Multiple-Life Theory

Christina Elliott

April 19, 2008

Abstract

This report will discuss the basic theory of multiple-life contracts and theeffect of dependency between lives on the net single premiums of multiple-life annuity contracts. Furthermore, models of dependency are assessedfor their relevance in actuarial calculations, in particular the copula andcommon shock models.

Contents

1 Introduction 31.1 An Overview of the Project . . . . . . . . . . . . . . . . . . . 4

2 Multiple-Life Contracts 52.1 The Joint-Life Status . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Joint-Life Annuities . . . . . . . . . . . . . . . . . . . 72.1.2 Joint-Life Immediate Insurances . . . . . . . . . . . . 92.1.3 Joint-Life Moment of Death Insurances . . . . . . . . 102.1.4 Joint-life Annual Premiums . . . . . . . . . . . . . . . 13

2.2 The Last-Survivor Status . . . . . . . . . . . . . . . . . . . . 142.2.1 Last-Survivor Annuities . . . . . . . . . . . . . . . . . 15

2.3 The General Symmetric Status . . . . . . . . . . . . . . . . . 172.3.1 Schuette-Nesbitt Formula . . . . . . . . . . . . . . . . 172.3.2 Using The Schuette-Nesbitt Formula . . . . . . . . . . 18

2.4 Asymmetric Statuses . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 Asymmetric Annuities . . . . . . . . . . . . . . . . . . 192.4.2 Asymmetric Insurances . . . . . . . . . . . . . . . . . 20

2.5 The General Two-Life Annuity Contract . . . . . . . . . . . . 212.6 The General Two-Life Insurance Contract . . . . . . . . . . . 222.7 Contingent Insurances . . . . . . . . . . . . . . . . . . . . . . 23

2.7.1 First-Death Contingent Insurances . . . . . . . . . . . 232.7.2 Second-Death Contingent Insurances . . . . . . . . . . 242.7.3 The General Contingent Insurance . . . . . . . . . . . 24

3 Copulas 253.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Sklar’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Quasi-Inverses and Constructing Copulas . . . . . . . . . . . 293.4 Survival Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Common Copulas . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5.1 Frank’s Copula . . . . . . . . . . . . . . . . . . . . . . 333.5.2 The Frechet-Hoeffding Bounds . . . . . . . . . . . . . 34

3.6 Using Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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3.6.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . 353.6.2 Maximum Lifetime of Model . . . . . . . . . . . . . . 363.6.3 The Method . . . . . . . . . . . . . . . . . . . . . . . 37

4 Analysis of Annuity Values 394.1 Whole Life Last-Survivor Annuity . . . . . . . . . . . . . . . 39

4.1.1 The Effect of Age . . . . . . . . . . . . . . . . . . . . . 414.1.2 The Effect of Interest Rate . . . . . . . . . . . . . . . 454.1.3 The Effect of the Alpha Parameter . . . . . . . . . . . 47

4.2 Whole-Life Joint-Life Annuity . . . . . . . . . . . . . . . . . . 494.3 Reversionary Annuity . . . . . . . . . . . . . . . . . . . . . . 50

5 Other Models of Dependency 525.1 The Common Shock Model . . . . . . . . . . . . . . . . . . . 52

5.1.1 Using the Common Shock Model . . . . . . . . . . . . 545.1.2 Analysis of Annuity Value using Common Shock Model 56

5.2 The Shared Frailty Model . . . . . . . . . . . . . . . . . . . . 58

6 Conclusion 60

Bibliography 62

A Schuette-Nesbitt Formula 64A.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64A.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

B Proofs of Lemmas from Chapter 3 66B.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . 66B.2 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . 67

C Life Tables 70

D Material from the 2H Course 73D.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.2 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74D.3 Force of Mortality . . . . . . . . . . . . . . . . . . . . . . . . 75D.4 Lifetime Models . . . . . . . . . . . . . . . . . . . . . . . . . . 75D.5 Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76D.6 Life Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . 76D.7 Annuities and Life Annuities . . . . . . . . . . . . . . . . . . 77D.8 Net Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . 78

E Does “Broken Heart Syndrome” Exist? 79

F Notation 81

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Chapter 1

Introduction

Actuarial Mathematics is an intriguing area of mathematics for me, as itprovides an opportunity to apply mathematical theory in practical situa-tions. The UK insurance industry was the largest in Europe and the thirdlargest worldwide in 2006, according to [ABI], showing this industry to bean important part of our economy. In the UK, 40% of households have someform of life insurance policy, and £69 billion was paid out in pension andlife insurance benefits in 2006. Clearly actuarial mathematics is central toour society.

I consider “Multiple-Life Theory” to be a particularly interesting areaof actuarial mathematics, as contracts can be manipulated to suit differentgroups of people. Multiple-life contracts can provide financial support dur-ing life, using annuities, as well as in death through insurance policies. In anaging population, pension schemes are becoming increasingly important tofinance our extending lives, and a reduced (e.g. 50%) spouse’s pension in theevent of the death of the policy holder is a practical multiple-life contract.

A central focus of this project is to determine the effect on contract costswhen considering dependent future lifetimes of the policyholders. Dependentlives have future lifetimes that cannot be assumed independent; the death ofone life affects the future lifetime of the second life. Groups of lives that pur-chase multiple-life contracts are likely to be dependent, for example marriedcouples and family members; however the industry assumes independencebetween all lives. This is a strong assumption and one I find intriguing.There are many contributing factors to dependency between lives, includingsimilar lifestyles, genetic diseases and exposure to illnesses within families.Alternatively frequently flying business partners have dependent lives dueto their equal exposure to plane crashes.

A controversial factor contributing to dependency between lives, partic-ularly married couples, is known as “Broken Heart Syndrome”. This theorysuggests that the grief caused by bereavement may result in an increasedmortality for the remaining life. The Young, Benjamin and Wallis (1963)

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study, mentioned in [CMP], claimed that the peak of mortality in widowersis during the first year after a bereavement, with a 40% increase in the deathrate amongst the widowers in this study during the first 6 months. Furtherdetails of studies on “broken heart syndrome” can be found in Appendix E.

This project assumes the understanding of concepts covered in the sec-ond year “Actuarial Mathematics” course. Appendix D contains a conciseexplanation of the relevant theory and formulae from this course. A con-tract considered throughout this report is a whole life annuity-due contract.For a single-life this is a sequence of regular payments to the policy holder,life (x), commencing immediately and ceasing at the moment of death of(x). The net single premium (NSP) is the cost of the contract to (x), andis calculated using kpx, the probability that life (x) will survive a further kyears, and v, the discounting factor, as follows (assuming unit benefits):

ax =∞∑k=0

vkkpx (1.1)

1.1 An Overview of the Project

In chapter 2, the basic statuses on which multiple-life contracts are condi-tioned are introduced, and multiple-life probability calculations and NSPcalculations for particular contracts are shown. Throughout this chapternumerical examples emphasise the theory explained.

In chapter 3, the topic of copulas is introduced as a method of modellingbivariate distributions. Using copulas with survival data in an actuarialcontext is explained and commonly used copulas within insurance are intro-duced. Also, a detailed explanation of how I have used copulas in creatingan annuity NSP calculator is given.

In chapter 4, I have used the annuity calculator to analyse the differencesin contract values when using assumptions of independence and dependence.In this chapter I have analysed the effect of age, interest rates and the levelof dependence on the differences between the two contract costs.

In chapter 5, I discuss briefly other dependency models used in the in-surance industry. These methods are the Common Shock Model and theFrailty Model, which model dependency in different situations. Also, I com-pare the annuity ratio value for the common shock model to the copularesults ascertained in chapter 4.

The sources of information used throughout this project are listed in thebibliography. At the start of each section I have indicated the sources fromwhich I have directly obtained formulas and information in that section. Ihave also indicated my own independent work in each chapter. This topicintroduces a range of notation so for ease of reading a notation summaryis located in Appendix F, containing explanations and section references towhere the notation is introduced.

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Chapter 2

Multiple-Life Contracts

I have constructed and calculated all examples in this chapter using myown knowledge and understanding of the topic. These examples use thelife tables located in Appendix C and assume 5% interest unless otherwisestated. All explanations in this chapter are my own contributuion using myunderstanding of the topic I have researched.

2.1 The Joint-Life Status

The basic theory and formula in this section has been mainly sourced from[PR] and [HG].

Consider two lives, (x) and (y), who wish to buy a multiple-life contract,for example a married couple or business partners. The joint-life statusfor these two lives is in a state of survival while both lives are alive, hencethe status failing on the first death. The joint-life status is denoted by x : y.

The probabilities relating to the joint-life status use the assumption thatlives (x) and (y) are independent, as is assumed throughout this chapter.The probability that (x) and (y) are both still alive in t years is the prob-ability that the contract remains in a state of survival for t years. Thisprobability is denoted by(T (x) is future lifetime of (x): See Appendix D):

tpx:y = P (T (x) > t, T (y) > t) (2.1)

Using independence in probability we can calculate this as the productof the probability that (x) is alive at time t and the probability that (y) isalive at time t, as follows:

tpx:y = tpx tpy (2.2)

Similarly, the probability that the status is in failure at time t is:

tqx:y = 1− tpx:y = 1− tpx tpy (2.3)= tqx + tqy − tqx tqy (2.4)

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This uses the inclusion-exclusion principle, as the probability that oneof the lives is dead is the probability that life (x) is dead and that life (y) isdead, subtracting the probability that both lives have ended.

Example 2.1

Consider a married couple, John and Madge. John is 53 years old andMadge is 49 years old. Using illustrative life tables from Appendix C, andassuming John and Madge’s lives are independent, compute the probabilitythat their joint-life status is in survival after 10 years.

The probability that John is alive in 10 years is:

10p53 =l63

l53=

78239598779258

= 0.891186817

Similarly, the probability that Madge is alive in 10 years is:

10p49 = 0.921406956

Hence, the probability that the joint-life status survives 10 years is:

10p53:49 = 0.891186817× 0.921406956 = 0.821145732

Also the probability of either John or Madge dying within 10 years is:

10q53:49 = 1−10 p53:49 = 0.178854268

The joint-life contract can be generalised for n lives. If we let s be thejoint-life status for n lives:

s = x1 : x2 : ... : xn (2.5)

Then we can use this variable to define the failure time of the contract, thetime when the first death occurs, as:

T (s) = minT1, T2, ..., Tn (2.6)

where Ti is the time of death of life (xi). Assuming independence, we canconclude that the probability that the joint-life status is in survival at timet is:

tpx1:x2:...:xn =n∏i=1

(tpxi) (2.7)

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2.1.1 Joint-Life Annuities

In this subsection I have proved the result stated in [PR] for the m-yeardeferred joint-life whole life annuity using purely my own understanding.

A joint-life whole life annuity-due contract for two people would payout a sum, bk, at the beginning of every year that lives (x) and (y) remainalive. The income from this annuity stops on the event of the first death.The probability that the annuity will pay out bk in year k is simply theprobability kpx:y. Hence, the NSP of the joint-life annuity contract is:

ax:y(b) =N−1∑k=0

bk vkkpx:y (2.8)

where N = minW − x,W − y and W is the maximum lifetime.

Example 2.2

John and Madge, aged 53 and 49 respectively, wish to take out a wholelife joint-life annuity-due. Using life tables with the maximum age W = 99,and with unit benefits the NSP of the contract is:

ax:y =N−1∑k=0

vkkpx:y =45∑k=0

vkkp53:49 =45∑k=0

vkkp53 kp49 = 12.34966524

An annuity with the payments made at the end of the year is called ajoint-life whole life immediate annuity, with an annual rate b(t), theNSP is:

ax:y(b) =∫ N

0b(t) vt tpx:y dt (2.9)

An n-year temporary joint-life annuity-due is an annuity contractwhere the income is paid at the start of each year for n years, as long as both(x) and (y) are still living. In a similar style to the single-life temporaryn-year annuity-due (see Appendix D), the NSP of this contract is:

ax:y:n (b) =n−1∑k=0

bk v(k) kpx:y (2.10)

Example 2.3

John and Madge, ages 53 and 49 respectively, wish to take out a 4-yearjoint-life annuity-due contract, paying £20,000 each year providing bothJohn and Madge are still alive. What is the present value of the contract,where v(k) = 2−k ∀ k.

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The discount factors are v(0) = 1, v(1) = 12 , v(2) = 1

4 , v(3) = 18 . And

the probabilities are calculated by:

0p53:49 = 0p53 0p49 = 1

p53:49 = p53p49 =l54

l53× l50

l49=

87127118779258

× 89509949000135

= 0.987001332

Similarly, 2p53:49 = 0.973073498 and 3p53:49 = 0.95816358. Hence thepresent value of the 4-year annuity contract is:

a53:49:4 (20000) = 20000[(1× 1) + (12× 0.987) + (

14× 0.973) + (

18× 0.958)]

= 20000× 1.856539488 = 37130.789756

So the NSP is £37130.79. If we calculate the NSP with the normal discountfactor, vk, we obtain:

a53:49:n (20000) = 200003∑

k=0

vk kp53:49 = 20000× 3.650305343 = 73006.10686

A joint-life annuity contract where payments start after a time of m yearsif both (x) and (y) are alive is called a whole life m-year deferred joint-life annuity-due, and is denoted by m|ax:y. We can consider this contractto be the same the difference between a whole life joint-life contract and atemporary m-year joint-life annuity-due. Hence:

m|ax:y = ax:y − ax:y:m (2.11)= mpx:y v

m a(x:y)+m (2.12)

Example 2.4

John and Madge wish to take out a 4-year deferred joint-life annuity-due. Using the results from Example 2.2 and Example 2.3, the NSP for thiscontract is:

4|a53:49 = a53:49 − a53:59:4

= 12.34966524− 3.650305343 = 8.699359897

The result in equation 2.12 was stated in [PR], using my own knowledgeI will now construct a proof of this result. Using equations 2.11, 2.8 and2.10:

m|ax:y = ax:y − ax:y:m

=∞∑k=0

vk kpx:y −m−1∑k=0

vk kpx:y =∞∑k=m

vk kpx:y

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The using a change of variables such that l = k −m, we achieve:

m|ax:y =∞∑l=0

vl+m l+mpx:y = vm∞∑l=0

vl l+mpx:y

Finally, using basic actuarial probability functions found in Appendix D,namely equation D.18, we know that:

l+mpx:y = lp(x:y)+m mpx:y

Therefore,

m|ax:y = vm∞∑l=0

vl (lp(x:y)+m mpx:y)

= mpx:y vm

∞∑l=0

vl lp(x:y)+m

= mpx:y vm a(x:y)+m as required

2.1.2 Joint-Life Immediate Insurances

In this subsection I have shown a proof for the relationship between the joint-life whole life annuity and the joint-life immediate insurance using only myown contribution.

A joint-life insurance policy pays out bk on the failure of the status. Foran immediate life insurance policy this is at the end of the year of the firstdeath, at time k+ 1; where k is the curtate future lifetime of the first life todie. Considering two lives, (x) and (y), the probability that both lives willsurvive k years but will not survive k + 1 years is just:

kpx:y − k+1px:y = kpx:y qx+k:y+k (2.13)

Hence, the joint-life whole life immediate insurance policy NSP is:

Ax:y(b) =N−1∑k=0

bk v(k + 1) (kpx:y − k+1px:y) (2.14)

=N−1∑k=0

bk v(k + 1) kpx:y qx+k:y+k (2.15)

Similar to the single life formula (equation D.40 in Appendix D) we canderive:

ax:y =1−Ax:y

d(2.16)

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where d = i1+i , the annual effective discount rate. This can be generalised

for n lives:ax1:x2:...:xn =

1−Ax1:x2:...:xn

d(2.17)

I will now show a proof of the relationship in equation 2.16 using my ownknowledge. Using equation 2.14:

Ax:y =N−1∑k=0

v(k + 1) (kpx:y − k+1px:y)

=N−1∑k=0

v(k + 1) kpx:y −N−1∑k=0

v(k + 1) k+1px:y

= v ax:y −N∑l=1

vl lpx:y (using change of variables l = k + 1)

= v ax:y − (ax:y − 1) (since 0px:y = 1 and Npx:y = 0)= ax:y(v − 1) + 1

⇒ (1− v)ax:y = 1−Ax:y

And since:

11− v

= 1− 11 + i

=1 + i− 1

1 + i=

i

1 + i= d

This concludes the proof of equation 2.16.

