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Unformatted text preview: Chapter 3 Life tables and selection Chapter 3 Life tables and selection
ACTSC 232 Introduction to Actuarial Mathematics
Tianxiang Shi
Department of Statistics and Actuarial Science
University of Waterloo Winter 2012 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection Outline 1 Life tables 2 Fractional age assumptions
UDD assumption
CFM assumption 3 Select and ultimate survival models Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Life tables Heuristic analysis on a life table
What a life table looks like?
Age x
0
1
2
.
.
. lx
100,000
99,329
99,285
.
.
. dx
671
44
30
.
.
. qx
0.006713
0.000444
0.000300
.
.
. ex
77.7
77.2
76.3
.
.
. l0 : initial number (say 100,000 newborns).
TEMPORARILY consider
lx : # of survivors up to age x
dx : # of death between age x and age x + 1
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Life tables US/Canada life tables 2006 US life tables
http:
//www.cdc.gov/nchs/data/nvsr/nvsr58/nvsr58_21.pdf
2002 Canada life tables
http://www.statcan.gc.ca/pub/84537x/
84537x2006001eng.htm
Some comments
tabulated at integer ages only, and containing lx , dx , qx and
other functions.
l0 : radix of the table. Arbitrary positive number, usually large.
the maximum age is NOT necessarily the limiting age. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Life tables Survivors to age x (Lx )
Expected number of survivors
Denote Lx as the number of survivors up to age x among l0
newborns. Let
Ik = 1, if the kth life survives to age x (with prob. x p0 )
0, otherwise (with prob. x q0 )
l0 Then, Lx = Ik . Deﬁne
k=1 lx := E[Lx ]
If Ik are independent, Lx ∼ Bin(l0 , x p0 )
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Life tables t px and lx
The mean of Lx
l0 E[Ik ] = l0 · x p0 lx = E[Lx ] =
k=1 In general
t px = x+t p0
x p0 = lx+t /l0
lx+t
=
lx /l0
lx or
lx+t = lx · t px
In particular
lx+1 = lx · px , and px =
Tianxiang Shi([email protected]) lx+1
lx Chapter 3 Life tables and selection
Life tables Death between ages x and x + n (n Dx )
Expected number of death between ages x and x + n
Denote n Dx as the number of death between ages x and x + n
among l0 newborns. Let
uk = 1, kth life dies b/t ages x & x + n (w.p.
0, otherwise (w.p. 1 − xn q0 )
l0 Then, n Dx = Lx − Lx+n = uk . Deﬁne
k=1 n dx := E[n Dx ] and denote: dx = 1 dx If uk are independent, Lx ∼ Bin(l0 , xn q0 )
Tianxiang Shi([email protected]) xn q0 ) Chapter 3 Life tables and selection
Life tables n qx , n dx and lx The mean of n Dx
n dx = E[n Dx ] = E[Lx − Lx+n ] = lx − lx+n or l0
n dx E[uk ] = l0 · xn q0 = lx · n qx =
k=1 In particular
lx = lx+1 + dx , and qx = dx
lx Other property
n dx Tianxiang Shi([email protected]) = dx + dx+1 + · · · + dx+n−1 Chapter 3 Life tables and selection
Life tables Other functions
kn qx in terms of lx
kn qx = k px − k+n px = lx+k − lx+k+n
lx Total expected # of years lived beyond age x by those
survivors of l0 newborns
∞ ∗
Tx = lx+t dt = lx · ˚x
e
0 ∗
Note: Tx corresponds to Tx in the US life table.
Total expected # of years lived b/t ages x and x + n of l0
newborns
n
n Lx =
0 Tianxiang Shi([email protected]) ∗
∗
lx+t dt = lx · ˚x:n = Tx − Tx+n
e Chapter 3 Life tables and selection
Life tables Example Example
x
lx
dx 30
1500
50 31
1450
(1) 32
1390
65 33
1025
75 34
950
90 35
(2) You are given the incomplete life table,
(i) Fill in the blanks (1) and (2)
(ii) Calculate 2 p30 , q34 and 3 q31
(iii) Calculate the probability that a life (31) dies b/t ages 33 and
35 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Life tables Life table under De Moivre’s law
Under the De Moivre’s law, Tx ∼ U (0, ω − x)
µx = 1
t
, and t px = 1 −
ω−x
ω−x for 0 ≤ t ≤ ω − x (x ≤ ω)
Life table under De Moivre’s law
lx = c · (ω − x),
where c is a constant.
For the generalized De Moivre’s law with µx =
lx = c · (ω x)α , α > 0
Tianxiang Shi([email protected]) α
ω−x Chapter 3 Life tables and selection
Fractional age assumptions Fractional age assumptions (FAA)
Life table tabulated at integer ages, which is not suﬃcient to
calculate probabilities involving fractional ages, e.g., 0.6 px or
7.2 q10 .
