𝑛
=
𝑛
∑ 𝐴𝑥:𝑡1
𝑡=1
𝑛
= ∑ 𝑡 𝐸𝑥 = ∑ 𝑣 𝑡 · 𝑡 𝑝𝑥
𝑡=1
𝑡=1
To analyze the random variable approach to the temporary immediate annuity, consider the
following diagram
FIGURE 8.4
If failure 𝑘 𝑡ℎ time interval, where 𝑘 ≤ 𝑛, then k – 1 payments are made, with the last payment
made at the end of the interval preceding the interval of failure, so the present value of payments
will be 𝑎̅̅̅̅̅̅̅
𝑘−1| . But if failure occurs after age x + n, so that 𝐾𝑥 ≥ 𝑛, then n payments will be
made and the present value of payments will be 𝑎̅̅̅
𝑛| .
Therefore the present value random variable 𝑌𝑥:𝑛|
̅̅̅ is defined as
𝑌𝑥:𝑛|
̅̅̅ = {
𝑎̅̅̅̅̅
𝐾𝑥 | 𝑓𝑜𝑟 𝐾𝑥 < 𝑛
𝑎̅̅̅
𝑛| 𝑓𝑜𝑟 𝐾𝑥 ≥ 𝑛
(8.22)
With expected value given by
𝑛−1
∞
𝐸[𝑌𝑥:𝑛|
̅̅̅ ] = ∑ 𝑎̅̅̅
𝑘| · 𝑃𝑟 ( 𝐾𝑥 = 𝑘) + ∑ 𝑎̅̅̅
𝑛| · 𝑃𝑟 (𝐾𝑥 = 𝑘)
𝑘=0
(8.23)
𝑘=𝑛
In contingent annuities-immediate we have the analogous concept of the actuarial accumulated
value (AAV), which is denoted by 𝑠𝑥:𝑛|
̅̅̅ and is related to the actuarial present value (APV) 𝑎𝑥:𝑛|
̅̅̅
by
𝑠𝑥:𝑛|
̅̅̅
(1 + 𝑖)𝑛 · 𝑙𝑥
1
1
= 𝑎𝑥:𝑛|
= 𝑎𝑥:𝑛|
= 𝑎𝑥:𝑛|
̅̅̅ ·
̅̅̅ · 𝑛
̅̅̅ ·
𝑣 𝑛 𝑝𝑥
𝑙𝑥+𝑛
𝑛 𝐸𝑥
(8.24)
FIGURE 8.5
Suppose each of the 𝑙𝑥+1 survivors deposits one unit of money in a fund at time t =1, each of
the 𝑙𝑥+2 survivors do the same at time t = 2, and so on, with each of the 𝑙𝑥+𝑛 survivors doing
the same time at time t = n. Suppose the fund accumulates at compound interest rate i, and the
total accumulated fund is then distributed equally among the 𝑙𝑥+𝑛 survivors at time n. The share
of each of the 𝑙𝑥+𝑛 survivors would then be
𝑋=
𝑙𝑥+1 (1 + 𝑖)𝑛−1 + 𝑙𝑥+2 (1 + 𝑖)𝑛−2 + ⋯ + 𝑙𝑥+𝑛
𝑙𝑥+𝑛
𝑋=
(1 + 𝑖)𝑛 · 𝑙𝑥 𝑙𝑥+1 (1 + 𝑖)𝑛−1 + 𝑙𝑥+2 (1 + 𝑖)𝑛−2 + ⋯ + 𝑙𝑥+𝑛
[
]
𝑙𝑥+𝑛
(1 + 𝑖)𝑛 · 𝑙𝑥
(1 + 𝑖)𝑛 · 𝑙𝑥 𝑣 · 𝑙𝑥+1 + 𝑣 2 · 𝑙𝑥+2 + ⋯ + 𝑣 𝑛 · 𝑙𝑥+𝑛
𝑋=
[
]
𝑙𝑥+𝑛
𝑙𝑥
𝑋=
(1 + 𝑖)𝑛 · 𝑙𝑥
(1 + 𝑖)𝑛 · 𝑙𝑥
[𝑣 · 𝑝𝑥 + 𝑣 2 · 2 𝑝𝑥 + ⋯ + 𝑣 𝑛 · 𝑛 𝑝𝑥 ] =
· 𝑎𝑥:𝑛|
̅̅̅
𝑙𝑥+𝑛
𝑙𝑥+𝑛
8.2.2 THE DUE CASE
If contingent payments are made at the beginning of each year instead of the end, but for n
years at the most and contingent on the continued survival of (x), then we have n-year
temporary annuity-due model.