Example 2.5

The joint-life whole life immediate insurance between John and Madgewith unit payments has NSP:

A53:49 =45∑k=0

vk+1 (kp53:49 − k+1p53:49) = 0.432507

2.1.3 Joint-Life Moment of Death Insurances

In this subsection I have produced a proof of the simplifications for the Gom-pertz and Makeham mortality laws which is entirely my own contribution.

For joint-life insurance policies which pay out at the moment of death,rather than at the end of year of death, we must define a function calledthe force of failure. Similar to the single-life function known as force ofmortality (see Appendix D), this function is:

µx:y(t) = − d

dtln(tpx:y) (2.18)

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Using the definition tpx:y = tpx tpy and the properties of the naturallogarithmic function, we can see that:

µx:y(t) = − d

dtln(tpx:y) = − d

dtln(tpx tpy) (2.19)

= − d

dt(ln(tpx) + ln(tpy)) (2.20)

= µx(t) + µy(t) (2.21)

Therefore, we can use the force of failure of the joint-life status in orderto compute the NSP of a joint-life whole life insurance, with unit paymentat moment of death.

Ax:y =∫ ∞

0vt tpx:y µx:y(t) dt (2.22)

Simplifications

Simplifications can be made to the joint-life formulae above when consideringlives which follow certain mortality laws. Such laws include the GompertzLaw and Makeham Law.

Gompertz Mortality LawHere the assumption and result have been obtained from [HG], however,

I have used my own knowledge to prove that the assumption is necessary.

The Gompertz Mortality law models the mortality of lives using the forceof mortality of (x) at age x+ t to be:

µx+t =gx(t)

1−Gx(t)(2.23)

= − d

dtln(tpx) = Bcx+t (2.24)

If we assume that cx1 + cx2 + cx3 + ... + cxn = cw and u is the joint-lifestatus for n lives, u = x1 : x2 : x3 : ... : xn, then the force of failure is:

µu(t) = µw+t (2.25)

where µw+t is the force of mortality for a single life (w). This result hasbeen stated in [HG], I will now use my own knowledge to show why theassumption is necessary to lead to this result. I will show this proof in thecase of two lives, however, it may be generalised to the case of n lives. Thejoint-life status will be u = x : y, using the equation for the force of mortality

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for a single life (see Appendix D), then:

µu(t) = µx:y(t) = − d

dtln(tpx:y) = − d

dtln(tpx tpy)

= − d

dt[ln(tpx) + ln(tpy)] (using properties of the ln function)

= − d

dtln(tpx)− d

dtln(tpy) (using the distributivity of differentiation)

= µx+t + µy+t = Bcx+t +Bcy+t = B[cx+t + cy+t]

If we require that the joint-life force of failure is equal to the force of mor-tality of a single life (w), then:

⇒ Bcw+t = B[cx+t + cy+t]⇒ Bctcw = Bct[cx + cy]⇒ cw = cx + cy

This is the assumption stated above, which is necessary to represent thejoint-life force of failure as that of a single (w), where w is the solution ofthis assumption. Hence this can easily be generalised to the case whereu = x1 : x2 : x3 : ... : xn, as above.

Therefore, we can see that when the lives within the joint-life status allfollow the same Gompertz mortality law, the status can be represented asa single life with initial age w. Hence, all calculations of probabilities andNSPs can be calculated in terms of single life (w).

Makeham Mortality LawHere the assumption and the result have been obtained from [HG], and thenI have proved why the assumption is necessary for this result using my ownknowledge.

The Makeham Mortality Law has force of mortality as follows:

µx+t = A+Bcx+t (2.26)

And solving this equation with respect to n lives, we see that:

cx1 + cx2 + cx3 + ...+ cxn = ncw (2.27)

which implies:µu(t) = nµw+t = µw:w:w:...:w(t) (2.28)

Therefore, it follows that n lives with varying ages can be replaced with nlives all with initial age w.

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To show that the assumption cx1+cx2+cx3+...+cxn = ncw is necessary torepresent the force of failure of the joint-life status to that of a single life forceof mortality, I will show the case for n lives (i.e. u = x1 : x2 : ... : xn), usingmy own knowledge. Following from the proof for the Gompertz Mortalitylaw, we have;

µu(t) = µx1+t + µx2+t + ...+ µxn+t

= A+Bcx1+t +A+Bcx2+t + ...+A+Bcxn+t

= nA+B[cx1+t + cx2+t + ...+ cxn+t]

Therefore, to achieve µu+t = nµw+t, we require:

n[A+Bcw+t] = nA+B[cx1+t + cx2+t + ...+ cxn+t]⇒ n[A+Bcwct] = nA+Bct[cx1 + cx2 + ...+ cxn ]

⇒ n[cw] = cx1 + cx2 + ...+ cxn

And, also:

nµw+t = −n ddt

ln(tpw)

= − d

dtln(tpw)− d

dtln(tpw)− ...− d

dtln(tpw) (n times)

= − d

dtln(tpwtpwtpw...tpw)

= − d

dtln(tpw:w:...:w) = µw:w:...:w(t)

Using the reverse argument as shown in the Gompertz mortality lawproof, this proves the result and assumption stated above.

2.1.4 Joint-life Annual Premiums

Considering a life insurance based on two lives, the premiums will be paid ona regular basis but will cease upon the first death, and hence the premiumpayments form a joint-life annuity. Consequently, the annual premium forthe life insurance contract can be calculated in the same way as a single-lifeannual premium, specifically:

π0 =Ax:y(b)ax:y

(2.29)

Example 2.6

Assuming unit payments, the annual premium for John and Madge is:

Π0 =A53:49

a53:49=

0.43250712.34966524

= 0.035021759

Using Example 2.2 and Example 2.5.

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2.2 The Last-Survivor Status

In this section the basic theory and formulae have been ascertained from[HG] and [PR].

The last-survivor status based on two independent lives, (x) and (y),is in survival as long as one person is still alive. Therefore, the status failson the death of the second person. We denote the last-survivor status byx : y to differentiate from the joint-life status. The probability of the statusbeing in survival after t years is:

tpx:y = P (T (x) > t or T (y) > t) (2.30)

Using independent probabilities and the inclusion-exclusion principle,the probability that the last-survivor status is in survival is:

tpx:y = tpx + tpy − tpx:y (2.31)

In words this is the probability that (x) is alive or that (y) is alive andsubtracting from this the probability that the joint-life status is in survival.As expected the probability that the status fails is:

tqx:y = 1− tpx:y (2.32)

If independence between the two lives can be assumed then the proba-bility of status failure is:

tqx:y = tqx tqy (2.33)

Example 2.7

The probability the last-survivor status for John and Madge will last 10years is:

10p53:49 = 0.891186817 + 0.921406952− 0.821145732 = 0.991448037

In comparison to the joint-life status this is a larger probability as both livesmust fail before the last-survivor status fails.

The last-survivor status can be generalised to n lives, and the status uis represented by:

u = x1 : x2 : x3 : ... : xn (2.34)

And as defined for two lives, this status remains in survival for as longas at least one of the n lives exists. Hence, the time of failure is upon theevent of the last death, T (u), where:

T (u) = max(T1, T2, ..., Tn) (2.35)

14

and Ti is the time of death of live (xi). An analogy to contrast the joint-lifeand last-survivor statuses is to imagine the joint-life status as the light bulbson your Christmas tree, and the last-survivor status as the light bulbs inyour house. If the one bulb breaks on your Christmas tree all the lights goout, however, if a bulb breaks in your house all the other lights in the housecontinue to work. Only when the last bulb in the house breaks is there nomore light in the house.

Calculating probabilities and net single premiums with the last-survivorstatus requires use of the inclusion-exclusion theory from probability. Thisstates that if Bk with k = 1, 2, ...., n are events then the probability thatat least one of these events occurs (i.e. the probability of the union of theevents) is:

P (B1 ∪B2 ∪ ... ∪Bn) = S1 − S2 + S3 − ...+ (−1)n−1Sn (2.36)

where:Sk = ΣP (Bl1 ∩Bl2 ∩ ... ∩Blk) (2.37)

And this summation ranges over all(nk

)subsets of k events. Using

the inclusion-exclusion principle we can form probabilities under the last-survivor status using certain joint-life statuses. If we define Bk to be theevent that live (xk) is living at time t and Stk = Σtpxl1 :xl2 :xl3 :...:xlk

, then:

tpx1:x2:x3:...:xn = S1t − S2

t + S3t − ...+ (−1)n−1Sn

t (2.38)

2.2.1 Last-Survivor Annuities

In this subsection I have shown how the generalisation to n lives correspondsto the two life situation using my own knowledge.

In a similar fashion to the annuity formula for a joint-life contract, wecalculate the NSP of the last-survivor whole life annuity-due for twolives using the formula:

ax:y =∞∑k=0

vk kpx:y (2.39)

For lifetime distributions with a maximum age, W , we change this to a finitesum as follows:

ax:y =N−1∑k=0

vk kpx:y (2.40)

where N = minW − x,W − y. In terms of this project we assume themaximum lifetime of our model is 99 for independent lives, as the life tableswe are using, see Appendix C, continue to age 99.

15

Example 2.8

John, (53), and Madge, (49), wish to take out a last-survivor wholelife annuity contract. What is the NSP for this contract? To calculate thepresent value we shall sum from k = 0 to k = min99−53, 99−49−1 = 45.

a53:49 =45∑k=0

vk kp53:49 =45∑k=0

vk (tpx + tpy − tpx:y)

=45∑k=0

vk(tpx + tpy − tpx tpy) = 16.43096

Therefore, for a unit benefit the present value of this contract is £16.43(4d.p.).

To calculate the NSP of a last-survivor whole life annuity-due for n livesusing equation 2.38, we have:

ax1:x2:x3:...:xn = S1a − S2

a + S3a − ...+ (−1)n−1San (2.41)

whereSk

a = Σaxl1 :xl2 :xl3 :...:xlk(2.42)

Two explain this more clearly I will show how this generalisation fitsthe two life situation using my own understanding. In the case of two livesequation 2.41 becomes:

ax:y = S1a − S2

a

where, using equation 2.42:

S1a = ax + ay

S2a = ax:y

Hence:

ax:y = ax + ay − ax:y (as expected)

Example 2.9

John and Madge wish to take out a last-survivor whole life annuity-duewith their son Frank, who is 23 years old. Let John, Madge and Frank berepresented by the letters J , M and F respectively. The NSP is:

aJ :M :F = Sa1 − Sa2 + Sa3

16

where:

Sa1 = aJ + aM + aF

Sa2 = aJ :M + aJ :F + aM :F

Sa3 = aJ :M :F

All the above joint-life whole life annuity-due NSPs can be calculatedusing the formulas described in section 2.1.1.

2.3 The General Symmetric Status

The main theory and formulae in this section have been sourced from [HG]and [W1].

The general symmetric status for n lives is defined to be:

u = x1 : x2 : x3 : ... : xnm (2.43)

This status is in survival as long as at least m of the n lives survive,therefore the failure of this status is at the moment of the (n − m + 1)th

death. From this we can see that both the joint-life and last-survivor sta-tuses are special cases of the general symmetric status. The joint-life statuscorresponds to when m = n and the last-survivor status is when m = 1.

A status which survives when exactly m of the initial n lives exists comesinto survival at the (n−m)th death and then fails at the (n−m+1)th death,and is represented by:

u = x1 : x2 : x3 : ... : xn[m] (2.44)

This particular status is not relevant in terms of insurances, however,it may be considered in the context of multiple-life annuities. To form ageneral solution for this status we require the Schuette-Nesbitt Formula, Iwill discuss this briefly.

2.3.1 Schuette-Nesbitt Formula

Consider the arbitrary events C1, ..., Cm and let N denoted the randomnumber of events which occur simultaneously. Define Sk = ΣP (Bl1∩...∩Bln)and S0 = 1. Also we need to define the shift operator, E and the differenceoperator,∆ by:

Eck = ck+1 (2.45)

E = 1 + ∆ (2.46)

17

From this we can find the Schuette-Nesbitt Formula:m∑n=0

P (N = n)En =m∑k=0

Sk∆k (2.47)

For the history and proof of this formula please see Appendix A.

2.3.2 Using The Schuette-Nesbitt Formula

Using the Schuette-Nesbitt Formula (S-N Formula) above we can concludethe general solution for the general symmetric status. Let c0, c1, .., cm bearbitrary chosen coefficients and use the S-N Formula to obtain:

m∑k=0

ck tpx1:x2:...:xm[k] =m∑j=0

∆jc0Stj (2.48)

and also:m∑k=0

ck ax1:x2:...:xm[k] =m∑j=0

∆jc0Saj (2.49)

where Stj = Σtpxk1:...:xkj

and Saj = Σaxk1:...:xkj

for j = 1, ....m. We alsodefine St0 = 1 and Saj = a∞ . With arbitrary coefficients d1, d2, ..., dm wherec0 = 0 and ck = d1 + ...+ dk we then can obtain:

m∑k=0

dk tpx1:x2:...:xm[k] =m∑j=0

∆j−1d1Stj (2.50)

and similarly:m∑k=0

dk ax1:x2:...:xm[k] =m∑j=0

∆j−1d1Saj (2.51)

The latter of these two expressions can be generalised to a solution for a lifeinsurance, like so:

m∑k=0

dk Ax1:x2:...:xmk=

m∑j=0

∆j−1d1SAj (2.52)

2.4 Asymmetric Statuses

In this section I have sourced the main theory and formulae from [HG] and[BOW].

An example of an asymmetric status is the status denoted by:

a : b : c : d (2.53)

18

This status is in a state of survival as long as one of lives (a) and(b) are alive and at least one of lives (c) and (d) are alive. Therefore,it is clear to see that the time of failure for this status can be calcu-lated using the future lifetimes of the lives involved in the contract, i.e.T = minmaxT (a), T (b),maxT (c), T (d).

2.4.1 Asymmetric Annuities

The asymmetric status can be viewed in terms of two separate last-survivorstatuses, one involving (a) and (b), called r, and one involving (c) and (d),called s. The probability that the joint-life status between r and s is insurvival at time t is:

tpr:s = tpr + tps − tpr:s (2.54)

As seen in section 2.1, this implies that:

ar:s = ar + as − ar:s (2.55)

Therefore, the NSP of an immediate annuity for the survival of thisstatus can be calculated using the following formula.

aa:b:c:d = aa:b:c + aa:b:d − aa:b:c:d (2.56)= aa:c + ab:c − aa:b:c + aa:d + ab:d − aa:b:d (2.57)− aa:c:d − ab:c:d + aa:b:c:d (2.58)

The reversionary annuity is a different type of annuity which is morerelevant for widows and orphans. Denoted by the symbol ax/y, this is NSPof a contract with unit payments that commences at the death of live (x)and ends with the death of live (y). For example, if a man and his wife takeout this annuity the payments start when the man dies and continue for therest of the duration of the widow’s life. This NSP can be calculated using:

ax/y = ay − ax:y (2.59)

This is obviously the present value of a whole life annuity for the bene-ficiary minus the present value of an annuity for the duration of survival ofthe joint-life status of (x) and (y).

Example 2.10

John and Madge wish to take out a reversionary annuity contract whichnames Madge as the beneficiary after John’s death. What is the presentvalue of this contract with unit benefit payments?

ax/y = ay − ax:y = a49 − a53:49

= 14.86592− 12.34867 = 2.516243

19

So the present value of this reversionary annuity is £2.52

What would be the effect of a greater age difference between the twolives on the present value of this contract?

Example 2.11

Madge’s father, Harold age 88, wishes to take out a reversionary annuitywith his daughter as the beneficiary of the contract. What is the NSP ofthis contract?

ax/y = ay − ax:y = a49 − a88:49

= 14.86592− 4.022515 = 10.843402

The present value £10.85 is a much greater value then the NSP of thereversionary annuity between John and Madge. This is due to the greaterprobability that the joint-life status of Madge with her father is likely to failthen the joint-life status of John and Madge. Also Madge is likely to outliveher father my a greater number of years the she is likely to exceed John’slife. Therefore, the annuity between Madge and her father is likely to payout for a greater number of years than it would if the contract was betweenJohn and Madge, causing the NSP to be greater.

