15.8. ABNORMAL INCIDENCE OF DECREMENT
261
Application to Profit-Testing
Let = withdrawal, and = mortality. It is assumed that withdrawals occur only at the end of a
policy year. That is, mode only operates at time k = 1.
In chapter 14, the following resul

262
CHAPTER 15. MULTIPLE-DECREMENT TABLES
Example 15.8.2. Among the employees of a certain firm retirement may take place at or after age
57, but is compulsory at age 60. The independent rates of mortality of the employees are those of
A1967-70 ultimate.

264
CHAPTER 15. MULTIPLE-DECREMENT TABLES
Solutions
15.1
1 is uniformly distributed, so
1 1
1
t px x+t = qx
2
x+t = c, so
Z
2
t px = exp[
Z
1
(aq)
x =
1
Z
t
0
=
0
0t1
ct
2
for 0 t 1
x+t dt] = e
1 1
2
t px x+t .t px
0
=
t
for
dt
qx1 ect dt
qx1
t=1
1 ct
e

15.9. EXERCISES
263
Exercises
15.1
Let 1 , 2 be the modes of decrement in a double-decrement table. Suppose that 1 is
uniformly distributed over the year of age from x to x + 1 in its associated single- decrement
2
table, and
x+t = c for 0 t 1.
2
1
2
1
F

15.8. ABNORMAL INCIDENCE OF DECREMENT
259
cannot be used.
To deal with this abnormal mode of decrement, we argue that from age x to x + k, mode
operates by itself. Then, at age x + k, there is a chance qx of exit by mode , so
(aq)
x = P rcfw_(x) survives

Chapter 16
FINANCIAL CALCULATIONS
USING
MULTIPLE-DECREMENT
TABLES
16.1
Principles
One may value cash flows, calculate premiums and find reserves using the same ideas as are used
when there is just one mode of decrement (death).
Notation is best considered

270CHAPTER 16. FINANCIAL CALCULATIONS USING MULTIPLE-DECREMENT TABLES
If the premiums are payable continuously, expression (16.3.2) can be approximated to a sum.
The m.p.v. is now
"
#
1 (al)x+ 1
3 (al)x+ 3
1 (al)x+n 1
2
2
2
P v2
+ v2
+ + v n 2
(al)x
(al)x

260
CHAPTER 15. MULTIPLE-DECREMENT TABLES
of injury or failed by one of the instructors and sent back to his base. The independent weekly
Hospitalised
Week through injury Being failed
rates of decrement are as follows:
1
0.078
0.132
2
0.102
0.092
3
0.058

256
15.6
CHAPTER 15. MULTIPLE-DECREMENT TABLES
Further Formulae
The Identity of the forces
Define
(a)
x = lim
h (aq)x
h0+
h
Theorem
(a)
x = x for all x
(15.6.1)
Proof
Assume that
x and x are continuous.
Then
h (aq)x
(a)
x = lim+
h0
Rh
h
(ap)x
x+t
0 t
=

268CHAPTER 16. FINANCIAL CALCULATIONS USING MULTIPLE-DECREMENT TABLES
F (t) = P rcfw_T t
= t (aq)x
Z t
=
r px x+r .r px dr
0
Since
(aq)x
+ (aq)x = 1,
we have
lim F (t) = lim t (aq)x < 1
t
t
A defective variable T may have a probability density function

15.7. GENERALIZATION TO 3 MODES OF DECREMENT
257
R1
m
x
0
=
lx+t
x+t dt
R1
l
dt
0 x+t
'
x+ 1
2
Therefore it is normally assumed that
(am)
x ' mx
Note
We may also construct tables using select rates of decrement, using
d
= q[x]+1 , .
qxd = q[x] , qx+1
fo

15.10. SOLUTIONS
265
r = remarriage
r
Notice that q51
= q[50]+1 , i.e. duration 1 year from age 50 at end of former marriage.
x
50
51
qxd
0.00728
0.00823
qxr
0.050
0.025
(aq)dx
0.00710
0.00813
(aq)rx
0.04982
0.02490
r
2 (aq)50
15.4
(al)x
100,000
94,308
91

16.3. EVALUATION OF MEAN PRESENT VALUES
269
(ii) Consider premiums of P per annum, payable continuously for at most n years while (x) remains
a member of the group under consideration. The mean present value is
Z n
Z n
(al)x+t
P
vt
dt = P
v t .t (ap)x dt

