7/26/2019 Models for Quantifying Risk http://slidepdf.com/reader/full/models-for-quantifying-risk 1/40 P REFACE The analysis and management of financial risk is the fundamental subject matter of the discipline of actuarial science, and is therefore the basic work of the actuary. In order to manage financial risk, by use of insurance schemes or any other risk management technique, the actuary must first have a framework for quantifying the magnitude of the risk itself. This is achieved by using mathematical models that are appropriate for each particular type of risk under consideration. Since risk is, almost by definition, probabilistic, it follows that the appropriate models will also be probabilistic, or stochastic, in nature. This textbook, appropriately entitled Models for Quantifying Risk , addresses the major types of financial risk analyzed by actuaries, and presents a variety of stochastic models for the actuary to use in undertaking this analysis. It is designed to be appropriate for a two- semester university course in basic actuarial science for third-year or fourth-year undergraduate students or entry-level graduate students. It is also intended to be an appropriate text for use by candidates in preparing for Exam MLC of the Society of Actuaries or Exam 3L of the Casualty Actuarial Society. One way to manage financial risk is to insure it, which basically means that a second party, generally an insurance company, is paid a fee to assume the risk from the party initially facing it. Historically the work of actuaries was largely confined to the management of risk within an insurance context, so much so, in fact, that actuaries were thought of as “insurance mathematicians ” and actuarial science was thought of as “insurance math.” Although the insurance context remains a primary environment for the actuarial management of risk, it is by no means any longer the only one. However, in recognition of the insurance context as the original setting for actuarial analysis and management of financial risk, we have chosen to make liberal use of insurance terminology and notation to describe many of the risk quantification models presented in this text. The reader should always keep in mind, however, that this frequent reference to an insurance context does not reduce the applicability of the models to risk management situations in which no use of insurance is involved. The text is written in a manner that assumes each reader has a strong background in calculus, linear algebra, the theory of compound interest, and probability. (A familiarity with statistics is not presumed.) This edition of the text is organized into three sections. The first, consisting of Chapters 1-4, presents a review of interest theory, probability, Markov Chains, and stochastic simulation, respectively. The content of these chapters is very much needed as background to later
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The analysis and management of financial risk is the fundamental subject matter of thediscipline of actuarial science, and is therefore the basic work of the actuary. In order to
manage financial risk, by use of insurance schemes or any other risk management technique,the actuary must first have a framework for quantifying the magnitude of the risk itself. This isachieved by using mathematical models that are appropriate for each particular type of risk
under consideration. Since risk is, almost by definition, probabilistic, it follows that theappropriate models will also be probabilistic, or stochastic, in nature.
This textbook, appropriately entitled Models for Quantifying Risk , addresses the major typesof financial risk analyzed by actuaries, and presents a variety of stochastic models for the
actuary to use in undertaking this analysis. It is designed to be appropriate for a two-semester university course in basic actuarial science for third-year or fourth-yearundergraduate students or entry-level graduate students. It is also intended to be anappropriate text for use by candidates in preparing for Exam MLC of the Society ofActuaries or Exam 3L of the Casualty Actuarial Society.
One way to manage financial risk is to insure it, which basically means that a second party,generally an insurance company, is paid a fee to assume the risk from the party initiallyfacing it. Historically the work of actuaries was largely confined to the management of risk
within an insurance context, so much so, in fact, that actuaries were thought of as “insurancemathematicians” and actuarial science was thought of as “insurance math.” Although theinsurance context remains a primary environment for the actuarial management of risk, it is
by no means any longer the only one.
However, in recognition of the insurance context as the original setting for actuarial analysisand management of financial risk, we have chosen to make liberal use of insuranceterminology and notation to describe many of the risk quantification models presented in this
text. The reader should always keep in mind, however, that this frequent reference to aninsurance context does not reduce the applicability of the models to risk management
situations in which no use of insurance is involved.
The text is written in a manner that assumes each reader has a strong background in calculus,linear algebra, the theory of compound interest, and probability. (A familiarity with statisticsis not presumed.)