2.4.2 Asymmetric Insurances

Insurance contracts exist considering n lives in a joint-life status, a deathbenefit ck(t) is paid if live k is the first of the n lives to die, and thisoccurs at time t. This is referred to as a first death insurance and thejoint-life status fails due to the cause k. The net single premium of theresulting contract is calculated as follows, assuming that the future lifetimesare independent:

m∑i=1

∫ ∞0

ck(t) vt tpx1:x2:...:xn µxk+t dt (2.60)

There exists a special case of this first death insurance where ck(t) = 1and cl(t) = 0 where k 6= l and hence the formula can be simplified. TheNSP of this formula is shown below:

Ax1:x2:...:xk−1:x1k:xk+1:...:xm =

∫ ∞0

vt tpx1:x2:...:xn µxk+t dt (2.61)

This can be simplified if it can be assumed that all n lives follow thesame Gompertz Law as described in Section 2.1.4. In this case:

µxk+t =cxk

cwµx1+t:x2+t:...:xn+t (2.62)

20

where w is the solution to the equation cx1 + cx2 + cx3 + ...+ cxn = cw andhence:

Ax1:x2:...:xk−1:x1k:xk+1:...:xn =

cxk

cwAx1:x2:...:xn =

cxk

cwAw (2.63)

A similar contract to this insurance is a contract which pays out on theevent of the death of life (x) provided that this is the rth death of the nlives. In order for the payment to be made in this contract n−r of the othern− 1 lives must still be in survival. The NSP for this contract is:

Ax1:x2:...:xk−1:xrk:xk+1:...:xn =∫ ∞

0vt tpx1:x2:...:xn[n−r] tpxk µxk+t dt (2.64)

2.5 The General Two-Life Annuity Contract

In this section the main formulae and theory have been gained from [PR]and [BOW].

Let us consider two independent lives (x) and (y), a general two-life an-nuity contract between these two lives can be represented by three annuitybenefit vectors. There are many situations in which this contract is useful,some of which are described below.

Example 2.12

1. John and Madge wish to take out an annuity for their retirement,where there will receive £30,000 each year while they are both aliveand once one of them dies the other will then receive only £20,000yearly until their demise.

2. John and Madge wish to take out an annuity which pays £20,000 everyyear as long as at least one of John and Madge are still alive and Johnis over 75 years old. If this is not the case then a benefit is only madeif Madge is alive and under the age of 69

3. Like the reversionary annuity, imagine John and Madge take out anannuity which does not start to pay out until after the first death, andthen pays out £10,000 every year until the seconds person’s death

The three annuity benefit vectors which represent the general contract are:

• f - where fi is the payment amount at time i if only (x) is alive

• g - where gi is the payment amount at time i if only (y) is alive

• h - where hi is the payment amount at time i if both (x) and (y) arealive

21

Now, we define j to be the vector equivalent to j = h− f− g. From thiswe can describe any general two-life contract to be the sum of three separateannuities; an annuity on life (x) with payment vector f , an annuity on life(y) with payment vector g and an annuity on the joint-life status betweenlives (x) and (y) with payment vector j.

ax(f) + ay(g) + ax:y(j) (2.65)

Hence, this much simplifies the calculations.

2.6 The General Two-Life Insurance Contract

In this section general formulae and concepts have been taken from [PR].

These insurance contracts are all based on the death benefit of the con-tract being paid at the moment of death, not end of year of death. Whetherthe lives in the contract wish the death benefit to paid on the first death, onthe second death or on both deaths we can consider all these contracts inthe general two-life insurance contract. If we define b(t) to be the value ofthe payment at the time of the first death and d(t) to be the amount paidat the time of the second death. In the same way as with the general two-life annuity, we consider the general two-life insurance contract in terms of asum of three different insurance contracts to simplify the calculations. First,a single-life insurance contract for life (x) with benefit function d. Then asingle-life insurance contract for life (y) with benefit function d. And finally,a joint-life insurance contract on (x) and (y) with benefit function b − d.This creates the present value of the benefits to be:

Ax(d) + Ay(d) + Ax:y(b− d) (2.66)

This clearly works if we look again at our example.

Example 2.13

John and Madge take out a two-life insurance policy which will pay outdiffering amounts on the event of both deaths. With a single contract basedon the two lives b(t) will be paid out on the first death and d(t) on the eventof the second death.

If we consider the three separate insurances described above and if Madgedies first then John will receive a payment of d(t) from the single-life contracton Madge and a payment of b(t)− d(t) from the joint-life contract.

When John then dies the single-life contract on John pays a further d(t).So in total the three contracts have paid out d(t) + d(t) + b(t) − d(t) =b(t) + d(t), the same as the single contract.

22

2.7 Contingent Insurances

In this section describing contingent insurances I have obtained the generaltheory and basic formalae from [PR].

Contingent insurances based on two lives are life insurance policies wherea selected person in the insurance must die first or second for the insuranceto pay out any benefits.

2.7.1 First-Death Contingent Insurances

This is an insurance policy where the designated life must die before theother person in the contract in order for the policy to pay out the deathbenefit at the end of the year of the first death.

Example 2.14

John and Madge take out a first-death contingent life insurance policy.In this policy they designate Madge to be the individual who must die first.If John dies before Madge then Madge receives no benefit. If Madge diesfirst but John dies before the end of the year of Madge’s death then theirfamily will still receive the payout of the insurance.

The present value of this contract is represented by A1x:y(d) where d is

the death benefit vector for the contract. To derive a formula for the presentvalue of this contract, we must define a new probability q1

x:y , which denotesthe probability that in the next year (x) will die and (y) must be alive atthe time of death of (x), but does not necessarily have to survive to the endof that year. The superscript indicates that life (x) is the first to die in thiscontract. This probability can be estimated from other probabilities and wewill show this in a intuitive manner here.

To derive this estimation we must consider two different cases for lives(x) and (y) which may occur:

Case 1: (x) dies within a year and (y) is alive at the time of death of (x)

Case 2: (x) dies within the year and (y) survives to the mid-year point

The probability for case 1 is q1x:y, and we claim that the probability of

case 2 is close to this. The probability of case 2 can be calculated, usingformulae from Appendix D and assuming a uniform distribution if deathsover a year, as:

12py = 1− qy

2(2.67)

23

Assuming independence and using basic probability theory, we can seethat the probability of both these independent events occurring is the prod-uct of the two probabilities.

q1x:y = qx −

12qxqy (2.68)

From this we can also see that:

qx:y = q1x:y + q1

y:x (2.69)

As the probability that either (x) or (y) die within the year is the sumof the probabilities of (x) dying in that year and (y) being alive at the timeor (y) dying in that year and (x) being alive at that time. These two eventscan not both occur and are therefore mutually exclusive events. If eitherone of these events occur the joint-life status fails that year.

q1x:y + q1

y:x = qx −12qxqy + qy −

12qyqx (2.70)

= qx + qy − qxqy = qx:y (2.71)

Since this probability has now been defined we can use it to calculatethe present value of the first-death contingent insurance policy. Using thepresent value for the joint-life insurance policy with payment vector b atend of year of death, we can see that:

A1x:y(b) =

N−1∑k=0

bk v(k + 1) kpx:y q1x+k:y+k (2.72)

And using equation 2.69 it follows that:

Ax:y(b) = A1x:y(b) +A1

y:x(b) (2.73)

2.7.2 Second-Death Contingent Insurances

This policy is one such that the benefit is paid only if the designated lifesurvives the other life. This contract is denoted by A2

x:y, where (x) is thedesignated life. Since this contract pays out if (x) dies after (y) and thefirst-death contingent policy pays out if (x) dies before (y) then the twopolicies cover all possibilities of when the payout will occur, hence:

Ax(b) = A1x:y(b) +A2

x:y(b) (2.74)

2.7.3 The General Contingent Insurance

Consider a general policy on lives (x) and (y). If (x) dies before (y) then itpays bk at the end of year of death of (x) and dk at the end of year of deathof (y). If (y) dies first the policy pays ck at the end of year of death of (x)and ak at the end of year of death of (y). The NSP of this contract is:

A1x:y(b) +A2

y:x(d) +A1y:x(c) +A2

x:y(a) (2.75)

24

Chapter 3

Copulas

In this chapter I have constructed and calculated all examples.

So far all the multiple-life contracts we have discussed in this report havebeen based on the assumption that the lives involved were independent.Realistically this may not be the case, and in this chapter we study usecopulas as a model of dependence.

3.1 Definition

The following definition and derivation have been obtained and understoodfrom [PR] and shown using my own explanation.

A “copula” is a “tool for understanding relationships among multivariateoutcomes”, that is to say a tool that can be used to show dependence betweenlives. In terms of this project we are interested in understanding and using ajoint distribution of two lifetimes. A joint distribution consists of two parts,the first is the distribution of the two lifetimes and the second is the waythese respective future lifetimes are linked together. This relationship can bedescribed by a copula, which also allows us to consider each life separately.If we consider the joint distribution (X,Y ), when X and Y are independentwe obtain the joint distribution as a product of the individual distributions:

FX,Y (x, y) = FX(x)FY (y) (3.1)

The next step is to consider whether we can replace the multiplicationon the right-hand side of equation 3.1 with other transformations. Thismeans we require a function C from [0, 1] × [0, 1] to itself so that a validjoint distribution is maintained. Hence:

FX,Y (x, y) = C(FX(x), FY (y)) (3.2)

25

To restrict C we take any point w in [0,1], then we assume that if X > w,so that FX(w) = 0, then for all values of t, FX,Y (w, t) = 0 (similarly for Y ).Therefore, for all a, b in [0,1], the condition forces:

C(0, b) = C(a, 0) = 0 (3.3)

Similarly, if X ≤ w so that FX(w) = 1, then ∀ t, FX,Y (w, t) = FY (t),leading to the condition that for all a, b in [0,1]:

C(1, b) = b (3.4)C(a, 1) = a (3.5)

A final condition can be assumed from knowing that for any smallerrectangle within [0,1] the probability that (X,Y ) lies within that smallerrectangle is non-negative. Also, we know this probability is the sum of thevalue of the joint distribution FX,Y at the northeast corner and southwestcorner of the rectangle minus the sum of the values at the other two corners,due to the cumulative nature of the values for the distribution. This givesthe condition for a1 ≤ a2, b1 ≤ b2:

C(a2, b2) + C(a1, b1)− C(a1, b2)− C(a2, b1) ≥ 0 (3.6)

We can now give a formal definition of a copula using these conditionson C, obtained from [PR].

Definition 3.1

A copula is a function C from [0, 1]× [0, 1] satisfying conditions 3.3-3.6.

If T1 and T2 are uniformly distributed on [0,1], then by definition thismeans that FTi(a) = a for i = 1, 2 and for all a ∈ [0,1], hence if follows thatthe joint distribution is:

FT1,T2(a, b) = C(a, b) (3.7)

3.2 Sklar’s Theorem

In this section I have used lemmas and theorems provided in [RN] to aid myown explanation of this topic.

Sklar’s theorem introduces a relationship between joint and marginaldistributions and the relevant copula. Firstly, we define these distributions.

Definition 3.2

A function F is a distribution function with domain R if:

26

• F is nondecreasing

• F (−∞) = 0 and F (∞) = 1

Definition 3.3

A function H is a joint distribution function with domain R2 if

• H is 2-increasing

• H(x,−∞) = H(−∞, y) = 0 and H(∞,∞) = 1

H has marginal distribution functions F (x) = H(x,∞) andG(y) = H(∞, y),which are both distribution functions. H is grounded and this is shownthrough the condition that H(x,−∞) = H(−∞, y) = 0.

Example 3.1

An example of a joint distribution function is Gumbel’s bivariate logisticdistribution

H(x, y) = (1 + e−x + e−y)−1

∀x, y ∈ R. We can see that this is indeed a joint distribution function sincethe function is 2− increasing and:

H(x,−∞) = (1 + e−x + e∞)−1 =1∞

= 0

Similarly for H(−∞, y) = 0 and:

H(∞,∞) = (1 + 2e−∞)−1 = 1−1 = 1 as required

It is also easy to show that H has marginal distribution functions F (x) andG(y). We can find these by the following:

F (x) = H(x,∞) = (1 + e−x + e−∞)−1 = (1 + e−x)−1

G(x) = H(∞, y) = (1 + e−∞ + e−y)−1 = (1 + e−y)−1

Definition 3.4

A function C ′ is a subcopula if the function has the following properties:

• If S1 and S2 are any subsets of [0,1] containing 0 and 1 thenDomC ′ = S1 × S2

• C ′ must be grounded and 2-increasing

27

• ∀a ∈ S1 and ∀b ∈ S2

C ′(a, 1) = a and C ′(1, b) = b

• RanC ′ is also [0,1]

From these definitions we can use some Lemmas found in [RN].

Lemma 3.1(corresponding to Lemma 2.3.4 in [RN])

Let H be a joint distribution function with margins F and G. Then thereexists a unique subcopula C ′ such that

• DomC ′ = RanF ×RanG

• ∀x, y ∈ R, H(x, y) = C ′(F (x), G(y))

For proof see Appendix B

Lemma 3.2(corresponding to Lemma 2.3.5 in [RN])

If C ′ is a subcopula. Then there exists a copula C such thatC(a, b) = C ′(a, b) ∀ (a, b) ∈ DomC ′.

For proof see Appendix B

We are now in a position to state and prove Sklar’s Theorem, obtainedfrom [RN] and fully understood and worked through.

Sklar’s Theorem

Let H be a joint distribution function with margins F and G. Thenthere exists a copula C such that ∀x, y ∈ R:

H(x, y) = C(F (x), G(y)) (3.8)

If F and G are continuous then C is unique, otherwise C is uniquely de-termined on RanF ×RanG. If C is a copula and F and G are distributionfunctions then H is a joint distribution function with marginals F and G.

Proof of Sklar’s Theorem

The existence of a copula C that holds ∀ x, y ∈ R follows from Lemmas3.1 and 3.2. These Lemmas show that equation 3.8 holds where C is asubcopula and x,y are in the DomC. If F and G are continuous the RanF =RanG = [0, 1], so that the unique subcopula is now actually a copula. Theconverse is a matter of straightforward verification.

28

3.3 Quasi-Inverses and Constructing Copulas

In this section I have obtained the main theory from [RN] and explained itusing my own understanding.

Definition 3.5

The quasi-inverse of a distribution function F , is a function with domain[0,1] represented by F (−1), such that:

1. F (F (−1)(t)) = t ∀t ∈ RanF

2. If t ∈ RanF , then

F (−1) = infx|F (x) ≥ t= supx|F (x) ≤ t

If the distribution function F is a strictly increasing function then it onlyhas one quasi-inverse, the ordinary inverse.

Example 3.2

In Example 3.1 we found the marginals F (x) = (1 + e−x)−1 and G(y) =(1 + e−y)−1 for Gumbel’s bivariate distribution. To find the quasi-inversesof F (x) and G(y), we set F (x) = u and rearrange the equation:

F (x) = u = (1 + e−x)−1

⇒ u−1 = 1 + e−x

⇒ u−1 − 1 = e−x

⇒ − ln(u−1 − 1) = x

Therefore the quasi-inverse of F (x) is F (−1)(u) = − ln(u−1− 1). And, simi-larly, for G(y) the quasi-inverse is G(−1)(v) = − ln(v−1 − 1).

Quasi-Inverses can be very useful in constructing copulas for joint dis-tribution functions, shown in this corollary.

Corollary 3.1This corollary is based on Corollary 2.3.7 from [RN].

Let H be a joint distribution function with marginals F (x) and G(y),and let C ′ be a subcopula where, from Lemma 3.1, H(x, y) = C ′(F (x), G(y))∀ x,y ∈ R. If F (−1)(u) and G(−1)(v) are the quasi-inverses of F (x) and G(y)respectively, then ∀ (u, v) ∈ DomC ′:

C ′(u, v) = H(F (−1)(u), G(−1)(v)) (3.9)

29

This provides a simple but effective method of constructing copulas fromjoint distribution functions.

Example 3.3

Gumbel’s bivariate logistic distribution is H(x, y) = (1 + e−x + e−y)−1.In Example 3.1 we found the marginals F (x) = (1 + e−x)−1 and G(y) =(1 + e−y)−1, and then in Example 3.2 we found the quasi-inverses of thesefunctions, namely F (−1)(u) = − ln(u−1 − 1) and G(−1)(v) = − ln(v−1 − 1).Now we can find the copula of Gumbel’s distribution using equation 3.9.

