# Health Insurance Lecture s notes (Syllabus) by Aldona Skučaitė

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1 Health Insurance Lecture s notes (Syllabus) by Aldona Skučaitė

2 2 Course content 1. Peculiarities of Health insurance markets (see [1], [4], [8], [9], p.p , [1]) (a) Preferences and choice of individuals under risk: Utility theory; Allais parado; Prospect theory (b) Comparison of Health care market and "ideal" market (see ([1])) (c) Asymmetric information and its consequences - moral hazard and adverse selection - in insurance markets (see ([8]), ([1])) 2. Health Care Financing (see [2], [7] ) (a) "Ideal" health care system: main goals, tasks and measures (b) Two main systems for financing of health care: public - mandatory and private - voluntary (c) Health care systems in European Union and USA 3. Actuarial Models for Health Insurance (see [5], p.p ; 8-142; ) (a) Multiple state model. Markov chain. Semi Markov model (b) Medical epenses insurance (c) Disability insurance (d) Critical illness insurance (e) Long term care insurance

3 Chapter 1 Health Insurance Markets 1.1 Preferences and choice of individuals under risk Utility Theory Probably, the first time suggested by D. Bernoulli, developed by J. von Neumann and O. Morgenstein, see Theory of Games and Economic Behaviour, 1944). Formal hypothesis and applications may be found, for eample, in [4]. Choices of individuals in riskless situation as well as under risk are eplained here. Let U() - utility function of individual, - wealth (income, assets, etc.), U is supposed to be non decreasing. Suppose that individual must choose between two alternatives - risky choice A, where: { I1, with probability < α < 1; (A) = I 2, with probability 1 α. Corresponding utilities are: for riskless alternative B - U(I ) for risky alternative (A) - Û(A) = αu(i 1) + (1 α)u(i 2 ) According to Formal hypothesis of Utility theory individual s choice will be: { A, if Û(A) > U(I ); B, if U(I ) > Û(A). 3

5 1.2. COMPARISON OF HEALTH CARE MARKET AND "IDEAL" MARKET5 Eperiments show that majority of individuals chooses F 1, that is F 1 F 2 (F 1 dominates (is better than F 2 ). Now, again suppose that individuals to asked to select either investment strategy F 3 or F 4. Again, possible wins (profits) are -, 1 millions or 3 millions and corresponding probabilities are as follows: \$ 1 mil \$ 3 mil \$ F F In this situation majority of individuals chooses F 3, that is F 3 F 4. It is easy to see that such choices contradict Utility theory, because since F 1 F 2 : U(1), 1U() +, 89U(1) +, 1U(3), (1.1.1) while from the F 3 F 4 we get:, 9U() +, 1U(3), 89U() +, 11U(1) (1.1.2) and it is easy to see that inequalities (1.1.1) and (1.1.2) are opposite. There are more similar eperiments which results contradicts Utility theory (see, for eample, [6]). Some methods for improvement of Utility theory are proposed (see [9]), however we will not study these theories in this course. 1.2 Comparison of Health care market and "ideal" market Some concepts from Microeconomy: Ideal market Market imperfections Information asymmetry: Adverse selection Moral hazard

6 6 CHAPTER 1. HEALTH INSURANCE MARKETS Comparison of Health care industry market and ideal market are presented in [1]. In this paper many features of health care industry most of which contradict theory of ideal market are described, for eample: Nature of demand and supply Epected behavior of physician as provider of services Product uncertainty Pricing process Moral hazard If we use Theory of Ideal insurance, then, under assumption that individuals are risk averse and average cost of medical care is m, individuals will prefer insurance for a price m. Actually due to risk avoidance individuals will agree to pay more than m for insurance provided than insurance premium is not "too unfair". Actually due to administrative costs insurance premium must be higher than actuarially fair premium. In such case optimal policy for insured is no longer full insurance. Two main theorems about optimal policy are proved in the paper Theorem. If an insurance company is willing to offer any insurance policy against loss desired by the buyer at a premium which depends only on the policy s actuarial value, then the policy chosen by a risk averting buyer will take the form of 1 percent coverage above the deductible minimum. Proof. Let W - initial wealth, X - possible random loss, I(X) - insurance benefit paid if loss X occurs, P - insurance premium, Y (X) - wealth of individual after loss occurred and insurance benefit was paid. So Y (X) = W P X + I(X). (1.2.1) Let utility function be U(y), where y - wealth. Then individual seek to maimize: E[U(Y (X))]. (1.2.2)