The present value random variable is denoted 𝑌̈𝑥:𝑛|
̅̅̅ and is given as a series of pure endowments
by
𝑛−1
1
𝑌̈𝑥:𝑛|
̅̅̅ = ∑ Z𝑥:𝑡
(8.25)
𝑡=0
The expected value of 𝑌̈𝑥:𝑛|
̅̅̅ denoted 𝑎̈ 𝑥:𝑛|
̅̅̅ is given by
𝑛−1
1
𝑎̈ 𝑥:𝑛|
̅̅̅ = 𝐸[𝑌̈𝑥:𝑛|
̅̅̅ ] = ∑ E[𝑍𝑥:𝑡 ]
𝑡=0
𝑛−1
𝑛−1
(8.26)
= ∑ 𝑡 𝐸𝑥 = ∑ 𝑣 𝑡 · 𝑡 𝑝𝑥
𝑡=0
𝑡=0
Comparing Equation (8.20) and (8.25) it is easy to see that 𝑌̈𝑥:𝑛|
̅̅̅ is related to 𝑌𝑥:𝑛|
̅̅̅ by
1
𝑌̈𝑥:𝑛|
̅̅̅ = 𝑌𝑥:𝑛|
̅̅̅ + 1 − 𝑍𝑥:𝑛
(8.27)
Since 𝑍𝑥:01 = 1 then it follows that
1
𝑎̈ 𝑥:𝑛|
̅̅̅ = 𝐸[𝑌̈𝑥:𝑛|
̅̅̅ ] = 𝐸[𝑌𝑥:𝑛|
̅̅̅ + 1 − Z𝑥:𝑛 ] = 𝑎𝑥:𝑛|
̅̅̅ + 1 − 𝑛 𝐸𝑥
(8.28)
𝑛
(Note the similarity of Equation (8.28) to the interest-only relationship 𝑎̈ ̅̅̅
̅̅̅ + 1 − 𝑣
𝑛| = 𝑎𝑛|
For the random variable approach in the annuity-due case we have:
𝑌̈𝑥:𝑛|
̅̅̅ = {
𝑎̈ ̅̅̅̅̅̅̅̅̅
𝐾𝑥 +1| 𝑓𝑜𝑟 𝐾𝑥 < 𝑛
𝑎̈ ̅̅̅
𝑓𝑜𝑟 𝐾𝑥 ≥ 𝑛
𝑛|
(8.29 a)
With expected value given by
𝑛−1
∞
𝐸[𝑌̈𝑥:𝑛|
̅̅̅ ] = ∑ 𝑎̈ ̅̅̅̅̅̅̅
̅̅̅ · 𝑃𝑟 (𝐾𝑥 = 𝑘)
𝑘+1| · 𝑃𝑟 ( 𝐾𝑥 = 𝑘) + ∑ 𝑎̈ 𝑛|
𝑘=0
(8.30)
𝑘=𝑛
By writing Equations (8.29a) as
𝑌̈𝑥:𝑛|
̅̅̅
1 − 𝑣 𝐾𝑥 +1
𝑑
=
1 − 𝑣𝑛
{
𝑑
𝑓𝑜𝑟 𝐾𝑥 < 𝑛
(8.29b)
𝑓𝑜𝑟 𝐾𝑥 ≥ 𝑛
It then follows that
𝑌̈𝑥:𝑛|
̅̅̅ =
1 − 𝑍𝑥:𝑛|
̅̅̅
(8.31)
𝑑
Where
𝑍𝑥:𝑛|
̅̅̅ = {
𝑣 𝐾𝑥 +1 𝑓𝑜𝑟 𝐾𝑥 < 𝑛
𝑣𝑛
𝑓𝑜𝑟 𝐾𝑥 ≥ 𝑛
Was given Equation (7.24). Taking the expectation in Equation (8.31) obtain
𝑎̈ 𝑥:𝑛|
̅̅̅ = 𝐸[𝑌̈𝑥:𝑛|
̅̅̅ ] =
1 − 𝐸[𝑍𝑥:𝑛|
̅̅̅ ]
𝑑
=
1 − 𝐴𝑥:𝑛|
̅̅̅
𝑑
(8.32a)
Which is often stated as
𝐴𝑥:𝑛|
̅̅̅ = 1 − 𝑑 · 𝑎̈ 𝑥:𝑛|
̅̅̅
(8.32b)
Taking the variance in Equation (8.31) we find
𝑉𝑎𝑟 = (𝑌̈𝑥:𝑛|
̅̅̅ ) =
𝑉𝑎𝑟 (𝑍𝑥:𝑛|
̅̅̅ )
𝑑2
2
=
2
𝐴𝑥:𝑛|
̅̅̅ − 𝐴𝑥:𝑛|
̅̅̅
𝑑2
(8.33)
Returning now to the temporary annuity-immediate case, we see that
𝑌𝑥:𝑛|
̅̅̅ = 𝑌̈𝑥:𝑛+1|
̅̅̅̅̅̅̅ − 1
(8.34)
The actuarial accumulated value in the annuity-due case is denoted by 𝑠̈ 𝑥:𝑛|
̅̅̅ and is given by
𝑆̈𝑥:𝑛|
̅̅̅ = 𝑎̈ 𝑥:𝑛|
̅̅̅ ·
1
(1 + 𝑖)𝑛 · 𝑙𝑥
=
· 𝑎̈ 𝑥:𝑛|
̅̅̅
𝑙𝑥+𝑛
𝑛 𝐸𝑥
(8.35)
8.2.3 THE CONTINUOUS CASE
As with the whole life continuous model in section 8.1.3, we return to the future lifetime
random variable 𝑇𝑥 in place of the discrete duration at failure random variable 𝐾𝑥 used thus far
in this section. We consider that payment is made continuously at annual rate 1 up to the time
of failure of status (x), but for n years at the most. If 𝑇𝑥 = 𝑡, for 𝑡 ≤ 𝑛, the present value of
payment is 𝑎̅𝑡|̅ . If 𝑇𝑥 = 𝑡 for 𝑡 > 𝑛, then the present value of payment is 𝑎̅̅̅̅
𝑛| . Together we
have
𝑌̈𝑥:𝑛|
̅̅̅ = {
𝑎̅ ̅̅̅̅
𝑓𝑜𝑟 𝑇𝑥 ≤ 𝑛
𝑇𝑥 |
𝑎̅̅̅̅
𝑛| 𝑓𝑜𝑟 𝑇𝑥 > 𝑛
(8.36)
The expected value of the continuous present value random variable, denoted 𝑎̅𝑥:𝑛|
̅̅̅ is therefore
𝑛
𝑎̅𝑥:𝑛|
̅̅̅
= 𝐸[𝑌̅𝑥:𝑛|
̅𝑡|̅ · 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + 𝑎̅̅̅̅
̅̅̅ ] = ∫ 𝑎
𝑛| · 𝑃𝑟( 𝑇𝑥 > 𝑛)
(8.37a)
0
Substituting 𝑎̅𝑡|̅ = (1 − 𝑣 𝑡 )/𝛿 and 𝑃𝑟( 𝑇𝑥 > 𝑛) =
𝑎̅𝑥:𝑛|
̅̅̅
𝑛 𝑝𝑥 ,
we have
1 𝑛
= [∫ (1 − 𝑣 𝑡 ) · 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 + (1 − 𝑣 𝑛 ) · 𝑛 𝑝𝑥 ]