Additional assumption on the distribution of the future
lifetime r.v. b/t integer ages is needed.
We use the term fractional age assumption” (FAA) refer to
these assumptions.
Common FAAs (based on diﬀerent interpolations)
Uniform distribution of deaths (UDD)
Constant force of Mortality (CFM)
Balducci (also known as Hyperbolic) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Fractional age assumptions
UDD assumption Two equivalent UDD assumptions
UDD 1
For x ∈ N and 0 ≤ s < 1, assume that
s qx = s · qx UDD 2
Let Rx = Tx − Kx be the fractional part of Tx . For x ∈ N assume
that
Rx ∼ U (0, 1), and Rx is independent of Kx
Immediate consequences
lx = lx+s + s · dx
Linear interpolation: lx+s = (1 − s) · lx + s · lx+1
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Fractional age assumptions
UDD assumption Probabilities under UDD assumption
For x ∈ N and 0 ≤ s < 1, we have
s qx = s · qx , and s px = 1 − s · qx = (1 − s)0 px + s · px We also have
fx (s) = qx , and µx+s =
(Question: What about fx (n + s) and
For t px+s , qx
1 − s · qx
n+s px ?) Transfer to an integer age and use prev. formula
t+s px
t px+s =
s px
Using life table function
t px+s
Tianxiang Shi([email protected]) = lx+t+s
lx+s Chapter 3 Life tables and selection
Fractional age assumptions
UDD assumption Example Example
Under UDD assumption, express following terms using life table
functions, i.e. lx and/or dx , for x ∈ N
(i) 0.6 p50 (ii) 0.3 q31.6 (iii) 6.7 q8.5 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Fractional age assumptions
CFM assumption CFM assumptions
CFM b/t ages
Recall that 1
0 px = e− µx+s ds for x ∈ N and 0 ≤ s < 1. Now we assume that
µx+s = c(x) for 0 ≤ s < 1
(Note that c(x) depends on x), then
px = e−c(x)
Immediate consequences
s px
Tianxiang Shi([email protected]) = e− s
0 µx+t dt = (px )s Chapter 3 Life tables and selection
Fractional age assumptions
CFM assumption Probabilities under CFM assumption
Exponential (Log linear) interpolation:
ln lx+s = (1 − s) ln lx + s ln lx+1
or
ln n+s px = (1 − s) ln n px + s ln n+1 px
for x ∈ N and 0 ≤ s < 1.
We also have
s qx = 1 − (px )s For t px+s , transfer to an integer age
t px+s Tianxiang Shi([email protected]) = t+s px
s px Chapter 3 Life tables and selection
Fractional age assumptions
CFM assumption Example Example
Under CFM assumption, express following terms using life table
functions, i.e. lx and/or dx , for x ∈ N
(i) 0.6 p50 (ii) 0.3 q31.6 (iii) 6.7 q8.5 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Fractional age assumptions
CFM assumption Balducci assumption and others Harmonic interpolation:
1
1−s
s
=
+
lx+s
lx
lx+1
or 1
1−s
=
+
n+s px
n px s
n+1 px for x ∈ N and 0 ≤ s < 1.
UDD v.s. CFM: s px under UDD is larger than one under CFM Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Select process
Underwriting
Consider an applicant who wants to buy a life insurance product.
Underwriting is the process of collecting the applicant’s relevant
information and evaluating related factors, such as age, gender,
smoking habits, occupation, dangerous hobbies, personal/family
health history, etc.. These factors are called rating factors. The
insurers may also require the applicant to do medical examinations.
Purpose of underwriting
to classify potential policyholders into broadly homogeneous
risk categories
to assess appropriate premiums for applicants
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Risk categories
Preferred lives: very low mortality risk based on the standard
information; no undesirable indicators and evidence.
Normal lives: have some higher rated risk factors than
preferred lives but still insurable at standard rates. Most
applicants fall into this category. (The above 2 categories
account for 95% for traditional life insurance)
Rated lives: one or more risk factors at raised levels. Can be
insured at a higher premium. e.g. a family history of heart
disease. (23%)
Uninsurable lives: signiﬁcant risk that insurer won’t sell
insurance to them. (23%) Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Underwriting eﬀect Adverse selection: If underwriting process is not strict, very
highrisk individuals will buy insurance in disproportionate
numbers, leading to excessive losses. When insurers increase
their rates, they tends to further lose preferred lives.
Selection eﬀect
Accepted applicants are expected to have lower mortality rates
than the general population (national life tables).
Selection eﬀect will wear oﬀ eventually. This leads us to study the select and ultimate survival models. Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Select and ultimate survival models
Model description
An individual who enters the group (e.g. who buys a term
insurance) at, say, age x, is said to be selected (or select) at age x.
A select and ultimate survival model follows:
future survival probabilities for a life in the group depends on
the current age and on the age at which s/he joined the
group.