7 1.2. COMPARISON OF HEALTH CARE MARKET AND "IDEAL" MARKET7 Insurance benefit must be nonnegative, so: I(X), X. (1.2.3) If policy is optimal then it must be the best from the set of all policies with the same actuarial value E[I(X)] in the sense of Consider two insurance policies. Initial policy is I(X) with I 1 (X) > and Y 1 (X 1 ) > Y 1 (X 2 ). Suppose δ > is sufficiently small that: and I 1 (X) >, if X 1 X X 1 + δ (1.2.4) Y 1 (X ) < Y 1 (X), if X 2 X X 2 + δ, X 1 X X 1 + δ (1.2.5) Define π 1 - probability that X lies in (X 1, X 1 + δ); and π 2 - probability that loss X lies in (X 2, X 2 + δ). Then we may choose sufficiently small ε >, such that from (1.2.4) and (1.2.5) we get: and I 1 (X) π 2 ε, if X 1 X X 1 + δ (1.2.6) Y 1 (X ) + π 1 ε < Y 1 (X) π 2 ε, if (1.2.7) X 1 X X 1 + δ, and X 2 X X 2 + δ Define new insurance policy I 2 (X), which is: I 1 (X) π 2 ε, [X 1, X 1 + δ]; I 2 () = I 1 (X) + π 1 ε, [X 2, X 2 + δ]; I 1 (X), elsewhere. From (1.2.6) we get that I 2 (X), so (1.2.3) is satisfied. Suppose that f() - is probability density function of X. Then:

8 8 CHAPTER 1. HEALTH INSURANCE MARKETS E[I 2 (X) I 1 (X)] = X1 +δ X 1 [I 2 () I 1 ()]f()d + = ( π 2 ε) X1 +δ X 1 = π 2 επ 1 + π 1 επ 2 =, f()d + π 1 ε X2 +δ X 2 X2 +δ X 2 [I 2 () I 1 ()]f()d f()d so actuarial values of both policies are the same. It is obvious that Y 2 (X) Y 1 (X) = I 2 (X) I 1 (X). From (1.2.7) we get: Y 1 (X ) < Y 2 (X ) < Y 2 (X) < Y 1 (X) if (1.2.8) X 2 X X 2 + δ X 1 X X 1 + δ, because: Y 2 (X ) = Y 1 (X ) + I 2 (X ) I 1 (X ) = Y 1 (X ) + π 1 ε > Y 1 (X ); Y 2 (X) > Y 2 (X ) ; Y 2 (X) = Y 1 (X) + I 2 (X) I 1 (X) = Y 1 (X) π 2 ε < Y 1 (X) Since Y 1 (X) Y 2 (X) = everywhere ecept [X 1, X 1 + δ] and [X 2, X 2 + δ] we have: and E[U(Y 2 (X)) U(Y 1 (X))] = (1.2.9) X1 +δ X 1 From the Mean value theorem we get: [U(Y 2 ()) U(Y 1 ())]f()d + X 2 +δ X 2 [U(Y 2 ()) U(Y 1 ())]f()d. U(Y 2 (X)) U(Y 1 (X)) = U (Y (X))(Y 2 (X) Y 1 (X)) (1.2.1) where Y (X) is between Y 1 (X) and Y 2 (X). So from (1.2.8) we get: = U (Y (X))(I 2 (X) I 1 (X)), Y (X ) < Y (X), if X 2 X X 2 + δ, X 1 X X 1 + δ.