𝛿 0
=
𝑛
𝑛
1
[ 𝑛 𝑝𝑥 − 𝑣 𝑛 · 𝑛 𝑝𝑥 + ∫ 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 − ∫ 𝑣 𝑡 · 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡]
𝛿
0
0
The first integral inside the bracket evaluates to
𝑛 𝑞𝑥
= 1 − 𝑛 𝑝𝑥
And the second integral evaluate (using integration by parts) to
𝑛
𝑛
1 − 𝑣 . 𝑛 𝑝𝑥 − 𝛿 ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝑑𝑡
0
Substituting we have
𝑎̅𝑥:𝑛|
̅̅̅
𝑛
𝑛
1
𝑛
𝑛
𝑡
= [ 𝑛 𝑝𝑥 − 𝑣 · 𝑛 𝑝𝑥 + 1 − 𝑛 𝑝𝑥 − 1 + 𝑣 · 𝑛 𝑝𝑥 + 𝛿 ∫ 𝑣 · 𝑡 𝑝𝑥 𝑑𝑡] = ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝑑𝑡
𝛿
0
0
8.37b
We recall from Equation (7.38a)
𝑣 𝑇𝑥 for
1
1
𝑍̅𝑥:𝑛|
̅̅̅ = 𝑍̅𝑥:𝑛 + 𝑍̅𝑥:𝑛 = { 𝑛
𝑣
for
𝑇𝑥 ≤ 𝑛
𝑇𝑥 > 𝑛
Comparing this with Equation (8.36) for 𝑌̅𝑥:𝑛|
̅̅̅ we observe that
𝑌̅𝑥:𝑛|
̅̅̅ =
1 − 𝑍̅𝑥:𝑛|
̅̅̅
(8.38)
𝛿
Taking the expectation in Equation (8.38)
̅ 𝑥:𝑛|
𝑎̅𝑥:𝑛|
̅̅̅ = 𝐸[𝑌
̅̅̅ ] =
1 − 𝐸[𝑍̅𝑥:𝑛|
̅̅̅ ]
𝛿
=
1 − 𝐴̅𝑥:𝑛|
̅̅̅
𝛿
(8.39a)
Which is often stated as
𝐴̅𝑥:𝑛|
̅𝑥:𝑛|
̅̅̅ = 1 − 𝛿 · 𝑎
̅̅̅
(8.39b)
Taking the variance in Equation (8.38) we find
𝑉𝑎𝑟(𝑌̅𝑥:𝑛|
̅̅̅ ) =
𝑉𝑎𝑟(𝑍̅𝑥:𝑛|
̅̅̅ )
𝛿2
2
=
2
𝐴̅𝑥:𝑛|
̅̅̅ − 𝐴̅𝑥:𝑛|
̅̅̅
𝛿2
(8.40)
̅
Finally, the actuarial accumulated value in the continuous case is denoted by 𝑆𝑥:𝑛|
and is given
̅̅̅̅̅̅
by
̅ ̅̅̅ = 𝑎̅𝑥:𝑛|
𝑆𝑥:𝑛|
̅̅̅ ·
1
𝑛 𝐸𝑥
(1 + 𝑖)𝑛 · 𝑙𝑥 𝑛 𝑡
=
∫ 𝑣 · 𝑡 𝑝𝑥 𝑑𝑡
𝑙𝑥+𝑛
0
𝑛
= ∫ (1 + 𝑖)𝑛−𝑡 ·
0
𝑙𝑥+𝑡
𝑑𝑡
𝑙𝑥+𝑛
(8.41)
𝑛
(1 + 𝑖)𝑛−𝑡
=∫
𝑑𝑡
𝑛−𝑡 𝑝𝑥+𝑡
0
8.3 DEFERRED WHOLE LIFE ANNUITY MODELS
8.3.1 THE IMMEDIATE CASE
Payments under the n-year deferred whole life annuity-immediate are illustrated in the
following diagram
FIGURE 8.6
Note that the first payment is at time t = n+1, provide (x) has not yet failed. The payments are
deferred for n years, so the first payment is for the (𝑛 + 1)𝑠𝑡 year. Since the annuity is
immediate, the first payment is therefore at t = n+1. The last payment is at the end of the year
preceding the year of failure, provided failure does not occur so early that payment never begins
at all.
The present value random variable, which by 𝑛 |𝑌𝑥 is given by
∞
𝑛 |𝑌𝑥
= ∑ 𝑍𝑥:𝑡1
(8.42)
𝑡=𝑛+1
Together these two models provide the same payments as the whole life model of section 8.1.
Therefore it follows that
𝑌𝑥 = 𝑌𝑥:𝑛|
̅̅̅ + 𝑛 |𝑌𝑥
(8.43a)
Or
𝑛 |𝑌𝑥
= 𝑌𝑥 − 𝑌𝑥:𝑛|
̅̅̅
(8.43)
Then the expected present value of the immediate deferred model is
∞
𝑛 |𝑎𝑥
𝑡
= 𝐸[ 𝑛 |𝑌𝑥 ] = 𝐸[𝑌𝑥 − 𝑌𝑥:𝑛|
̅̅̅ ] = 𝑎𝑥 − 𝑎𝑥:𝑛|
̅̅̅ = ∑ 𝑣 · 𝑡 𝑝𝑥
(8.44)
𝑡=𝑛+1
Using the change of variable 𝑠 = 𝑡 − 𝑛, so that 𝑡 = 𝑠 + 𝑛, we can rewrite Equation (8.44) as
∞
𝑛 |𝑎𝑥
= ∑𝑣
𝑠=1
∞
𝑠+𝑛
·
𝑠+𝑛 𝑝𝑥
𝑛
= 𝑣 · 𝑛 𝑝𝑥 ∑ 𝑣 𝑠 ·𝑠 𝑝𝑥+𝑛 =
𝑠=1
𝑛 𝐸𝑥
· 𝑎𝑥+𝑛
(8.45)
𝑛
Note the analogy with annuities-certain, where we have 𝑛 |𝑎̅̅̅̅
̅̅̅̅ . There we discount
𝑚| = 𝑣 · 𝑎𝑚|
the value 𝑎̅̅̅̅
𝑚| back n years at interest only. Here we discount 𝑎𝑥+𝑛 back n years for both interest
and probability of survival.