This edition of the text is organized into three sections. The first, consisting of Chapters 1-4, presents a review of interest theory, probability, Markov Chains, and stochastic simulation,respectively. The content of these chapters is very much needed as background to later
material. They are included in the text for readers needing a comprehensive review of thetopics. For those requiring an original textbook on any of these topics, we recommend either
Broverman [6] or Kellison [16] for interest theory, Hassett and Stewart [12] for probability,Ross [26] for Markov Chains, and Herzog and Lord [13] for simulation.
The second section, made up of Chapters 5-14, addresses the topic of survival-contingent
payment models, traditionally referred to as life contingencies. The third section, consisting ofChapters 15-17, deals with the topic of models for interest rate risks.
The new material appearing in this edition has been added to the text to meet the new set oflearning objectives for Exam MLC, to be effective with the May 2012 administration of theexam. There are several major areas of expanded material:
(1) Part Three (Chapters 15-17) has been added to address the topic of interest rate risk,including an introduction to variable, or interest-sensitive , insurance and annuities.
(2) The new learning objectives provide that many of the models described in the text shouldalso be presented in the context of multi-state models, using the theory of Markov Chains.
Accordingly, we have inserted new Chapter 3 to provide background on Markov Chains,and have then inserted numerous applications of this theory to our actuarial modelsthroughout the text.
(3) In the prior edition of the text, the topic of contingent contract reserves was presentedin one large chapter. In the new edition, we have separated the topic into two chapters:
Chapter 10 covers net level premium benefit reserves only, and Chapter 11 addressesthe accounting notion of reserves as financial liabilities, as well as other policy values,
as suggested by the new learning objectives.
(4) Similarly, the topic of multiple decrements is now presented in two chapters: Chapter
13 addresses multiple decrement theory and Chapter 14 presents a number of applica-
tions of the theory, many in the context of multi-state models.
(5) Certain actuarial risk models are most efficiently evaluated through simulation, ratherthan by use of closed form analytic solutions. We have inserted new Chapter 4 to
provide theoretical background on the topic of stochastic simulation, and thendeveloped a number of applications of that theory to our actuarial models. Readerswho are preparing for SOA Exam MLC should be aware that this topic is not coveredon that examination. Accordingly we have placed all of the simulation applications inAppendix B in this edition.
(6) A number of minor topics have been deleted from the prior edition, including (a) the
central rate of failure, (b) use of the population functions , , x x
L T and , x
Y and (c) our
presentation of Hattendorf’s Theorem. Several notational changes have also beenmade.
The writing team would like to thank a number of people for their contributions to thedevelopment of this text.
The original manuscript was thoroughly reviewed by Bryan V. Hearsey, ASA, of LebanonValley College and by Esther Portnoy, FSA, of University of Illinois. Portions of the
manuscript were also reviewed by Warren R. Luckner, FSA, and his graduate student LuisGutierrez at University of Nebraska-Lincoln. Kristen S. Moore, ASA, used an earlier draft asa supplemental text in her courses at University of Michigan. Thorough reviews of theoriginal edition were also conducted by James W. Daniel, ASA, of University of Texas,
Professor Jacques Labelle, Ph.D., of Université du Québec à Montréal, and a committeeappointed by the Society of Actuaries. A number of revisions in the Second Edition werealso reviewed by Professors Daniel and Hearsey; Third Edition revisions were reviewed byProfessors Samuel A. Broverman, ASA (University of Toronto), Matthew J. Hassett, ASA(Arizona State University), and Warren R. Luckner, FSA (University of Nebraska-Lincoln).All of these academic colleagues made a number of useful comments that have contributed
to an improved published text.
The new topics contained in this edition were researched for us by actuaries with consider-able experience in their respective fields, and we wish to acknowledge their valuable
contributions. They include Ronald Gebhardtsbauer, FSA (Penn State University) forSection 14.5, Ximing Yao, FSA (Hartford Life) for Chapter 16, and Chunhua (Amy) Meng,
FSA (Yindga Taihe Life) for Chapter 17.
