Multiple Life Models. Lecture: Weeks Lecture: Weeks 9-10 (STT 456) Multiple Life Models Spring Valdez 1 / 38

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1 Multiple Life Models Lecture: Weeks 9-1 Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 1 / 38

2 Chapter summary Chapter summary Approaches to studying multiple life models: define multiple states traditional approach (use joint random variables) Statuses: joint life status last-survivor status Insurances and annuities involving multiple lives evaluation using special mortality laws Simple reversionary annuities Contingent probability functions Dependent lifetime models Chapter 9 (Dickson et al.) Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 2 / 38

3 Approaches multiple states States in a joint life and last survivor model x alive y alive () µ 1 x+t:y+t x alive y dead (1) µ 2 x+t:y+t µ 13 x+t x dead y alive (2) µ 23 y+t x dead y dead (3) Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 3 / 38

4 Approaches joint future lifetimes Joint distribution of future lifetimes Consider the case of two lives currently ages x and y with respective future lifetimes T x and T y. Joint cumulative dist. function: F TxT y (s, t) = Pr[T x s, T y t] independence: F TxT y (s, t) = Pr[T x s] Pr[T y t] = F x (s) F y (t) Joint density function: f TxT y (s, t) = 2 F TxTy (s,t) s t independence: f TxT y (s, t) = f x (s) f y (t) Joint survival dist. function: S TxT y (s, t) = Pr[T x > s, T y > t] independence: S TxT y (s, t) = Pr[T x > s] Pr[T y > t] = S x (s) S y (t) Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 4 / 38

5 Approaches illustration Illustrative example 1 Consider the joint density expressed by f TxTy (s, t) = 1 (s + t), for < s < 4, < t < Prove that T x and T y are not independent. 2 Calculate the covariance of T x and T y. 3 Evaluate the probability (x) outlives (y) by at least one year. Solution to be discussed in lecture. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 5 / 38

6 Statuses joint life status The joint life status This is a status that survives so long as all members are alive, and therefore fails upon the first death. Notation: (xy) for two lives (x) and (y) For two lives: T xy = min(t x, T y ) Cumulative distribution function: F Txy (t) = t q xy = Pr[min(T x, T y ) t] = 1 Pr[min(T x, T y ) > t] = 1 Pr[T x > t, T y > t] = 1 S TxT y (t, t) = 1 p t xy where p t xy = Pr[T x > t, T y > t] = S Txy (t) is the probability that both lives (x) and (y) survive after t years. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 6 / 38

7 Statuses joint life status The case of independence Alternative expression for the distribution function: F Txy (t) = F x (t) + F y (t) F TxT y (t, t) In the case where T x and T y are independent: and tpxy = Pr[T x > t, T y > t] = Pr[T x > t] Pr[T y > t] = p t x p t y tqxy = t q x + t q y t q x t q y Remember this (even in the case of independence): tqxy t q x t q y Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 7 / 38

8 Statuses last-survivor status The last-survivor status This is a status that survives so long as there is at least one member alive, and therefore fails upon the last death. Notation: (xy) For two lives: T xy = max(t x, T y ) General relationship among T xy, T xy, T x, and T y : for any constant a >. T xy + T xy = T x + T y T xy T xy = T x T y a Txy + a T xy = a Tx + a Ty For each outcome, note that T xy is equal either T x or T y, and therefore, T xy equals the other. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 8 / 38

9 Statuses distribution Distribution of T xy Recall method of inclusion-exclusion of probability: Pr[A B] + Pr[A B] = Pr[A] + Pr[B]. Choose events A = {T x t} and B = {T y t} so that A B = {T xy t} and A B = {T xy t}. This leads us to the following useful relationships: F Txy (t) + F Txy (t) = F x (t) + F y (t) S Txy (t) + S Txy (t) = S x (t) + S y (t) tpxy + t p xy = t p x + t p y f Txy (t) + f Txy (t) = f x (t) + f y (t) These relationships lead us to finding distributions of T xy, e.g. F Txy (t) = F x (t) + F y (t) F Txy (t) = F TxT y (t, t) which is obvious from F Txy (t) = Pr[T x t T y t]. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 9 / 38

10 Statuses distribution Interpretation of probabilities Note that: tpxy is the probability that both lives (x) and (y) will be alive after t years. tpxy is the probability that at least one of lives (x) and (y) will be alive after t years. In contrast: tqxy is the probability that at least one of lives (x) and (y) will be dead within t years. tqxy is the probability that both lives (x) and (y) will be dead within t years. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 1 / 38