𝑛 |𝑌𝑥
0
= {𝑣 𝑛 · 𝑎
for 𝐾𝑥 < n
for 𝐾𝑥 ≥ n
̅̅̅̅̅̅̅̅̅
𝐾𝑥 −𝑛|
8.46
∞
8.47
𝑛
𝑛 |𝑎𝑥 = 𝐸[ 𝑛 |𝑌𝑥 ] = ∑ 𝑣 · 𝑎̅̅̅̅̅̅̅
𝑘−𝑛| · 𝑃𝑟 (𝐾𝑥 = 𝑘)
𝑘=𝑛
8.3.2 THE DUE CASE
For the n-year deferred whole life annuity-due, the first payment is at t = n, provided (x) has
survived to that point. The present value random variable is
∞
̈ = ∑𝑍 1
𝑥:𝑡
𝑛 |𝑌𝑥
(8.48)
𝑡=𝑛
With expected value given by
∞
𝑛 |𝑎̈ 𝑥
= 𝐸[ 𝑛 |𝑌𝑥̈ ] =
∞
∑ 𝐸[𝑍𝑥:𝑡1 ]
𝑡=𝑛
= ∑ 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.48b)
𝑡=𝑛
It should
̈ = 𝑌𝑥̈ − 𝑌𝑥:𝑛
̈ ̅|
𝑛 |𝑌𝑥
(8.49)
So that
𝑛 |𝑎̈ 𝑥
= 𝑎̈ 𝑥 − 𝑎̈ 𝑥:𝑛̅|
∞
𝑛 |𝑎̈ 𝑥
= ∑𝑣
(8.50)
∞
𝑠+𝑛
·
𝑠+𝑛 𝑝𝑥
𝑛
𝑠
= 𝑣 · 𝑛 𝑝𝑥 ∑ 𝑣 · 𝑠 𝑝𝑥+𝑛 =
𝑠=0
(8.51)
𝑛 𝐸𝑥
· 𝑎̈ 𝑥+𝑛
𝑠=0
8.3.3 THE CONTINUOUS CASE
If failure occurs at time 𝑇𝑥 = 𝑡, for 𝑡 > 𝑛, then continuous payment will be made from time n
to time t. The present value of this payment at 𝑡 = 0 is 𝑣 𝑛 · 𝑎̅𝑡−𝑛
̅̅̅̅̅̅| . The present value random
variable is
̅ ={
𝑛 |𝑌𝑥
0
𝑓𝑜𝑟 𝑇𝑥 ≤ 𝑛
𝑛
𝑣 . 𝑎̅ 𝑇̅̅̅̅̅̅̅
𝑓𝑜𝑟 𝑇𝑥 > 𝑛
𝑥 −𝑛|
(8.52a)
With expected value given by
∞
̅𝑥 = 𝐸[ 𝑛 |𝑌̅𝑥 ] = ∫ 𝑣 𝑛 · 𝑎̅𝑡−𝑛
̅̅̅̅̅̅| · 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
𝑛 |𝑎
(8.52b)
0
Using the variable change 𝑠 = 𝑡 − 𝑛, so that 𝑡 = 𝑠 + 𝑛, we have
∞
̅𝑥
𝑛 |𝑎
∞
𝑛
= ∫ 𝑣 · 𝑎̅𝑠̅| ·
𝑠+𝑛 𝑝𝑥
𝑛
𝜇𝑥+𝑠+𝑛 = 𝑣 · 𝑛 𝑝𝑥 ∫ 𝑎̅𝑠̅| · 𝑠 𝑝𝑥+𝑛 𝜇𝑥+𝑛+𝑠 𝑑𝑠 =
0
8.53
̅𝑥+𝑛
𝑛 𝐸𝑥 . 𝑎
0
Returning to Equation (8.52b) we can write
̅𝑥 =
𝑛 |𝑎
=
1 ∞ 𝑛
∫ 𝑣 (1 − 𝑣 𝑡−𝑛 ). 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡
𝛿 𝑛
∞
1 𝑛 ∞
[𝑣 ∫ 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡 − ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝜇𝑥+𝑡 𝑑𝑡]
𝛿
𝑛
𝑛
∞
1 𝑛
𝑛
= [𝑣 · 𝑛 𝑝𝑥 − 𝑣 . 𝑛 𝑝𝑥 + 𝛿 ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝑑𝑡]
𝛿
𝑛
(8.54)
∞
= ∫ 𝑣 𝑡 . 𝑡 𝑝𝑥 𝑑𝑡
𝑛
From the now-familiar integration by parts technique. It is clear that
̅ = 𝑌̅𝑥 − 𝑌̅𝑥:𝑛̅|
𝑛 |𝑌𝑥
(8.55)
So that
̅𝑥
𝑛 |𝑎
= 𝑎̅𝑥 − 𝑎̅𝑥:𝑛̅|
(8.56)
The variance of 𝑛 |𝑌̅𝑥 is given by
2
𝑉𝑎𝑟[ 𝑛 |𝑌̅𝑥 ] = 𝐸 [ 𝑛 |𝑌̅𝑥 ] − (𝐸[ 𝑛 |𝑌̅𝑥 ])2 =
2 2𝑛
· 𝑣 · 𝑛 𝑝𝑥 (𝑎̅𝑥+𝑛 − 2 𝑎̅𝑥+𝑛 )−( 𝑛 |𝑎̅𝑥 )2
𝛿
(8.57)
8.4 SUMMARY OF ANNUAL PAYMENT ANNUITIES
TABLE 8.1
Annuity Function
Immediate
Due
Continuous
Whole Life APV
𝑎𝑥
𝑎̈ 𝑥
𝑎̅𝑥
Temporary APV
𝑎𝑥:𝑛|
̅̅̅
𝑎̈ 𝑥:𝑛|
̅̅̅
𝑎̅𝑥:𝑛|
̅̅̅
Temporary AAV
𝑆𝑥:𝑛|
̅̅̅
𝑆̈𝑥:𝑛|
̅̅̅
̅ ̅̅̅̅
𝑆𝑥:𝑛|
Deferred APV
𝑛 |𝑎𝑥
𝑛 |𝑎̈ 𝑥
̅𝑥
𝑛 |𝑎
8.5 LIFE ANNUITIES PAYABLE 𝒎𝒕𝒉𝒍𝒚
8.5.1 THE IMMEDIATE CASE
The whole life 𝑚𝑡ℎ𝑙𝑦 annuity-immediate pays an amount 1/m at the end of each (1/𝑚 )𝑡ℎ of a
year, provided the status (x) continues to survive. This is illustrated in the following diagram
specifically for m = 4.
