HEALTH INSURANCE: ACTUARIAL ASPECTS

Transcription

1 HEALTH INSURANCE: ACTUARIAL ASPECTS Ermanno Pitacco University of Trieste (Italy) p. 1/152

2 Agenda 1. The need for health-related insurance covers 2. Products in the area of health insurance 3. Between Life and Non-Life insurance: the actuarial structure of sickness insurance 4. Indexation mechanisms 5. Individual experience rating: some models 6. The (aggregate) longevity risk in lifelong covers p. 2/152

3 1 THE NEED FOR HEALTH-RELATED INSURANCE COVERS 1. Individual flows 2. Aims of health insurance products 3. Risks inherent in the random lifetime p. 3/152

4 1.1 INDIVIDUAL FLOWS The following flows are considered inflows: earned income (wage / salary) pension (+ possible life annuities) outflows: health-related costs medical expenses (medicines, hospitalization, surgery, etc.) expenses related to long-term care loss of income because of disability (caused by sickness or accident) p. 4/152

5 Individual flows (cont d) amount EXPECTED COSTS 0 n x x+n time age Health-related expected costs p. 5/152

6 Individual flows (cont d) amount LIKELY VARIABILITY RANGE EXPECTED COSTS 0 n x x+n time age Health-related expected costs and their variability p. 6/152

7 Individual flows (cont d) amount NATURAL PREMIUMS not applicable EXPECTED COSTS 0 n x x+n time age Health-related expected costs and natural premiums (including safety loading) p. 7/152

8 Individual flows (cont d) amount 0 n x x+n time age start of working period retirement Income profile p. 8/152

9 Individual flows (cont d) amount WHOLE - LIFE LEVEL PREMIUMS NATURAL PREMIUMS 0 n x x + n time age Health-related expected costs and whole-life level premiums p. 9/152

10 Individual flows (cont d) amount TEMPORARY LEVEL PREMIUMS NATURAL PREMIUMS 0 n time x x + n age Health-related expected costs and temporary level premiums p. 10/152

11 Individual flows (cont d) amount TEMPORARY STEP - WISE LEVEL PREMIUMS NATURAL PREMIUMS 0 n x x + n time age Health-related expected costs and temporary st-wise level premiums p. 11/152

12 Individual flows (cont d) Level premiums vs natural premiums, and the reserving process p. 12/152

13 1.2 AIMS OF HEALTH INSURANCE PRODUCTS 1. Rlace random costs with deterministic costs (insurance premiums) risk coverage 2. Limit the consequences of time mismatching between income and health costs pre-funding and risk coverage pre-funding long term products (possibly lifelong) p. 13/152

14 1.3 RISKS INHERENT IN THE RANDOM LIFETIME Random lifetime random duration of income (working period and retirement) health costs premiums?? Randomness in lifetime Possible assessment via probability distribution of the lifetime p. 14/152

15 Risks inherent in the random lifetime (cont d) amount PROBABILITY DISTRIBUTION OF THE RANDOM LIFETIME 0 n x x+n time age RANDOM TIME Probability distribution of the random lifetime p. 15/152

16 Risks inherent in the random lifetime (cont d) 5000 dx SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM age Probability distributions of the random lifetime (Source: ISTAT - Italian Males) p. 16/152

17 Risks inherent in the random lifetime (cont d) Difficulties originated by coexistence of: random fluctuations of numbers of survivors around expected values individual longevity risk and, more critical: systematic deviations of numbers of survivors from expected values, because of uncertainty in future mortality trend aggregate longevity risk p. 17/152

18 2 PRODUCTS IN THE AREA OF HEALTH INSURANCE 1. General aspects 2. Main products p. 18/152

19 2.1 GENERAL ASPECTS Health insurance : in several countries, a large set of insurance products providing benefits in the case of need arising from: accident illness and leading to: loss of income (partial or total, permanent or non-permanent) expenses (hospitalization, medical and surgery expenses, nursery, etc.) p. 19/152

20 General aspects (cont d) Area: health insurance belongs to the area of insurances of the person, which includes life insurance (in a strict sense): benefits are due dending on death and survival only, i.e. on the insured s lifetime health insurance: benefits are due dending on the health status, and relevant economic consequences (and dending on the lifetime as well) other insurances of the person: benefits are due dending on events such as marriage, birth of a child, education and professional training of children, etc. Health insurance (in broad sense) products are usually shared by life and non-life branches dending on national legislation and regulation p. 20/152

21 General aspects (cont d) NON-LIFE LIFE other classifications As Health Insurance Protection Insurances of the person Health insurance in the context of insurances of the person p. 21/152

22 2.2 MAIN PRODUCTS Types of benefits Reimbursement benefit: to meet (totally or partially) health costs, e.g. medical expenses Forfeiture allowance: amounts stated at policy issue, e.g. to provide an income when the insured is prevented by sickness or injury from working annuity lump sum Service benefit: care service, e.g. hospital, CCRC (Continuing Care Retirement Communities), etc. p. 22/152

23 Main products (cont d) Classification of products Accident insurance Sickness insurance Health benefits as riders to a basic life insurance cover Critical Illness (or Dread Disease) insurance Disability annuities Long Term Care insurance Remark In the following (see products listed in Sect. 3.2) we focus on sickness insurance p. 23/152

24 3 BETWEEN LIFE AND NON-LIFE INSURANCE: THE ACTUARIAL STRUCTURE OF SICKNESS INSURANCE 1. Introduction 2. One-year covers 3. Multi-year covers 4. From the basic model to more general models p. 24/152

25 3.1 INTRODUCTION Life insurance aspects mainly concerning medium and long term contracts: disability annuities, LTC insurance, some types of sickness insurance products survival modeling benefits are due in case of life to be on the safe side, survival probabilities should not be underestimated financial issues asset accumulation (backing technical reserves), return to policyholders p. 25/152

26 Introduction (cont d) Non-Life insurance aspects claim frequency concerns all types of covers problems: availability, data format, experience monitoring and experience rating claim size concerns insurance covers providing reimbursement (e.g. medical expenses), and covers in which benefits dend on some health-related parameter, e.g. the degree of disability expenses ascertainment and assessment of claims checking the health status in case of non-necessarily permanent disability p. 26/152