The new material added to this Fourth Edition was also reviewed by Professor Luckner, as well
as by Tracey J. Polsgrove, FSA (John Hancock USA), Link Richardson, FSA (AmericanGeneral Life), Arthur W. Anderson, ASA, EA (The Arthur W. Anderson Group), Cheryl AnnBreindel, FSA (Hartford Life), Douglas J. Jangraw, FSA (Massachusetts Mutual Life), Robert
W. Beal, FSA (Milliman Portland), Andrew C. Boyer, FSA (Milliman Windsor), and MatthewBlanchette, FSA (Hartford Life), who also contributed to the Chapter 17 exercises.
Special thanks goes to the students enrolled in Math 287-288 at University of Connecticutduring the 2004-05 academic year, where the original text was classroom-tested, and to
graduate student Xiumei Song, who developed the computer technology material presented in
Appendix A.
Thanks also to the folks at ACTEX Publications, particularly Gail A. Hall, FSA, the projecteditor, and Marilyn J. Baleshiski, who did the typesetting and graphic arts for all editions.
Kathleen H. Borkowski designed the cover for the first two editions, and Christine Phelpsdid the same for the latter two editions.
Finally, a very special acknowledgment is in order. When the Society of Actuaries publishedits textbook Actuarial Mathematics in the mid-1980s, Professor Geoffrey Crofts, FSA, then
at University of Hartford, made the observation that the authors’ use of the generic symbol Z as the present value random variable for all insurance models and the generic symbol Y asthe present value random variable for all annuity models was confusing. He suggested that
the present value random variable symbols be expanded to identify more characteristics ofthe models to which each related, following the principle that the present value randomvariable be notated in a manner consistent with the standard International Actuarial Notation
used for its expected value. Thus one should use, for example, : x n Z in the case of the
All the discussion to this point in the text has focused on the complementary concepts of sur-vival and failure of a single entity, such as a single person under a life insurance policy or a lifeannuity. We consistently referred to the entity whose survival was being observed as our statusof interest , and we defined what constituted survival (and therefore failure) in each particularcase. This generic approach to the concept of survival will be continued in this chapter.
We now consider the case where the status of interest is itself made up of two or more entities,such as two separate individual lives. In actuarial science such models are said to involve multi-
ple lives and are known as multi-life models. We will develop the theory of multi-life models inthe two-life case first, and then show how it is easily extended to more than two lives.
We will also need to distinguish whether the two individual lives comprising a multi-lifestatus have independent or dependent future lifetimes. The assumption of independence willsimplify our work to some extent, and is presumed wherever needed throughout the chapter.Independence is not presumed for the discussion in Sections 12.5 and 12.6.
12.1 THE JOINT-LIFE MODEL
A joint -life status is one for which survival of the status requires the survival of all (or both,
in the two-life case) of the individual members making up the status. Accordingly, the status
fails upon the first failure of its component members.
12.1.1 THE TIME-TO-FAILURE R ANDOM VARIABLE FOR A JOINT-LIFE STATUS
Consider a two-life joint status, made up of lives that are ages x and y as of time 0. We use
the notation ( ) xy to denote such a status,1 and we use xyT to denote the random variable for
the future lifetime (or time-to-failure) of the status. From the definition of failure it is clear
that xyT will be the smaller of the individual future lifetimes denoted by xT and . yT That is,
min( , ). xy x yT T T (12.1)
Our analysis of the future lifetime random variable xy
T will parallel that for the individual
life xT presented in Section 5.3. Indeed, by using the generic concept of a status or entity,
the two cases of xyT and xT are really the same except for the different notation.
1 If numerical ages are used instead of the letters x and y, such as if 20 x and 25, y for example, the status is
denoted (20:25). The colon would also be used to denote the status ( : ). x n y n
We begin our analysis of the random variable xyT with its SDF, given by
( ) ( ) xy xyS t Pr T t (12.2)
and denoted by t xy p in standard actuarial notation. Since survival of the status itself requiresthe survival of both component members of the status, then assuming independence of theindividual lifetimes we have
,t xy t x t y p p p (12.3)
which is the bridge between joint-life and individual life functions. Equation (12.3) will al-
low us to evaluate many joint-life functions from a single-life tabular survival model.