C ′(u, v) = H(F (−1)(u), G(−1)(v))

= (1 + eln(u−1−1) + eln(v−1−1))−1

= (1 + u−1 − 1 + v−1 − 1)−1

= (u−1 + v−1 − 1)−1

This form of the copula may be simplified by multiplying by uvuv .

C ′(u, v) =uv

u+ v − uvExample 3.4

In this example we consider the Type B bivariate extreme value distributions(Johnson and Kotz-1972):

HΘ(x, y) = exp[−(e−Θx + e−Θy)1Θ ]

∀x, y ∈ R, where Θ ≥ 1. The first step is to find the marginals:

F (x) = HΘ(x,∞) = exp[−(e−Θx + e−Θ∞)1Θ ]

= exp[−(e−Θx)1Θ ] = exp[−e−x]

Similarly, G(y) = HΘ(∞, y) = exp[−e−y]. now we can find the quasi-inverses of F (x) and G(y) through rearranging the formulae:

u = exp[−e−x]lnu = −e−x

− ln(− lnu) = x

So F (−1)(u) = − ln(− lnu) and similarly G(−1)(v) = − ln(− ln v). Usingequation 3.9, we can calculate the copula of this joint distribution function.

CΘ(u, v) = HΘ(F (−1)(u), G(−1)(v))= HΘ(− ln(− lnu),− ln(− ln v))

= exp[−(e−Θ ln(− lnu) + e−Θln(− ln v))1Θ ]

= exp[−((− lnu)Θ + (− ln v)Θ)1Θ ]

This is known as the Gumbel-Hougaard parametric family of copulas.

30

3.4 Survival Copulas

This section contains theory obtained from [RN] explained using my ownunderstanding of the topic.

In the context of this project we are considering the future lifetimes ofpeople, for this reason we will be dealing with survival functions rather thandistribution functions. We can equate what we have established so far interms of survival functions and joint survival functions.

The survival function, F (x) of a life (y) is the probability of the in-dividual living beyond a time x, we already represent this probability to bexpy in standard actuarial notation. So F (x) = xpy = 1− F (x), where F (x)is the distribution function of X.

The joint survival function is represented by H(x, y). For a pair ofrandom variable (X,Y ), with the joint distribution function H(x, y), H(x, y)is the probability that X will survive x years and Y will survive y years, i.e.H(x, y) = P [X > x, Y > y]. The marginals of H(x, y) are the univariatesurvival functions F (x) = H(x,−∞) and G(y) = H(−∞, y).

To find a relationship between the joint survival function and a copulawe must use the relationships we know between distribution functions andsurvival functions, as follows:

H(x, y) = 1− F (x)−G(y) +H(x, y) (3.10)= F (x) +G(y)− 1 + C(F (x), G(y)) (3.11)= F (x) +G(y)− 1 + C(1− F (x), 1−G(y)) (3.12)

From this let us define a survival copula as a function C from [0, 1]× [0, 1]to [0,1], with form:

C(u, v) = u+ v − 1 + C(1− u, 1− v) (3.13)

And using this together with equation 3.12 we can conclude that:

H(x, y) = C(F (x), G(y)) (3.14)

Using the inverse forms of the survival functions we find:

C(u, v) = H(F (−1)(u), G(−1)(v)) (3.15)

I now show how to construct survival copulas using joint survival functions.

Example 3.5

LetX and Y be random variables with the joint survival functionH(x, y) =(ex+ey−1)−1 ∀x, y ≥ 0. First we must find the univariate survival functionsF (x) and G(y).

F (x) = H(x,−∞) = (ex + e−∞ − 1)−1 = (ex − 1)−1

31

G(y) = H(−∞, y) = (e−∞ + ey − 1)−1 = (ey − 1)−1

Secondly, we must inverse these functions.

u = (ex − 1)−1

⇒ u−1 = (ex − 1)⇒ ln(u−1 + 1) = x

So F(−1)(x) = ln(u−1 + 1). And similarly, G

(−1)(y) = ln(v−1 + 1). To findthe copula for this joint survival function we must use equation 3.15.

C(u, v) = H(F (−1)(u), G(−1)(v))= H(ln(u−1 + 1), ln(v−1 + 1))

= (eln(u−1+1) + eln(v−1+1) − 1)−1

= (u−1 + 1 + v−1 + 1− 1)−1 = (u−1 + v−1 + 1)−1

When dealing with a pair of random variables X and Y , with C as thecopula, there are two other functions which represent probabilities relatedto these random variables. The first of these functions is the dual of thecopula, given by C, this is the probability that X does not survive x yearsor Y does not survive y years.

C(F (x), G(y)) = P [X ≤ x or Y ≤ y] (3.16)

The dual of a copula can be calculated directly from the copula, however,this function is not a copula itself.

C(u, v) = u+ v − C(u, v) (3.17)

The second of the functions is the co-copula, given by C∗, which is not acopula itself either but can also be calculated directly from the copula. Theco-copula represents the probability that X survives x years or Y survivesy years.

C∗(F (x), G(y)) = P [X > x or Y > y] (3.18)

C∗(u, v) = 1− C(1− u, 1− v) (3.19)

3.5 Common Copulas

In this section I have referenced [PR] and [BOW] to aid my explanations.

Using the work from Chapter 2 it is easy to see that the copula for anindependent joint distribution must be C(a, b) = ab, and there are manyother examples of copulas including Frank’s Family of Copulas.

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3.5.1 Frank’s Copula

The parametric family of copulas, known as Frank’s Copulas, has the form:

Cα(a, b) =1α

log(1 +(eαa − 1)(eαb − 1)

eα − 1) (3.20)

where the parameter α is any non-zero real number. We can see that Frank’sfamily of copulas has an interesting limit as α tends to zero. This is:

limα→0

Cα(a, b) = ab (3.21)

For all a, b ∈ [0, 1].

The basics of the following proof have been taken from [BOW], using myown knowledge I have elaborated on the explanations and calculation stepsprovided in [BOW], making them more accessible to my target reader.

First we define FT (x)T (y)(s, t) to be Frank’s copula.

FT (x)T (y)(s, t) =1α

log(1 +(eαFT (x)(s) − 1)(eαFT (y)(t) − 1)

eα − 1) (3.22)

Then by differentiating by each variable we can find:

δ2

δsδtFT (x)T (y)(s, t) = fT (x)T (y)(s, t)

=αfT (x)(s)fT (y)(t)[eα(FT (x)(s)+FT (y)(t))]

[(eα − 1) + (eαFT (x)(s) − 1)(eαFT (y)(t) − 1)]2(eα − 1)

≥ 0

So:

fT (x)T (y)(s, t) =αfT (x)(s)fT (y)(t)[eα(FT (x)(s)+FT (y)(t))]

[(eα − 1) + (eαFT (x)(s) − 1)(eαFT (y)(t) − 1)]2(eα − 1) (3.23)

If we take the limit of the function above as α tends to zero, we canseparate the function into a product of smaller functions and consider thelimit of each individual function.

limα→0

fT (x)T (y)(s, t) = fT (x)(s)fT (y)(t) limα→0

[A(α)B(α)C(α)] (3.24)

where,A(α) = eα[FT (x)(s)+FT (y)(t)] (3.25)

B(α) =(eα − 1)α(eα − 1)2

(3.26)

33

C(α) =1

1 + [ (eαFT (x)(s)−1)(e

αFT (y)(t)−1)(eα−1) ]2

(3.27)

To find the limit of equation 3.23 we must find the limits of the func-tions A, B and C using the fact that e0 = 1, from our basic mathematicalknowledge. Therefore:

limα→0

A(α) = limα→0

eα[FT (x)(s)+FT (y)(t)] = e0 = 1

Since as α tends to zero the value of the power tends to zero. Also:

limα→0

B(α) = limα→0

(eα − 1)α(eα − 1)2

= 1

The limit of function C(α) depends on the limit of its denominator only.If we define the denominator of C(α) to be 1 +D(α) it follows:

limα→0

D(α) = limα→0

[(eαFT (x)(s) − 1)(eαFT (y)(t) − 1)

ealpha]

= limα→0

[FT (x)(s)eαFT (x)(s)(eαFT (y)(t) − 1) + F (t)eαFT (y)(t)(eαFT (x)(s) − 1)

eα

=FT (x)(s)e0(e0 − 1) + FT (y)(t)e0(e0 − 1)

e0=

0 + 01

= 0

Therefore,

limα→0

C(α) =1

(1 + 0)2= 1 (3.28)

and so,

limα→0

fT (x)T (y)(s, t) = fT (x)(s)fT (y)(t)[1× 1× 1] = fT (x)(s)fT (y)(t) (3.29)

Therefore, T (x) and T (y) are independent as α → 0. Since the futurelifetimes are independent then Franks’s copula tends to the independencecopula as α→ 0.

limα→0

FT (x)T (y)(s, t) = FT (x)(s)FT (y)(t) as required (3.30)

3.5.2 The Frechet-Hoeffding Bounds

The Frechet-Hoeffding bounds are functions that provided the lower andupper bounds for the values of any copula C. The Frechet-Hoeffding lowerbound, ∀ u,v ∈ [0,1] is:

W (u, v) = maxu+ v − 1, 0 (3.31)

The Frechet-Hoeffding upper bound for copula values ∀ u,v ∈ [0,1] is:

M(u, v) = minu, v (3.32)

34

Hence, we have the inequality that for any copula, C, and ∀ u,v ∈ [0,1]:

W (u, v) = maxu+ v − 1, 0 ≤ C(u, v) ≤ minu, v = M(u, v) (3.33)

The Frechet-Hoeffding bounds and the independence copula are exam-ples of very simple copulas. These copulas are not useful when studyingdependent lives as the Frechet-Hoeffding bounds give very extreme valuesand C(u, v) = uv is only relevant for independent lives. Frank’s family ofcopulas is a widely used choice when studying dependent lifetimes since itis a parametric family, for which we can vary the value of α to fit the dataobserved.

3.6 Using Copulas

In this section I have used the calculation model from [FCV] and explana-tions in [SOA2] to aid the creation of my own annuity calculator. Otherwise,all work in this section is my own contribution.

I shall give a short explanation on how I have used copulas to find thevalue of a whole life annuity contract for dependent lives. In my calculationsI am using data presented in an article by Frees, Carriere and Valdez (1996),reference [FCV], which studied annuity valuation with dependent mortality.

3.6.1 The Model

The Frees et al. article obtained data from a major insurance company onwhich they based their study, and from this data they found the Gompertzdistribution to be a suitable distribution to model the male and female lives,using this expression:

F (x) = 1− exp(e−m/σ(1− ex/σ)) (3.34)

The parameters in the expression are m and σ, where m is the mode ofthe data and σ is the scale measure. The Gompertz distribution is deemedto be a suitable distribution to model the lives in the study since it is asmooth function which closely replicates the non-parametric fit, as seen inFigure 1 in [FCV], and only has two parameters required to be estimated. Ihave opted to use the Gompertz distribution as it is a function which I amfamiliar with since using it within my 2H studies.

I will use the Frank’s family of copulas to model the dependence betweenlives as it is the most widely used copula for this purpose. Now I have definedthe model which I intend to use I require estimates for the parameters onwhich I can base my calculations. If j = 1 represents the males lives andsimilarly j = 2 the female lives, the estimates required are for the parameters

35

Figure 3.1:

mj , σj and α. These estimates have been taken from “Table 2” in [FCV],as shown in Figure 3.1:

Having used these estimates of the parameters I have then created anannuity NSP calculator in a spreadsheet. When entering the age of themale and female clients this calculates the NSP of a last-survivor whole lifeannuity-due contract based on the Gompertz model with Frank’s Copula.From this spreadsheet I will obtain values for contracts with dependent livesand compare these to values we found for independent lives. Furthermore,the spreadsheet calculates the NSP for contracts using the independencecopula, and both of Frechet-Hoeffding’s copula bounds. When calculatingvalues using the independence copula we must use the parameters corre-sponding to the second column of the table above, as by definition theunivariate distributions of the lives will differ from the dependent case.

3.6.2 Maximum Lifetime of Model

The NSPs produced by the annuity calculator that I have formed are sumsup to k = N−1, where N = min110−x, 110−y and 110 is the maximumlifetime modelled. The calculator is for lives aged 40 and above as it hasbeen assumed that the values added after k = 69 are so small that theymake an insignificant difference to the total of the sum, and for ages greaterthan 40 these values become even less significant.

This is a short numerical explanation as to why this assumption is viable.Taking the youngest age this calculator is designed for, age 40, the proba-bilities of that life not living to the age of 110 from birth, e.g. k = 69, beingeither male or female using the Gompertz distribution are shown below:

Male : F1(110) = 0.99996292 (3.35)

Female : F2(110) = 0.999985997 (3.36)

If the male and female in the contract are both aged 40, then the depen-dent probability that the status will be in survival at age 110 is:

1−HT (69, 69) = 5.16783E − 05

36

The discounting for k = 69 years with a 5% interest rate is:

vk = 0.034509476

So the amount added to the total sum at this stage is:

vk(1−HT (69, 69)) = 1.78339E − 06

The greater k becomes the smaller this value becomes, so we can findan approximation for the NSP by ignoring these negligible amounts, andsumming from k = 0 to k = N − 1 for all ages.

3.6.3 The Method

The calculations involved in the annuity calculator find the NSP of thecontract using a method briefly outlined in [FCV]. I will described thismethod in more detail now. Firstly, to define the notation being used,F1(x) is the probability of a male dying before age x from birth using theGompertz distribution function in equation 3.34 with parameters m1 and σ1

from Figure 3.1 above. Similarly, F2(y) is the corresponding probability for afemale with parameters m2 and σ2. To consider the bivariate distribution ofthe future lifetimes of males and females, we define H(x, y) = P (X ≤ x, Y ≤y), where X and Y are the future lifetime random variables of the male andfemale respectively. H(x, y) is the probability that the male does not surviveto age x and the female dies before age y. C represents Frank’s copula forthe dependent model, the independence copula for the independence modeland the Frechet-Hoeffding bounds in two more calculations. Regardless ofthe copula used the method remains the same, only we must use the secondcolumn of variables in Figure 3.1 for the Gompertz distribution when usingthe independence assumption. From Sklar’s Theorem:

H(x, y) = C(F1(x), F2(y)) (3.37)

The Gompertz distribution function finds the probability of survival ofa life to an age from birth, however, when finding the NSP for a maleand female, (x) and (y), we know these lives have already reached theircontract initiation ages, x and y respectively. Therefore, we must conditionthe probability of the last-survivor status surviving by the fact the lives havealready reached their contract initiation ages. To do this, we introduce thefuture lifetime random variables, conditonal on the survival of lives to theircontract initiation ages, T1 = X − x and T2 = Y − y, assuming T1 > 0 andT2 > 0. Letting HT (a, b) be the conditional distribution function of T1 andT2, we can see that:

HT (a, b) = P (T1 ≤ a, T2 ≤ b|T1, T2 > 0) (3.38)

=P (0 < T1 ≤ a, 0 < T2 ≤ b)

P (T1 > 0, T2 > 0)(3.39)

37

HT (a, b) =H(x+ a, y + b)−H(x, y + b)−H(x+ a, y) +H(x, y)

1−H(x,∞)−H(∞, y) +H(x, y)(3.40)

We can see here that the denominator of this equation actually con-ditions the probability on the fact that the lives have reached ages x andy respectively. H(x,∞) and H(∞, y) are the respective marginal distri-bution functions for males and females, therefore, H(x,∞) = F1(x) andH(∞, y) = F2(y). Also, when assessing an annuity based on a last-survivorstatus we are required to find kpx:y for all k, which is the probability thestatus will survive k years, hence we require a = b = k. Using these obser-vations we can alter equation 3.40 to obtain:

HT (k, k) =H(x+ k, y + k)−H(x, y + k)−H(x+ k, y) +H(x, y)

1− F1(x)− F2(y) +H(x, y)(3.41)

Since F1(x) = xq0 and F2(y) = yq0, the function HT (k, k) is actuallythe probability that the last-survivor status will not survive k years, namelyHT (k, k) = kqx:y. Hence we take:

kpx:y = 1−HT (k, k) (3.42)

We then find this value for each k, from 0 to N − 1, find the product ofthis with vk and then sum these values for all k to find the NSP required.

