Actuarial Science 2427A/B Lecture Notes - Lecture 13: Rolladen-Schneider Ls1, Linear Interpolation, Perinatal Mortality
14 Jun 2018
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Department
Professor
AS2427b Chapter 3 Lecture Notes : Jan 29, 2018
1
3.1 Intro/Overview
3.2 Life Tables
3.3 Fractional Age assumptions
3.4 National Life Tables
3.5 Survival Models for Life Insurance Policyholders
3.6 Life Insurance Underwriting
3.7 Select and Ultimate Survival Models
3.8 Notation & formulae for select survival models
3.9 Select Life tables
3.10 Heterogenity in mortality
3.11 Mortality trends
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AS2427b Chapter 3 Lecture Notes : Jan 29, 2018
2
•
Life tables a useful way of summarizing a lifetime distribution and
contain tabulations by ages including basic functions such as qx, dx,
and lx
•
tables defined from an initial age x0 (most often 0) to a limiting age
•
It is quite common to present tables for integer values only (and often
values are only defined at integer ages only)
•
tables can be constructed in two different ways
using a survival model, OR
based on a mortality study(defined for integer ages only)
•
if constructing a table using a survival model, you start with an
original group of lives called the radix(this number is arbitrarily set)
and use the survival model to determine expected number of
lives(from the radix) at future ages
o
info can be used to determine many different survival probabilities
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AS2427b Chapter 3 Lecture Notes : Jan 29, 2018
3
Working page for Life Tables and notation definitions
lx+t expected # of survivors that reach age (x+t) from the lx
independent individuals aged x
dx+t expected # of deaths between age (x+t) and (x+t+1) from the lx
independent individuals aged x
when t=1
l
x+t
= l
x
(
t
p
x
) (1) or
t
p
x
= l
x+t
/ l
x
(2)
p
x
= l
x+1
/ l
x
t
q
x
= 1-
t
p
x
or
t
q
x
= (l
x
l
x+t
)/ l
x
(3) q
x
=
d
x
/l
x
=
(l
x
l
x+1
)/ l
x
Also,
u
t
q
x
=(
u
p
x
u+t
p
x
)
or
u
t
q
x
= (l
x+u
l
x+t+u
)/ l
x
(4)
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Document Summary
As2427b chapter 3 lecture notes : jan 29, 2018 (cid:7) (cid:1)(cid:2)(cid:3)(cid:4)(cid:3)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:7)(cid:12)(cid:13)(cid:14)(cid:15)(cid:9)(cid:14)(cid:16)(cid:11)(cid:14)(cid:17)(cid:9)(cid:11)(cid:18)(cid:7)(cid:19)(cid:7) (cid:12)(cid:20)(cid:21)(cid:7)(cid:22)(cid:7)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:7)(cid:23)(cid:6)(cid:24)(cid:25)(cid:11)(cid:18)(cid:7)(cid:6)(cid:14)(cid:26)(cid:7)(cid:2)(cid:11)(cid:25)(cid:11)(cid:17)(cid:15)(cid:9)(cid:13)(cid:14)(cid:7)(cid:7) 3. 8 notation & formulae for select survival models. Also, u(cid:2)t qx =( u px (cid:1) u+t px ) or u(cid:1)t qx = (lx+u (cid:1) lx+t+u )/ lx (4) Table 3. 1: extract from a life table (generated from a survival model) x lx dx. Note: l30 set at 10,000, lx values and dx values then generated using survival model (and p. 3 formulas) It then follow that spx = e- s for 0 (cid:4) s < 1 or spx = (px)s. Given p40 = 0. 999473 calculate 0. 4 q40. 2 under each of the fractional age assumptions (a) uniform distribution of deaths(udd) within each year of age (b) constant force of mortality(cf) within each year of age. Example 1(working page) q40 = 1- p40 =0. 000527. First, can rewrite 0. 4 q40. 2 = 1 0. 4 p40. 2 = 1 0. 6 p40. 0. 2 p40 (a) udd: sqx (cid:3) s(qx) (cid:2)(cid:2)(cid:2)(cid:2) spx (cid:3) 1 (cid:1) s(qx)