27 Introduction (cont d) Non-life insurance features: claim frequency, claim severity, ascertainment and assessment of claims, etc. Life insurance features: life table, interest rate, indexing, etc. TYPE OF BENEFIT FIXED AMOUNT REIMBURSEMENT DURATION & PREMIUM ARRANGEMENT n YEARS NATURAL PREMIUMS random frequencies random durations random frequencies random amounts random durations ONE YEAR mathematical reserves LEVEL PREMIUMS Life and Non-life aspects in health insurance products p. 27/152

28 3.2 ONE-YEAR COVERS Products 1. medical expense reimbursement 2. forfeiture daily allowance for hospitalization 3. forfeiture daily allowance for short-term disability General features Random number N of claims for the generic insured (N = 0, 1,... ) Insurer s payment: Y j for the j-th claim Total annual payment to the generic insured: S S = { 0 if N = 0 Y 1 + Y Y N if N > 0 p. 28/152

29 One-year covers (cont d) Premium calculation: equivalence principle Net premium Π = E[S] or (to approx take into account timing of payments) where i = interest rate Π = E[S] (1 + i) 1 2 Hypotheses (realistic?) for any N = n, stochastic indendence and identical probability distribution or random variables (r.v.) Y 1, Y 2,...,Y n stochastic indendence of r.v. N, Y 1, Y 2,... Hypotheses factorizing the expectation of S E[S] = E[Y ] E[N] with Y random variable distributed as the Y j s p. 29/152

30 One-year covers (cont d) Statistical estimation Estimate the quantities E[Y ], E[N] (technical basis) Assumption: analogous risks, in terms of amounts (maximum amounts) and exposure time Portfolio of medical expense reimbursement policies data r = number of insured risks m = number of claims in the portfolio y 1, y 2,...,y m = amounts paid average claim amount per claim ȳ = y 1 + y y m m average number of claims per policy ( claim frequency index) φ = m r p. 30/152

31 One-year covers (cont d) estimates: φ E[N], ȳ E[Y ] premium Π = ȳ φ (1 + i) 1 2 Portfolio of forfeiture daily allowance policies data r = number of insured risks m = number of claims in the portfolio g 1, g 2,...,g m = claim lengths in days average length per claim ḡ = g 1 + g g m m average number of claims per policy ( claim frequency index) φ = m r p. 31/152

32 One-year covers (cont d) estimates: φ E[N], ḡ E[Y ] (for a unitary daily allowance) premium (for a daily allowance d) Π = d ḡ φ (1 + i) 1 2 morbidity coefficient = average length of claim per policy ḡ φ = g 1 + g g m r A more general (and realistic) setting allowing for: amounts exposed to risk (annual maximum amounts) exposure time (within 1 observation year) p. 32/152

33 One-year covers (cont d) Risk factors Split a population into risk classes, according to values assumed by risk factors Risk factors objective: physical characteristics of the insured (age, gender, health records, occupation) subjective: personal attitude towards health, which determines the individual demand for medical treatments and, consequently, the application for insurance benefits Incidence of age: see the following Table p. 33/152

34 One-year covers (cont d) Example x 100 φ x x 100 φ x φ = Average number of claims as a function of the age; males (Source: ISTAT) φ = overall average p. 34/152

35 One-year covers (cont d) Premiums Age as a risk factor probability distribution of the random variable S dending on age In particular: estimated values ȳ x, φ x, ḡ x as functions of age x Premiums Π x = ȳ x φ x (1 + i) 1 2 Π x = d ḡ x φ x (1 + i) 1 2 or, considering just the average number of claims as a function of the age Π x = ȳ φ x (1 + i) 1 2 Π x = d ḡ φ x (1 + i) 1 2 p. 35/152

36 One-year covers (cont d) Multiplicative model Assume φ x = φ t x ȳ x = ȳ u x ḡ x = ḡ v x where - quantities φ, ȳ, ḡ do not dend on age - coefficients t x, u x, v x express the age effect (aging coefficients) Practical interest: assuming that the specific age effect does not change throughout time, claim monitoring can be restricted to quantities φ, ȳ, ḡ observed over the whole portfolio more reliable estimates p. 36/152

37 One-year covers (cont d) Example Forfeiture daily allowance (d = 100) Assumptions (ISTAT data, graduated by ANIA): φ x = } {{} } {{ e x } φ t x ḡ x = } {{} e }{{ x } ḡ v x p. 37/152

38 One-year covers (cont d) x φ x ḡ x Π x Average number of claims, average time (days) per claim, equivalence premium p. 38/152

39 3.3 MULTI-YEARS COVERS Premiums Medical expense reimbursement or forfeiture daily allowance Age x at policy issue, term m years Single premium Π x,m = m 1 h=0 hp x (1 + i) h Π x+h with h p x probability, for a person age x, of being alive at age x + h Natural premiums: Π x, Π x+1,...,π x+m 1, with (see table above) Π x < Π x+1 < < Π x+m 1 p. 39/152

40 Multi-year covers (cont d) Single premium in a multiplicative model For example, if Π x = ȳ x φ x (1 + i) 1 2 = ȳ φ u x t x (1 + i) 1 2 then Π x,m = m 1 h=0 hp x (1 + i) h ȳ x+h φ x+h (1 + i) 1 2 = ȳ φ }{{} K (ind. of age) = K m 1 h=0 w x,h m 1 h=0 hp x (1 + i) h 1 2 u x+h t x+h }{{} w x,h (dendent on age) = K π x,m p. 40/152

41 Multi-year covers (cont d) Annual level premium (payable for m years) we have P x,m = P x,m = Π x,m ä x:m m 1 h=0 hp x (1 + i) h Π x+h m 1 h=0 hp x (1 + i) h thus: annual level premium = arithmetic weighted average of the natural premiums Consequence: mathematical reserve Annual level premiums vs natural premiums, and mathematical reserve p. 41/152

42 Multi-year covers (cont d) Example Hospitalization daily benefit Data: SIM1992; i = 0.03; d = 100; φ x, ḡ x as above x m = 5 m = 10 m = 15 m = Single premiums p. 42/152

43 Multi-year covers (cont d) x m = 5 m = 10 m = 15 m = Annual level premiums p. 43/152

44 Multi-year covers (cont d) Natural premium Level premium Natural premiums and annual level premiums; x = 45, m = 15 Natural premiums for various ages at policy issue; m = 15 p. 44/152