Obviously, when considering the joint-life status the condition of statussurvival changes and so to must the calculations involved. In chapter 2we saw from equation 2.31 that tpx:y = tpx + tpy − tpx:y and so kpx:y, theprobability the joint-life status will survive k years, is:

kpx:y = kpx + kpy − kpx:y (3.43)= 1−HT (k,∞)−HT (∞, k) +HT (k, k) (3.44)

This is the conditional probability that both lives will survive a furtherk years. From equation 3.41 it is easy to see that HT (k,∞) and HT (∞, k)are as follows:

HT (k,∞) =F1(x+ k)− F1(x)−H(x+ k, y) +H(x, y)

1− F1(x)− F2(y) +H(x, y)(3.45)

HT (∞, k) =F2(y + k)−H(x, y + k)− F2(y) +H(x, y)

1− F1(x)− F2(y) +H(x, y)(3.46)

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Chapter 4

Analysis of Annuity Values

In this chapter, I will assess the values of annuity contracts with dependentlives using the annuity calculator I have created, and compare these valuesto those found in chapter 2, where independence is assumed. All contractsdiscussed in this chapter are annuity-due contracts, where payments ceaseat the moment of death, and 5% interest is assumed unless stated otherwise.

4.1 Whole Life Last-Survivor Annuity

Example 4.1

Recall John, (53), and Madge, (49). The couple wish to take out awhole life annuity conditioned on the last-survivor status. In Example 2.8we calculated the NSP of this contract, assuming independence, to be £16.43for a unit benefit. The values obtained from the annuity calculator are:

Copula NSP Cost (£)Frank 17.43337325 17.43Independence 17.99606572 18.00Lower Bound 17.27563682 17.28Upper Bound 18.0674048 18.07

The value for the contract calculated in Example 2.8 used probabilitiesgained from life tables, see Appendix C. Through using these life tables tomodel future lifetimes the NSP achieved was much lower than when usingthe Gompertz model. This may be explained by the fact that life tables aredesigned by segregating the survival times into discrete intervals, whereasthe Gompertz distribution is continuous. Life tables are used mainly be-cause of the simplifications they allow in calculations; however the accuracyof the values in life tables can be improved upon. In practice life tables areoverlooked in favour of the more accurate approximation of the Gompertzdistribution, which closely fits the general observed distribution of survival

39

times and is a smooth function. For these reasons, it is clear that the valuescalculated using life tables and the Gompertz distribution are likely to vary.We can see that using the independence copula lends to a much higher NSPthan using life tables, even though both calculations use the independenceassumption. This is a consequence of the Gompertz distribution producinghigher survival probabilities than life tables (highlighted in Example 4.2).

Example 4.2

The probability that John will survive to age 99, a further 46 years, usinglife tables is:

46p53 =l53+46

l53=l99

l53= 0.002703

However, the Gompertz distribution, for the univariate distribution, finds:

46p53 = 0.027962047

And for the bivariate distribution:

46p53 = 0.024513879

Example 4.2 has shown clearly that using the Gompertz distributionthe probability the life (53) will survive to age 99 is almost ten times morelikely than when using life tables. These differences in the extreme ageprobabilities impact the overall present value of the contract significantly.A further explanation for the life table calculation providing a much smallerNSP is due to the difference in the maximum lifetimes of the models, thelife tables continue to age 99, whereas we have defined our Gompertz modelto have the maximum lifetime of 110. As the probabilities of survival arerelatively small at these ages this difference does not have a profound effectoverall, however, it is a contributing factor to the difference between the twovalues using the independence assumption.

For a fair comparison of the effect of dependence on the NSP of a last-survivor whole life annuity we must compare the values using Frank’s copulaand the independence copula, both with the Gompertz distribution. InExample 4.1, the independence copula produced a larger NSP by more than0.3 units. In fact when using this annuity calculator, I have found that theindependence copula creates a greater NSP than the dependence assumptionfor most calculations. Since the industry assumes independence, in this isthe case the industry are charging more than necessary for this contract. Theindependence assumption producing greater present values for this contractis to be expected since the parameter used for Frank’s copula models apositive dependence, which results in the death of one life reducing theexpected future lifetime of the surviving life.

40

Throughout this chapter, in order to effectively compare the NSP of con-tracts for independent or dependent lives I will be considering the AnnuityRatio value. This is:

Annuity Ratio =NSP with the dependence assumption

NSP with the independence assumption(4.1)

Example 4.3

The annuity ratio value for the contract in Example 4.1 is:

Annuity Ratio =17.4333732517.99606572

= 0.968732473

Since this value is less than one it is clear that the independence assumptioncreates a greater NSP for this contract.

The assumption of independence results in a higher probability that thestatus will still be in survival when nearing the maximum lifetime, whencompared to the assumption of dependence. If the first death has no effecton the future lifetime of the second person it can be assumed that they willlive longer than if the lives were positively dependent. If the probability ofstatus survival is greater than in the dependent situation, then there arelikely to be a greater number of years where then contract would be payingout causing an increase in the cost of the contract. For the same reason theFrechet-Hoeffding bounds create a maximum and minimum value for thecontract, by modelling opposite types of dependency.

From the quantities produced for the NSP of the annuity, for the inde-pendence copula and Frank’s copula, we can see that there is not a vastdifference in the values. The NSP of the contract for the dependent livesis actually numerically closer to the present value when using the indepen-dence copula then it is to the lower bound for the contract. This signifiesthat the α parameter used must model the lives to have a small amount ofdependency. If data had been observed which resulted in a more negativeestimate for α then lower values for the NSP would be observed.

4.1.1 The Effect of Age

We have seen how dependency affected the NSP of the contract for Johnand Madge, but is this the case for all ages? To evaluate this we will firstconsider the situation where we assume the annuitants are of the same age,the annuity ratio for this scenario is shown in Figure 4.1.

In Figure 4.1 we can see that when the annuitants are assumed to bethe same age the annuity ratio value is never greater than one, showingthe industry assumption of independence always produces a greater costto the policyholders. This is a direct result of Frank’s copula modelling

41

Figure 4.1: Plot Comparing Annuity Ratio for Varying Ages ofSame Age Annuitants (5% interest assumed).

the dependence to be positive, and therefore reducing the probability of thelast-survivor status survival. Consequently, this reduction in probability willreduce the cost of the contract, as discussed above.

A further observation we can make from Figure 4.1 is that as the ages ofthe annuitants increase, from 40 to 70 years old, the annuity ratio value de-creases in a relatively linear manner. This indicates that as the ages increasethe difference between the contract evaluation with the independence anddependence assumption increases. As the annuitants ages increase the valueof the NSP of the contract with dependence falls at a greater rate to thatof the independence assumption. Subsequently, the annuity ratio value fallsbetween ages 40 and 70. However, it is interesting to notice that the annuityratio value does not continue to fall, as between the ages of approximately70 and 88 the graph shows a rise in annuity ratio. This corresponds to thedifference between the two NSPs beginning to reduce, and this is due to thepresent value with the independence value starting to decrease more rapidly,at a similar rate to the NSP with the dependence assumption. This can beseen in Figure 4.2, where the plots of NSP for independence and dependenceare in juxtaposition. Between ages 70 and 88 the gradient of the indepen-dent graph becomes more negative, a similar slope to that of the dependencegraph. This reflects the sudden reduction in probabilities of survival for livesof these ages, a property shown in the Gompertz distribution.

In Figure 4.1 we can see that at approximately age 88 this increase stopsand then there is a dramatic decrease in the annuity ratio until age 100.This again can been seen as a consequence of the graphs in Figure 4.2. Thegraph corresponding to dependent lives maintains a steady slope; however,the gradient of the independent graph reduces significantly after age 88,

42

Figure 4.2: Plots Comparing NSP for dependenceand independence, for Varying Ages of Same AgeAnnuitants (5% interest assumed).

causing the difference between the two NSP values to increase until theend of the graph. This is a direct consequence of Frank’s copula producinga more steady reduction in the probability of status survival over the agegroups than the independence copula.

To further investigate the effect of age on the contract cost with depen-dency I will next consider the situation where the annuitants are not thesame age.

Figure 4.3: Plot Comparing Annuity Ratio for Varying Ages ofMale and Female Annuitants (5% interest assumed).

As we can see from the plot above this is a significantly more complicatedsituation to consider. In Figure 4.3 the minimum corresponds to the valuesseen in Figure 4.1, and we can clearly see from Figure 4.3 that as the ages

43

of the male and female vary from being the same age the effect on theannuity ratio can be considerable, especially for older lives. Unlike in Figure4.1 the annuity ratio value is no longer always less than one, showing thedependence NSP to be greater than the independence NSP in some pairs ofages. In these cases the industry will consequently not be charging enoughfor a contract between potentially dependent lives. It is clear from Figure 4.3that the maximum difference between the two contract values occurs whenconsidering a very old female age and a middle-aged male, and occurs whenthe dependent NSP is greater. The graph rises to this peak, approximately1.25, for a woman aged 100 from lower male ages, all values being greaterthan one. After this peak the annuity ratio value then falls to its lowest valuefor same aged annuitants. This relationship is constant as the female andmale ages vary, always with the lowest difference in contract cost occurringwhen the annuitants are the same age. On Figure 4.3, the annuity ratiovalue appears only to be less then one in the nearest corner, where bothannuitants are relatively young and are of similar ages, for example, whenthe ages of the annuitants vary between 40 and 70 years of age.

The relationship between male and female ages and the annuity ratiois due to the effects of age on Frank’s copula, in comparison to the inde-pendence copula. As expected and seen in Figure 4.2, as the age of thepolicyholder increases the NSP of this contract decreases. This is a resultof the expected length of time the annuity will be paying out is shorter dueto the future lifetimes being expected to be shorter. If we keep a constantage for the male and increase the female age, a greater rate of reduction inthe NSP is apparent for the independence copula rather than the depen-dence copula. This is a result of the positive dependency modelled by theparameter in Frank’s copula. In addition to dependency reducing the futurelifetime of a life after the death of a partner, the copula also models a posi-tive effect on the future lifetimes of both lives while they are both still alive.Therefore, by increasing the age of the female annuitant, the probabilityof status survival falls more rapidly when working with the independenceassumption. Hence the NSP falls at a greater rate with independence untilan aged is reached which causes the NSP to be smaller than the dependentNSP. This is similar to maintaining a constant female age and increasing themale age ut to a smaller extent, reflected by the difference in the height ofthe two peaks on Figure 4.3. Figure 4.3 is almost symmetric, showing thatthe difference in marginal distributions for the males and females does noteffect the annuity ratio in a significant way.

To portray the effect of age on the annuity ratio value I have createdFigure 4.4 to produce a more accessible view. To simplify the plot I havechosen to focus on annuitants between the ages of 50 and 80.

From Figure 4.4 we can conclude that for younger females the annuityratio increases with male age, and for older females the annuity ratio de-creases as male age increases. Figure 4.4 also portrays that in the event of

44

Figure 4.4: Multiple Scatter Plot Comparing Annuity Ratio andMale Age over Various Female Ages (5% interest assumed).

extreme differences between the ages of the male and female annuitants, theannuity ratio is always greater than one, and hence independence has a lowerNSP. This could result in a loss for insurance companies from policies be-tween these age groups. However, it would be questionable as to whether a50 year old and an 80-year old would take out an annuity contract together.Considering policies between parents and children, which would fund thechild after the parent’s death, a contract between these ages would be areal-life possibility. Although, the dependency between a parent and childmay not fit the model and parameter we have used.

4.1.2 The Effect of Interest Rate

When considering the effect of interest rate on the NSP of a contract weknow that as interest rate (i) increases, the discounting factor (v) decreases,due to the relationship v = 1

1+i . Hence, as the interest increases the NSP ofthe contract always decreases. But how does interest rate affect the annuityratio for this multiple-life contract? In Figure 4.5, the relationship betweenannuity ratio and interest rate is shown as the age of the annuitants vary,assuming the annuitants are the same age.

As expected the annuity ratio never exceeds one, as the effect of interestnever causes the independence assumption to produce a smaller NSP. Thisis a result of the effect of interest being proportional for each calculation.The relationship between joint ages of the annuitants and the annuity ratio,shown in Figure 4.1, remains constant and this relationship can be observedin Figure 4.5. This shows that interest rate does not alter the overall effectof the ages of the annuitants. However, Figure 4.5 does show, for all ages,

45

Figure 4.5: Plot Comparing Annuity Ratio and Interest Rate, forVarying Ages of Same Age Annuitants (5% interest assumed).

that as the interest rate increases from 0% towards 60% the annuity ratiovalue also increases towards one. This means that as the interest rate in-creases the difference between the NSP with dependence and the NSP withindependence becomes less significant. This is due to a large interest rateforcing a very small discount factor value, meaning each value in the sum-mation of the annuity (vk kpx:y) becomes less significant as the interest rateincreases. Hence any difference between the probability of status survivalfor independent lives and dependent lives then becomes an insignificant dif-ference in the overall summation, causing the annuity ratio to tend to one.Consequently, dependence between lives becomes increasingly significant intimes of low interest rates. Recently, the interest rates in the UK have fallenand as a result of this adults taking out this contract now will not only bepaying a greater NSP than they would have earlier in the year, but also theycould have saved greater amount if dependence between their lives had beenmodelled using Frank’s copula. So in times of low interest rates the industryassumption of independence produces more inaccurate contract costs, andencourages a greater profit margin for the insurer.

Figure 4.6 shows more clearly the effect of interest rates on the annuityratio value as the age of the annuitants increases. The lines on this graphnever intercept, which is to be expected, as the interest rate is inverselyproportional to the NSP of the contract for each age. This chart does show,however, that as the interest rate increases, for any particular age, the effectof the increase becomes less significant, i.e. the lines become closer on thegraph. This is a result of the decrease in v, as i increases, becoming smallerfor every increase in i. Figure 4.6 also reiterates the observation that asthe interest rate increases so to does the annuity ratio value, causing the

46

Figure 4.6: Multiple Scatter Plot Comparing Annuity Ratio andInterest Rates, for Varying Ages of Same Aged Annuitants (5%interest assumed).

dependence and independence assumptions to produce increasingly similarvalues for the NSP of the last-survivor annuity contract.

4.1.3 The Effect of the Alpha Parameter

In Chapter 3 we saw that the alpha parameter in Frank’s copula capturesthe dependency between lives, and that as alpha tends to zero the limit ofthe copula is the independence copula. However, we do not know if alphais directly related to dependence, i.e. if positive numbers model negativedependence and negative values model positive dependence. To investigatethe relationship between alpha and the annuity ratio I have created the twoplots in Figures 4.7, showing the effect of negative and positive alpha values.As when alpha is zero we assume independence, we can assume an annuityratio of one when alpha is zero.

In Figure 4.7 it is noticeable that alpha directly lower than zero, for allages, produces an annuity ratio much less than one. This conveys a reduc-tion in the value of the NSP in the dependent situation. A lower NSP inthe dependency model is a result of positive dependency between the lives,and as all annuity ratio values for negative alpha are less than one, we canconclude that all negative values of alpha model positive dependency. Inaddition, Figure 4.7 portrays varying effects of negative alpha on the annu-ity ratio value as the age of the annuitants varies. When considering olderpolicyholders, as alpha becomes more negative the annuity ratio increasesin a relatively linear fashion, however, for younger clients the annuity ratiodecreases as alpha decreases. A higher level of dependency between the lives

47

Figure 4.7: Plots Showing the Effect of Positive and NegativeValues of the Alpha Parameter on the Annuity Ratio, for VaryingAges of Same Aged Annuitants (5% interest assummed).

is modelled as alpha becomes more negative, and this is clear from the rela-tionships described between age and alpha. For annuitants of a younger agethe dependency between lives results in the expected future lifetimes beingshorter than with the independence assumption, as the death of one lifewould have an increasingly negative effect on the future lifetime of the re-maining life. This would reduce the probabilities of status survival and hencereduce the time the annuity is expected to payout, and consequently causesa decrease in the NSP for dependent lives. This subsequently increases thedifference between the two NSP values, hence increasing the annuity ratiovalue. In comparison, for older lives the independence assumption producesvery small probabilities of survival for these lives, however, dependency in-creases the probability of survival slightly for these ages due to the positiveeffect of both lives still surviving. This relationship becomes stronger as al-pha becomes more negative, causing a slight increase in the dependent NSPand hence the annuity ratio increases a small amount as alpha decreases.