FIGURE 8.7
(𝑚)
The actuarial present value of the annuity, which we denoted by 𝑎𝑥 , is given by
∞
(𝑚)
𝑎𝑥
1
= · ∑ 𝑣 𝑡⁄𝑚 ·𝑡⁄𝑚 𝑝𝑥
𝑚
(8.58)
𝑡=1
(𝑚)
Remember that the base symbol for the APV, 𝑎𝑥
in this case, represents a payment amount
of one unit per year, but payable 𝑚𝑡ℎ𝑙𝑦 within the year, so that each actual payment is amount
1/m.
For the temporary 𝑚𝑡ℎ𝑙𝑦 annuity-immediate, payment is made at once unit per year for n years
(𝑚)
at the most, but payable 𝑚𝑡ℎ𝑙𝑦 within the year. The APV is denoted by 𝑎𝑥:𝑛 is given by
𝑚𝑛
(𝑚)
𝑎𝑥:𝑛
1
= · ∑ 𝑣 𝑡⁄𝑚 ·𝑡⁄𝑚 𝑝𝑥
𝑚
(8.59)
𝑡=1
Note that the final payment (contingent on survival) is made at time t = n years, or 𝑡 = 𝑚𝑛 𝑚𝑡ℎ𝑠
of a year. As before, The actuarial accumulated value is
(𝑚)
(𝑚)
𝑆𝑥:𝑛 = 𝑎𝑥:𝑛 ·
1
𝑛 𝐸𝑥
(8.60)
The n – years deferred whole life 𝑚𝑡ℎ𝑙𝑦 annuity-immediate would make its first payment at the
end of the first 𝑚𝑡ℎ following the n – years deferral period, contingent on the survival of (x).
Its actuarial value would therefore be
(𝑚)
𝑛 |𝑎𝑥
1
= ·
𝑚
∞
∑ 𝑣 𝑡⁄𝑚 ·𝑡⁄𝑚 𝑝𝑥
𝑡=𝑚𝑛+1
(8.61)
8.5.2 THE DUE CASE
In the whole 𝑚𝑡ℎ𝑙𝑦 annuity-due case, the first payment of 1/m is made at time 0 (age x) itself,
and is made with probability 1 since the status (x) is known exist at that time. For the temporary
𝑚𝑡ℎ𝑙𝑦 annuity-due, the final payment (contingent on the survival (x), of course) is at the
beginning of the final 𝑚𝑡ℎ in the 𝑛𝑡ℎ . For the deferred 𝑚𝑡ℎ𝑙𝑦 annuity-due, the first payment is
at time n, the beginning of the first 𝑚𝑡ℎ following the deferral period. Thus the APVs in the
annuity-due case are
∞
(𝑚)
𝑎̈ 𝑥
1
= · ∑ 𝑣 𝑡⁄𝑚 ·
𝑚
𝑡/𝑚 𝑝𝑥
(8.62)
𝑡=0
𝑚𝑛−1
1
· ∑ 𝑣 𝑡⁄𝑚 · 𝑡/𝑚 𝑝𝑥
𝑚
(𝑚)
𝑎̈ 𝑥:𝑛 =
(8.63)
𝑡=0
∞
(𝑚)
𝑛 |𝑎̈ 𝑥
1
= · ∑ 𝑣 𝑡⁄𝑚 ·
𝑚
𝑡/𝑚 𝑝𝑥
(8.64)
𝑡=𝑚𝑛
For the whole life, temporary, and deferred whole file models, respectively. As before, the
actuarial accumulated value of the temporary 𝑚𝑡ℎ𝑙𝑦 annuity-due is
̈ (𝑚) = 𝑎̈ (𝑚) ·
𝑆𝑥:𝑛
𝑥:𝑛
1
𝑛 𝐸𝑥
(8.65)
An analogous set of identities to those developed in the annual payment case exist in the 𝑚𝑡ℎ𝑙𝑦
payment case as well, such as
(𝑚)
𝑛 |𝑎𝑥
=
𝑛 𝐸𝑥
· 𝑎𝑥+𝑛
(𝑚)
(8.66)
(𝑚)
𝑛 |𝑎̈ 𝑥
=
𝑛 𝐸𝑥
· 𝑎̈ 𝑥+𝑛
(𝑚)
(8.67)
(𝑚)
(𝑚)
(𝑚)
(𝑚)
(𝑚)
= 𝑎𝑥:𝑛 + 𝑛 |𝑎𝑥
(𝑚)
= 𝑎̈ 𝑥:𝑛 + 𝑛 |𝑎̈ 𝑥
(8.69)
1
𝑚
(8.70)
𝑎𝑥
𝑎̈ 𝑥
(8.68)
It should also be clear that
(𝑚)
𝑎̈ 𝑥
(𝑚)
= 𝑎𝑥
+
8.5.3 RANDOM VARIABLE ANALYSIS
Contingent 𝑚𝑡ℎ𝑙𝑦 annuity models can be analyzed in a random variable framework totally
(𝑚)
parallel to that presented for the annual payment cases. Recall the random variable 𝐾𝑥
defined in Section 7.3.4 as the 𝑚𝑡ℎ𝑙𝑦 curtate duration at which the status of interest fails. Then
(𝑚)
the event 𝐾𝑥
= 𝑘 denotes failure in the (𝑘 + 1)𝑠𝑡 𝑚𝑡ℎ𝑙𝑦 time interval. In this case,
𝑘 𝑚𝑡ℎ𝑙𝑦 payments are made under an annuity-immediate and (𝑘 + 1)𝑠𝑡 𝑚𝑡ℎ𝑙𝑦 payments are
made under an annuity-due, and the present value of the payments is then (1⁄𝑚) · 𝑎̅̅̅
𝑘| or
(1⁄𝑚) · 𝑎̈ 𝑘+1|
̅̅̅̅̅̅̅ respectively. Note that the interest rate contained in 𝑎̅̅̅
𝑘| and 𝑎̈ ̅̅̅̅̅̅̅
𝑘+1| is effective
over (1⁄𝑚)
𝑡ℎ
of a year.
