# Insurance Benefits. Lecture: Weeks 6-8. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 1 / 36

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1 Insurance Benefits Lecture: Weeks 6-8 Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 1 / 36

2 An introduction An introduction Central theme: to quantify the value today of a (random) amount to be paid at a random time in the future. main application is in life insurance contracts, but could be applied in other contexts, e.g. warranty contracts. Generally computed in two steps: 1 take the present value (PV) random variable, b T v T ; and 2 calculate the expected value E[b T v T ] for the average value - this value is referred to as the Actuarial Present Value (APV). In general, we want to understand the entire distribution of the PV random variable b T v T : it could be highly skewed, in which case, there is danger to use expectation. other ways of summarizing the distribution such as variances and percentiles/quantiles may be useful. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 2 / 36

3 An introduction A simple illustration Consider the simple illustration of valuing a three-year term insurance policy issued to age 35 where if he dies within the first year, a \$1,000 benefit is payable at the end of his year of death. If he dies within the second year, a \$2,000 benefit is payable at the end of his year of death. If he dies within the third year, a \$5,000 benefit is payable at the end of his year of death. Assume a constant discount rate of 5% and the following extract from a mortality table: Calculate the APV of the benefits. x q x Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 3 / 36

4 An introduction chapter summary Chapter summary Life insurance benefits payable contingent upon death; payment made to a designated beneficiary actuarial present values (APV) actuarial symbols and notation Insurances payable at the moment of death continuous level benefits, varying benefits (e.g. increasing, decreasing) Insurances payable at the end of year of death discrete level benefits, varying benefits (e.g. increasing, decreasing) Chapter 4 (Dickson, et al.) - both 1st/2nd ed. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 4 / 36

5 The present value random variable The present value random variable Denote by Z, the present value random variable. This gives the value, at policy issue, of the benefit payment. Issue age is usually denoted by x. In the case where the benefit is payable at the moment of death, Z clearly depends on the time-until-death T. For simplicity, we drop the subscript x for age-at-issue. It is Z = b T v T where: b T is called the benefit payment function v T is the discount function In the case where we have a constant (fixed) interest rate, then v T = v T = (1 + i) T = e δt. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 5 / 36

6 Policy types term life Fixed term life insurance An n-year term life insurance provides payment if the insured dies within n years from issue. For a unit of benefit payment, we have { 1, T n b T = 0, T > n and v T = v T. The present value random variable is therefore { v T, T n Z = 0, T > n = vt I(T n) where I( ) is called indicator function. E[Z] is called the APV of the insurance. Actuarial notation: n Ā 1 x: n = E[Z] = v t f x (t)dt = 0 n 0 v t p t x µ x+t dt. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 6 / 36

7 Policy types term life Rule of moments The j-th moment of the distribution of Z can be expressed as: E [ Z j] = n 0 v tj p t x µ x+t dt = n 0 e (jδ)t p t x µ x+t dt. This is actually equal to the APV but evaluated at the force of interest jδ. In general, we have the following rule of moment: E [ Z δ t = jδt. For example, the variance can be expressed as Var[Z] = 2 Ā x: 1 n ( Ā 1 ) 2 x: n. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 7 / 36

8 Policy types whole life Whole life insurance For a whole life insurance, benefits are payable following death at any time in the future. Here, we have b T = 1 so that the present value random variable is Z = v T. APV notation for whole life: Ā x =E[Z] = Variance (using rule of moments): Var[Z] = 0 2 Ā x ( ) 2 Ā x. v t p t x µ x+t dt. Whole life insurance is the limiting case of term life insurance as n. Note also that if the benefit amount is not 1, but say b T = b, then E[Z] = bāx and that Var[Z] = b 2 [ Ā 2 x ( Ā x ) 2 ]. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 8 / 36