45 Multi-year covers (cont d) Reserves Prospective mathematical reserve (or aging reserve, or senescence reserve) with From ( ) we find V t = Π x+t,m t P x,m ä x+t:m t ; t = 0, 1,...,m ( ) V 0 = V m = 0 V t = Π x+t,1 P x,m + 1 p x+t (1+i) 1 (Π x+t+1,m t 1 P x,m ä x+t+1:m t 1 ) and, as Π x+t,1 = Π x+t, we have the recursion V t + P x,m = Π x+t + 1 p x+t (1 + i) 1 V t+1 technical balance in year (t, t + 1) p. 45/152

46 Multi-year covers (cont d) Example Hospitalization daily benefit. Data: as above Reserves for two ages at policy issue; m = 15 Reserves for various policy terms; x = 35 p. 46/152

47 3.4 FROM THE BASIC MODEL TO MORE GENERAL MODELS Basic model: a static approach, under an individual perspective a portfolio (or population) perspective Possible generalizations, in particular allowing for dynamic features: claim frequency and claim cost dynamics at portfolio level individual claim experience longevity dynamics and related consequences in lifelong sickness covers p. 47/152

48 From the basic model to more general models (cont d) Changes in: overall claim frequency overall claim costs Indexation mechanisms [Chapter 4] Basic model: a static approach Individual experience rating Premium adjustments [Chapter 5] (Aggregate) longevity risk In lifelong covers Comparing premium arrangements [Chapter 6] Introducing dynamic aspects p. 48/152

49 4 INDEXATION MECHANISMS 1. Introduction 2. The adjustment model p. 49/152

50 4.1 INTRODUCTION Refer, for example, to medical reimbursement policies Possible changes, at a portfolio level (or population level), in claim frequency average cost per claim (e.g. because of inflation) throughout the policy duration Approaches: 1. change policy conditions, so that the actuarial value of future benefits kes constant throughout time; in particular (a) raise the deductible (if any) (b) lower the maximum amount 2. allow for variations in actuarial values of benefits because of change in claim frequency and / or average cost per claim indexing policy elements (future premiums and / or reserve) to ke the equivalence principle fulfilled p. 50/152

51 Introduction (cont d) In what follows, we focus on approach 2 (assuming increase in the actuarial value of benefits) Refer, for example, to hospitalization benefits Interest in keing constant the purchasing power of the daily allowance; then indexation of benefits need for approach 2 p. 51/152

52 4.2 THE ADJUSTMENT MODEL Actuarial model equivalence at time t (see the definition of the reserve ( )) assume the multiplicative model V t + P x,m ä x+t:m t = Π x+t,m t Π x+t,m t = K π x+t,m t assume that changes only concern the factor K (whilst do not concern the specific effect of age) change in the factor K K (1 + j [K] ) p. 52/152

53 The adjustment model (cont d) example: medical expense reimbursement K = ȳ φ change in the average cost per claim because of inflation K = ȳ φ K (1 + j [K] ) = ȳ (1 + j [K] ) }{{} φ example: hospitalization benefit (daily allowance) K = d ḡ φ change in the daily allowance to ke the purchasing power K = d ḡ φ K (1 + j [K] ) = d (1 + j [K] ) }{{} ḡ φ p. 53/152

54 The adjustment model (cont d) change in the actuarial value Π x+t,m t Π x+t,m t (1 + j [K] ) = K (1 + j [K] ) π x+t,m t new equivalence condition at time t: (V t + P x,m ä x+t:m t )(1 + j [K] ) = Π x+t,m t (1 + j [K] ) ( ) or, in more general terms: V t (1+j [V] )+P x,m (1+j [P] ) ä x+t:m t = Π x+t,m t (1+j [K] ) ( ) with j [V], j [P] fulfilling equation ( ) equivalence condition on the increments: V t j [V] + P x,m j [P] ä x+t:m t = Π x+t,m t j [K] ( ) p. 54/152

55 The adjustment model (cont d) from ( ) we find: and then: j [K] = V t j [V] + P x,m j [P] ä x+t:m t Π x+t,m t j [K] = V t j [V] + P x,m j [P] ä x+t:m t V t + P x,m ä x+t:m t relation among the three adjustment rates: j [K] is the weighted arithmetic mean of j [V], j [P] usually, application of ( ) each year, to express an annual adjustment of the actuarial value of the insured benefits adjustment rates at time t: j [K] t, j [V] t, j [P] t p. 55/152

56 The adjustment model (cont d) in pratice: increase in the reserve (rate j [V] t ) financed by the insurer (profit participation) increase in premiums (rate j [P] t ) paid by the policyholder in general: if j [V] t if j [P] t (because j [K] t < j [K] t j [P] t < j [K] t j [V] t > j [K] t > j [K] t is a weighted arithmetic mean of j [V] t, j [P] t ) Example Medical expense reimbursement policy x = 50, m = 15 annual level premiums payable for the whole policy duration p. 56/152

57 The adjustment model (cont d) t j [K] t j [V] t j [P] t Table 1 - Benefit adjustment maintained via reserve increment only p. 57/152

58 The adjustment model (cont d) t j [K] t j [V] t j [P] t Table 2 - Only premium increment to maintain a given benefit adjustment p. 58/152

59 The adjustment model (cont d) t j [K] t j [V] t j [P] t Table 3 - Premium increment, given the reserve increment, to maintain a chosen benefit adjustment p. 59/152

60 The adjustment model (cont d) Remark Sickness insurance policies (in particular temporary policies) are not accumulation products the mathematical reserve is small (see numerical examples in the previous section), provided that the policy duration is not too long Then: the only increment of the reserve cannot maintain the raise in the actuarial value of future benefits (see Table 1) the raise in the actuarial value of future benefits can be financed by a reasonable increment of future premiums only (see Table 2) p. 60/152

61 5 INDIVIDUAL EXPERIENCE RATING: SOME MODELS see: E. Pitacco (1992), Risk classification and experience rating in sickness insurance, Transactions of the 24th International Congress of Actuaries, Montreal, vol. 3: Introduction 2. The inference model 3. The experience-rating model 4. Some particular rating systems 5. Numerical examples p. 61/152