Proceeding in a manner totally parallel to the annual payment case, we define, in the immediate
case, the present value random variable
(𝑚)
(𝑚)
𝑌𝑥
1
1 1 − 𝑣 𝐾𝑥
= · 𝑎̅̅̅̅̅̅̅
=
(
)
(𝑚) |
𝑚 𝐾𝑥 | 𝑚
𝑖
(8.71)
Where i is an effective 𝑚𝑡ℎ𝑙𝑦 interest rate, with expected value
∞
(𝑚)
𝑎𝑥
=
(𝑚)
𝐸[𝑌𝑥 ]
1
1 − 𝑣𝑘
(𝑚)
= ·∑
· 𝑃𝑟(𝐾𝑥 = 𝑘)
𝑚
𝑖
(8.72)
𝑘=0
(𝑚)
For 𝐾𝑥
= 0,1,2, … ., note that
(𝑚)
(𝑚)
𝑌𝑥
1 1 − (1 + 𝑖) · 𝑣 𝐾𝑥
= (
𝑚
𝑖
(𝑚)
Where 𝑍𝑥
(𝑚)
= 𝑣 𝐾𝑥
(𝑚)
𝑎𝑥
+1
+1
1 1
(𝑚)
) = ( ) ( ) [1 − (1 + 𝑖) · 𝑍𝑥 ]
𝑚 𝑖
(8.73)
is defined by Equation (7.39), with expected value
(𝑚)
= 𝐸[𝑌𝑥
1 1
(𝑚)
] = ( ) ( ) [1 − (1 + 𝑖) · 𝐴𝑥 ]
𝑚 𝑖
(8.74)
And variance
(𝑚)
𝑉𝑎𝑟 =
(𝑚)
(𝑌𝑥 )
(𝑚)
(𝑚)
(1 + 𝑖)2 · 𝑉𝑎𝑟(𝑍𝑥 ) 𝑉𝑎𝑟(𝑍𝑥 )
1 − (1 + 𝑖) · 𝑍𝑥
= 𝑉𝑎𝑟 [
]=
=
𝑚·𝑖
𝑚2 · 𝑖 2
𝑚2 · 𝑑 2
Where d is the effective 𝑚𝑡ℎ𝑙𝑦 discount rate so that m·d is the nominal annual discount rate
(𝑚)
𝑑 (𝑚) . Thus we can write the variance of 𝑌𝑥
as
(𝑚)
(𝑚)
𝑉𝑎𝑟(𝑌𝑥
)=
(𝑚)
(𝑚) 2
2
𝑉𝑎𝑟(𝑍𝑥 )
𝐴𝑥 − 𝐴𝑥
=
(𝑑 (𝑚) )2
(𝑑 (𝑚) )2
(8.75)
In the annuity-due case the present value random variable is
1
1
(𝑚)
· 𝑎̈ ̅̅̅̅̅̅̅̅̅̅̅|
= 𝑌𝑥 +
(𝑚)
𝑚 𝐾𝑥 +1|
𝑚
(𝑚)
𝑌𝑥̈
=
(8.76)
From which, by taking expectations, we verify that
(𝑚)
𝑎̈ 𝑥
(𝑚)
= 𝑎𝑥
+
1
𝑚
(8.70)
(𝑚)
(𝑚)
𝑉𝑎𝑟(𝑌𝑥̈ ) = 𝑉𝑎𝑟 (𝑌𝑥 )
8.5.4 NUMBERICAL EVALUATION THE 𝒎𝒕𝒉𝒍𝒚 AND CONTINUOUS CASES
If annuity functions are being evaluated from a parametric survival model, then it is no more
difficult to evaluate 𝑚𝑡ℎ𝑙𝑦 functions than to evaluate the annual functions, since 𝑟 𝑝𝑥 can be
calculated for fractional as well as integral values of r. The challenge arises when we seek to
evaluate 𝑚𝑡ℎ𝑙𝑦 annuity functions from a life table showing values of 𝑟 𝑝𝑥 only for integral value
of r.
For the 𝑚𝑡ℎ𝑙𝑦 life insurance models, we will first show how to approximate 𝑚𝑡ℎ𝑙𝑦 life annuity
functions from a life table under the UUD assumption.
Along with identity 𝐴𝑥 = 1 − 𝑑 · 𝑎̈ x and 𝐴̅x = 1 − 𝛿 · 𝑎̅x we also have
(𝑚)
𝐴𝑥
(𝑚)
= 1 − 𝑑 (𝑚) · 𝑎̈ 𝑥
(8.77)
In the discrete 𝑚𝑡ℎ𝑙𝑦 case. Then we can obtain the UDD-based approximation for the APV of
the 𝑚𝑡ℎ𝑙𝑦 whole life annuity-due as
(𝑚)
1 − 𝐴𝑥
=
𝑑 (𝑚)
𝑖
1 − (𝑚) · 𝐴x
(𝑚)
𝑖
𝑎̈ 𝑥 =
𝑑 (𝑚)
𝑖
1 − (𝑚) (1 − 𝑑 · 𝑎̈ x )
(𝑚)
𝑖
𝑎̈ 𝑥 =
𝑑 (𝑚)
1
𝑖
𝑖𝑑
(𝑚)
𝑎̈ 𝑥 = (𝑚) (1 − (𝑚) ) + (𝑚) (𝑚) · 𝑎̈ 𝑥
𝑑
𝑖
𝑖
𝑑
(𝑚)
𝑎̈ 𝑥
(8.78a)
(𝑚)
𝑎̈ 𝑥
𝑖𝑑
𝑖 − 𝑖 (𝑚)
= (𝑚) (𝑚) · 𝑎̈ 𝑥 − (𝑚) (𝑚)
𝑖
𝑑
𝑖
𝑑
𝑖𝑑
𝑖−𝑖 (𝑚)
For convenience of notation, let 𝛼(𝑚) = 𝑖 (𝑚) 𝑑(𝑚) and 𝛽(𝑚) = 𝑖 (𝑚) 𝑑(𝑚) . Then we have
(𝑚)
𝑎̈ 𝑥
= 𝛼(𝑚) · 𝑎̈ 𝑥 − 𝛽(𝑚)
(8.78b)
UUD-based approximations for the 𝑚𝑡ℎ𝑙𝑦 annuities-immediate can be found by first expressing
the 𝑚𝑡ℎ𝑙𝑦 annuity-immediate functions in term of the corresponding 𝑚𝑡ℎ𝑙𝑦 annuity-due
functions and the substituting the UUD-based approximations for the 𝑚𝑡ℎ𝑙𝑦 annuity-due
𝑑(𝑚) −𝑑
functions. Where 𝛾(𝑚) = 𝑖 (𝑚)𝑑(𝑚).