9 Policy types pure endowment Pure endowment insurance For an n-year pure endowment insurance, a benefit is payable at the end of n years if the insured survives at least n years from issue. { 0, T n Here, we have b T = 1, T > n and v T = v n so that the PV r.v. is { 0, T n Z = v n, T > n. APV for pure endowment: Ax: 1 n = n E x = v n p Variance (using rule of moments): n x. Var[Z] = v 2n n p x nqx = 2 Ax: 1 n ( Ax: 1 ) 2 n. Sometimes, we can also express the present value random variable based on an indicator function: Z = v n I(T x > n), where I(E) is 1 if the event E is true, and 0 otherwise. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 9 / 36

10 Policy types endowment insurance Endowment insurance For an n-year endowment insurance, a benefit is payable if death is within n years or if the insured survives at least n years from issue, whichever occurs first. { v T, T n Here, we have b T = 1 and v T = v n so that the PV r.v. is, T > n { v T, T n Z = v n, T > n. It is easy to see that we can re-write Z as Z = v min(t,n). APV endowment: Ā x: n = Ā 1 x: n + A 1 x: n. Variance (using rule of moments): Var[Z] = 2 Ā x: n ( ) 2 Ā x: n. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 10 / 36

11 Policy types deferred insurance Deferred insurance For an n-year deferred whole insurance, a benefit is payable if the insured dies at least n years following issue. { 0, T n Here, we have b T = 1, T > n and v T = v T so that the PV r.v. is { 0, T n Z = v T, T > n. APV for deferred insurance: n Ā x = Variance (using rule of moments): Var[Z] = n v t p t x µ x+t dt. ( ) 2 n 2 Ā x n Āx. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 11 / 36

12 Special cases constant force Constant force of mortality - all throughout life Assume mortality is based on a constant force, say µ, and interest is also based on a constant force of interest, say δ. Find expressions for the APV for the following types of insurances: whole life insurance; n-year term life insurance; n-year endowment insurance; and m-year deferred life insurance. Check out the (corresponding) variances for each of these types of insurance. [Details in class] Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 12 / 36

13 Special cases De Moivre s law De Moivre s law Find expressions for the APV for the same types of insurances in the case where you have: De Moivre s law. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 13 / 36

14 Special cases whole life - illustration Illustrative example 1 For a whole life insurance of \$1,000 on (x) with benefits payable at the moment of death, you are given: { 0.04, 0 < t 10 δ t = 0.05, t > 10 and µ x+t = { 0.006, 0 < t , t > 10 Calculate the actuarial present value for this insurance. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 14 / 36

15 Special cases whole life Equivalent probability calculations We can also compute probabilities of Z as follows. Consider the present value random variable Z for a whole life issued to age x. For 0 < α < 1, the following is straightforward: Pr[Z α] = Pr[e δtx α = Pr[ δt x log(α)] = Pr[T x > (1/δ) log(α)] = p u x, where u = (1/δ) log(1/α) = log(1/α) 1/δ. Consider the case where α = 0.75 and δ = Then u = log(1/0.75) 1/0.05 = Thus, the probability Pr[Z 0.75] is equivalent to the probability that (x) will survive for another years. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 15 / 36

16 Varying benefits Insurances with varying benefits Type b T Z APV Increasing whole life T + 1 T + 1 v T ( I Ā ) x Whole life increasing m-thly T m + 1 /m v T T m + 1 /m ( I (m) Ā ) x Constant increasing whole life Decreasing n-year term T T v T (ĪĀ ) x { n T, T n 0, T > n { (n T ) v T, T n 0, T > n ( D Ā ) 1 x: n * These items will be discussed in class. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 16 / 36

17 Varying benefits varying benefits - illustration Illustrative example 2 For a whole life insurance on (50) with death benefits payable at the moment of death, you are given: Mortality follows De Moivre s law with ω = 110. b t = 10000(1.10) t, for t 0 δ = 5% Z denotes the present value random variable for this insurance. Calculate E[Z] and Var[Z]. Can you find an explicit expression for the distribution function of Z, i.e. Pr[Z z]? Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 17 / 36