As the annuity ratio value would be one for alpha as zero, we wouldexpect for all positive values of alpha that the annuity ratio would be greaterthan one, increasing as alpha increased. Yet as the positive plot in Figure4.7 represents, alpha equal to zero is not the turning point between positiveand negative dependence models. In this graph the annuity ratio value,directly after alpha equal to zero, is less than one for all ages. This showsthat after the independence assumption the dependency model reverts backto positive dependence. For the younger ages we can see that as alphaincreases the annuity ratio increases to a value greater than one, so themodel alters to negative dependence. It is particularly interesting that anincrease in alpha has the opposite effect for older ages, as the annuity ratiodecreases quite rapidly. For example, if we consider an age of approximately

48

85, the annuity ratio falls to a value of almost 0.5 as alpha increases to 10,as shown for positive alpha in Figure 4.7. Figure 4.7 shows that up until anage of approximately 65 the increase in alpha causes the dependent NSP tobecome greater than the independent NSP, however, after this age the effectof increasing alpha is to make the dependent NSP a much smaller value.Hence, the relationship between alpha and the annuity ratio is not as simpleas we would have first expected.

4.2 Whole-Life Joint-Life Annuity

The joint-life status fails on the event of the first death of the two lives, asexplained at the beginning of Chapter 2. Our interest is to establish the dif-ference dependency makes on the NSP of a joint-life whole life annuity-due.

Example 4.4

In Example 2.2 we found the present value of a whole life joint-life annu-ity for John and Madge to be 12.349665 using life tables and the assumptionof independence. Using an annuity calculator we find:

Copula NSP Cost (£)Frank 14.7501486 14.75Independence 14.72351411 14.72Lower Bound 13.90471141 13.90Upper Bound 15.114071 15.11

The immediate characteristic of these values which I notice is the differ-ence between these NSPs and the values in Example 4.1, for the last-survivorannuity. The values in Example 4.4 are noticeable smaller, and this is to beexpected due to the nature of the statuses. The period of time of survivalfor a last-survivor status must be either the same, if both lives die at exactlythe same time, or longer than the period of survival of a joint-life status.This is since the last-survivor status fails on the second death, whereas thejoint-life status fails at the first death. Hence, the joint-life annuity is likelyto pay out for a shorter time period than the last-survivor annuity, resultingin a lower NSP.

Considering the issue of dependency between the lives, we see thatFrank’s copula in this situation produces a higher present value than theindependence copula, unlike in Example 4.1. In this case, the industryassumption of independence will not be charging enough for a contract be-tween dependent lives. Due to the joint-life status failing on the first death,if there is a dependency between the lives, then while the status is in survivalit is more likely to remain in survival than if the lifetimes were independent.Thus Frank’s copula yields greater probabilities that the status is in survival

49

than the independence copula, resulting in a higher NSP for the contract. Inthis situation it is not beneficial to the insurance company to model lives asindependent, however, the joint-life annuity contract is not a very commonchoice of annuity as after the first death the surviving partner is left withno source of income.

4.3 Reversionary Annuity

A common annuity contract that is considered by dependent lives is the re-versionary annuity, see Asymmetric Annuities in Chapter 2. This contractcommences paying benefits after the first death and continues paying annu-ally until the second death.

Example 4.5

In Example 2.10 we calculated the present value of a reversionary annuitycontract for John and Madge using Equation 2.59, as follows:

ax/y = ay − ax:y = 2.516243

Using the revised annuity calculator we found the NSP values to be:

Copula NSP Cost (£)Frank 2.229137211 2.23Independence 2.734597625 2.73Lower Bound 1.865214815 1.87Upper Bound 3.074574407 3.07

Since the lower bound copula creates a lower bound for the joint-life annuitypresent value, it therefore maximises the present value for the reversionaryannuity. The converse is true for the upper bound copula.

We can see from this example that for the reversionary annuity, Frank’scopula produces a NSP for the contract more similar in value to the NSPproduced by the independence copula than it is to lower bound value forthe contract. The present values for the reversionary contract also followthe trend of producing a higher value using the independence copula thanthe dependency model produces, as seen with the last-survivor contract.Since the joint-life status fails after the first death, the NSP of the joint-life annuity is lower using the independence copula, rather than Frank’sCopula. This is because the first death is more likely to occur sooner in anindependent situation, whereas if the lives are dependent, the first death ismore likely to occur later since the partners are together. If the independencecopula creates a lower NSP for the joint-life annuity contract then the overallvalue of the reversionary contract will be higher, since the whole life annuity

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for the beneficiary is a constant value in all of these calculations. Thisshows that once again the insurance companies are more likely to modellives independently to cover the company from unexpected losses, and makean extra profit.

In the case of Example 4.5, the NSP that has been calculated using lifetables and assuming independence gives a relatively low estimate. Thereis only a difference of approximately 0.56 between this value and the oneobtained from using the independence copula, showing the life table model tobe fairly realistic. This maybe a repercussion of this contract using the joint-life status compared to the last-survivor status in Example 4.1. A cause forthe large difference in between the two values produced from independenceassumptions will be an effect of the difference in distribution models used.The life tables give much lower probabilities for the future lifetimes of thefemale than the Gompertz distribution gives. When comparing the life tablecalculation with the value obtained through Frank’s copula we can see thatthere is only a difference of approximately 0.3 between the values, Frank’scopula producing the greater of the two values.

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Chapter 5

Other Models of Dependency

5.1 The Common Shock Model

In this section I will briefly describe other models of dependency used inthe insurance industry, using theory from [PR] and [BOW], together withparameters estimated in [FCV]. Using this information I have created myown explanation, expanding the theory with my own knowledge to create acommon shock annuity calculator.

This model is used to introduce dependence between lives which wouldotherwise have independent lifetime distributions. The common shock modelfinds dependence between lives due to the risk that they face from a catas-trophe, such as a plane crash, a hurricane or an earthquake. These livesare dependent simply due to the location they are in at a certain time,and the risk that the group of people face from a catastrophe in that area.For example, business partners may take out a joint-life insurance policy,and they may lead otherwise independent lives except for the times whenthey travel by plane together, and in this case the dependency betweenthe lives would be modelled using the common shock model. The commonshock model takes into account no other causes of dependency between liveswhich results in the usefulness of this model being limited for the majorityof multiple-life contracts. However, this model is particularly useful for in-surance companies when reducing the risk off loss from multiple claims, asa catastrophe can cause a large number of unexpected payouts at one timefor the company and so they must incorporate this into their risk analysis.

So how does this model introduce dependency between the lives?

If we consider independent future lifetimes T1(x) and T2(y), let GT1:T2

be the joint distribution of the future lifetimes, and let GTi be the marginalfuture lifetime distribution for random variable Ti. Using the independence

52

assumption we know that:

GT1(x):T2(y)(s, t) = GT1(x)(s)GT2(y)(t) (5.1)

The common shock model then introduces a common shock randomvariable, Z, this random variable is independent of the future lifetimes ofthe two lives but is associated with the time of the catastrophe. This randomvariable can affect the joint distribution of the time-until-death variables ofthe lives x and y, and has an exponential distribution as follows:

GZ(z) = e−λz (5.2)

Where z > 0 and λ ≥ 0, λ is defined as the common shock parame-ter. In terms of a life annuity or a life insurance policy, it is of interestwhether natural death or death as a result of the catastrophe will occurfirst. Now we define new random variables T (x) = minT1(x), Z andT (y) = minT2(y), Z, and now consider the joint survival function of T (x)and T (y).

GT (x):T (y)(s, t) = P (minT1(x), Z > s ∩ minT2(y), Z > t)= P ([T1(x) > s ∩ Z > s] ∩ [T2(y) > t ∩ Z > t])= P (T1(x) > s ∩ T2(y) > t ∩ Z > maxs, t)

Since we know that the random variables of T1(x), T2(y) and Z are allindependent it follows that:

GT (x):T (y)(s, t) = GT1(x)(s)GT2(y)(t)e−λmaxs,t (5.3)

The marginal distribution functions for the survival of the lives followfrom equation 5.3 and are given by:

GT (x)(s) = P (T (x) > s ∩ T (y) > 0) (5.4)

= GT1(x)(s)e−λs (5.5)

GT (y)(t) = P (T (x) > 0 ∩ T (y) > t) (5.6)

= GT2(y)(t)e−λt (5.7)

Throughout this report we have been interested in the probabilities ofsurvival of the multiple-life statuses, in particular the joint-life status andthe last-survivor status. So how can we relate the common shock modelto these statuses in order to calculate the probabilities of status survival,and hence determine the NSPs of contracts conditional on these statuses?Firstly, when considering the joint-life status, we must define the joint-liferandom variable T (x : y) = minT (x), T (y), as the joint-life status fails onthe first death. Using equation 2.1, which defines the probability of survival

53

of a joint-life status using the independence assumption, equation 5.3 andthe fact that T1(x) and T2(y) are independent, then:

GT (x:y)(s) = GT1(x)(s)GT2(y)(s)e−λs , s > 0 (5.8)

Similarly, if we define T (x : y) to be the random variable relating to thelast-survivor status, it is clear that T (x : y) = maxT (x), T (y). Henceusing equations 2.31, 5.8, 5.7 and 5.5 we find:

GT (x:y)(s) = [GT1(x)(s) +GT2(y)(s)−GT1(x):T2(y)(s, s)]e−λs , s > 0 (5.9)

As the common shock parameter varies we can notice the effect of thedependency on the lives. If λ = 0, then e−λz = 1 and hence equations 5.8and 5.9 become the joint-life and last-survivor distributions and regress tothe independent forms as described in chapter 2. However, when λ > 0 thisresults in e−λz < 1, and hence the probabilities of status survival are lessthan in the independent case. Therefore, the effect of the common shockmodel is to reduce the NSPs of contracts.

5.1.1 Using the Common Shock Model

To be able to compare the effects of modelling dependency using the commonshock model, rather than the copula model, I have devised a common shockannuity calculator. The annuity calculator finds the NSP of the last-survivorwhole life annuity, ax:y. The method used to calculate the NSP of thiscontract is similar to the method used with the copulas. By taking themarginal survival functions of the male, (x), and female, (y) lives to beF1(x) and F2(y) respectively, where the function Fi relates to the Gompertzdistribution function found below:

Fi(x) = 1− exp(e−mi/σi(1− ex/σi)) (5.10)

The estimates for parameters mi and σi can be found in the bivariate dis-tribution column of Figure 5.1 below. Figure 5.1 corresponds to “Table 7”in [FCV], I will be using the parameter estimates found in this article toaid my calculations. We define X and Y to be the age at death for the twolives, whereas, x and y are their contract initiation ages.

Now, we defined the common shock random variable to be an indepen-dent, exponential variable with parameter λ. The common shock variable isassociated with the time of an event, so we define the probability that theevent occurs before time t to be:

P (Z ≤ t) = 1− e−λt (5.11)

If we define two new lifetime random variables associated with age at deathto be X∗ and Y ∗, where:

X∗ = T (x) + x = minX,Z + x (5.12)Y ∗ = T (y) + y = minY,Z + y (5.13)

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Figure 5.1:

This leads to a bivariate distribution function:

H(x+ a, y + b) = P (X∗ ≤ x+ a, Y ∗ ≤ y + b) (5.14)= 1− e−λa(1− F1(x+ a)) (5.15)− e−λb(1− F2(y + b)) (5.16)+ e−λmaxa,b(1− F1(x+ a))(1− F2(y + b)) (5.17)

It is interesting to note that the marginals are not the Gompertz distri-bution function as would be expected, but they are a function of λ.

H(x+ a,∞) = 1− e−λa(1− F1(x+ a)) (5.18)H(∞, y + b) = 1− e−λb(1− F2(y + b)) (5.19)

And so this implies that:

kpx =1−H(x+ k,∞)

1−H(x,∞)(5.20)

= e−λk1− F1(x+ k)

1− F1(x)(5.21)

Hence the last-survivor survival probability is kpx:y = kpx+kpy−eλk kpx kpy.It is clear that the common shock model lends itself to a much computa-tionally simpler formula than the copula method. By using this probabilitywe can calculate the NSP of the last-survivor whole life annuity.

In the case of independence between the lives the parameters can befound in the univariate distribution column of Figure 5.1. Therefore, we cancalculate the NSP of the contract, as independence corresponds to λ = 0,using the corresponding equation to 5.21:

kpx =1− F1(x+ k)

1− F1(x)(5.22)

and:kpx:y = kpx + kpy − kpx kpy (5.23)

As we would expect for the independence assumption.

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5.1.2 Analysis of Annuity Value using Common Shock Model

Example 5.1

In Example 4.1, we saw the NSP of the last-survivor annuity usingFrank’s copula for John and Madge to be £17.43. Now using the commonshock annuity calculator, we find that a53:49 = 17.92897393, approximately£17.93.

With the independence assumption the present value achieved is 17.99607437,approximately £18.00, the same value found when using the copula method,as we should expect. Therefore, the annuity ratio value is 0.996271385.

From Example 5.1 we can see that the common shock model creates agreater NSP for the last-survivor whole life annuity. This is supported byFigure 5.2, a plot showing the annuity ratio values for the common shockmodel and the copula in juxtaposition. This is a consequence of the re-striction in the causes of dependency taken into account by the commonshock model. As described in Section 5.1, the common shock model onlytakes into account the risk the lives face from a catastrophe, and does notconsider other causes of dependency, such as lifestyle. However, the cop-ula model does take into account other causes of dependency, and thereforemodels a greater level of dependency between the lives considered in [FCV].A larger NSP for the contract when using the common shock model, andhence a greater annuity ratio value, is a direct result of the lower level ofdependency modelled. A smaller dependency between lives increases thelikelihood of survival for the lives, and hence a greater cost for the contract.

Figure 5.2: Plot Comparing Annuity Ratio for Same Age Annu-itants for the Common Shock Model and the Copula Model (5%interest assumed).

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Figure 5.2 shows the annuity ratio value to always be less than one forthe copula model, assuming same aged male and female annuitants, whereasthe common shock model produces an annuity ratio value which is mainlygreater than one. Thus it is clear that the common shock model creates agreater NSP than the independence assumption for most ages, unlike thecopula model. The independence assumption corresponds to λ = 0, and soas the value of λ increases (from 0 to 0.00054) a small decrease in the NSP ofthe contract results. This can be observed from expressing ax:y differently.

ax:y =∞∑k=0

e−(λ+δ)k (kpx + kpy − kpx kpy) (5.24)

Where kpx = eλkkpx and δ is the force of interest (see Appendix D). Fromthis rearrangement of the equation it is clear that as λ increases the NSPdecreases. However, this is outweighed by the larger increase in the NSPcaused by the greater Gompertz parameters in the bivariate distribution.The larger values of the parameters for the Gompertz distribution producegreater probabilities for survival for lives in the bivariate distribution, thusincreasing the NSP when using the common shock model. Hence, althoughthe increase in λ should produce a reduction in the NSP, when movingfrom the independent to the bivariate situation, the change is insignificantin comparison to the increase as a result of the large parameter values.Subsequently, the annuity ratio value for most ages is greater than one forthe common shock model.

Figure 5.3: Plot Comparing Annuity Ratio for Varying Ages ofAnnuitants for the Common Shock Model (5% interest assumed).

Furthermore, Figure 5.3 shows the effect of varying the ages of the an-nuitants on the annuity ratio for the common shock model. Figure 5.3 is

57

completely different in structure to Figure 4.3, the corresponding plot forthe copula model. The annuity ratio steadily increases as either one of theannuitant’s ages increase, and the highest annuity ratio values correspondto when both annuitants are nearing the maximum lifetime age. The com-mon shock model creates relatively much higher survival probabilities thanthe independence assumption, due to the higher Gompertz parameters, andhence a greater NSP. Additionally, Figure 5.3 also supports the observationof the annuity ratio being greater than one for most ages, as this holds foreven extreme combinations of ages.

5.2 The Shared Frailty Model

In this section I have used basic theory obtained from [PH] and [AW], I haveused my own knowledge and understanding to explain these concepts andtransfer them to an actuarial context.