62 5.1 INTRODUCTION In several countries, many policies provide a one-year cover The insurer is not obliged to renew the policy In the case of (too many) claims no renewal What is better: no cover or higher (experience-based) premium? Ratemaking according to individual characteristics a-priori classification based on observable risk factors (age, current health conditions, profession, gender (?),... ) experience-based classification claim experience providing information, in order to partially rlace risk characteristics which are unobservable at policy issue p. 62/152

63 Introduction (cont d) In this chapter we define: a Bayesian inference model fitting the particular characteristics of sickness insurance (see Sect. 5.2), which in particular provides a straight experience rating model (Sect. 5.3) some practical rating systems (see Sect. 5.4), such as Bonus Malus (BM) and No-Claim Discount (NCD), relying on the inference model p. 63/152

64 5.2 THE INFERENCE MODEL Notation x = insured s age at policy issue, i.e. time 0 m = policy term N x+h = random number of claims between age x + h and x + h + 1, h = 0, 1,...,m 1 N x (k) = time k k 1 h=0 N x+h = cumulated random number of claims up to Θ = random parameter in the probabilistic structure of N x, N x+1,...,n x+m 1 θ = generic outcome of Θ p. 64/152

65 The inference model (cont d) Hypotheses given Θ = θ, the random numbers N x, N x+1,...,n x+m 1 are indendent ( conditional indendence) the probability distribution of N x+h, h = 0, 1,...,m 1, is Poisson with parameter t x+h θ, briefly Pois(t x+h θ): P[N x+h = n Θ = θ] = e t x+hθ (t x+h θ) n ; n = 0, 1,... n! then: E[N x+h Θ = θ] = t x+h θ t x+h expresses the age effect; in practice t x < t x+1 < t x+2 <... p. 65/152

66 The inference model (cont d) the probability distribution of Θ is Gamma with given (positive) parameters α, β, briefly Gamma(α, β) probability density function (pdf) given by with g(θ) = βα Γ(α) θα 1 e β θ E[Θ] = α β Var[Θ] = α β 2 p. 66/152

67 The inference model (cont d) Some results Unconditional distribution of N x+h, h = 0, 1,..., m 1 P[N x+h = n] = = + 0 ( β Γ(α) n! P[N x+h = n Θ = θ] g(θ) dθ ) α t x+h Γ(α + n) ( β t x+h + 1 ) α+n that is, a negative binomial: ( NegBin α, β ) t x+h β t x+h + 1 p. 67/152

68 The inference model (cont d) Then: E[N x+h ] = α β = t x+h E[Θ] Var[N x+h ] = t x+h α ) t x+h + 1 ( ) 2 β t x+h ( β Given Θ = θ, the probability distribution of N x (k) is Remark Pois ( θ ) k t x+h 1 h=1 The expression E[N x+h ] = t x+h E[Θ] for the expected value corresponds to φ x+h = t x+h φ used in Chap. 3 p. 68/152

69 The inference model (cont d) Then, the unconditional distribution of N x (k) is P[N x (k) = n] = = + 0 P[N x (k) = n Θ = θ] g(θ) dθ ( ) α β kh=1 t Γ(α + n) x+h 1 ( ) α+n ; n = 0, 1,... β Γ(α) n! kh=1 + 1 t x+h that is, NegBin α, β kh=1 t x+h 1 β k h=1 t x+h p. 69/152

70 The inference model (cont d) The inference procedure Claim record (k < m) n x, n x+1,...,n x+k 1 Posterior distribution of the parameter Θ: g(θ n x, n x+1,...,n x+k 1 ) g(θ) P[(N x = n x ) (N x+1 = n x+1 ) (N x+k 1 = n x+k 1 ) Θ = θ] e θ (β+ k 1 h=0 t x+h) θ α+ k 1 h=0 n x+h 1 that is, Gamma ( α + k 1 h=0 n x+h, β + ) k 1 h=0 t x+h, with E[Θ n x, n x+1,...,n x+k 1 ] = α + k 1 h=0 n x+h β + k 1 h=0 t x+h p. 70/152

71 The inference model (cont d) Unconditional distribution of N x+j, j k, calculated by using g(θ n x, n x+1,...,n x+k 1 ) (instead of g(θ)) In particular: E[N x+j n x, n x+1,...,n x+k 1 ] = t x+j α + k 1 h=0 n x+h β + k 1 h=0 t x+h ( ) Remark sufficient statistics given by Eq. ( ) credibility formula E[N x+j n x, n x+1,..., n x+k 1 ] = t x+j α β β β + k 1 h=0 t x+h ( k 1 h=0 t x+h, ) k 1 h=0 n x+h + k 1 h=0 n x+h k 1 h=0 t x+h k 1 h=0 t x+h β + k 1 h=0 t x+h }{{} credibility factor z x,k p. 71/152

72 The inference model (cont d) Example 1 Assume: E[N y ] = y (y 20) ( ) Let x = 40 = age at policy issue We find: h E[N 40+h ] Expected number of claims p. 72/152

73 The inference model (cont d) Example 2 Purpose: to determine the t s (useful in inference procedures) We know that E[N y ] = t y E[Θ] Assume y as reference age, and set t y = 1 Then: t y = E[N y] E[N y ] For example, with y = 20 and the assumption ( ) we find the following Table p. 73/152

74 The inference model (cont d) y t y Ageing parameters p. 74/152

75 The inference model (cont d) Example 3 Assume parameters of the gamma distribution: α = 1.1; β = age at policy issue x = 40 We find the following credibility factors: k z x,k Credibility factors p. 75/152

76 The inference model (cont d) We find the following expected values of N 45, dending on the previous claim experience 4 n 40+h E[N 45 n 40,...,n 44 ] h= Expected number of claims according to claim experience p. 76/152

77 5.3 THE EXPERIENCE-RATING MODEL Annual level premium, payable for m years, if no experience rating is adopted m 1 h=0 hp x (1 + i) h Π x+h P = ä x:m where, for a medical expenses insurance cover: Assuming ȳ = 1, we have: Π x+h = ȳ E[N x+h ] (1 + i) 1 2 P = m 1 h=0 hp x (1 + i) h 1 2 E[N x+h ] ä x:m (in line with an experience rating system based on the observed number of claims) p. 77/152