12.1.3 THE CUMULATIVE DISTRIBUTION FUNCTION OF xyT
The CDF of xyT is given by
( ) 1 ( ) ( ), xy xy xy F t S t Pr T t (12.4)
and is denoted by t xyq in actuarial notation. It follows that
1t xy t xyq p (12.5a)
for all t . If the individual lifetimes are independent, then we can write
1
1 (1 )(1 ) .
t xy t x t y
t x t y t x t y t x t y
q p p
q q q q q q
(12.5b)
Equation (12.5b) illustrates a very basic concept in probability. t xyq denotes the probability
that the joint status fails before (or at) time t , which occurs if either or both of the individual
lives fail before (or at) time t . Since the events ( ) xT t and ( ) yT t are not mutually exclu-
sive, the probability of the union event, which is ,t xyq is given by the general addition rule
reflected in Equation (12.5b).
As in the individual life case discussed earlier in the text, the pre-subscript t is suppressed in
Recall that a hazard rate function (HRF) measures the conditional instantaneous rate of fail-
ure at precise time t , given survival to time t . For individual lives in a life insurance context
we refer to the hazard rate as the force of mortality; in the context of a joint status it is more
appropriate to view the hazard rate as a force of failure rather than a force of mortality. Re-
gardless of the terminology used, the HRF is defined as
( )( ) .
( )
xy xy
xy
f t t
S t (12.9a)
In the special case of independent lives we have
( )( ) .
t xy x t y t
xy x t y t t xy
pt
p
(12.9b)
In standard actuarial notation the HRF is denoted by : , x t y t 2 so we have
: . x t y t x t y t (12.9c)
This shows that the force of failure acting on the joint status is the sum of the forces of fail-
ure (or forces of mortality) acting on the individual components (lives) in the case of inde-
pendent lives.
12.1.6 CONDITIONAL PROBABILITIES
The conditional failure probability, denoted by |n xq in the single-life case, has its counter-
part in the joint-life case. We define
| ( 1),n xy xyq Pr n T n (12.10)
the probability that the time of failure of the joint status occurs in the ( 1) st n time interval.3
Since failure of the status occurs with the first failure of the individual components, then
|n xyq denotes the probability that the first failure occurs in the ( 1) st n interval. Thus we
have
1
:
|
(1 )
n xy n xy n xy
n xy x n y n
q p p
p p
(12.11a)
: .n xy x n y n p q (12.11b)
2 Note that the joint status HRF is a function of t , with the identifying characteristics of its component members( x) and ( y) fixed at time 0.t To reinforce the idea that the joint status HRF is a function of t only, some texts
prefer the notation ( ). xy t (See, for example, Section 9.3 of Bowers, et al . [4] .)3 Recall the discrete random variable *, x K defined in Chapter 5 as the time interval of failure for the status ( x). Ifwe now define *
xy K as the random variable for the interval of failure of the joint status ( ), xy we have*
e represents the average number of whole years of survival within the next n
years for the joint status ( ). xy
EXAMPLE 12.3
Let xT and yT be independent time-to-failure random variables, each with exponential dis-
tributions with hazard rates x and , y respectively. Find an expression for . xye
SOLUTION
We know that xt t x p e and ,
t t y p e
so under independence we have
( ). x yt t xy t x t y p p p e
This shows us that xyT has an exponential distribution with hazard rate , x y so
1[ ] . xy xy x y
e E T
12.2 THE LAST-SURVIVOR MODEL
A last - survivor status is one for which survival of the status requires the survival of any one
(or more) of its component members. That is, the status is said to survive as long as at least
one of its members survives, so that it fails only when all of its members have failed. Then thetime of failure of the status is the time of the last failure among its components, or the second
failure in the two-life case. Note that the n-year certain and continuous annuity, defined in Ex-
ample 8.14, is a special case of a last-survivor status, since the annuity pays until the second
failure out of (40) and 10 . The APV of this annuity is denoted40:10
a in actuarial notation.