9 1.2. COMPARISON OF HEALTH CARE MARKET AND "IDEAL" MARKET9 Since individuals are risk averse U (y) is decreasing function of y, so U (Y (X )) > U (Y (X)). We may find u such that U (Y (X )) > u, if X 2 X X 2 + δ (1.2.11) U (Y (X)) < u, if X 1 X X 1 + δ. Putting (1.2.1) into (1.2.9), we get: E[U(Y 2 (X)) U(Y 1 (X))] = X1 +δ π 2 ε +π 1 ε X 1 X2 +δ X 2 U (Y ())f()d U (Y ())f()d. And finally from (1.2.11) we obtain: E[U(Y 2 (X)) U(Y 1 (X))] > π 2 εuπ 1 + π 1 εuπ 2 =, so second policy is better than initial policy and risk averse individual will choose the second one. So, the policy I 1 (X) cannot be optimal if one can find two values of X - X 1 and X 2 for which: I(X 1 ) > and Y (X 1 ) > Y (X 2 ). So, suppose that Y min - minimal value of wealth (Y (X)) under optimal policy. Then according to the theorem I(X) = if Y (X) > Y min. So, insurance benefit is not paid until wealth of individual does not reach Y min. This is insurance policy with 1 percent coverage above stated deductible Remark. Alternative proof may be found in [9] Theorem. If insurance company and insured person are both risk averters and there no other costs ecept of coverage of losses then any non trivial Pareto optimal policy I(X) has the property < di < 1. Such type dx of policies is called coinsurance policies.

10 1 CHAPTER 1. HEALTH INSURANCE MARKETS The proof may be found in [1]. Problem of moral hazard is addressed in more detail in [8]. In this paper moral hazard is treated more like rational economical behavior rather than "moral" problem. Some suggestions how to deal with consequences of moral hazard are provided (deductibles and coinsurance). 1.3 Asymmetric information in insurance markets Probably best known work describing the impact of asymmetric information to insurance markets is [1] by M. Rothschild and J. Stiglitz (1976). It is said that insurance policy belong to equilibrium set if under assumption that individuals buy maimum one insurance policy (which must maimize their epected utility): no insurance contract in the equilibrium make negative epected profits there is no contract outside the equilibrium which if offered will make a non negative profit. Authors show that if two groups of individuals eist in the market - the low risk group and the high risk group - then there may be no equilibrium in the market. Moreover they show that eistence of high risk individuals eerts a negative eternality on low risk individuals - there are losses for low risk individuals if high risk individuals are present. However, high risk individuals are not be in a better position than they would be in isolation. For complete proof see [1]. 1.4 Sample eercises 1 Eercise. Suppose that risk averse individual whose utility function is u() is considering whether to insure against random risk or not. Individual may choose not to insure at all; to buy full insurance or to buy partial insurance. Individual s choice must maimize his / her epected utility. { Suppose that initial wealth of individual is w. Random loss is d < w, with probability p; ξ =, 1-p. Show that:

11 1.4. SAMPLE EXERCISES If insurance premium is actuarially fair individual will choose full insurance. 2. If insurance premium is no longer actuarially fair and equal (1+ρ)Eξ), ρ >, then individual will not choose full insurance. Interpret (comment) these situations. 2 Eercise. Individual whose initial wealth is w = 2 and utility function is u() = 4 2 ; 2 faces random loss: ξ U[; 2]. Suppose that insurance premium is actuarially fair but individual decides to insure only against 8% of loss. Calculate his / her epected utility if: 1. Benefit paid is always equal to 8% of loss (proportional insurance) {, ξ < d; 2. Deductible d is set so that insurance benefit is : ξ d, ξ d.. It is insurance with deductible. Deductible d must be such that epected benefit I is 8% of possible loss, that is E[I] =.8E[ξ] = Maimum coverage s is set such that: to be E[I] =.8E[ξ] = 8. { ξ, ξ < s; s, ξ s. Again calculate s Which option is best for individual? Interpret results. 3 Eercise. Suppose that risk { averse individual with initial wealth w = 2 2, p =, 25; may suffer random loss ξ =, 1 p =, 75. Policies of proportional insurance are traded in the market. Under insurance conditions k ( k 1) percent of loss is compensated while insurance premium is calculated: G = (1 + ρ) k E[ξ]. Find optimal (best) insurance policy and ρ =, 2 and utility function of individual is: 1. u() = u() = ln( + 1) In both cases calculate lowest level of ρ for which optimal decision of individual will be not to insure (to accept loss himself). Compare results. Interpret them using concept of risk aversion.