The shared frailty model can be viewed as a model incorporating thetypes of dependence modelled in both the copula model and the commonshock model. The method by which the frailty model assesses dependenceis by organising the population into smaller groups, and subsequently mod-elling the dependency between the groups and the dependency within thegroups. To aid the explanation of this model I we use the example of con-sidering the population of a small town. In this example we could define thegroups to be the individual families or households in the town. The depen-dency between all the households in the town would be due to a catastropheas in the common shock model, and the dependency between the lives ineach group would be factors incorporated in the copula model, such as healthrisks, lifestyles and “broken heart syndrome”. The shared frailty model isa specific branch of the common shock model, and is a model based on theconditional independence that all time observations are independent giventhe relevant frailty values.

To implement this model we define the frailty random variable to beY , which describes the risks to the lives. The value of the random variableY remains constant over time and is common to all the lives in a group.The frailty random variable in the example of the town is constant overtime and is the same for all members of each household. We can denotethe frailty random variable for group i to be Yi, and the variation in Yover the groups represents the different risks to the groups. If there is novariation in the random variable Y implies there is independence betweenthe groups, whereas if Y is degenerate implies there is positive dependence.In our town, we will have for example n households and we will assume thateach household contains m people, so i = 1, .., n.

As the frailty variable finds the variation between the groups, the hazard

58

function, µ(t), represents the variation between the individuals in eachgroup. So the hazard function will find the differences in risk between theindividual lives in each household in the town.

The frailty is a random value, for which we will assume a Gamma dis-tribution for this value. From previous studies I know that the gammadistribution function, Γ(a, b), with parameter a and inverse scale parameterb, is as follows:

f(y) =1

Γ(a)baya−1e−by (5.25)

The multivariate survival function S(t1, t2) = P (T1 > t1, T2 > t2), where

Mj(t) =∫ t

0µj(u) du for j = 1, 2, and using the Gamma distribution in

equation 5.25 is:

S(t1, t2) =ba

(b+M1(t1) +M2(t2))a(5.26)

From equation 5.26 the inverse relation, where S1(t1) = S(t1, 0) is themarginal function:

M1(t1) = b(S1(t1)−1a − 1) (5.27)

Using equations 5.27 and 5.26 we can now find the bivariate distribution tobe:

S(t1, t2) = (S1(t1)−1a + S2(t2)−

1a − 1)−a (5.28)

We can see from equation 5.28 that the scale parameter b is no longerpart of the equation, this makes the calculations simpler. To find the densityof the lifetimes in this bivariate distribution we can find the equation below.

µ1(t1)µ2(t2)(b+M1(t1) +M2(t2))−a−2ba(a+ 1)a (5.29)

Using the Gamma distribution for this model can have some problems.These problems can be overcome by using a larger family of distributionsto produce a better fit to the expected lifetimes and factors effecting de-pendency. A common family of distributions which is used is the powervariance function (PVF) as it is a very large family of distributions. ThePVF family includes the gamma distributions, inverse gaussian distributionsand the positive stable distributions. Also, when considering the bivariatecase this model has some drawbacks as it models long-term dependence.This suits the lives we have considered so far, e.g. relatives and marriedcouples. However, there are cases where this is not acceptable. Using anexample of short-term dependence from [PH], if twins were born and onewas strangled by the umbilical cord during birth then the death of this twinhas no effect on the future lifetime of the second twin.

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Chapter 6

Conclusion

Multiple-Life theory is presently an important part of the insurance indus-try, as a significant number of life insurance, annuity and pension policiesare conditional on multiple-life survival statuses. Also, multiple-life policies,such as the reversionary annuity, cannot be related to any single-life pol-icy, highlighting the opportunity to develop more detailed contracts whenconsidering multiple lives. An interesting complication in multiple-life the-ory is the possibility of dependent lives in contrast with the assumption ofindependence throughout the industry.

In chapter 2, we introduced the notion of multiple-life statuses and howto find the NSP of contracts conditional on the survival of these statuses.Additionally, using the industry assumption of independence and suitablelife tables, I was able to clearly show how to calculate the NSP of multiple-life contracts using numerical examples.

In chapter 3, I began to question the industry assumption of indepen-dence by introducing copulas, and showing how these are using to modeldependence. We discussed various common copulas in this chapter, includ-ing Frank’s parametric family of copulas which we later use to model depen-dency between lives. Copula functions can be very complicated, however,we saw that they allowed us to model all types of dependency between lives,with fewer restrictions than when using other models.

Analysing the NSP of multiple-life annuity contracts, in chapter 4, al-lowed us to clearly show the effect of the independence assumption in com-parison to modelling dependent lives. To show the effects we studied the an-nuity ratio value, the ratio of the dependence NSP to the independence NSP.Through the effective use of plots and graphs, we were enabled to concludethat for many ages of annuitants the independence assumption produceda greater cost for the contract than the dependence model. Therefore, wecan see that for many contracts the industry is charging more for contractsthan they should dependent lives. As the ages of the annuitants increasethe relative difference between the two NSPs becomes greater. Therefore,

60

the effect of dependency is more significant as age increases, hence the in-dustry is making a greater profit from the elderly. Morally, whether thisprocedure is justified is questionable, charging more for those with limitedincome. However, neglecting the ethics of the industry, there is a consider-able difference in contract costs for the two assumptions, so why does theindustry assume independence?

In chapter 5, we introduced the common shock and frailty dependencymodels, and compared the annuity ratio value for an annuity when usingthe common shock model to that when using copulas. This highlights therestrictions in the types of dependency taken into account by the commonshock model, in comparison to copulas. Also, when using the common shockmodel, we observed that the industry assumption of independence does notgenerally produce a higher NSP for the contract. In comparison to the cop-ula method, the industry would be making a loss if the lives were dependent.Hence, the conclusion as to whether the independence assumption creates agreater NSP than the dependence assumption depends on the model used.

In conclusion, we have seen the industry assumption of independencecan cause higher costs to annuitants than necessary for dependent lives, andtherefore questioned why the industry makes this assumption. The industryassumes independence not only to make a profit but to protect the insurancecompanies from a loss. If the companies were to model a general level ofdependency for all lives then some may not fit this general dependencymodel. If two lives had a lower dependency level than modelled the insurerwould be at risk of a loss. This is a result of the lives surviving longer thanexpected by the model, and therefore, an annuity contract would be payingout longer than expected, causing a loss to the insurer.

In this project I have relied heavily on the data attained in [FCV] andthe parameters they have estimated from this data. Therefore, the precisionof the results I have calculated from the NSP calculator in chapters 4 and 5incorporate any errors in the results found in [FCV]. Also, through the useof this data I have assumed that all lives fit the levels of dependency foundin [FCV], and I have not considered the case where dependency between thelives maybe greater or smaller than the models found in this article. Hence,I would be able to extend this project to a higher level by studying variouslevel of dependency for varying groups of people, such as married couples,business partners, relatives and others. Additionally, to further this projectit may be useful to study many other types of dependency models to ascer-tain the most effective model for each group of lives studied.

Acknowledgements

I would like to thank Professor Frank Coolen, for his advice and sup-port throughout this project, and Dr. Robert Johnson, for the invaluableguidance notes on using Latex.

61

Bibliography

[CMP] Colin Murray Parkes,Bereavement: Studies of Grief in Adult Life (Penguin Books)(ISBN 0-14-025754-3). Chapter 2: pgs 14-17 , Figure 1: pg 212

[HG] Hans U. Gerber,Life Insurance Mathematics: Third Edition (Springer)(ISBN 3-540-62242-X). Chapter 8: pgs 83-92

[RN] Roger B. Nelsen,An Introduction to Copulas: Second Edition (Springer)(ISBN 0-387-28659-4). Chapter 2: pgs 7-49,Chapter 3 : pgs 51-108, Chapter 5 : pgs 157-167

[PR] S. David Promislow,Fundamentals of Actuarial Mathematics (Wiley)(ISBN 0-470-01689-2). Chapter 10: pgs 135-150,Chapter 16: pgs 237-239

[BOW] Bowers, Gerber, Hickman, Jones, Nesbitt,Actuarial Mathematics (SOA)(ISBN 0-938959-46-8). Chapter 9: pgs 274-300

[PH] Philip Hougaard,Statistics for Biology and Health: Analysis of Multivariate SurvivalData (Springer)(ISBN 0-387-98873-4). Chapter 7: pgs 215-236

[FCV] Edward W. Frees; Jacques Carriere; Emiliano ValdezAnnuity Valuation with Dependent Mortalityfrom The Journal of Risk and Insurance, Vol. 63, No. 2 (Jun, 1996)pgs.229-261

[AW] Andreas WienkeFrailty Models: MPIDR Working Paper WP 2003-032,September 2003

62

[2H] Supplied by Professor Frank Coolen “Actuarial Mathematics” 2HCourse Notes

[SOA] http://www.soa.org/about/membership /about-deceased-members-2001-cecil-nesbitt.aspxfor a biography of Cecil J. Nesbitt in ‘Deceased Members-2001’(visited Nov 2007)

[SOA2] http://www.soa.org/files/pdf/naaj9801 1.pdf‘Understanding Relationships Using Copulas’(visited Oct 2007)

[W1] http://en.wikipedia.org/wiki/SchuetteE28093Nesbitt formuladescription of the ‘Schuette-Nesbitt Formula’(visited Nov 2007)

[ABI] http://www.abi.org.uk/BookShop/ResearchReports/UK%20Insurance%20-%20Key%20Facts%202007.pdf‘Association of British Insurers: UK Insurance - Key Facts’(visited March 2008)

[ABI2] http://www.abi.org.uk/Display/File/524/Annual Overview 2006.pdf‘Association of British Insurers: Statistical Overview of UK Insur-ance in 2006’(visited March 2008)

[G1] http://www.econ.kuleuven.ac.be/tew/academic/actuawet/pdfs/DVW(multilife).pdf‘A note on dependencies in multiple-life statuses’(Dhaene, Vanneste, Wolthuis) via Google Scholar (visited Feb 2008)

[G2] http://www.actuaries.org.uk/files/pdf/library/JIA-079/0323-0335.pdf‘Faculty and Institute of Actuaries : The Valuation of Last-SurvivorAnnuities’(Bailey) (visited Feb 2008)

[G3] http://www.math.uni-leipzig.de/ tschmidt/TSchmidt Copulas.pdf‘Coping with Copulas’ (Thorsten Schmidt)via Google Scholar (visited Jan 2008)

63

Appendix A

Schuette-Nesbitt Formula

In this description of the origins of the Schuette-Nesbitt Formula I have usedinformation from [W1] and [SOA].

A.1 History

The first appearance of the Schuette-Nesbitt Formula was in 1959, in theTransactions of Society of Actuaries, in a discussion by Donald R. Schuetteand Cecil J. Nesbitt on a paper written by Robert P. White and T.N.E.Greville.

Cecil J. Nesbitt, born in America in 1912, graduated from the Universityof Toronto with a PH.D in Mathematics, and later moved on to a career inteaching at the University of Michigan. He served a long career at thisinstitution, whilst being innovative in research in the actuarial field as wellas others. He died in 2001, at the age of eighty-nine. During Nesbitt’s careerhe over saw a particular Ph.D student, namely Donald Richard Schuette.

A.2 Proof

This proof has been taken from [W1]

To proof equation 2.47 we must first verify the operator equation:

m∑n=0

1N=nEn =

m∏j=1

(1AcjI + 1AjE) (A.1)

This involves the indicator functions of the events A1, ..., Am and theircomplements with respect to Ω, the probability set. Let ω ∈ Ω, so that ωbelongs to exactly k of the events out of A1, .., Am, where k is non-negativeand k ≤ m. Hence, we can say, for notation purposes, that ω belongs to

64

A1, ..., Ak. Then on the left hand side of the equation we get Ek. On theright hand side the first k factors in the multiplication are identically E, andthe remaining factors are equal to I, the identity operator. The product ofthe factors equal to I is also Ek, hence the equation above holds.If the difference operator is ∆ = E − I then:

1AcjI + 1AjE = 1ΩI − 1AjI + 1AjE = 1ΩI + 1Aj∆ (A.2)

where j = 0, ..,m. If we insert this into equation A.1 and expand theproducts we get:

m∑n=0

1N=nEn =

m∑n=0

∑J⊂1,..,m,|J |=n

1∩j∈JAj∆n (A.3)

And taking the expectation of this we acheive the Schuette-Nesbitt formula.

65

Appendix B

Proofs of Lemmas fromChapter 3

B.1 Proof of Lemma 3.1

Lemma 3.1

Let H be a joint distribution function with margins F and G. Thenthere exists a unique subcopula C ′ such that

• DomC ′ = RanF ×RanG

• ∀ x, y ∈ R, H(x, y) = C ′(F (x), G(y))

ProofThis proof has been based on the proof of Lemma 2.3.4 from [RN]

Since H is a joint distribution function we know from the definition thatthe function is grounded and 2-increasing, with margins F and G. Let thedomain of H be S1 × S2, where S1 and S2 are non-empty subsets of R.Then ∀(x1, y1) and (x2, y2) ∈ S1×S2, from Lemma 2.1.5 from [RN] and thetriangle inequality we have:

|H(x2, y2)−H(x1, y1)| ≤ |F (x2)− F (x1)|+ |G(y2)−G(y1)| (B.1)

If this instance S1 = S2 = R. Thus it follows that if F (x1) = F (x2) andG(y1) = G(y2), then H(x1, y1) = H(x2, y2). Therefore, the set of orderedpairs ((F (x), G(y)), H(x, y))| x,y ∈ R defines a 2-place real function C ′

whose domain is RanF × RanG. To show that this function is indeed asubcopula we use the conditions from Definition 3.4, as shown below:

Definition 3.4

A function C ′ is a subcopula if the function has the following properties:

66

• If S1 and S2 are any subsets of [0,1] containing 0 and 1 thenDomC ′ = S1 × S2

• C ′ must be grounded and 2-increasing

• ∀ a ∈ S1 and ∀ b ∈ S2;C ′(a, 1) = a and C ′(1, b) = b

• RanC ′ is also [0,1]

Clearly, this function is grounded and 2-increasing, DomC ′ has alreadybeen shown and to show C ′(a, 1) = a and C ′(1, b) = b is as follows. For eacha ∈ RanF there is an x ∈ R such that F (x) = a, and, with DomC ′ = [0, 1],we see that:

C ′(a, 1) = C ′(F (x), G(∞)) = H(x,∞) = F (x) = a

And similarly for C ′(1, b) = b.

B.2 Proof of Lemma 3.2

Lemma 3.2(corresponding to Lemma 2.3.5 in [RN])

If C ′ is a subcopula. Then there exists a copula C where C(a, b) = C ′(a, b),∀ (a, b) ∈ DomC ′.

ProofLet DomC ′ = S1 × S2 and state the following theorem (Theorem 2.2.4 in[RN]) that:

Theorem: If C ′ is a subcopula. Then for every (u1, u2), (v1, v2) ∈ DomC ′

|C ′(u2, v2)− C ′(u1, v1)| ≤ |u2 − u1|+ |v2 − v1| (B.2)

Hence C ′ is uniformly continuous on its domain

Using this and the fact that C ′ is nondecreasing in each place, we canextend C ′ by continuity to a function C ′′ with domain S1 × S2, where Siis the closure of Si. We can see that C ′′ is also a subcopula, and next weextend C ′′ to a function C with domain [0, 1] × [0, 1]. If (a, b) is any pointin [0, 1] × [0, 1] and a1 and a2 are the greatest and least elements of S1 re-spectively that satisfy a1 ≤ a ≤ a2, and similarly for the elements b1 and b2∈ S2. If a ∈ S1, then a1 = a = a2 and, if b ∈ S2, then b1 = b = b2.

67

Define:

λ1 =(a− a1)(a2 − a1)

if a1 < a2

= 1 if a1 = a2

µ1 =(b− b1)

(b2 − b− 1)if b1 < b2

= 1 if b1 = b2

and:

C(a, b) = (1− λ1)(1− µ1)C ′′(a1, b1) + (1− λ1)µ1C′′(a1, b2)

+ λ1(1− µ1)C ′′(a2, b1) + λ1µ− 1C ′′(a2, b2)

This is bilinear interpolation since λ1 and µ1 are linear in a and b re-spectively. Clearly, DomC = [0, 1] × [0, 1], C(a, b) = C ′′(a, b) for any (a, b)in DomC ′′ and C satisfies C(a, 0) = 0 = C(0, b) and C(a, 1) = a andC(1, b) = b. Now we need to verify that C satisfies:

C(a2, b2)− C(a2, b1)− C(a1, b2) + C(a1, b1) ≥ 0 (B.3)

We now define some new variables, if (c, d) is another point in [0, 1]×[0, 1]such that c ≥ a and d ≥ b, and let c1, d1, c2, d2, λ2, µ2 have the same re-lationships to c and d as a1, b1, a2, b2, λ1, µ1 did with a and b. Considerthe rectangle B = [a, c] × [b, d], there are several cases to consider whenevaluating VC(B) such as whether the points in S1 fall strictly between aand c, and whether the points in S2 fall strictly between b and d.