78 The experience-rating model (cont d) In presence of experience rating in principle: in every year different premiums should be determined and charged according to each individual claim record in practice: a too complex premium system would be generated To obtain an applicable premium system, we have to state: times at which premium adjustments may occur the number of different premiums at each adjustment time relationships between claim experience and adjusted premiums See following notation and Figure 1 p. 78/152

79 The experience-rating model (cont d) Notation r = number of premium adjustments k 1,...,k r = times of premium adjustments; k = k 1 if r = 1 ν max = number of premiums in the experience rating system ν = index of premium (ν = 1, 2,...,ν max ) k(ν) = adjustment time at which premium ν may be charged σ(ν) = a set of outcomes of N x (k(ν)): N x (k(ν)) σ(ν) premium ν will be charged (at time k(ν)) q(x, h, n) = P[N x (h) = n] = probability of n claims up to time h s(ν) = q(x, k(ν), n) = probability that premium ν will be n σ(ν) charged (at time k(ν)) P(ν) = amount of premium ν p. 79/152

80 The experience-rating model (cont d) = 2 P(2) n(k) = 0 = 1 P(1) = 3 P(3) n(k) 1 0 k m time Figure 1 An experience-based rating system; 1 adjustment time p. 80/152

81 The experience-rating model (cont d) = 1 P(1) = 2 P(2) n(k 1 ) = 0 = 3 P(3) n(k 1 ) 1 = 4 P(4) n(k 2 ) = 0 = 5 P(5) 1 n(k 2 ) 2 = 6 P(6) n(k 2 ) 3 0 k 1 k 2 m time Figure 2 An experience-based rating system; 2 adjustment times p. 81/152

82 The experience-rating model (cont d) Premiums P(1) = hp x (1 + i) h 1 2 E[Nx+h ] k 1 1 h=0 ä x:k1 ( ) P(ν) = k j+1 1 ν = 2,...,ν max ; h k j p x+kj (1 + i) h kj 1 2 E [N x+h h=k j ä x+kj :k j+1 k j j = 1,...,r, with k r+1 = m n σ(ν) ] (N x (k j ) = n) ( ) p. 82/152

83 The experience-rating model (cont d) Note that: Expected values in ( ) calculated before any specific experience; then E[N x+h ] = t x+h E[Θ] = t x+h α β Conditional expected values in ( ) dend on the specific information provided by the adoption of premium P(ν), i.e. by the set of outcomes of N x (k j ) which imply P(ν). We have: [ E N x+h = n σ(ν) = 1 s(ν) n σ(ν) ] N x (k j ) = n E[N x+h N x (k j ) = n] n σ(ν) q(x, k j, n) n σ(ν) q(x, k j, n) E[N x+h N x (k j ) = n] q(x, k j, n) p. 83/152

84 The experience-rating model (cont d) As N x (k j ) = k j 1 i=0 N x+i, we have (according to ( )): E[N x+h N x (k j ) = n] = t x+h α + n β + k j 1 i=0 t x+i By using the equations above, we can calculate P(1), P(2),..., P(ν max ) experience rating system fully defined p. 84/152

85 5.4 SOME PARTICULAR RATING SYSTEMS Let Π x,m denote the single premium for a m-year insurance cover: Π x,m = m 1 h=0 hp x (1 + i) h Π x+h = m 1 h=0 hp x (1 + i) h 1 2 E[Nh ] It can be proved that the set of premiums P(1), P(2),..., P(ν max ) (see ( ), ( ) in Sect. 5.3) fulfills the equivalence principle, that is ν max ν=1 s(ν) P(ν) ä x+kj :k j+1 k j = Π x,m Now consider the ν max amounts P(1), P(2),..., P(ν max ) p. 85/152

86 Some particular rating systems (cont d) We say that the P(ν) are equivalence premiums if and only if they fulfill the equivalence principle, i.e. ν max s(ν) P(ν) ä x+kj :k j+1 k j = Π x,m ( ) Note that: ν=1 A particular solution of ( ) is given by P(1), P(2),..., P(ν max ) Other particular solutions of ( ) can be found by stating specific relationships among the premiums, e.g. in order to smooth the sequences of premiums implied by the various claim records For example set P(ν) = f ν P(1); solve ( ) with respect to P(1) ν = 2, 3,..., νmax for given f ν s, calculate P(2),..., P(ν max ) p. 86/152

87 Some particular rating systems (cont d) Alternative approach define P as a reference premium (not necessarily charged to the contract, whatever the node) set P(ν) = f ν P; ν = 1, 2,..., ν max solve ( ) with respect to P for given f ν s, calculate P(1), P(2),..., P(ν max ) Any premium system P(1), P(2),..., P(ν max ) (other than P(1), P(2),..., P(ν max )) implies a solidarity effect among insureds p. 87/152

88 Some particular rating systems (cont d) Remarks 1. Note that, when the approach based on the reference premium is adopted, we may find, because of the choice of the reference premium P and the parameters f s, P(1) < P(1) where P(1) is the initial premium in a straight experience-rating model Then the insured is not fully financed throughout the first period, i.e. (0, k 1 ) loss in case of lapses p. 88/152

89 Some particular rating systems (cont d) 2. As regards the mathematical reserve: (a) in the straight experience rating model, the P(ν) s fulfill the equivalence principle in each period, i.e. (0, k 1 ), (k 1, k 2 ),..., then a small reserve required in each period because of the annual increase in natural premiums reserve = 0 at times k 1, k 2,... (b) in other experience rating systems, the P(ν) s only ensure the equivalence over the cover period (0, m) considered as a whole, then a higher reserve may be required in each period reserve 0 at times k 1, k 2,... p. 89/152

90 Some particular rating systems (cont d) NCD systems A no-claim discount (NCD) system can be defined as a solution of ( ) For example (see Figure 3): r = 1 k = time of premium adjustment ν max = 3 P(1) = initial premium P(2) = f 2 P(1); P(3) = P(1) 0 < f 2 < 1 σ(2) = {0}; σ(3) = {1,2,... } p. 90/152

91 Some particular rating systems (cont d) _ = 2 _ P(2) = 2 f P(1) n(k) = 0 _ = 1 P(1) _ = 3 _ P(3) = P(1) n(k) 1 Figure 3 NCD system: example with 1 adjustment time p. 91/152