12 12 CHAPTER 1. HEALTH INSURANCE MARKETS 4 Eercise. Suppose that: Individual with utility{ function u() = ln and 2,.25; initial wealth w = 21 may suffer random loss ξ =,.75. Insurance policies (α 1 ; α 2 ) are traded in the market, where α 1 - insurance premium; α 2 - insurance benefit minus insurance premium (see model in [1]. Suppose that due to perfect competition profits of insurers is zero. In the system of coordinates (W 1 ; W 2 ) (W 1 - wealth in the case of no loss; W 2 - wealth after loss) show the point of initial wealth status (without any insurance) and break even line. a) Suppose that all individuals are identical (their utility functions and accident probabilities are the same). Write down the equation of individual utility indifference curve (the one that maimizes individual s utility). Using MsEcel in the system of coordinates (W 1 ; W 2 ) graph break even line, "fair odds" line and utility curve of individual. Eplain choice of individual in this situation. b) Suppose that probability,25 is mean value of accident probabilities in the market. Actually, half (5%) of individuals belong to high risk group (p H =.35), while the other - to low risk group (p L =.15). Insurance companies know the proportion of high and low risk individuals in the market, but are unable to distinguish to which group individual belong. In this situation show the set of contracts which belong to possible separating equilibrium (You are not supposed to eplore whether or not equilibrium eist) and eplain Your decision.

13 Chapter 2 Health Care Financing 2.1 "Ideal" health care system Ehaustive eplanation of features of Ideal health care system is given in [2]. According to World Health Organization health is - "a state of complete physical, mental and social well-being and not merely the absence of disease or infirmity". Main goals of any health care system are - Cost, Quality and Access. Main problems of health care financing, tasks of any health care system and comparison of two main ways of financing - public mandatory and private voluntary - are presented in the paper. 2.2 Health care systems in European Union and USA Main issues: Financing system: public, private or mi of both; share of public and private ependitures in total health ependitures Role of Voluntary health insurance Achievement of Cost, Quality and Access goals See [2], [7] and / or any other reliable source that you will be able to achieve. 13

14 14 CHAPTER 2. HEALTH CARE FINANCING 2.3 Sample eercises 5 Eercise. Select any European country and make short presentation (5-1 min.) about its Health care system.

15 Chapter 3 Actuarial Models for Health Insurance The content of this chapter is based on material from [5]. 3.1 Multiple state models and Markov process The basis of majority actuarial models used for health insurance is Multiple state models. Let I - set of possible states (we will suppose that it is finite set), I = {1, 2,..., N} while J - set of direct transitions, J {(i, j) i j; i, j I}. Suppose that all states are reachable from initial state via direct or indirect transitions. The pair (I, J) - is called Multiple state model. Define S(t) - state of a system at time t and let S() = 1. We will call the process {S(t), t } - time continuous stochastic process any realization of this process - s(t) is called trajectory of the process. Stochastic process {S(t), t } is called time continuous Markov chain if n and any finite sets t < t 1 < < t n < u and i, i 1,..., i n, j, satisfying condition: P r(s(t ) = i,..., S(t n 1 ) = i n 1, S(t n ) = i n, S(u) = j) > (3.1.1) the following relation hold: P r(s(u) = j S(t ) = i, S(t 1 ) = i 1,..., S(t n ) = i n ) = P r(s(u) = j S(t n ) = i n ). (3.1.2) 15

16 16 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE (3.1.2) is called Markov property. Obvious that for any w > u > t n : P (S(u) = j, S(w) = k S(t ) = i,..., S(t n 1 ) = i n 1, S(t n ) = i n ) = P (S(u) = j S(t ) = i,..., S(t n 1 ) = i n 1, S(t n ) = i n ) P (S(w) = k S(t ) = i,..., S(t n 1 ) = i n 1, S(t n ) = i n, S(u) = j) = P (S(u) = j S(t n ) = i n ) P (S(w) = k S(u) = j). In more general case when S(z) = s(z), z t Markov property is defined: τ, u : τ < u and integer numbers i, j, s(z) ( z < τ) such that: P ([S(z) = s(z), z < τ], [S(τ) = i], [S(u) = j]) > the following must hold: P (S(u) = j [S(z) = s(z), z < τ], [S(τ) = i]) (3.1.3) = P (S(u) = j S(τ) = i). (3.1.4) Probabilites (3.1.2), e.g. P r(s(u) = j S(t) = i), t < u and i, j I are called transition probabilities and defined P ij (t, u) = P r(s(u) = j S(t) = i). Obvious: P ij (t, t) = δ ij, t, where δ ij = { if i j; 1 if i = j. If pairs t, u, t < u and i, i, j I probabilities P ij (t, u) depend only on the difference u t, the process is called homogeneous. In most actuarial applications processes are not homogeneous. Obviuos that: P ij (t, u) 1 i, j t u; (3.1.5) P ij (t, u) = 1 i t u. j I Chapman - Kolmogorov equations hold: P ij (t, u) = P ik (t, w)p kj (w, u), (3.1.6) k I