11 Statuses illustration Illustrative example 2 For independent lives (x) and (y), you are given: and q x =.5 and q y =.1, q x+1 =.6 and q y+1 =.12. Deaths are assumed to be uniformly distributed over each year of age. Calculate and interpret the following probabilities: 1 q.75 xy 2 q 1.5 xy Solution to be discussed in lecture. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 11 / 38

12 Force of mortality joint life Force of mortality of T xy Define the force of mortality (similar manner to any random variable): µ x+t:y+t = f T xy (t) 1 F Txy (t) = f T xy(t) S Txy (t) = f T xy(t). p t xy We can then write the density of T xy as f Txy (t) = p t xy µ x+t:y+t In the case of independence, we have: t µ x+t:y+t = p x t p y (µ x+t + µ y+t ) tpx t p y = µ x+t + µ y+t. The force of mortality of the joint life status is the sum of the individuals force of mortality, when lives are independent. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 12 / 38

13 Force of mortality last-survivor Force of mortality for T xy The force of mortality for T xy is defined as µ x+t:y+t = f Txy (t) 1 F Txy (t) = f T (t) xy S Txy (t) = f x(t) + f y (t) f Txy (t) tpxy t = p x µ x+t + p t y µ y+t t p xy µ x+t:y+t p t xy Indeed we have the density of T xy expressed as f Txy (t) = p t xy µ x+t:y+t. Check what this formula gives in the case of independence. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 13 / 38

14 Insurance benefits discrete Insurance benefits - discrete Consider an insurance under which the benefit of $1 is paid at the EOY of ending (failure) of status u. Status u could be any joint life or last survivor status e.g. xy, xy. Then the time at which the benefit is paid: K u + 1 the present value (at issue) of the benefit: Z = v Ku+1 APV of benefits: E[Z] = A u = v k+1 Pr[K u = k] variance: Var[Z] = 2 Au (A u ) 2 k= Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 14 / 38

15 Insurance benefits continuous Insurance benefits - continuous Consider an insurance under which the benefit of $1 is paid immediately of ending (failure) of status u. Status u could be any joint life or last survivor status e.g. xy, xy. Then the time at which the benefit is paid: T u the present value (at issue) of the benefit: APV of benefits: E[Z] = Āu = variance: Var[Z] = 2Ā u (Āu) 2 Z = v Tu v t p t u µ u+t dt Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 15 / 38

16 Insurance benefits continuous Some illustrations For a joint life status (xy), consider whole life insurance providing benefits at the first death: A xy = v k+1 k qxy = v k+1 k p xy q x+k:y+k Ā xy = k= k= v t p t xy µ x+t:y+t dt For a last-survivor status (xy), consider whole life insurance providing benefits upon the last death: A xy = v k+1 k qxy = v k+1 ( k qx + k q y k q xy ) Ā xy = = k= k= v t p t xy µ x+t:y+t dt v t ( p t x µ x+t + p t y µ y+t p t xy µ x+t:y+t ) dt Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 16 / 38

17 Insurance benefits continuous - continued Useful relationships: A xy + A xy = A x + A y Ā xy + Āxy = Āx + Āy Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 17 / 38

18 Annuity benefits discrete Annuity benefits - discrete Consider an n-year temporary life annuity-due on status u. Then the present value (at issue) of the benefit: Y = {ä Ku+1, K u < n ä n, APV of benefits: E[Y ] = ä u: n = n 1 k= ä q k+1 k u + ä n n p u variance: Var[Y ] = 1 [ 2 d 2 A u: n ( ) ] 2 A u: n Other ways to write APV: n 1 ä u: n = v k k p u = 1 ( ) 1 Au: d n. k= K u n Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 18 / 38

19 Annuity benefits continuous Annuity benefits - continuous Consider an annuity for which the benefit of $1 is paid each year continuously for years so long as a status u continues. Then the present value (at issue) of the benefit: Y = ā Tu APV of benefits: E[Y ] = ā u = variance: Var[Y ] = 1 [ 2 Ā δ 2 u ( ) ] 2 Ā u ā t p t u µ u+t dt = Note that the identity δā Tu + v Tu = 1 provides the connection between insurances and annuities. v t p t u dt Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 19 / 38