92 Some particular rating systems (cont d) Another example (see Figure 4): r = 2 k 1, k 2 = times of premium adjustment ν max = 5 P(1) = initial premium P(2) = f 2 P(1); P(3) = P(1); P(4) = f 4 P(1); P(5) = P(1) 0 < f 4 < f 2 < 1 σ(2) = {0}; σ(3) = {1,2,... }; σ(4) = {0}; σ(5) = {1, 2,... } p. 92/152

93 Some particular rating systems (cont d) _ = 2 _ P(2) = 2 f P(1) n(k 1 ) = 0 _ = 4 _ P(4) = 4 f P(1) n(k 2 ) = 0 _ = 1 P(1) _ = 3 _ P(3) = P(1) n(k 1 ) 1 _ = 5 _ P(5) = P(1) n(k 2 ) 1 Figure 4 NCD system: example with 2 adjustment times p. 93/152

94 Some particular rating systems (cont d) BM systems A bonus-malus (BM) system can be defined as a solution of ( ) For example (see Figure 5): r = 1 k = time of premium adjustment ν max = 5 P(1) = initial premium P(2) = f 2 P(1); P(3) = P(1); P(4) = f4 P(1); P(5) = f5 P(1) 0 < f 2 < 1 < f 4 < f 5 σ(2) = {0}; σ(3) = {1}; σ(4) = {2}; σ(5) = {3,4,... } p. 94/152

95 Some particular rating systems (cont d) _ = 2 _ P(2) = 2 f P(1) n(k) = 0 _ = 1 P(1) _ = 3 _ P(3) = P(1) n(k) = 1 _ = 4 _ P(4) = 4 f P(1) n(k) = 2 _ = 5 _ P(5) = 5 f P(1) n(k) 3 Figure 5 BM system: an example p. 95/152

96 Some particular rating systems (cont d) AD systems An advance-discount (AD) system can be defined as a solution of ( ) For example (see Figure 6): r = 1 k = time of premium adjustment ν max = 3 P = reference premium P(1) = P(2) = f P ; P(3) = g P f < g p. 96/152

97 Some particular rating systems (cont d) _ = 2 _ P(2) f P n(k) = 0 _ = 1 _ P(1) f P _ = 3 _ P(3) g P n(k) 1 Figure 6 AD system: example with 1 adjustment time p. 97/152

98 Some particular rating systems (cont d) Another example (see Figure 7): r = 2 k 1, k 2 = times of premium adjustment ν max = 5 P = reference premium P(1) = f 1 P ; P(2) = f 2 P ; P(3) = f 3 P ; P(4) = f 4 P ; P(5) = f 5 P f 4 f 2 = f 1 < f 3 = f 5 σ(2) = {0}; σ(3) = {1,2,... }; σ(4) = {0}; σ(5) = {1, 2,... } p. 98/152

99 Some particular rating systems (cont d) _ = 2 _ P(2) 2 f P n(k 1 ) = 0 _ = 4 _ P(4) 4 f P n(k 2 ) = 0 _ = 1 _ P(1) 1 f P _ = 3 _ P(3) 3 f P n(k 1 ) 1 _ = 5 _ P(5) 5 f P n(k 2 ) 1 Figure 7 AD system: example with 2 adjustment times p. 99/152

100 5.5 NUMERICAL EXAMPLES The following examples are based on: E[N y ] = y (y 20) Ageing coefficients t y given by the previous table Let x = 40 = age at policy issue Parameters of the gamma distribution of Θ: α = 1.1; β = The following arrangements are considered: straight experience rating (Examples 1, 2, 3, 4) NCD (Examples 5, 6, 7) BM (Example 8) AD (Examples 9, 10, 11) p. 100/152

101 Numerical examples (cont d) Example 1 Straight experience rating m = 5 k = 2 (see Figure 1) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 101/152

102 Numerical examples (cont d) Example 2 Straight experience rating m = 5 k = 3 (see Figure 1) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 102/152

103 Numerical examples (cont d) Example 3 Straight experience rating m = 5 k = 3 observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 103/152

104 Numerical examples (cont d) Example 4 Straight experience rating m = 10 k 1 = 3, k 2 = 7 observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 104/152

105 Numerical examples (cont d) Example 5 NCD system m = 5 k = 3 (see Figure 3) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 105/152

106 Numerical examples (cont d) Example 6 NCD system m = 5 k = 3 f 2 = 0.70 (see Figure 3) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 106/152

107 Numerical examples (cont d) Example 7 NCD system m = 10 k 1 = 3, k 2 = 7 f 2 = 0.75; f 4 = 0.60 (see Figure 4) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 107/152

108 Numerical examples (cont d) Example 8 BM system m = 5 k = 3 f 2 = 0.75; f 4 = 1.30; f 5 = 1.60 (see Figure 5) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 108/152

109 Numerical examples (cont d) Example 9 AD system m = 5 k = 2 f = 0.90; g = 1.20 (see Figure 6) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 109/152

110 Numerical examples (cont d) Example 10 AD system m = 5 k = 2 f = 0.80; g = 1.20 (see Figure 6) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 110/152

111 Numerical examples (cont d) Example 11 AD system m = 5 k = 2 (see Figure 6) observed s(ν) = probability time k number of node ν premium P(ν) of charging claims n(k) the premium P(ν) p. 111/152

112 6 THE (AGGREGATE) LONGEVITY RISK IN LIFELONG COVERS see: A. Olivieri, E. Pitacco (2002), Premium systems for post-retirement sickness covers, Belgian Actuarial Bulletin, 2: Available at: 1. Introduction 2. Sickness insurance and longevity risk 3. Loss functions 4. Premium systems 5. The process risk 6. The uncertainty risk 7. Premium loadings p. 112/152

113 6.1 INTRODUCTION Focus on premium systems for lifelong insurance covers providing sickness benefits (viz reimbursement of medical expenses) Causes of risk affecting lifelong sickness covers: (a) random number of claim events in any given insured period (b) random amount (medical expenses refunded) relating to each claim (c) random lifetime of the insured Causes (a) and (b): common to all covers in general insurance safety loading difficulties in lifelong sickness covers because of paucity of data Cause (c): biometric risk, and in particular longevity risk impact related to the premium system adopted p. 113/152