17 3.1. MULTIPLE STATE MODELS AND MARKOV PROCESS 17 where t w u. (3.1.6) is derived using Markov property: P ij (t, u) = P r(s(u) = j S(t) = i) = P r(s(u) = j, S(w) = k S(t) = i) k I = P r(s(w) = k S(t) = i) P r(s(u) = j S(t) = i, S(w) = k) k I = P r(s(w) = k S(t) = i) P r(s(u) = j S(w) = k) k I = P ik (t, w)p kj (w, u). k I Define occupancy probabilities: P ii (t, u) = P r(s(z) = i, z [t; u] S(t) = i). (3.1.7) Obvious that: t w u. P ii (t, u) = P ii (t, w) P ii (w, u), (3.1.8) Classification of states: Absorbing state i - P ii (t, u) = 1, t u; Transient state i - P ii (t, ) =, t ; Strictly transient state i - P ii (t, u) = P ii (t, u) < 1, t u. Define intensity of transition µ ij (t): µ ij (t) = lim u t P ij (t, u) u t. (3.1.9) For homogeneous Markov process: µ ij (t) = lim u t P ij (t, u) u t = µ ij.

18 18 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE We will define: µ i (t) = j:j i µ ij (t). (3.1.1) Then: µ i (t) = j:j i lim u t P ij (t, u) u t 1 P ii (t, u) = lim. u t u t = lim u t j:j i P ij(t, u) u t Kolmogorov forward differential equations: d dt P ij(z, t) = k:k j P ik (z, t)µ kj (t) P ij (z, t)µ j (t), (3.1.11) where t, z, z t - any time moments and i, j - possible states of model; boundary condition P ij (z, z) = δ ij. Obviuos that (3.1.11) may be presented: dp ij (z, t) = P ik (z, t)µ kj (t)dt P ij (z, t)µ j (t)dt. (3.1.12) k:k j Derivation of (3.1.11). Using (3.1.6) we have: P ij (z, t + t) = P ik (z, t)p kj (t, t + t) + P ij (z, t)p jj (t, t + t) then: k:k j P ij (z, t + t) P ij (z, t) t = k:k j P ik (z, t) P kj(t, t + t) t +P ij (z, t) P jj(t, t + t) 1. t Moreover: so 1 P jj (t, t + t) = P jk (t, t + t), k:k j

19 3.2. SEMI MARKOV PROCESSES AND MARKOV PROCESSES OF IIND ORDER19 P ij (z, t + t) P ij (z, t) t = k:k j P ik (z, t) P kj(t, t + t) P ij (z, t) P jk (t, t + t) t t k:k j Taking lim t + we get (3.1.11), or dp ij (z, t) = P ik (z, t)µ kj (t)dt P ij (z, t)µ j (t)dt. (3.1.13) k:k j Kolmogorov backward differential equations: d dz P ij(z, t) = P ij (z, t)µ i (z) P kj (z, t)µ ik (z) k:k i For derivation see [5]. Occupancy probabilities satisfy: with boundary condition P ii (z, z) = 1. From (3.1.14) we get: d dt P ii(z, t) = P ii (z, t)µ i (t), (3.1.14) d dt ln P ii(z, t) = µ i (t). (3.1.15) And from (3.1.15), using boundary condition P ii (z, z) = 1, we get: P ii (z, t) = ep[ t z µ i (u)du]. 3.2 Semi Markov processes and Markov processes of IInd order See [5]. In most actuarial models for health insurance it is possible to split states and use Markov process instead of semi Markov process, see [5].