16.7. EXERCISES
275
Exercises
16.1
The following is an extract from a multiple-decrement table referring to mortality and withdrawal from certain life assurance contracts, these modes being referred to as d and w
respectively.
age,
x
(al)x
(ad)w
x
(ad)dx

16.5. EXTRA RISKS TREATED AS AN ADDITIONAL MODE OF DECREMENT
273
A1967-70 select,
4% interest,
expenses are ignored.
A proposer, aged 45, for temporary assurance ceasing at age 65 is subject to an extra occupational
hazard which is considered to be equiva

272CHAPTER 16. FINANCIAL CALCULATIONS USING MULTIPLE-DECREMENT TABLES
16.4
Benefits on Death by a Particular Cause
Sometimes a policy will provide death benefits if (x) dies from a particular cause. One may write
x+t = force of mortality from cause , and

258
CHAPTER 15. MULTIPLE-DECREMENT TABLES
In the practical construction of a multiple-decrement table, we find the values of (al)x , (ad)
x,
(ad)x , (ad)x and use
(al)x+1 = (al)x [(ad)
x + (ad)x + (ad)x ].
Example 15.7.1. The members of a large companys m

16.3. EVALUATION OF MEAN PRESENT VALUES
271
Mortality: the independent rates of mortality of those not permanently disabled are those of
A1967-70 ultimate; the permanently disabled are subject to the mortality of English Life Table
No.12 - Males with the

44
3.6
CHAPTER 3. ASSURANCES
Assurances payable at the end of the 1/m of a year of
death.
Suppose that 1 is payable at the end of the 1/m year (measured from the issue date) following the
death of (x); for example, if m = 12 , the sum assured is payable a

2.3. THE CONSTRUCTION OF A1967-70.
31
(which are found by replacing x by [x] + t in the formulae of chapter 1) are true. We may omit
the square brackets enclosing x in expressions such as h p[x]+t and l[x]+t+h if t s or t + h s
respectively.
We now give f

52
CHAPTER 3. ASSURANCES
If the benefits are payable at the end of the year of death, their M.P.V. is
X
(S + B)(1 + b)k+1 v k+1 k |qx+t
k=0
=(S + B)
=(S +
k+1
X
1+b
k=0
B)Ax+t
where A is at rate of interest
1+i
k |qx+t
(3.10.9)
ib
1+b
(3.10.10)
If the b

3.10. VALUING THE BENEFITS UNDER WITH PROFITS POLICIES
51
(1) Simple bonuses. In this system, bonuses are calculated only on the basic sum assured (B.S.A.),
which we shall denote by S. Consider a whole life with profits policy issued t years ago to a life

42
CHAPTER 3. ASSURANCES
where
Ax =
X
v k+1 k |qx
(3.5.2)
k=0
We define the commutation functions
Cy = v y+1 dy
X
Cy+k
My =
(3.5.3)
(3.5.4)
k=0
and find that
P
k=0
Ax =
=
v x+k+1 dx+k
Dx
Mx
Dx
Example 3.5.1. Show that the variance of Z (= Sv K+1 ) is
h
i

46
CHAPTER 3. ASSURANCES
and
n |Ax
X
=
t=n
X
v t+1 t| qx
(3.7.10)
Cx+t
t=n
=
Dx
Mx+n
Dx
=
(3.7.11)
In view of the relationship between Cx and Cx , we have the approximations:
; (1 + i) 12 n |Ax ; i n |Ax
(3.7.12)
The evaluation of term and deferred assur

3.8. PURE ENDOWMENTS AND ENDOWMENT ASSURANCES
47
n years, whichever occurs first. It follows that an endowment assurance is a combination of a term
assurance and a pure endowment (of the same term). If we assume that the death benefit is payable
immediate

32
CHAPTER 2. SELECT LIFE TABLES
Table 2.3.1
2.4
Some formulae for the force of mortality.
We assume that l[x]+t and [x]+t are continuous in t (t 0).
h q[x]+t
[x]+t = lim
h0+
h
l[x]+t l[x]+t+h
= lim
h0+
hl[x]+t
d
l[x]+t
= dt
(t 0)
l[x]+t
(2.4.1)
(the der

2.7. SOLUTIONS
35
who was selected t years ago will die within a year , etc. The select period, s years, is such
that
q[x]+t = qx+t
for t s
i.e. for lives who were selected at least s years ago (these being called ultimate lives) mortality
depends only on

3.3. COMMUTATION FUNCTIONS
3.3
39
Commutation functions
Functions such as Ax may easily be evaluated by numerical integration, and are often tabulated
directly at various rates of interest. (In fact, a modern computer can easily compute Ax directly at
any