114 Introduction (cont d) Premium systems considered in the following; (1) single premium at retirement age, meeting all expected costs (2) sequence of level premiums (3) sequence of natural premiums (4) mixtures of (1) and (2) upfront premium + sequence of level premiums (5) mixtures of (1) and (3) upfront premium + sequence of premiums proportional to natural premiums In particular: system (1) policyholder s point of view: interesting if a lump sum is available at retirement insurer s point of view: high risk, related to longevity p. 114/152

115 Introduction (cont d) system (3) policyholder s point of view: dramatic increase of premiums at very old ages insurer s point of view: lowest risk related to longevity system (4) an interesting compromise adopted by Continuous Care Retirement Communities (CCRC) advance fee (upfront premium), plus sequence of periodic fees (periodic premiums), possibly adjusted for inflation p. 115/152

116 6.2 SICKNESS INSURANCE AND LONGEVITY RISK Main aspects of mortality trends (a) decrease in annual probabilities of death (b) increasing life expectancy (c) increasing concentration of deaths around the mode of the curve of deaths (rectangularization of the survival curve) (d) shift of the mode of the curve of deaths towards older ages (expansion) Need for projected life tables when living benefits are concerned (in particular benefits provided by health insurance products) Whatever life table is used, future trend is random risk of systematic deviations from expected values p. 116/152

117 Sickness insurance and longevity risk (cont d) Mortality trends at old ages (e.g. beyond age 65) (a) decrease in annual probabilities of death (b) increasing life expectancy (c) absence of concentration of deaths around the mode of the curve of deaths (d) shift of the mode of the curve of deaths towards older ages (expansion) Because of (c) and (d), coexistence of random fluctuations around expected values (individual longevity risk) systematic deviations from expected values (aggregate longevity risk) p. 117/152

118 Sickness insurance and longevity risk (cont d) 6000 dx SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM age Curves of deaths SIM 1881 SIM 1901 SIM 1931 SIM 1951 SIM 1961 SIM 1971 SIM 1981 SIM 1992 SIM 2002 Me[T 65 ] x 25 [T 65 ] x 75 [T 65 ] IQR[T 65 ] Markers of T 65 p. 118/152

119 Sickness insurance and longevity risk (cont d) In the context of living benefits, the possibility of facing the (aggregate) longevity risk is strictly related to the type of benefits; in particular immediate post-retirement life annuity single premium high longevity risk borne by the annuity provider post-retirement sickness benefits possible premium systems including periodic premiums lower longevity risk borne by the insurer p. 119/152

120 6.3 LOSS FUNCTIONS Notation, definitions y = insured s age at policy issue (= retirement age) N = random number of claims from the time of retirement on T y = future lifetime of the insured K y = curtate future lifetime of the insured C h = random payment for the h-th claim T h = random time of payment of the h-th claim Random present value of the payments of the insurer, Y, at the time of retirement (time 0): N Y = C h v T h h=1 where v = 1 1+i = discount factor, i = interest rate p. 120/152

121 Loss functions (cont d) Random present value, Y k+1, at time k of payments in year (k + 1)-th: Hence Y k+1 = h:k T h <k+1 Y = K y k=0 Y k+1 v k link between Y and K y (or T y ) appears Assume: C h v T h k claims are uniformly distributed over each year number of claims and claim costs are indendent claim costs are equally distributed p. 121/152

122 Loss functions (cont d) Let φ y+k = expected number of claims in year (k, k + 1) c y+k = expected payment for each claim in the same year Under the assumptions, the expected present value at time k of payments in year (k + 1)-th is: E[Y k+1 ] = c y+k φ y+k or E[Y k+1 ] = c y+k φ y+k v 1/2 The natural premium is of course P [N] k = E[Y k+1 ] p. 122/152

123 Loss functions (cont d) Loss function definition Let X = random present value at time 0 of premiums Loss function: or Random items in ( ): future lifetime L = random number of claims costs of claims L = Y X K y k=0 Y k+1 v k X ( ) In the following main interest in consequences of the longevity risk instead of ( ), we adopt the following definition: p. 123/152

124 Loss functions (cont d) L = K y k=0 E[Y k+1 ] v k X = K y k=0 P [N] k v k X Mortality assumption Assume for the random variable T 0 the Weibull distribution, with mortality intensity µ(x) = b ( x ) b 1 (a, b > 0) a a Survival function: S(x) = P[T 0 > x] = e (x/a)b Density function ( curve of deaths"): f 0 (x) = ds(x) dx = S(x) µ(x) = b a ( x a) b 1 e (x/a) b p. 124/152

125 Loss functions (cont d) Mode (Lexis point): ξ = a ( b 1 b ) 1/b Expected lifetime: E[T 0 ] = a Γ ( ) 1 b + 1 Variance: Var[T 0 ] = a 2 (Γ ( ) 2 b + 1 ( ( )) ) 2 1 Γ b + 1 where Γ denotes the complete Gamma function p. 125/152

126 6.4 PREMIUM SYSTEMS Whatever the premium system, we adopt the equivalence principle, i.e. E[L] = 0 hence E[X] = E[Y ] Loss function dends on the premium system Single premium Π L = K y k=0 P [N] k v k Π p. 126/152

127 Premium systems (cont d) Lifelong annual premiums, π k paid at time k (k = 0, 1,... ); we have and then L = L = X = K y K y k=0 (P [N] k k=0 K y k=0 P [N] k π k v k π k ) v k v k Π Premiums π k for k = 0, 1,... can be, for example level premiums: π k = π natural premiums: π k = P [N] k... p. 127/152

128 Premium systems (cont d) Mixtures of up-front premium and annual premiums; then X = Π + K y k=0 π k v k Let Π = α E[Y ]; 0 α 1 Equivalence principle fulfilled if K y k=0 π k v k = (1 α) E[Y ] We denote premiums with Π(α) and π k (α) for k = 0, 1,... In particular: α = 1 single premium Π(1) = Π α = 0 premiums π k (α), k = 0, 1,... Π(0) = 0 p. 128/152

129 Premium systems (cont d) Loss function: L(α) = K y k=0 ( P [N] k ) π k (α) v k Π(α) Note that L(α) rresents the loss function in the general case, 0 α 1 p. 129/152