20 2 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE 3.3 Actuarial values of premiums and reserves We will suppose that process satisfy Markov property. Define: p i (t) - rate of continuous premium paid at the moment t if S(t) = i, then p i (t)dt - premiums paid during the interval [t, t + dt). b j (t) - rate of annuity benefit paid at moment t if S(t) = j. c ij (t) - single (lump sum) benefit paid at moment t if transition from i to j occurs. d j (t) - single (lump sum) benefit paid at moment t if S(t) = j. { 1, if E - is true; Let: v = e δ and I E =, otherwise. Present value at the moment t of continuous annuity - b j (u) paid during interval [u, u + du) is: So: Y t (u, u + du) = v u t I S(u)=j b j (u)du. Y t (u 1 ; u 2 ) = u2 u 1 And actuarial present values are: v u t I S(u)=j b j (u)du. and E[Y t (u, u + du) S(t) = i] = v u t P ij (t, u)b j (u)du E[Y t (u 1 ; u 2 ) S(t) = i] = If b j (u) 1 then define: a ij (t, n) = t u2 u 1 v u t P ij (t, u)b j (u)du. v u t P ij (t, u). For lump sum benefit c jk (u) we have present value at moment t: Y t (u) = v u t I {S(u )=j S(u)=k}c jk (u)

21 3.3. ACTUARIAL VALUES OF PREMIUMS AND RESERVES 21 and actuarial present value: E[Y t (u) S(t) = i] = v u t P ij (t, u)µ jk (u)c jk (u)du. If c ij (u) = 1 then define: Ā ijk (t, n) = v u t P ij (t, u)µ jk (u)du t Ā i.k (t, n) = Ā ijk (t, n) j:j k Ā ij. (t, n) = Ā ijk (t, n) k:k j For lump sum benefit d j (u) present value at the moment t: and actuarial present value: If d j (u) = 1, then define: Y t (u) = v u t I S(u)=j d j (u), E[Y t (u) S(t) = i] = v u t P ij (t, u)d j (u). Ē ij (t, u) = v u t P ij (t, u). If c jk (u) = 1, u (t, n] and d j (m) = 1, then: Ā ij. (t, n, m) = Āij.(t, n) + Ēij(t, m). For n = m we get generalized formula of endowment insurance. Actuarial present value of benefits at the moment t if S(t) = i is: B i (t, n) = + t t v u t ( j v u t ( j = v u t ( u:u t j P ij (t, u)b j (u))du + P ij (t, u)µ jk (u)b jk (u))du + k:k j P ij (t, u)d j (u)). While actuarial present value of premiums is :

22 22 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE P i (t, n) = t v u t ( j P ij (t, u)p j (u))du, Then according to Equivalence principle: P 1 (, n) = B 1 (, n). Prospective reserve at time t (if S(t) = i) is calculated: V i (t) = B i (t, n) P i (t, n) Since S() = 1, then V 1 () =. If only two types of benefits b j (u) and c jk (u) are used then: V i (t) = t t t v u t j v u t j v u t j P ij (t, u)b j (u)du + (3.3.1) P ij (t, u)µ jk (u)c jk (u)du k:k j P ij (t, u)p j (u)du and: d dt V i (t) = δ V i (t) b i (t) + p i (t) j:j i µ ij (t)[c ij (t) + V j (t) V i (t)] (Use backward Kolmogorov differential equations for derivation of above formula). So: or d V i (t) = δ V i (t)dt + p i (t)dt b i (t)dt (3.3.2) j:j i µ ij (t)[c ij (t) + V j (t) V i (t)]dt (3.3.3)

23 3.3. ACTUARIAL VALUES OF PREMIUMS AND RESERVES 23 p i (t)dt = Investment premium { }} { [d V i (t) δ V i (t)dt] + Risk premium { }} { + b i (t) + µ ij (t)[c ij (t) + V j (t) V i (t)]dt j:j i = I + b i (t) + II Discrete time Markov processes Let I - set of possible states, S(t); t =, 1, stochastic process. Process S(t) is called discrete time Markov chain, if n and for any finite sets t < t 1 < < t n < u and i, i 1,..., i n the following holds: P r(s(u) = j S(t ) = i, S(t 1 ) = i 1,..., S(t n 1 = i n 1, S(t n ) = i n ) (3.3.4) = P r(s(u) = j S(t n ) = i n ), if P r(s(t ) = i, S(t 1 ) = i 1,..., S(t n 1 = i n 1, S(t n ) = i n, S(u) = j) >. In such case: P ij (t, u) = k I P ik (t, w)p kj (w, u), where t, w, u - discrete time moments (t w u). Obvious that: P ij (t, u) = k I P ik (t, t + 1)P kj (t + 1, u).