18 Discrete insurances term insurance Insurances payable at EOY of death For insurances payable at the end of the year (EOY) of death, the PV r.v. Z clearly depends on the curtate future lifetime K x. It is Z = b K+1 v K+1. To illustrate, consider an n-year term insurance which pays benefit at the end of year of death: { 1, K = 0, 1,..., n 1 b K+1 =, v K+1 = v K+1, 0, otherwise and therefore Z = { v K+1, K = 0, 1,..., n 1 0, otherwise. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 18 / 36

19 Discrete insurances term insurance - continued APV of n-year term: n 1 A 1 x: n = E[Z] = n 1 v k+1 k qx = k=0 k=0 v k+1 p k x q x+k Rule of moments also apply in discrete situations. For example, Var[Z] = 2 Ax: 1 n ( A 1 ) 2 x: n, where 2 1 A x: n = E [ Z 2] n 1 = e 2δ(k+1) k p x q x+k. k=0 Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 19 / 36

20 Discrete insurances whole life (Discrete) whole life insurance Consider a whole life insurance which pays benefit at the end of year of death (for life): b K+1 = 1, v K+1 = v K+1, and Z = v K+1. APV: A x = E[Z] = k=0 vk+1 q k x = k=0 vk+1 q k x q x+k Applying rule of moments, Var[Z] = 2 Ax (A x ) 2, where 2 A x = E [ Z 2] = k=0 e 2δ(k+1) p k x q x+k. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 20 / 36

21 Discrete insurances endowment (Discrete) endowment life insurance The APV of a (discrete) endowment life insurance is the sum of the APV of a (discrete) term and a pure endowment: A x: n = A 1 x: n + A 1 x: n The policy pays a death benefit of \$1 at the end of the year of death, if death is prior to the end of n years, and a benefit of \$1 if the insured survives at least n years. { v K+1, K n 1 In effect, we have b K+1 = 1 and v K+1 = v n so, K n { v K+1, K n 1 that the PV r.v. is Z = v n., K n Here Z = v min(k+1,n) and one can also apply the rule of moments to evaluate the corresponding variance. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 21 / 36

22 Discrete insurances recursive relationships Recursive relationships The following will be derived/discussed in class: whole life insurance: A x = vq x + vp x A x+1 term insurance: A 1 x: n = vq x + vp x A 1 x + 1: n 1 endowment insurance: A x: n = vq x + vp x A x+1: n 1 Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 22 / 36

23 Discrete insurances interest rate sensitivity Ax Makeham parameters: A = , B = , c = i = 1% i = 5% i = 10% i = 15% i = 20% x Figure : Actuarial Present Value of a discrete whole life insurance for various interest rate assumptions Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 23 / 36

24 Discrete insurances mortality rate sensitivity qx Makeham parameters: A = , B = , c varying c = c = c = c = c = Ax Makeham parameters: A = , B = , c varying c = c = c = c = c = x x Figure : Actuarial Present Value of a discrete whole life insurance for various mortality rate assumptions with interest rate fixed at 5% Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 24 / 36

25 Discrete insurances illustrative example Illustrative example 3 For a whole life insurance of 1 on (41) with death benefit payable at the end of the year of death, let Z be the present value random variable for this insurance. You are given: i = 0.05; p 40 = ; A 41 A 40 = ; and 2A 41 2 A40 = Calculate Var[Z]. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 25 / 36

26 Discrete insurances other forms Other forms of insurance Deferred insurances Varying benefit insurances Very similar to the continuous cases You are expected to read and understand these other forms of insurances. It is also useful to understand the various (possible) recursion relations resulting from these various forms. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 26 / 36