12.2.1 THE TIME-TO-FAILURE R ANDOM VARIABLE
FOR A LAST-SURVIVOR STATUS
For a two-life last-survivor status composed of the individual lives ( ) x and ( y), we denote
the status itself by ( ) xy and the random variable for the future lifetime of the status by . xyT
Since the status fails on the last failure, then xyT will be the larger of the individual futurelifetimes xT and . yT That is
if ( ) y has already failed and only ( ) x survives, or
t xy y t xy y t V A P a (12.53c)
if ( ) x has already failed and only ( ) y survives.
12.4.4 R EVERSIONARY ANNUITIES
A special type of two-life annuity is one that pays only after one of the lives has failed, and
then for as long as the other continues to survive. Such annuities are called reversionary an-
nuities.
If payment is made to the status ( y), provided it has not failed, but only after the failure ofthe status ( x), then the total condition for payment at time k is that ( x) has failed but ( y) has
not. The probability of this is
(1 ) ,k x k y k y k x k y k xyq p p p p p (12.54)
so the actuarial present value of a reversionary annuity payable under such circumstances
would be
|
1
( ) .k x y k y k xy y xy
k
a v p p a a
(12.55)
The result is intuitive. ya represents the APV of payment made for the lifetime of ( ), y and
xya is the APV of payment made for the joint lifetime of ( xy). We can read Equation (12.55)
as providing payment as long as ( ) y survives, but taking it away while ( ) x also survives, with
the net effect being payment made while ( ) y survives but after the failure of ( ). x
If the payment is made for n years at most, with the requirement that ( ) y has failed but ( ) x
If a reversionary annuity is funded by annual premiums, the length of the premium-paying
period would be the joint lifetime, since upon failure of the joint status either payments
begin or the contract expires without value. If payments are to be made to ( x) after the fail-
ure of ( y), then use of the equivalence principle leads to the annual benefit premium
|
|( ) . y x x xy
y x xy xy
a a a
P a a a
(12.58)
The benefit reserve at duration t would again depend on what combination of ( x) and ( y) stillsurvive. If both are alive, the reversionary annuity contract is still in premium-paying status
so the reserve is
| | | :( ) ( ) .t y x y t x t y x x t y t V a a P a a (12.59a)
If ( x) only is alive the contract is beyond the premium-paying period and in payout status, so
the reserve is simply
|( ) .t y x x t V a a (12.59b)
If ( y) only is alive, no payment will ever be made so the contract has expired and the reserve
is zero.
EXAMPLE 12.9
Show that | | . xy x y y x xya a a a
SOLUTION
The result is intuitive. | x ya represents payments made if ( y) is alive but ( x) is not, | y xa repre-
sents payments made if ( x) is alive but ( y) is not, and xya represents payments made if both
are alive. The three cases are mutually exclusive. Together they provide payments if either
( x) or ( y) is alive, which is represented by . xya Mathematically,
| |
,
x y y x xy y xy x xy xy
x y xy
xy
a a a a a a a a
a a a
a
as required.
12.4.5 CONTINGENT INSURANCE FUNCTIONS
A contingent insurance is one for which payment of the unit benefit depends on the order of
failure among its component members. The insurance functions follow from the probability
functions introduced in Section 12.3. Here we consider only insurances with immediate
A contingent insurance benefit paid at the failure of ( x) only if ( x) fails before ( y) has APV
given by
1
0.t
xy t xy x t A v p dt
(12.60)
If the benefit is paid at the failure of ( x) only if ( x) fails after ( y), the APV is
2 1
0(1 ) .t
xy t x x t t y x xy A v p p dt A A
(12.61)
EXAMPLE 12.10
Show that1 1
xy xy xy A A A
and2 2 . xy xy xy A A A
SOLUTION
1 1
0 0
:0
t t xy xy t xy x t t xy y t
t t xy x t y t xy
A A v p dt v p dt
v p dt A
2 2 11
1 1
xy xy x xy y xy
x y xy xy
x y xy xy
A A A A A A
A A A A
A A A A
12.5 MULTI-STATE MODEL R EPRESENTATION
The multi-life models presented thus far in this chapter can be easily represented as multi-
state models. We illustrate this idea in the two-life case, with extension to three or more
lives being apparent.