24 24 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE 3.4 Disability insurance Generally - 3 state model: 1st state - "active", 2nd state - "invalid", 3rd - "dead" Hamza notation P 11 (, t) t p aa P 11 (z, t) t z p aa +z P 11 (, t) t p aa P 12 (, t) t p ai µ 12 (t) µ ai +t a 11 (, n) = a aa :n a 11 (, ) = a aa a 22 (t, n) = a ii +t:n t Suppose that age of insured person at start of policy is, S( + t) - state of policy at the moment + t, t >, S() = a. Disability annuity is paid, its present value: Define: Then: Y = v u I {S(+u)=i} du φ(, u) = P (S( + u) = i S() = a) ā ai = E(Y S() = a) = Let Γ - set of policy conditions: where: Γ = [n 1, n 2, f, m, r], v u φ(, u)du (3.4.1) n 1 n 2 - waiting period (measured from start of policy) - end of policy period

25 3.4. DISABILITY INSURANCE 25 f - waiting period (measured from inception of illness) m - maimum annuity payment period (in years from start of payment) r - last moment when annuity is paid Define: φ Γ (, u) - probability that individual who bought insurance at () is ill and is paid annuity at time + u (according to policy conditionsγ). Obviuos: φ Γ (, u) φ(, u). And: For eample: φ [,,,, ] (, u) = φ(, u) φ [,n,,,n] (, u) = { φ(, u), if u < n;, if u n, and: ā ai,γ = ā ai : n = v u φ [,n,,,n] (, u)du = v u φ(, u)du We will suppose that conditions of Markov process are satisfied. Let: tp gh y = P (S(y + t) = h S(y) = g); h = a, i, d; g = a, i, and µ gh y tp gh y t t = lim ; h = a, i, d; g = a, i; h g, tp hh y = P (S(y + u) = h, u [, t] S(y) = h); h = a, i. Define probability: tp ai y (τ) = P (S(y + u) = i, u [t τ, t] S(y) = a); τ t Obviuos that: tp ai y () = t p ai y tp ai y (τ) =, τ t

26 26 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE or: If τ t, then: It is easy to show that: From (3.4.2) get: t τ tp ai y (τ) = up aa y µ ai y+u t u p ii y+udu. (3.4.2) ā ai = tp ai v t dt (3.4.3) t tp ai y = up aa y µ ai y+u t u py+udu. ii (3.4.4) Then from (3.4.3) and (3.4.4) we have: ā ai = t v t up aa After changing integration order: Define: ā ai = Define function: v u up aa µ ai +u t u p ii +ududt. (3.4.5) [ ] µ ai +u v t u t up+udt ii du. (3.4.6) u ā ii y = zpy ii v z dz. ψ(, u, t) = u p aa ψ(, t z, t) = t z p aa Then, for eample, if Γ = [, n,,, ]: µ ai +u t u p ii +u, µ ai +t z z p ii +t z, a ai,γ = = = tp ai t t v t dt + tp ai n ψ(, u, t)v t dudt + (t n)v t dt ψ(, t z, t)v t dzdt + n t n t n ψ(, u, t)v t dudt ψ(, t z, t)v t dzdt.

27 3.4. DISABILITY INSURANCE 27 If insurance premium is paid continuously at the rate P (u), if S(u) = a, then: For constant premium rate: v u up aa P (u)du = ā ai,γ. P = āai,γ. ā aa :m Let Γ = [, n,,, n]. Prospective reserve for active members at time moment + t is: V a +t,γ = t P m t v u up aa +t µ ai +t+ua ii +t+u:n t u du up aa +tv u du, and reserve for disabled members: V i +t,γ = t m t v u up ii +tdu P Eample of approimation formula: From (3.4.6)we have: ā ai = v u up aa up ia +tv u du. [ ] µ ai +u v t u t up ii +udt du u ā ai = Define: h= 1 h+up aa µ ai +h+uv h+u [ t h up ii +h+u vt h u dt]du (3.4.7) h+u Then: a +h+u = t h up ii +h+u vt h u dt. h+u