The initial selection eﬀect is assumed to have worn oﬀ after d
years, where d is called the selection period (generally a
positive integer). After d years, the mortality is called
ultimate mortality.
Remark: A select and ultimate survival model is also shorten for
select survival model.
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Notations under select survival models
[x]: a life selected at age x
([x] + t): a life selected at age x and currently aged at x + t
t p[x]+s := S[x]+s (t): Pr(a life ([x] + s) will survive t years) t q[x]+s = 1 − t p[x]+s : Pr(a life ([x] + s) will die within t years) ut q[x]+s : Pr(a life ([x] + s) will survive u years and then die
within the next t years) the force of mortality at age x + t for a life selected at age x
µ[x]+t := lim h→0+ 1
Pr(T[x]+t ≤ h),
h where T[x]+t is the future lifetime of ([x] + t)
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Formulas under select survival models
Under a select survival model,
T[x]+t = T[x] − tT[x] ≥ t
From this assumption, we have
t+u p[x]+s = u p[x]+s · t p[x]+s+u , or t p[x]+s+u = In particular,
t p[x]+u = t+u p[x]+s
u p[x]+s t+u p[x]
u p[x] BUT, the select age can not be broken in the formula
t p[x+u] Tianxiang Shi([email protected]) = t+u p[x]
u p[x] Chapter 3 Life tables and selection
Select and ultimate survival models Aggregate/Select and ultimate survival models
Aggregate (Ch. 2) v.s. Select and ultimate
Aggregate survival models (the model in Ch. 2)
all functions such as t px , t qx , and lx depend only on current
age x
no selection eﬀect
assume Tx = T0 − xT0 ≥ x Select and ultimate survival models
all functions such as s p[x]+t , s q[x]+t , and l[x]+t depends on
current age x + t and select age x
assume T[x]+t = T[x] − tT[x] ≥ t All the formulas regarding t px , t qx , ut qx and µx can be
transplanted from the aggregate survival models to the select and
ultimate survival models, as long as we DO NOT break the select
age x.
Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Select periods Select period d and probabilities
Select part: within the selection period e.g., for t < d,
s p[x]+t = s px+t , and s q[x]+t = s qx+t Ultimate part: selection eﬀect will wear oﬀ after d years, that
is T[x]+t and Tx+t are identically distributed for t ≥ d. e.g.,
s p[x]+t Tianxiang Shi([email protected]) = s px+t , and s q[x]+t = s qx+t Chapter 3 Life tables and selection
Select and ultimate survival models Example Example
Assume that the selection period is 2 years. Compare (i.e. >, =,
or <)
(i) s q[x] v.s. s qx
(ii) s q[x]+1 v.s. s q[x+1]
(iii) s q[x]+2 v.s. s q[x+1]+1 Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Example Example
x
30
31
32
33
34 1000q[x]
0.222
0.234
0.250
0.269
0.291 1000q[x]+1
0.330
0.352
0.377
0.407
0.441 1000qx+2
0.422
0.459
0.500
0.545
0.596 x+2
32
33
34
35
36 From the above 2year select and ultimate life table, ﬁnd out
and 3 q[31]+1 5 p[30] , 12 q[31] Tianxiang Shi([email protected]) Chapter 3 Life tables and selection
Select and ultimate survival models Select and ultimate life tables Construct dyear select life tables
l[x]+t : expected number of survivors at age x + t for those
individuals selected at age x. Similarly, we can deﬁne n d[x]+t .
For t ≥ d, the selection eﬀect has worn oﬀ
l[x]+t = lx+t
For 0 ≤ t < d, we deﬁne in a backward way
l[x]+t = Tianxiang Shi([email protected]) lx+d
d−t p[x]+t Chapter 3 Life tables and selection
Select and ultimate survival models Probabilities: l[x]+t and n d[x]+t
For t p[x]+s
l[x]+t = l[x] · t p[x] , or t p[x] = l[x]+t
l[x] and in general
t p[x]+s = l[x]+s+t
l[x]+s For t q[x]+s
t q[x]+s For = t d[x]+s l[x]+s , and t q[x] = ut q[x]+s
ut q[x]+s Tianxiang Shi([email protected]) = t d[x]+s+u l[x]+s t d[x] l[x] Chapter 3 Life tables and selection
Select and ultimate survival models Example
Example
x l[x] l[x]+1 20
21
.
.
. 995
970
.
.
. 973
948
.
.
. lx+2
1000
975
949
923
.
.
. x+2
20
21
22
23
.
.
. From the above 2year select and ultimate life table, ﬁnd out
(i) 3 p[20]
(ii) 12 q[20] (iii) 2 q[20]+1
(iv) 0.9 q[20]+0.6 Tianxiang Shi([email protected]) ...
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This note was uploaded on 03/30/2012 for the course ACTSC 232 taught by Professor Matthewtill during the Winter '08 term at Waterloo.
 Winter '08
 MATTHEWTILL

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