12.5.1 THE GENERAL MODEL
Consider two persons alive at ages x and y, respectively, at time t . The model is in State 1 as
long as both lives continue to survive. Since both lives are surviving at time t , it follows thatthe process begins in State 1 at that time. The model is illustrated in Figure 12.1 on the fol-
Since the only decrement is failure, or death, it follows that any state, once left, cannot be
reentered. Therefore the event of being in State 1 at time ,t r given in State 1 at time t , is
the same as the event of never leaving State 1 over that time interval.
Similarly, the event of being in State i, for 2i or 3, at any time after entering that state is
the same as the event of never leaving that state once it has been entered. Note that transition
from State 2 to State 3, or from State 3 to State 2, is not possible.
Clearly State 4 is an absorbing state, which can never be left once it has been entered. We
also assume, in this model, that the simultaneous failure of ( ) x and ( ) y is not possible, so
direct transition from State 1 to State 4 cannot occur.4
12.5.2 THE JOINT-LIFE MODEL
By definition, the joint-life status continues to survive as long as the process is in State 1,and fails when the process transitions to either State 2 or State 3. For a discrete process
known to be in State 1 at time n, the probability of being in State 1 at time n r is
( ) *( )11 11[ 1 | 1] ,n n
n r n r r Pr X X p p (12.62a)
and for a continuous process known to be in State 1 at time t , the probability of being in
State 1 at time t r is
( ) *( )11 11[ ( ) 1 | ( ) 1] .t t
r r Pr X t r X t p p (12.62b)
In the single-life case, for a person alive (i.e., in State 1) at age x at discrete time n (orcontinuous time t ), the probability of being in State 1 at discrete time n r (or continuous
In the multi-life case, assuming independence of ( ) x and ( ), y the corresponding probability
is .r xy r x r y p p p That is,
( ) *( ) ( ) *( )11 11 11 11 .n n t t
r xy r x r y r r r r p p p p p p p (12.63b)
For the joint-life model, we can write the APVs for annuities and insurances in multi-state
model notation by analogy to the single-life model as presented in Section 8.6 (for annuities)
and Section 7.6 (for insurances). Two examples of this are presented in Example 12.11, with
others left to the exercises (see Exercise 12-18).
EXAMPLE 12.11
Write the APVs for (a) the joint-life annuity-due and (b) the continuous joint-life insurance
in multi-state model notation.
SOLUTION
(a) For two persons alive at ages x and y, respectively, at time 0, the APV of a unit annuity-
due, by analogy to Equation (8.96), is
(0)11
0
,k xy k
k
a v p
(12.64)
where (0)11 ,k k xy k x k y p p p if ( ) x and ( ) y are independent.
(b) The unit continuous joint-life insurance is payable at the instant of transition from State1 to State 2 or from State 1 to State 3. These events are mutually exclusive, so the APV,
by analogy to Equation (7.80), is
(0)12 13110
( ) ( ) ,r xy r A v p r r dr
(12.65)
where (0)11r is defined in part (a), 12 ( ) , y r r and 13( ) . x r r
12.5.3 R EVERSIONARY ANNUITIES
Recall from Section 12.4.4 that the reversionary annuity with APV given by | y xa is payableto ( ) x after the failure of ( ). y In the multi-state model context, such an annuity is payable
13 ,r r y r x p q 24 ( ) , x r r and 34 ( ) . y r r
12.5.6 SOLVING THE K OLMOGOROV FORWARD EQUATION
In this section we consider the Kolmogorov differential equation presented in Section 3.2.2
as it applies to the multi-life model of Section 12.5.1. As shown by the arrows in Figure12.1, not all transitions between states are possible, so the only non-zero forces of transition
are 12 ( ), s 13( ), s 24( ), s and 34 ( ). s
Suppose ( ) x and ( ) y are both alive at those ages at general time t , so that the process is in
State 1. Then ( )11
t r p denotes the probability that the process is still in State 1 (i.e., both are
still surviving) at time ,t r and ( )12
t r p denotes the probability that the process is in State 2
(i.e., ( ) x is surviving but ( ) y has failed) at time .t r (Note that ( )
13
t
r p is the same as ( )
12,t
r p
with the roles of ( ) x and ( ) y reversed.)
