Introduction to Actuarial Studies Week 3: Survival Models and
the Life TableACTL 1101 Introduction to Actuarial StudiesMichael
SherriscUniversity of New South Wales (2015)School of Risk and
Actuarial Studies, UNSW Business SchoolARC Centre of Excellence in
Population Ageing [email protected] 3:Survival
Models and the Life Table1/47Introduction to Actuarial Studies Week
3: Survival Models and the Life Table1 Survival models2 The life
table2/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival models1 Survival models2 The life
table3/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsSurvival ModelsProbability model
to calculatethe probability that a life will surviveexpected
payments for insurance and annuity contractsTwo important
tools:survival functionhazard rate4/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableSurvival
modelsSurvival FunctionContinuous random variableXthat denotes the
age-at-deathCumulative distribution function ofXis denoted
byFX(x)Probability density function ofXis denoted byfX (x) = F
X(x)5/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableSurvival modelsSurvival Function for a
New-BornThe survival function for a new-born iss(x) = Pr (X> x)
= 1 FX(x), x 0FX(0) = 0, so thats(0) = 16/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life
TableSurvival modelsSurvival Function for a Life AgedxConditional
probability - Probability that two eventsA andBoccur is equal to
the probability that one occurs times theprobability that the other
occurs given that one has alreadyoccurredPr (A andB) = Pr (A) Pr
(B|A)= Pr (B) Pr (A|B)where Pr (A|B) is the probability thatA
occurs given thatBhas already occurredPr (B|A) =Pr (A andB)Pr
(A)7/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsSurvival Function for a Life
Agedx(Contd)Given that life is agedx, the probability that the life
willsurvive to agez(z> x) is:Pr (X> z|X> x) =Pr (X>
zandX> x)Pr (X> x)=Pr (X> z)Pr (X>
x)=s(z)s(x)8/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableSurvival modelsSurvival Function for a
Life Agedx- ExampleExample 3.1: Ifs (x) = 1 x100for 0 x< 100.
Determine theprobability that a life aged 20 will survive to age
65.Solution:The required probability iss (65)s (20)=1 651001 20100=
0.43759/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsDeath Probability for a Life
AgedxThe probability that a life agedxwill die between the ages
ofyandz(z> y> x) is:Pr (y< X z|X> x) =Pr (y< X
zandX> x)Pr (X> x)=Pr (X> y) Pr (X> z)Pr (X> x)=s(y)
s(z)s(x)10/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableSurvival modelsFuture Lifetime of a Life
AgedxA life aged x is denoted by the symbol(x)The future lifetime
random variable is denoted byT (x)Remember thatXdenotes the
age-at-death, so we have:T(x) = X x11/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableSurvival
modelsInternational Actuarial NotationDeath probabilities:tqx= Pr
[(x) will die withintyears]= Pr [T (x) t] t 0In particular, whent=
1qx= Pr [(x) will die within a year]= Pr [T (x) 1]12/47Introduction
to Actuarial Studies Week 3: Survival Models and the Life
TableSurvival modelsInternational Actuarial Notation
(Contd)Survival probabilities:tpx= Pr [(x) will survive at
leasttyears]= Pr [T (x) > t]= 1 tqxIn particular, whent= 1px= Pr
[(x) will survive at least a year]= Pr [T (x) > 1]= 1
qx13/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsInternational Actuarial Notation
(Contd)Link actuarial notation to survival function:xp0=
s(x)tpx=x+tp0xp0=s(x + t)s(x)tqx= 1 s(x + t)s(x)14/47Introduction
to Actuarial Studies Week 3: Survival Models and the Life
TableSurvival modelsInternational Actuarial Notation (Contd)The
probability that a life agedxsurvivestyears and dies inthe nextu
years:t|uqx= Pr_t< T (x) t + u_=s(x + t) s(x + t + u)s (x)=_1
s(x + t + u)s (x)__1 s(x + t)s (x)_=t+uqx tqx=tpx
t+upx15/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsInternational Actuarial Notation -
ExampleExample 3.2: Ifs (x) =_1 x100_2for 0 x 100,
calculate5p35.Solution:The required probability is5p35=s (40)s
(35)=_1 40100_2_1 35100_2= 0.8520716/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableSurvival
modelsHazard RateA key feature of survival models is the hazard
rate or failurerate functionIn actuarial studies and demography,
for a survival function,the hazard rate or failure rate is referred
to as the force ofmortalityDenition:(x) =ddxs (x)s (x)=s
(x)s (x)=fX (x)1 FX(x)17/47Introduction to Actuarial Studies
Week 3: Survival Models and the Life TableSurvival modelsHazard
Rate: Actuarial NatationThe force of mortality is denoted byxxis a
conditional probability density function - the probabilitydensity