27 Discrete insurances illustration of varying benefit amounts Illustration of varying benefits For a special life insurance issued to (45), you are given: Death benefits are payable at the end of the year of death. The benefit amount is \$100,000 in the in the first 10 years of death, decreasing to \$50,000 after that until reaching age 65. An endowment benefit of \$100,000 is paid if the insured reaches age 65. There are no benefits to be paid past the age of 65. Mortality follows the Illustrative Life Table at i = 6%. Calculate the actuarial present value (APV) for this insurance. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 27 / 36

28 Discrete insurances illustrative example Illustrative example 4 For a whole life insurance issued to age 40, you are given: Death benefits are payable at the moment of death. The benefit amount is \$1,000 in the first year of death, increasing by \$500 each year thereafter for the next 3 years, and then becomes level at \$5,000 thereafter. Mortality follows the Illustrative Life Table at i = 6%. Deaths are uniformly distributed over each year of age. Calculate the APV for this insurance. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 28 / 36

29 Insurance payable m-thly Insurances payable m-thly Consider the case where we have just one-year term and the benefit is payable at the end of the m-th of the year of death. We thus have A (m) 1 x: 1 = m 1 r=0 v (r+1)/m r/mpx 1/m q x+r/m. We can show that under the UDD assumption, this leads us to: A (m) 1 x: 1 In general, we can generalize this to: A (m) 1 x: n = i i (m) A1 x: 1. = i i (m) A1 x: n. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 29 / 36

30 Insurance payable m-thly Other types of insurances with m-thly payments For other types, we can also similarly derive the following (with the UDD assumption): whole life insurance: A (m) x = i i (m) A x deferred life insurance: endowment insurance: A (m) n x = i A i (m) n x A (m) x: n = i i (m) A1 x: n + A 1 x: n Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 30 / 36

31 Relationships Relationships - continuous and discrete For some forms of insurances, we can get explicit relationships under the UDD assumption: whole life insurance: Ā x = i δ A x term insurance: Ā 1 x: n = i δ A1 x: n increasing term insurance: ( I Ā ) 1 x: n = i δ (IA)1 x: n Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 31 / 36

32 Relationships illustrative example Illustrative example 5 For a three-year term insurance of 1000 on [50], you are given: Death benefits are payable at the end of the quarter of death. Mortality follows a select and ultimate life table with a two-year select period: [x] l [x] l [x]+1 l x+2 x Deaths are uniformly distributed over each year of age. i = 5% Calculate the APV for this insurance. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 32 / 36

33 Illustrative examples Illustrative example 6 Each of 100 independent lives purchases a single premium 5-year deferred whole life insurance of 10 payable at the moment of death. You are given: µ = δ = F is the aggregate amount the insurer receives from the 100 lives. The 95th percentile of the standard Normal distribution is Using a Normal approximation, calculate F such that the probability the insurer has sufficient funds to pay all claims is Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 33 / 36

34 Illustrative examples Illustrative example 7 Suppose interest rate i = 6% and mortality is based on the following life table: x l x Calculate the following: (a) A 94 (b) A 1 90: 5 (c) A (4) 3 92, assuming UDD between integral ages (d) A 95: 3 Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 34 / 36

35 Illustrative examples Illustrative example 8 A five-year term insurance policy is issued to (45) with benefit amount of \$10,000 payable at the end of the year of death. Mortality is based on the following select and ultimate life table: x l [x] l [x]+1 l [x]+2 l x+3 x Calculate the APV for this insurance if i = 5%. Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 35 / 36

36 Other terminologies Other terminologies and notations used Expression Actuarial Present Value (APV) basis interest rate (i) benefit amount (b) Expected value of Z Other terms/symbols used Expected Present Value (EPV) Net Single Premium (NSP) single benefit premium assumptions interest per year effective discount rate sum insured (S) death benefit E(Z) Variance of Z Var(Z) V [Z] Lecture: Weeks 6-8 (STT 455) Insurance Benefits Fall Valdez 36 / 36

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