130 6.5 THE PROCESS RISK Portfolio valuations: moments of the loss function For a given survival function S(x) and related probability of death q, the expected value is: E[L(α) S] = + t=1 [ t 1 1q y ( t 1 k=0 ( P [N] k ) π k (α) v k Π(α) Note that, if S(x) is also adopted for premium calculation, the equivalence principle implies E[L(α) S] = 0 )] Variance: Var[L(α) S] = + t=1 ( t 1 t 1 1 q y k=0 ( P [N] k ) π k (α) v k Π(α) ) 2 E[L(α) S] 2 p. 130/152

131 The process risk (cont d) Let S (x) = survival function used to calculate premiums (can in particular coincide with S (x)) Focus on two premium systems Upfront premium + annual premiums proportional to natural premiums; α = quota pertaining to the upfront premium; then: Π(α) = α E[Y S ] π k (α) = (1 α) P [N] k ; k = 0, 1,... Loss function: and then: L 1 (α) = L 1 (α) = α K y k=0 K y k=0 α P [N] k P [N] v k Π(α) k v k E[Y S ] p. 131/152

132 The process risk (cont d) in particular we find: Var[L 1 (α)] α 2 Note that the variance increases with α, i.e. with the amount of the upfront premium no upfront premium paid (α = 0) Var[L 1 (0] S) = 0 balance between expected costs and premiums in each year and absence of mortality / longevity risk for the insurer Upfront premium + annual level premiums; then: Π(α) = α E[Y S ] π k (α) = π(α); k = 0, 1,... Loss function: L 2 (α) = K y k=0 ( P [N] k ) π(α) v k Π(α) p. 132/152

133 The process risk (cont d) Denote with π the annual premium corresponding to α = 0; then π(α) = (1 α) π and hence L 2 (α) = K y k=0 ( P [N] k ) π v k α E(Y S ) π K y k=0 v k We find: E[L 2 (α) S] = [ + t=1 t 1 1q y ( t 1 k=0 ( P [N] k ) π v k α ( E[Y S ] π t 1 k=0 v k ))] p. 133/152

134 The process risk (cont d) Var[L 2 (α) S] = ( + t 1 = t 1 1 q y t=1 ( + [ t=1 k=0 t 1 1q y ( t 1 k=0 ( P [N] k ( P [N] k ) π v k α ( E(Y S ) π ( ) π v k α E(Y S ) π t 1 k=0 t 1 k=0 )) 2 v k v k ))]) 2 p. 134/152

135 The process risk (cont d) Moments of the loss function at portfolio level Loss functions at portfolio level, for a portfolio of (initially) N risks: L i (α) = N j=1 L (j) i (α); i = 1, 2 where L (j) i (α) denotes the loss function of the insured j In a portfolio of N homogeneous and (conditionally) indendent risks: E[L i (α) S] = N E[L i (α) S] Var[L i (α) S] = N Var(L i (α) S] p. 135/152

136 The process risk (cont d) Portfolio valuations: riskiness and the portfolio size Let Y = random present value of the benefits at portfolio level Risk index (or coefficient of variation): r = σ[y S] E[Y S] For a portfolio of homogeneous and indendent risks: E[Y S] = N E[Y S] Var[Y S] = N Var[Y S] Hence: r = 1 N σ[y S] E[Y S] riskiness decreases as the portfolio size increases p. 136/152

137 The process risk (cont d) Examples Mortality assumptions: S [min] (x), S [med] (x), S [max] (x) (see the following table) Assume: age at retirement y = 65 expected number of claims in the year of age (x, x + 1) φ x = e x expected cost per claim at age x, c x = c = 1 rate of interest i = 0.03 mortality assumption for premium calculation S = S [med] p. 137/152

138 The process risk (cont d) S [min] (x) S [med] (x) S [max] (x) a b ξ E[T 0 ] Var[T 0 ] Three projected survival functions The following tables show: variance of the individual loss function conditional on S [med] riskiness for portfolio size N = 100 and N = p. 138/152

139 The process risk (cont d) α Var[L 1 (α) S] Var[L 2 (α) S] Variance of the loss function p. 139/152

140 The process risk (cont d) N = 100 N = E[Y S] Var[Y S] r = σ[y S] E[Y S] E[Y S] Var[Y S] r = σ[y S] E[Y S] S [min] (x) S [med] (x) S [max] (x) Riskiness for two portfolio sizes p. 140/152

141 6.6 THE UNCERTAINTY RISK Portfolio valuations: moments of the loss function Assign the probabilities ρ [min], ρ [med], ρ [max] to the survival functions S [min] (x), S [med] (x), S [max] (x) respectively Unconditional expected value and variance of loss function E[L i (α)] = E ρ [E[L i (α) S)) = N E ρ [E[L i (α) S]] = N E[L i (α)]; i = 1, 2 Var[L i (α)] = E ρ [Var[L i (α) S)]] + Var ρ [E[L i (α) S]] = N E ρ [Var[L i (α) S]] + N 2 Var ρ [E[L i (α) S]]; i = 1, 2 }{{}}{{} random fluctuations systematic deviations p. 141/152

142 The uncertainty risk (cont d) If N = N, with N = E ρ[var[l i (α) S]] Var ρ [E[L i (α) S]] ; i = 1, 2 the two terms of the variance are equal Portfolio valuations: riskiness and the portfolio size Risk index: r = σ[y] E[Y] = 1 N E ρ [Var[Y S]] E 2 [Y ] } {{ } diversifiable + Var ρ[e[y S]] E 2 [Y ] }{{} non-diversifiable 1/2 p. 142/152

143 The uncertainty risk (cont d) Examples Assume ρ [min] = 0.2, ρ [med] = 0.6, ρ [max] = 0.2 Other data as in the previous example The following tables show: Expected value, variance and relevant components, in the case of premiums proportional to annual expected costs Expected value, variance and relevant components, in the case of level premiums Expected value, variance and risk index as functions of the portfolio size p. 143/152

144 The uncertainty risk (cont d) ] α E[L 1 (α)] Var[L 1 (α)] E ρ [Var[L 1 (α) S] ] Var ρ [E[L 1 (α) S] N Expected value, variance and relevant components (premiums proportional to annual expected costs) p. 144/152

145 The uncertainty risk (cont d) ] α E[L 2 (α)] Var[L 2 (α)] E ρ [Var[L 2 (α) S] ] Var ρ [E[L 2 (α) S] N Expected value, variance and relevant components (level premiums) p. 145/152