of the age-at-death(X) given that the life has survivedto
agex18/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableSurvival modelsHazard Rate: MathematicsBy
denition(y) = ddy s(y)s(y)= ddy ln s (y), so that(y) dy= d ln s
(y)Integrating fromxtox + twe have_x+tx(y) dy=_x+txd ln s (y) =
ln_s (x + t)s (x)_19/47Introduction to Actuarial Studies Week 3:
Survival Models and the Life TableSurvival modelsHazard Rate:
Mathematics (Contd)Solving the equation, we have:s (x + t)s (x)=
exp__x+tx(y) dy_=tpxands(x) = exp__x0(y) dy_20/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life
TableSurvival modelsHazard Rate: Mathematics (Contd)Cumulative
distribution function of the age-at-deathX:FX(x) = 1 s(x)= 1
exp__x0(y) dy_Probability density function of the age-at-deathX:fX
(x) =ddxFX(x) = (x) exp__x0(y) dy_= (x) s(x)21/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life
TableSurvival modelsHazard Rate: Mathematics (Contd)Cumulative
distribution function of future lifetimeT(x):FT(x)(t) = 1 s(x +
t)s(x)= 1 exp__x+tx(y) dy_Probability density function of the
age-at-deathX:fT(x)(t) =ddtFT(x)(t) = (x + t) exp__x+tx(y) dy_= (x
+ t) tpx22/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableSurvival modelsHazard Rate -
ExampleExample 3.3: Evaluate _0(x) s(x)dx.Solution:(x) s(x) = fX(x)
is the probability density function, so theintegral is equal to
1This is the probability that a new-born will die between theages
of 0 and 23/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableThe life table1 Survival models2 The life
table24/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableThe Life TableThe life table is a
table showing for each agex:qx- the probability of death between
agexandx + 1dx- the (expected) number of deaths agedxlast
birthdaylx- the (expected) number alive at exact
agex25/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableThe Life Table (Contd)Lx- the
number living aged betweenxandx + 1 (xlastbirthday)Lxis also the
(expected) number of years lived by thelxlivesagedxover the year
from agextox + 1Lxis a measure of the number of lives
exposed-to-risk of dyingagedxlast birthdayAny life in this group
who dies will be classied as a death atagexLxis often calculated as
follows:Lx=_10lx+tdt lx+ lx+1226/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableThe life tableThe
Life Table (Contd)Tx- the number of lives agedxor greaterTxis also
the (expected) total future lifetime of thelxlivesagedxTx=
t=0Lx+tThe average number of years lived by thelxlives isTxlx
ex- the average future lifetime of a life agedx exis also referred
to as the complete residual life expectancy27/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life TableThe
life tableExamples of Life Tables: IA64-70IA64-70 Table for
Australian Insured LivesThis table is based on the mortality of
whole-of-life (withoutterm riders) policies for mainly male lives
from datacontributed by 14 Life OcesThe data used was for the
period January 1964 to December1970The table was published by The
Institute of Actuaries ofAustraliaThe table shows only the ultimate
mortality rates - lives withpolicies for more than 2
years28/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableExamples of Life Tables:
A90-92A90-92 Table for Australian MalesPopulation mortality table
based on all male lives in Australiain the 1991 census and male
deaths over the period 1990 to1992The death rates are determined as
the average deaths of malesbetween 1990-92 divided by the 1991
census gures for eachage.These rates are smoothed using an
actuarial technique calledgraduation (covered in later actuarial
courses)29/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableThe life tableConstruction of Life
TableInsurance company, population census and deaths data
providethe information to calculateqxTable starts with a radix
which is the base for the tableFor the population mortality table
(A90-92 males) the radix isl0= 1, 000, 000In the case of the
IA64-70 table the radix chosen wasl10= 1, 000, 00030/47Introduction
to Actuarial Studies Week 3: Survival Models and the Life TableThe
life tableConstruction of Life Table (Contd)Proceed as follows:dx=
qxlxlx+1= lx dxLx=lx+ lx+12Tx= Tx+1 + Lx ex=Txlx31/47Introduction
to Actuarial Studies Week 3: Survival Models and the Life TableThe
life tableConstruction of Life Table (Contd)lxvalues are
interpreted as the number of lives exact agexsurviving from the
initiall0 livesConsider a group ofl0 newborns then the probability
that anyone of these will survive to agexiss(x)Assuming lives are
independent then the number of survivorsto agex(denoted byV) has a
binomial distribution withn = l0 andp= s(x) since each life can