The continuous annuity functions are limiting cases corresponding 𝑚𝑡ℎ𝑙𝑦 functions as 𝑚 → ∞,
so the UUD-based approximations for the continuous functions follow from the
approximations for the corresponding 𝑚𝑡ℎ𝑙𝑦 functions.
Consider a continuous function g(t) for which a number of successive derivatives exist. The
Woolhouse formula that
∞
∞
∑ 𝑔(𝑡⁄𝑚 ) = 𝑚 [∑ 𝑔(𝑡) −
𝑡=1
𝑡=1
𝑚−1
∞ 𝑚2 − 1 ′ ∞
· 𝑔(𝑡) −
· 𝑔 (𝑡) + ⋯ ]
2𝑚
0
12𝑚2
0
(8.79)
Where 𝑔′ (𝑡)denotes the first derivites of 𝑔(𝑡)
To apply this formula to 𝑚𝑡ℎ𝑙𝑦 annuity functions, we let 𝑔(𝑡) = 𝑣 𝑡 · 𝑡 𝑝𝑥 so that
𝑑 𝑡
(𝑣 · 𝑡 𝑝𝑥 )
𝑑𝑡
𝑑
𝑑 𝑡
= 𝑣𝑡 ·
𝑣
𝑡 𝑝𝑥 + 𝑡 𝑝𝑥 ·
𝑑𝑡
𝑑𝑡
𝑔′ (𝑡) =
= 𝑣 𝑡 (− 𝑡 𝑝𝑥 𝜇𝑥+𝑡 ) + 𝑡 𝑝𝑥 (−𝛿 · 𝑣 𝑡 )
= −𝑣 𝑡 · 𝑡 𝑝𝑥 (𝜇𝑥+𝑡 + 𝛿)
Then
(8.80)
∞
∞
= 𝑣 𝑡 . 𝑡 𝑝𝑥 = 0 − 1 = −1
0
0
∞
∞
𝑔′ (𝑡)
= −𝑣 𝑡 · 𝑡 𝑝𝑥 (𝜇𝑥+𝑡 + 𝛿)
= 0 − [−(𝜇𝑥 + 𝛿)] = 𝜇𝑥 + 𝛿
0
0
𝑔(𝑡)
So we have
∞
∞
𝑚 − 1 𝑚2 − 1
(𝜇 + 𝛿)+. . . ]
−
2𝑚
12𝑚2 𝑥
(8.81a)
1
𝑚 − 1 𝑚2 − 1
(𝜇 + 𝛿) + ⋯
· ∑ 𝑣 𝑡⁄𝑚 ·𝑡⁄𝑚 𝑝𝑥 = ∑ 𝑣 𝑡 · 𝑡 𝑝𝑥 +
−
𝑚
2𝑚
12𝑚2 𝑥
(8.81b)
∑𝑣
𝑡⁄𝑚
·𝑡⁄𝑚 𝑝𝑥 = 𝑚 [∑ 𝑣 𝑡 · 𝑡 𝑝𝑥 +
𝑡=1
𝑡=1
Or
∞
∞
𝑡=1
𝑡=1
Which, by Equations (8.58) and (8.2b), gives
(𝑚)
𝑎𝑥
= 𝑎𝑥 +
𝑚 − 1 𝑚2 − 1
(𝜇 + 𝛿) + ⋯
−
2𝑚
12𝑚2 𝑥
(8.81c)
Historically the formula has often been applied using only two terms, so that
(𝑚)
𝑎𝑥
≈ 𝑎𝑥 +
𝑚−1
2𝑚
(8.82a)
And
(𝑚)
𝑎̈ 𝑥
1
𝑚
𝑚−1 1
≈ 𝑎𝑥 +
+
2𝑚
𝑚
𝑚−1 1
≈ 𝑎̈ 𝑥 − 1 +
+
2𝑚
𝑚
𝑚−1
≈ 𝑎̈ 𝑥 +
2𝑚
(𝑚)
= 𝑎𝑥
+
(8.82b)
8.5.5 SUMMARY OF 𝒎𝒕𝒉𝒍𝒚 PAYMENTS ANNUITIES
Annuity Function
Immediate
Due
(𝑚)
𝑎̈ 𝑥
𝑎𝑥:𝑛
(𝑚)
𝑎̈ 𝑥:𝑛
𝑆𝑥:𝑛
(𝑚)
̈ (𝑚)
𝑆𝑥:𝑛
(𝑚)
𝑛 |𝑎𝑥
(𝑚)
𝑛 |𝑎̈ 𝑥
Whole Life APV
𝑎𝑥
Temporary APV
Temporary AAV
Deferred APV
(𝑚)
(𝑚)
8.6 NON – LEVEL PAYMENT ANNUITY FUNCTIONS
The level annuity-immediate APVs given by Equation (8.2b) in the whole life case and
Equation (8.21) in the temporary case can be modified to incorporate the case of non-level
payment. If the payment made at time t is denoted by 𝑟𝑡 , then we have, in general
∞
𝐴𝑃𝑉 = ∑ 𝑟𝑡 · 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.85)
𝑡=1
In the whole life case and
𝑛
𝐴𝑃𝑉 = ∑ 𝑟𝑡 · 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.84)
𝑡=1
In the n-year temporary case. In particular if 𝑟𝑡 = 𝑡, so the payment sequence is increasing we
have
∞
(𝐼𝑎)𝑥 = ∑ 𝑡 · 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.85)
𝑡=1
∞
𝑡
(𝐼𝑎)𝑥:𝑛|
̅̅̅ = ∑ 𝑡 · 𝑣 · 𝑡 𝑝𝑥
(8.86)
𝑡=1
The unit decreasing n-year temporary life annuity- immediate has APV given by
𝑛
𝑡
(𝐷𝑎)𝑥:𝑛|
̅̅̅ = ∑(𝑛 + 1 − 𝑡) · 𝑣 · 𝑡 𝑝𝑥
𝑡=1
The comparable expressions in the annuity-due case would be
(8.87)
∞
(𝐼𝑎̈ )𝑥 = ∑(𝑡 + 1) · 𝑣 𝑡 · 𝑡 𝑝𝑥
(8.88)
𝑡=0
𝑛−1
𝑡
(𝐼𝑎̈ )𝑥:𝑛|
̅̅̅ = ∑(𝑡 + 1) · 𝑣 · 𝑡 𝑝𝑥
(8.89)
𝑡=0
𝑛−1
𝑡
(𝐷𝑎̈ )𝑥:𝑛|
̅̅̅ = ∑(𝑛 − 𝑡) · 𝑣 · 𝑡 𝑝𝑥
(8.90)
𝑡=0
In the case of continuous payment, with payment made at rate r(t) at time t, we would have
∞
𝐴𝑃𝑉 = ∫ 𝑟(𝑡) . 𝑣 𝑡 ·𝑡 𝑝𝑥 𝑑𝑡
(8.91)
0
In the whole life case and
𝑛
𝐴𝑃𝑉 = ∫ 𝑟(𝑡) . 𝑣 𝑡 ·𝑡 𝑝𝑥 𝑑𝑡
(8.92)
0