Case 1:If no point falls strictly between a and c or strictly between b and d in

S1 and S2 respectively. This results in c1 = a1, c2 = a2, d1 = b1 and d2 = b2so that the expression yields:

VC(B) = VC([a, c]× [b, d]) = (λ2 − λ1)(µ2 − µ1)VC([a1, a2]× [b1, b2]) (B.4)

Since c ≥ a, d ≥ b, λ2 ≥ λ1 and µ2 ≥ µ1 it follows that VC(B) ≥ 0.

Case 2:The most complicated case is when at least one point falls strictly be-

tween a and c in S1, and when at least one point in S2 falls strictly betweenb and d. This means that a < a2 ≤ c1 < c and b < b2 ≤ d1 < d. This yields

68

the equation:

VC(B) = (1− λ1)µ2VC([a1, a2]× [d1, d2]) + µ2VC([a2, c1]× [d1, d2])+ λ2µ2VC([c1, c2]× [d1, d2]) + (1− λ1)VC([a1, a2]× [b2, d1])+ VC([a2, c1]× [b2, d1]) + λ2VC([c1, c1]× [b2, d1])+ (1− λ1)(1− µ1)VC([a1, a2]× [b1, b2])+ (1− µ1)VC([a2, c1]× [b1, b2]) + λ2(1− µ1)VC([c1, c2]× [b1, b2])

This is non-negative since it is made up of nine non-negative quantitieswith non-negative coefficients. The remaining cases are similar, all non-negative, hence concluding the proof.

69

Appendix C

Life Tables

These Life Tables have been taken from Appendix E of [HG] for theoreti-cal use only. The Society of Actuaries has granted permission to use thesetables. They were produced for educational purposes and may not be ap-propriate for practical work.

70

Figure C.1:

71

Figure C.2:

72

Appendix D

Material from the 2H Course

In this appendix I have used [2H] together with my own understanding ofthe topic to explain these concepts.

In this appendix I have explained concisely the key concepts covered inthe 2H course that are essential to the material in this project.

D.1 The Basics

Interest rates are percentages stated together with a basic time unit, forexample “interest rate of 4% monthly”. Here, 4% is the interest rate anda month is the basic time unit. The conversion period is a time interval atthe end of which the interest is credited. In this project we assume the theconversion period to be a year and to be the same as the basic time unit,this is known as effective interest. We denote i to be the annual effectiveintertest rate (AER), where i is a positive constant.

The accumulation value of an initial amount, C, after n years is theincreased value due to the percentage increase. The is calculated as:

(1 + i)nC (D.1)

The present value is the value now of the amount which is C in nyears, and the discount factor (v = 1

1+i) is the percentage decrease of thefuture capital over 1 year. To calculate the present value we use:

vnC (D.2)

When the interest rate is not effective, i.e. when the basic time unitdiffers from the conversion period, the interest rate is called nominal. Ifthe nominal interest rate is compounded m times per year, at the end of mequal length time periods, we denote i(m) to be the nominal interest rate:

i(m) = m[(1 + i)1m − 1] (D.3)

73

If the interest rate is added continuously throughout the year we say theinterest is continuously compounded. In this case we can find the forceof the interest equivalent to i (δ) by taking the limit of the nominalinterest rate as m tends to infinity, shown in equation D.3, hence:

δ = limm→∞

i(m) (D.4)

= limm→∞

(1 + i)1m − (1 + i)0

1m

(D.5)

=d(1 + i)x

dx‖x=0 =

dex ln (1+i)

dx‖x=0 (D.6)

= ex ln (1+i) ln (1 + i)‖x=0 = (1 + i)x ln(1 + i)‖x=0 (D.7)= ln(1 + i) (D.8)

Therefore, it follows that:

eδ = 1 + i (D.9)⇒ v = e−δ (D.10)⇒ vn = e−nδ (D.11)

If we were to consider the interest to be paid at the beginning of theconversion period, rather than at the end, then we are interested in theannual effective discount rate (d) rather than the interest rate, where0 < d < 1.

d =i

1 + i= i v (D.12)

D.2 Probability

Here we introduce actuarial notation and formulae for survival probabilities.We firstly denote a person aged x, by (x), and the future lifetime of this lifeby T (x). The future lifetime of (x) is a random quantity as the age of deathof a life is not known. T (x) has the cumulative distribution function GX(t).

GX(t) = PX(T (x) ≤ t) t ≥ 0 (D.13)

This is the probability that (x) will not survive a further t years from now,denoted by tqx in actuarial mathematics. The probability that (x) willsurvive a further t years is denoted by tpx. Hence:

tpx = 1− tqx = PX(T (x) > t) (D.14)

The probability that (x) lives a further s years but not s+ t years is:

s|tqx = PX(s < T (x) < s+ t) = s+tqx − sqx (D.15)

74

Similarly, the probability that (x) lives to age x+ s+ t given that it lives tox+ s is:

tpx+s = PX(T (x) > s+ t|T (x) > s) = s+tpx

spx(D.16)

For the probability that (x) does not live a further s+ t years given it livess years we use:

s|tqx = spx tqx+s (D.17)

For life (0) the probability that the life survives a further t years after itreaches age s is:

tps = s+tp0

sp0(D.18)

For the probability that (x) survives one more year we use the special nota-tion px, and for the probability that it does not survive one more year, qx.If we are interested in whether a life does not survive a proportion of a fullyear, we can find uqx where u ∈ [0, 1]. If we assume that deaths in a yearare uniformly distributed over that year then:

uqx = u qx (D.19)

D.3 Force of Mortality

The force of mortality of (x) at age x+ t is:

µx+t =gX(t)

1−GX(t)= − d

dtln (1−GX(t)) (D.20)

= − d

dtln (tpx) (D.21)

Where gX(t) = ddtGX(t). The force of mortality measures the failure rate of

the life at this age. When considering fractions of a whole year, assumingthe deaths per year are uniformly distributed and u ∈ [0, 1], then the forceof mortality at an age x+ u of life (x) is:

µx+u =qx

1− u qx(D.22)

D.4 Lifetime Models

Some specific lifetime models commonly used include:

• De Moivre Model

gX(t) =1

W − x0 ≤ t ≤W − x

= 0 elsewhere

75

where W is the maximum possible lifetime. And

µx+t =1

W − x− tfor t ∈ [0,W − x] (D.23)

• Gompertz Model, where B and c are parameters:

µx+t = B cx+t for c ≥ 1 (D.24)

• Makeham Model, where A, B, and c are parameters:

µx+t = A+B cx+t for c ≥ 1 (D.25)

• Weibull Model, where D and n are parameters:

µx+t = D(x+ t)n (D.26)

D.5 Life Tables

The life tables found in Appendix C show the discretised information on aproportion of a large population reaching certain ages, assuming i = 0.05.In these tables l0 represents the newborns in the population, and lx is thenumber of such newborns that are still alive at age x. The number of livesthat die at age x is dx, so:

dx = lx − lx+1 (D.27)

We can use the life table functions to calculate survival probabilies, such as:

tpx =lx+t

lx(D.28)

D.6 Life Insurance

The basic concept of a whole life insurance policy is that the policyholderpays the present value of the contract on the promise of a payout, b, on theevent of their death. If firstly we consider the payment at the end of yearof death, this payment occurs at time K(x) + 1, where K(x) is the curtatefuture lifetime; the number of full years (x) lives. Then, the net singlepremium (NSP), cost of the contract, is:

Ax(b) =∞∑k=0

b vk+1kpx qx+k (D.29)

The NSP for a moment of death payment, with T (x) the future lifetime, is:

Ax(b) =∫ ∞

0b vt tpx µx+t dt (D.30)

76

Two approximations we learnt were:

Ax ≈ (1 + i)12Ax and Ax ≈

i

δAx (D.31)

The n-year term life insurance pays out amount b at the end of yearof death if the life (x) dies within n years from the policy start date. TheNSP for this contract is:

A1x:n (b) =

n−1∑k=0

b vk+1kpx qx+k (D.32)

D.7 Annuities and Life Annuities

An annuity is a sequence of regular payments from a fund or an account.An annuity-due is an annuity contract with n annual payments of b, atthe beginning of each year, with NSP:

an (b) = b[1 + v + v2 + v3 + ....+ vn−1] (D.33)

An immediate annuity is the same sequence payments but with thepayments at the end of each year, with NSP:

an (b) = b[v + v2 + v3 + ....+ vn] = b v an (D.34)

Life annuities are annuities conditional on the survival of a life, (x).The whole life annuity-due starts payments now and continues until thedeath of the policyholder. The NSP of this contract is:

ax(b) =∞∑k=0

b vk kpx (D.35)

A relationship between this contract and the whole life insurance policy is:

ax(b) = b1−Axd

(D.36)

where d is the annual effective discount rate. The contract with the pay-ments at the end of each year is the whole life immediate annuity:

ax(b) = ax(b)− b (D.37)

A n-year temporary life annuity-due has payments at the start ofeach year for n years, provided the life does not die within those n years.The payments stop at the moment of death of the life if the life dies in then years. The NSP is:

ax:n (b) =n−1∑k=0

b vk kpx (D.38)

77

Similarly, the immediate n-year temporary life annuity, with paymentsat the end of year is:

ax:n (b) = ax:n+1 (b)− b (D.39)

The m-year deferred whole life annuity-due is a whole life annuity-due which starts m years after the policy start date. If (x) dies within thesem years they will receive no payments. The NSP of this contract is:

m|ax(b) = ax(b)− ax:m (b) (D.40)

D.8 Net Premiums

Net Premiums are the amounts paid by the policyholder for a contractdifferent from paying a lump sum at time 0. These may be periodic pay-ments, with regular installments at the start of each period. Let L denotethe present value of the total loss to the insurer, this is the difference be-tween the present value of the contract to the insured and the present valueof all premiums paid. The equivalence principle states that:

E(L) = 0 (D.41)

The equivalence principle must hold in order for premiums to be netpremiums. If the net premiums are of value Π and are paid annually forn years, as long as (x) lives, then the sequence of payments to the insurerforms an annuity-due, Πan . If value Π depends on the contract bought, so ifwe denote the net single premium of the contract by R, using the equivalenceprinciple, we can find:

E(L) = 0 = R−Π an nPx

⇒ Π =R

an nPx

This can be altered and simplified depending on the type of contract beingpurchased.

78

Appendix E

Does “Broken HeartSyndrome” Exist?

In this appendix I have used facts from various reports, the results of whichhave all been stated in [CMP].

The death of a “loved one” or partner can be a significant emotionalstrain, however, whether bereavement reduces the future lifetimes of peopleis questionable. If “Broken Heart Syndrome” really does effect future life-times then this would be a legitimate cause of dependency between lives. In1657, on Dr Heberden’s Bill listing the official causes of death in London,it was stated that 10 people died of grief! There have been various studiessince then into the effect of bereavement on future lifetimes, however, griefis no longer considered an official cause of death.

In an article by Young, Benjamin and Wallis in 1963 it was reportedthat, from their study of 4486 widowers over the age of 54, the peak of mor-tality in widowers is during the first year after a bereavement. In fact, theyfound that the death rate among these widowers increased by almost 40%during the first 6 months after the bereavement. A similar study into theeffect on widows (Mellstrom et al, 1982) showed that it was the first threemonths which showed a significant increase in the mortality of widows. Be-ing particularly accurate by stating that the future lifetime of widows wasreduced by 6 months after a bereavement, whereas the corresponding valuefor widowers was one and a half years. The mortality rate of the widowersfrom Young et al. is shown in Figure E.1 below.

From the study by Young et al., a further study by Parkes Benjaminand Fitzgerald (1969) found that the causes of death of the lives who diedin the first six months after a bereavement could be mostly attributed toheart disease. This implied that the emotional strain of a bereavement can

79

Figure E.1:

increase the risk of heart failure amongst the bereaved. So we can see wherethe term “broken heart syndrome” originates.

A “broken heart” need not only be the result of the loss of a partner,the death of a child could also be a factor causing dependency. Rees andLutkins (1967) showed an increased mortality among parents who have losta child. Throughout [CMP], it is claimed that grief is not a direct causeof increased mortality, however an event such as the death of a relative canput excess strain on the heart, increasing the risk of heart failure and thusincreasing the mortality rate amongst the bereaved.

80

Appendix F

Notation

Listed in this appendix is the important notation in this project with a shortdefinition and a section reference to where the notation is first introduced.Similar notation symbols are grouped together approximately alphabetically.

Symbol Definition Section Ref.an net single premium (NSP) of annuity-due contract D.7an immediate annuity NSP D.7ax whole life annuity-due NSP D.7.1 / 1.0ax whole life immediate annuity NSP D.7.1ax:n n-year temporary life annuity-due NSP D.7.1ax:n immediate n-year temporary life annuity NSP D.7.1m|ax m-year deferred whole life annuity-due NSP D.7.1ax:y joint-life whole life annuity-due NSP 2.1.1ax:y joint-life whole life immediate annuity NSP 2.1.1ax:y:n n-year temporary joint-life annuity-due NSP 2.1.1m|ax:y m-year deferred joint-life annuity-due NSP 2.1.1ax:y last-survivor whole life annuity-due NSP 2.2.1ar:s asymmetric whole life immediate annuity NSP 2.4.1ax/y reversionary annuity NSP 2.4.1Ax whole life insurance NSP (end of year of death) D.6Ax whole life insurance NSP (moment of death) D.6A1x:n n-year term life insurance NSP D.6

Ax:y joint-life whole life immediate life insurance NSP 2.1.2Ax:y joint-life whole life insurance NSP (moment of death) 2.1.3Ax1:.:x1

k:.:xm special case asymmetric insurance NSP 2.4.2A1x:y first-death contingent insurance NSP 2.7.1

A2x:y NSP of insurance which benefits (x) dies after (y) 2.7.2

a : b : c : d asymmetric status 2.4C copula function 3.1

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Symbol Definition Section Ref.C ′ subcopula function 3.2C survival copula 3.4C dual of a copula function 3.4C∗ co-copula 3.4d = i

1+i annual effective discount rate D.1dx no. that die at age x (life tables) D.5δ = ln (1− i) force of interest D.1FX:Y (x, y) joint distribution function 3.1FX(x) marginal distribution function of X 3.1F (−1) quasi inverse of distribution function F 3.3F (x) survival function 3.4GX(t) cumulative distribution function of T (x) D.2Γ(a, b) gamma distribution, parameters a,b 5.2H(x, y) joint distribution function 3.2H(x, y) joint survival function 3.4i interest rate D.1lx no. lives still alive at age x (life tables) D.5λ common shock parameter 5.1tpx P (T (x) > t) D.2px P (T (x) > 1) = 1− qx D.2tpx:y P (T (x) > t, T (y) > t) (joint-life) 2.1tpx:y P (T (x) > t or T (y) > t) (last-survivor) 2.2tpr:s asymmetric survival probability 2.4.1tqx P (T (x) ≤ t) D.2s|tqx P (s < T (x) < s+ t) D.2tqx:y P (T (x) ≤ t or T (y) ≤ t)(joint-life) 2.1tqx:y P (T (x) ≤ t, T (y) ≤ t) (last-survivor) 2.2q1x:y P((x) dies next year and (y) is alive at time) 2.7.1T (x) expected future lifetime of (x) D.2µx+t = − d

dt ln (tpx) force of mortality D.3/2.1.3µx:y(t) force of failure 2.1.3v = 1

1+i discount factor D.1(x) life aged x D.2/1.0x : y joint-life status (2 lives) 2.1x1 : x2 : . : xn joint-life status (n lives) 2.1x : y last-survivor status (2 lives) 2.2x1 : . : xn last-survivor status (n lives) 2.2x1 : . : xnm general symmeteric status 2.3x1 : . : xn[m] status survives when exactly m lives survive 2.3X∗ lifetime random variable of (x) 5.1.1Y frailty random variable 5.2Z common shock random variable 5.1

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