either survive to agexwith probabilityp or die before agexwith
probability(1 p):V Binomial(l0, s(x))32/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableThe life
tableConstruction of Life Table (Contd)Expected number of lives out
of the initiall0 lives who surviveto exact agexwill be given by the
expected value of thebinomial distribution, i.e.:lx= E(V) =
l0s(x)The variance of the number of lives surviving to agexwill
beVar (V) = l0s(x) [1 s (x)]33/47Introduction to Actuarial Studies
Week 3: Survival Models and the Life TableThe life
tableConstruction of Life Table (Contd)Survival probabilities are
calculated as follows:tpx=s (x + t)s (x)=l0s (x + t)l0s
(x)=lx+tlx34/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableThe life tableUsing the Life Table -
ExampleExample 3.4: Use the IA64-70 Life Table to calculate1 the
expected age at death of a life aged 202 the probability that a
life aged 20 will survive to age 403 the probability that a life
aged 20 will die within the next 10years35/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life TableThe
life tableUsing the Life Table - SolutionSolution to Example 3.4:1
20 + e20= 20 + 53.71 = 73. 71.2 The required probability
is:20p20=l40l20=966752991729= . 974 813 The required probability
is:10q20=l20l30l20=991729 978901991729= 1. 293 5
10236/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableLaws of MortalitySimple models have
been proposed for the hazard rate for humanlives - referred to as
laws of mortalityDe Moivres LawGompertz LawMakehams
Law37/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableLaws of Mortality: De Moivres LawIn
1725, Abraham de Moivre assumed the number aliveaccording to the
life table decreased in arithmetical progressionlxis a linear
function ofxs (x) = 1 x, 0 x< Hazard rate (force of mortality)
is given by(x) = s
(x)s (x)=1 x, 0 x< 38/47Introduction to Actuarial Studies
Week 3: Survival Models and the Life TableThe life tableLaws of
Mortality: Gompertz LawIn 1825, Benjamin Gompertz hypothesized that
the force ofmortality increases with age in geometrical
progressionHazard rate(x) = Bcx,whereB> 0,c> 1, andx
039/47Introduction to Actuarial Studies Week 3: Survival Models and
the Life TableThe life tableLaws of Mortality: Gompertz Law
(Contd)Chance of death increases more and more rapidly with ages
(x) = exp__x0(y) dy_= exp__x0Bcydy_= exp_Bln ccy_x0= exp_Bln
c(cx1)_40/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableThe life tableFitting Gompertz Law to Life
Table DataParameters of the Gompertz curve areBandcLeast squares
selects the numerical values of the parameters byminimizing the sum
of the squared dierences between theLaw values and the actual
values41/47Introduction to Actuarial Studies Week 3: Survival
Models and the Life TableThe life tableFitting Gompertz Law to Life
Table Data (Contd)Denote actual values fors(x) from the life table
bysa(x)Over the age rangexltoxuselect values ofBandcthatminimize
the functionxu
x=xl_sa(x) exp_Bln c(cx1)__2Use a spreadsheet such as Excel and
the Solver, or othersoftware packages such as R, MATLAB, SAS,
etc.42/47Introduction to Actuarial Studies Week 3: Survival Models
and the Life TableThe life tableFitting Gompertz Law to Life Table
Data (Contd)Australian IA64-70 Life Table over the ages 10 to
110Gompertz curve using least squares was given by(x) = 0.000046
(1.100837)xUp to around age 80 the Gompertz curve provides
areasonable t to the IA64-70 data43/47Introduction to Actuarial
Studies Week 3: Survival Models and the Life TableThe life
tableLaws of Mortality: Makehams LawIn 1867, Makeham suggested an
addition of a constant toaccount for accidents and infections as
well as an increase inhazard geometrically with ageHazard rate(x) =
A + Bcx,whereA > B,B> 0,c> 1, andx 044/47Introduction to
Actuarial Studies Week 3: Survival Models and the Life TableThe
life tableFitting Makehams Law to Life Table DataIA64-70 Life Table
over the ages 10 to 110Best t Makeham curve was(x) = 0.0003225 +
0.000031 (1.1066413)x45/47Introduction to Actuarial Studies Week 3:
Survival Models and the Life TableThe life tableSummaryThis
weekSurvival models, including survival function, hazard rate,
andinternational actuarial notationThe life table, including each
variable in the life table, how toconstruct and interpret a life
table, and three laws of mortalityRead Chapter 3 of the textbookDo
exercises in Chapter 346/47Introduction to Actuarial Studies Week
3: Survival Models and the Life TableThe life tableSummary - Next
WeekNext weekAttend tutorials, prepare to discuss exercises from
Chapter 3and Tutorial Exercises Week 4Read Chapter 4 (just to get
the main ideas and identify newtopics)Basic nancial
mathematics47/47