20 Annuity benefits continuous Some illustrations For joint life status (xy), consider a whole life annuity providing benefits until the first death: ä xy = v k k p xy and ā xy = v t t p xy dt k= For last survivor status (xy), consider a whole life insurance providing benefits upon the last death: ä xy = k= Useful relationships: v k p k xy and ā xy = ä xy + ä xy = ä x + ä y ā xy + ā xy = ā x + ā y v t p t xy dt Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 2 / 38

21 Annuity benefits continuous Comparing benefits - annuities Type of life annuity Single life x Joint life status xy Last survivor status xy Whole life a-due ä x ä xy ä xy Whole life a-immediate a x a xy a xy Temporary life a-due ä x: n ä xy: n ä xy: n Temporary life a-immediate a x: n a xy: n a xy: n Whole life a-continuous ā x ā xy ā xy Temporary life a-continuous ā x: n ā xy: n ā xy: n Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 21 / 38

22 Annuity benefits continuous Comparing benefits - insurances Type of life insurance Single life x Joint life status xy Last survivor status xy Whole life - discrete A x A xy A xy Whole life - continuous Ā x Ā xy Ā xy Term - discrete A 1 x: n A 1 xy : n A 1 xy: n Term - continuous Ā 1 x: n Ā 1 xy : n Ā 1 xy: n Endowment - discrete A x: n A xy: n A xy: n Endowment - continuous Ā x: n Ā xy: n Ā xy: n Pure endowment Ax: 1 n or n E x Axy: 1 n or n E xy Axy: 1 n or n E xy Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 22 / 38

23 Annuity benefits continuous Illustrative example 3 You are given: (45) and (65) have independent future lifetimes. Mortality for either life follows demoivre s law with ω = 15. δ = 5% Calculate Ā45:65. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 23 / 38

24 Contingent functions Contingent functions It is possible to compute probabilities, insurances and annuities based on the failure of the status that is contingent on the order of the deaths of the members in the group, e.g. (x) dies before (y). These are called contingent functions. Consider the probability that (x) fails before (y) - assuming independence: Pr[T x < T y ] = = = f Tx (t) S Ty (t)dt tpx µ x+t t p y dt tpxy µ x+t dt The actuarial symbol for this is qxy. 1 It should be obvious this is the same as qxy. 2 Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 24 / 38

25 Contingent functions - continued The probability that (x) dies before (y) and within n years is given by q 1 n xy = n tpxyµ x+t dt. Similarly, we have the probability that (y) dies before (x) and within n years: n nqxy 1 = tpxyµ y+t dt. It is easy to show that q 1 n xy + q 1 n xy = q n xy. One can similarly define and interpret the following: nqxy 2 and nqxy, 2 and show that nqxy 2 + nqxy 2 = q n xy. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 25 / 38

26 Contingent functions illustration Illustrative example 4 An insurance of $1 is payable at the moment of death of (y) if predeceased by (x), i.e. if (y) dies after (x). The actuarial present value (APV) of this insurance is denoted by Ā 2 xy. Assume (x) and (y) are independent. 1 Give an expression for the present value random variable for this insurance. 2 Show that 3 Prove that Ā 2 xy = and interpret this result. Ā 2 xy = Āy Ā 1 xy. v t Ā y+t p t xy µ x+t dt, Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 26 / 38

27 Reversionary annuities Reversionary annuities A reversionary annuity is an annuity which commences upon the failure of a given status (u) if a second status (v) is then alive, and continues thereafter so long as status (v) remains alive. Consider the simplest form: an annuity of $1 per year payable continuously to a life now aged x, commencing at the moment of death of (y) - briefly annuity to (x) after (y). APV for this reversionary annuity: ā y x = v t p t xy µ y+t ā x+t dt. One can show the more intuitive formula (using current payment technique): ā y x = v t p t x ( 1 p t y ) dt = āx ā xy. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 27 / 38

28 Reversionary annuities Present value random variable For the reversionary annuity considered in the previous slides, one can also write the present-value random variable at issue as: Z = { T ā, y T x T y T y T x, T y > T x Can you explain the last line? {ā = Tx ā Ty, T y T x, T y > T x = ā Tx ā Txy. By taking the expectation of Z, we clearly have ā y x = ā x ā xy. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 28 / 38