28 28 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE So: ā ai = h= 1 h+up aa µ ai +h+uv h+u a +h+udu (3.4.8) ā ai v h+1/2 a +h+1/2 h= 1 From Chapman - Kolmogorov equation: h+up aa µ ai +h+udu (3.4.9) and then h+up aa = h p aa up aa +h + h p ai up ia +h So finally: h+up aa h p aa up aa +h ā ai = = = v h+1/2 a +u+1/2 h= v h+1/2 a +u+1/2 h= h= h= 1 1 h+up aa µ ai +h+udu hp aa up aa +hµ ai +h+udu 1 v h+1/2 a +u+1/2 hp aa up aa +hµ ai +h+udu v h+1/2 a +u+1/2 hp aa ω +h. +udu - probability for inception of illness for indi- where ω = 1 vidual aged (). u p aa µ ai 3.5 Critical illness insurance Define mortality probabilities µ ai ; µ ad = µ ad(o) ; µ id(o) ; µ id(d). Sometimes duration of illness r is important, then use probabilities µ id(o),r ; µ id(d),r. Then

29 3.5. CRITICAL ILLNESS INSURANCE 29 tp aa = t p aa = ep [ [ τp ii +u, = ep or, if we ignore illness duration r, : [ tp ii = ep t t (µ ai τ y+u + µ ad(o) y+u ] )du (µ id(o) y+u+r,r + µ id(d) y+u+r,r)dr (µ id(o) y+u ] + µ id(d) y+u )du Stand alone, or additional, benefit If transition a i occurs benefit of 1 is paid, define its present value by Ā (DD),n, then Ā (DD),n = up aa µ ai +uv u du. For continuous premium we have: P,n = Ā(DD),n, ā aa,n Reserve for active members: V a +t,n t = Ā(DD) +t,n t P,n ā aa +t,n t, where: Ā (DD) +t,n t = u tp aa +tµ ai +uv u t du. t In this case reserve in state i is not needed. Suppose that benefit is paid in installments α, β, 1 α β at time moments + u; + u + τ 1 ; + u + τ 1 + τ 2, second and third parts of benefit are paid only if insured person is alive at benefit payment moment. Then: [ ] Ā (DD;α,β),n = up aa µ ai +u αv u + τ1 p ii +u,βv u+τ 1 + τ1 +τ 2 p+u,(1 ii α β)v u+τ 1+τ 2 du, so premium is : P,n = Ā(DD;α,β),n ā aa,n ],

30 3 CHAPTER 3. ACTUARIAL MODELS FOR HEALTH INSURANCE and reserves: V a +t,n t = Ā(DD;α,β) +t,n t P,n ā aa +t,n t V i +t = u+τ1 tp ii +t,t uβv u+τ 1 t + u+τ1 +τ 2 tp ii +t,t u(1 α β)v u+τ 1+τ 2 t ; u t < τ 1 V i +t = u+τ2 tp ii +t,t u(1 α β)v u+τ 2 t ; u + τ 1 t < u + τ 1 + τ Accelerated insurance Suppose that fraction λ 1 of benefit is paid upon diagnosis of illness and rest part - (1 λ) on death of individual (if death occurs during policy period). Then: [ Ā (D+DD;λ),n = up aa µ ad(o) +u v u + µ ai +u(λv u + (1 λ) Define: Ā (DD;λ),n = Ā (D;λ),n = so: If λ = 1, then up aa µ ai up aa [ +uλv u du µ ad(o) +u v u + µ ai +u(1 λ) Ā (D+DD;λ),n Ā (DD;1),n = Ā (D;1),n = = Ā(D;λ),n u + Ā(DD;λ),n up aa µ ai +uv u du up aa µ ad(o) +u Formulas for discrete case may be found in [5]. 3.6 Long term care insurance v u du u ] v u+r rp ii +u(µ id(o) +u+r + µ id(d) +u+r)dr) d ] v u+r rp ii +u(µ id(o) +u+r + µ id(d) +u+r)dr du, Again, multiple state model is used to model LTC insurance policy conditions. For types of cover and calculation of premiums and / or reserves see [5].

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