In the n-year temporary case. In particular, if r(t) =t we have the increasing continuous models
with
∞
̅ ̅)𝑥 = ∫ 𝑡 . 𝑣 𝑡 ·𝑡 𝑝𝑥 𝑑𝑡
(𝐼 𝑎
(8.93)
0
In the whole life case and
𝑛
𝑡
̅ ̅)𝑥:𝑛|
(𝐼 𝑎
̅̅̅ = ∫ 𝑡 · 𝑣 ·𝑡 𝑝𝑥 𝑑𝑡
(8.94)
0
In the n-year temporary case. If r(t)=n-t we have the decreasing n-year temporary case with
𝑛
𝑡
̅ 𝑎̅)𝑥:𝑛|
(𝐷
̅̅̅ = ∫ (𝑛 − 𝑡) · 𝑣 ·𝑡 𝑝𝑥 𝑑𝑡
(8.95)
0
Another type of non-level payment annuity is one in which the payments vary in a geometric,
rather than arithmetic.
8.7 MULTI-STATE MODEL REPRESENTATION
The life annuities presented in this chapter can now be described as a sequence of payments
made while the process remain in State 0. We will merely write annuity APVs using multi-
state model notation rather than standard actuarial notation, with the multi-state model notation
reflecting the fact that the annuity is payable only while the process remains in States 0.
In the discrete case, a payment is made at time k if the process, known to have started in State
̅̅̅̅
0 at time 0 for a person age x, is still in State 0. The probability of this is 𝑘 𝑝𝑥00 or 𝑘 𝑝𝑥00
, which
are the same because State 1 is an absorbing state. Then the APV for a whole life annuity-due
is
∞
𝑎̈ 𝑥00
= ∑ 𝑣 𝑘 · 𝑘 𝑝𝑥00
(8.96)
𝑘=0
Where 𝑘 𝑝𝑥00 =
00
𝑟 𝑝𝑥
𝑟 𝑝𝑥 .
In the continuous case, the probability of still being in State 0 at time r is
00
or 𝑟 𝑝𝑥̅̅̅̅
and the APV for a whole life continuous annuity is
∞
𝑎̅𝑥00 = ∫ 𝑣 𝑟 ·
00
𝑟 𝑝𝑥
𝑑𝑟
(8.97)
0
8.8 MORTALITY IMPROVEMENT PROJECTION
Survival rate improvement is of particular concern in pricing annuity contracts. If the APV of
an annuity has been calculated from a model that reflects current survival rates, but those rates
then increase over time so that annuitants live longer than contemplated by the model, the price
of the annuity would turn out to be inadequate and the insurer would lose money on the annuity
contracts. To guard against this, the annuity price could be calculated using a survival model
that reflects a projected mortality improvement. Note that “mortality improvement” means
increased values of 𝑝𝑥 and decreased, or reduced, values of 𝑞𝑥
There are two different ways to define the mortality improvement projection factors. In both
cases we begin with the values of 𝑝𝑥 and 𝑞𝑥 assumed to apply age x in year M, denoted by 𝑝𝑥𝑀
and 𝑞𝑥𝑀 . Then we could define either a constant annual survival increase factor or a constant
annual mortality reduction factor.
If the mortality reduction factor for age x denoted 𝑓𝑥 then we would calculate
𝑞𝑥𝑀+1 = 𝑞𝑥𝑀 · 𝑓𝑥
(8.97a)
𝑞𝑥𝑀+2 = 𝑞𝑥𝑀 · 𝑓𝑥2
(8.98b)
𝑞𝑥𝑀+𝑘 = 𝑞𝑥𝑀 · 𝑓𝑥𝑘
(8.98c)
A mortality reduction factor of 𝑓𝑥 is sometimes stated as a mortality improvement projection
factor 1 − 𝑓𝑥 . The result are the same under each way of defining the factor, of course, but
readers of the literature should take care to note the form of the presentation.