If the process is in State 2 at time t , then both ( )22t
r p and ( )24t
r p denote concepts of interest to
us. (Note that ( ) ( )21 23 0.)t t
r r p p Furthermore, both ( )33t
r p and ( )34t
r p are also concepts of
interest, but they are the same as ( )22t
r p and ( )24 ,t
r respectively, with the roles of ( ) x and
( ) y reversed.
It is clear that, for a process in State 4 at time t , ( ) ( ) ( )41 42 43 0t t t
r r r p p p and ( )44 1t
r p for all r .
The above analysis suggests that we need solve the Kolmogorov equation only for ( )11 ,t n p ( )12 ,t
n p ( )22 ,t
n p and ( )24 .t
n p We do this for ( )12
t n p in the following example, and defer the others
three cases to Exercises 12-22 and 12-23.
EXAMPLE 12.13
The symbol ( )12
t n denotes the probability that a process in State 1 at time t will be in State 2
at time ,t n where State 1 means both ( ) x and ( ) y are alive at those ages. To be in State 2
at time t n requires that ( ) y fails in, and ( ) x survives over, the interval ( , ].t t n (In actu-
arial notation this is denoted by ,n x n y p q assuming that ( ) x and ( ) y are independent
lives.) Show that the Kolmogorov equation can be solved for this result.
SOLUTION
In Equation (3.14a), 1i and 2 j so k takes on the values 1, 3, 4 in the summation. Then
Because of the common hazard factor, such as travel accidents, there is a non-zero probabil-ity of simultaneous failure of ( ) x and ( ). y When represented as a multi-state model, theearlier Figure 12.1 is modified as shown in Figure 12.4.
Figure 12.4
The only change in this model from our Figure 12.1 model is that 14 ( ), s the force of transi-
tion function for transition directly from State 1 to State 4 is now non-zero. This means that
the probability value
( )14 ( ) 4 | ( ) 1t
r p Pr X t r X t
can be satisfied by direct transition from State 1 to State 4. In the Figure 12.1 model, ( )14
t r p
could be satisfied only by transition from State 1 to State 2 to State 4 or by transition from
12-24 Consider the joint density function of xT and yT given by
, 3
4( , ) ,(1 2 )
x y x y x y
f t t t t
for 0 xt and 0. yt Show that xT and yT are not independent.
12-25 For the survival model of Exercise 12-24, find ( ), xS t the marginal survival function
of . xT
12-26 For the survival model of Exercise 12-24, evaluate , (1,2). x y F
12-27 For the survival model of Exercise 12-24, evaluate , (1,2). x yS
12-28 For the survival model of Exercise 12-24, find an expression for .n xy
12-29 For the survival model of Exercise 12-24, find an expression for .n xyq
12.7 Common Shock – A Model for Lifetime Dependency
12-30 The APV for a last-survivor whole life insurance on ( ), xy with unit benefit paid at
the instant of failure of the status, was calculated assuming independent future life-
times for ( ) x and ( ) y with constant hazard rate .06 for each. It is now discovered
that although the total hazard rate of .06 is correct, the two lifetimes are not inde- pendent since each includes a common shock hazard factor with constant force .02.
The force of interest used in the calculation is .05. Calculate the increase in the
APV that results from recognition of the common shock element.
12-31 Lives ( ) x and ( ) y have independent future lifetime exponential random variables
* xT and *
yT with respect to risk factors unique to ( ) x and ( ), y respectively. As well,
both ( ) x and ( ) y are subject to a constant common hazard rate .01. Given that
.96 x p and .97, y p calculate the value of 5 . xy
12-32 For two persons alive at ages x and y at time t , show that the Kolmogorov differen-
tial equation for ( )14
t r p solves for
( )14 ,t
xyn n xy p q e
where is the constant common shock hazard described in Section 12.7.