29 Reversionary annuities Reversionary annuities - discrete In general, an annuity to any status (u) after status (v) is a v u = a u a uv where a is any annuity which takes discrete, continuous, or payable m times a year. Consider the discrete form of reversionary annuity: $1 per year payable to a life now aged x, commencing at the EOY of death of (y). APV for this reversionary annuity: a y x = v k ( ) k p x 1 kpy = ax a xy. k=1 If (v) is the term-certain ( n ) and (u) is the single life (x), then a n x = a x a x : n which is indeed a single-life deferred annuity. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 29 / 38

30 Multiple state framework probabilities Back to multiple state framework Translating the probabilities/forces earlier defined, the following should now be straightforward to verify: tpxy = tp xy tqxy = tp 1 xy + tp 2 xy + tp 3 xy tpxy = t p xy + tp 1 xy + tp 2 xy q t xy = p3 t xy q t xy = t p3 xy t q 1 t xy = q 2 t xy = t sp xy µ 2 x+s:y+sds sp 1 xy µ 13 x+sds Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 3 / 38

31 Multiple state framework annuities Annuities In terms of the annuity functions, the following should also be straightforward to verify: ā xy = ā xy = e δt p t xydt ā xy = ā xy + ā 1 xy + ā 2 xy = ā x y = ā 2 xy = e δt p 2 t xydt The following also holds true (easy to verify): ā xy = ā x + ā y ā xy ā x y = ā y ā xy e δt( p t xy + p 1 t xy + p 2 t xy) dt Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 31 / 38

32 Multiple state framework insurances Insurances In terms of insurance functions, the following should also be straightforward to verify: Ā xy = Ā xy = Ā 1 xy = Ā 2 xy = e δt p t xy ( µ 1 x+t:y+t + µ 2 x+t:y+t) dt e δt( p 1 t xy µ 13 x+t + p 2 t xy µ 23 y+t) dt e δt p t xy µ 2 x+t:y+tdt e δt p 1 t xy µ 13 x+tdt The following also holds true (easy to verify): Ā xy = Āx + Āy Āxy and ā xy = 1 ( ) 1 Ā xy δ Ā 1 xy + Ā2 xy = Āx Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 32 / 38

33 Multiple state framework case of independence The case of independence x alive y alive () µ f y+t x alive y dead (1) µ m x+t µ m x+t x dead y alive (2) µ f y+t x dead y dead (3) Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 33 / 38

34 Illustrations Illustrative example 5 Suppose that the future lifetimes, T x and T y, of a husband and wife, respectively are independent and each is uniformly distributed on [, 5]. Assume δ = 5%. 1 A special insurance pays $1 upon the death of the husband, provided that he dies first. Calculate the actuarial present value for this insurance and the variance of the present value. 2 An insurance pays $1 at the moment of the husband s death if he dies first and $2 if he dies after his wife. Calculate the APV of the benefit for this insurance. 3 An insurance pays $1 at the moment of the husband s death if he dies first and $2 at the moment of the wife s death if she dies after her husband. Calculate the APV of the benefit for this insurance. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 34 / 38

35 Illustrations Illustrative example 6 For a husband and wife with ages x and y, respectively, you are given: µ x+t =.2 for all t > µ y+t =.1 for all t > δ =.4 1 Calculate ā xy: 2 and ā xy: 2. 2 Rewrite this problem in a multiple state framework and solve (1) within this framework. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 35 / 38

36 Illustrations Illustrative example 7: SOA Fall 213 Question # 2 For (x) and (y) with independent future lifetimes, you are given: ā x = 1.6 ā y = ā xy = Ā 1 xy =.9 δ =.7 Calculate Ā1 xy. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 36 / 38

37 Common shock model The model with a common shock x alive y alive () µ 1 x+t:y+t x alive y dead (1) µ 3 x+t:y+t µ 2 x+t:y+t µ 13 x+t x dead y alive (2) µ 23 y+t x dead y dead (3) Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 37 / 38

38 Common shock model Illustrative example 8: SOA Spring 214 Question # 7 The joint mortality of two lines (x) and (y) is being modeled as a multiple state model with a common shock (see diagram in the previous page). You are given: µ 1 =.1 µ 2 =.3 µ 3 =.5 δ =.5 A special joint whole life insurance pays 1 at the moment of simultaneous death, if that occurs, and zero otherwise. Calculate actuarial present value of this insurance. Lecture: Weeks 9-1 (STT 456) Multiple Life Models Spring Valdez 38 / 38