Pure risk premiums under deductibles

Transcription

1 Pure risk premiums under deductibles K. Burnecki J. Nowicka-Zagrajek A. Wy lomańska Hugo Steinhaus Center Wroc law University of Technology hugo/

2 Pure risk premiums under deductibles 1 Introduction The idea of a deductible is, firstly, to reduce claim handling costs by excluding coverage for the often numerous small claims and, secondly, to provide some motivation to the insured to prevent claims, throught a limited degree of participation in claim costs. The reasons for introducing deductibles: (i) loss prevention (ii) loss reduction (iii) avoidance of small claims where administration costs are dominant (iv) premium reduction.

3 Pure risk premiums under deductibles 2 General formulae for premiums under deductibles Let X be a risk. A premium calculation principle is a rule saying what premium should be assigned to a given risk. We consider the simplest premium which is called pure risk premium, namely the mean of X. It is often applied in life and some mass lines of business in non-life insurance. The pure risk premium can be of practical use because, for one thing, in practice the planning horizon is always limited and for another, because there are indirect ways of loading a premium.

4 Pure risk premiums under deductibles 3 Let X a non-negative random variable describing the size of claim (assume EX exists), F (t) distribution of X, f(t) probability function, h(x) the payment function corresponding to a deductible. The pure risk premium P is then equal to the expectation (P =EX). We will express formulae for premiums under deductibles in terms of so-called limited expected value function E(X, x) = x 0 yf(y)dy + x(1 F (x)), x > 0.

5 Pure risk premiums under deductibles 4 Franchise deductible One of the deductibles that can be incorporated in the contract is a co-called franchise deductible. Under the franchise deductible of a, if the loss is less than a amount the insurer pays nothing, but if the loss equals or exceeds a amount claim is paid in full.the payment function can be described by h F D(a) (x) = 0, x < a, x, otherwise.

6 Pure risk premiums under deductibles 5 Franchise deductible cont. The pure risk premium under the franchise deductible can be expressed in terms of the premium in the case of no deductible and the corresponding limited expected value function: P F D(a) = P E(X, a) + a(1 F (a)). This premium is a decreasing function of a, when a = 0 the premium is equal to the one in the case of no deductible and if a tends to infinity the premium tends to zero.

7 Pure risk premiums under deductibles 6 Figure 1: The payment function under the franchise deductible (solid line) and no deductible (dashed line). STFded01.xpl

8 Pure risk premiums under deductibles 7 Fixed amount deductible An agreement between the insured and the insurer incorporating a deductible b means that the insurer pays only the part of the claim which exceeds the amount b; if the size of the claim falls below this amount, then the claim is not covered by the contract and the insured receives no indemnification. The payment function is thus given by h F AD(b) (x) = max(0, x b).

9 Pure risk premiums under deductibles 8 Fixed amount deductible cont. The premium in the case of fixed amount deductible has the following form in terms of the premium under the franchise deductible. P F AD(b) = P E(X, b) = P F D(b) b(1 F (b)). The premium is a decreasing function of b, for b = 0 it gives the premium in the case of no deductible and if b tends to infinity, it tends to zero.

10 Pure risk premiums under deductibles 9 Figure 2: The payment function under the fixed amount deductible (solid line) and no deductible (dashed line). STFded02.xpl

11 Pure risk premiums under deductibles 10 Proportional deductible In the case of the proportional deductible of c, where c (0, 1), each payment is reduced by c 100% (the insurer pays 100%(1 c) of the claim). The payment function is given by h P D(c) (x) = (1 c)x. The relation between the premium under the proportional deductible and the premium in the case of no deductible has the following form: P P D(c) = (1 c)ex = (1 c)p. The premium is a decreasing function of c, P P D(0) = P and P P D(1) = 0.

12 Pure risk premiums under deductibles 11 Figure 3: The payment function under the proportional deductible (solid line) and no deductible (dashed line). STFded03.xpl

13 Pure risk premiums under deductibles 12 Limited proportional deductible The proportional deductible is usually combined with a minimum amount deductible so the insurer does not need to handle small claims and with maximum amount deductible to limit the retention of the insured. For the limited proportional deductible of c with minimum amount m 1 and maximum amount m 2 (0 m 1 < m 2 ) the payment function is given by 0, x m 1, x m 1, m 1 < x m 1 /c, h LP D(c,m1,m 2 )(x) = (1 c)x, m 1 /c < x m 2 /c, x m 2, otherwise.

14 Pure risk premiums under deductibles 13 Limited proportional deductible cont. The following formula expresses the premium under the limited proportional deductible in terms of the premium in the case of no deductible and the corresponding limited expected value function { ( P LP D(c,m1,m 2 ) = P E(X, m 1 ) + c E X, m ) ( 1 E X, m )} 2. c c

15 Pure risk premiums under deductibles 14 Figure 4: The payment function under the limited proportional deductible (solid line) and no deductible (dashed line). STFded04.xpl

16 Pure risk premiums under deductibles 15 Disappearing deductible In the case of disappearing deductible the payment depends on the loss in the following way: if the loss is less than an amount of d 1, the insurer pays nothing, if the loss exceeds d 2 (d 2 > d 1 ) amount, the insurer pays the loss in full, if the loss is between d 1 and d 2, then the deductible is reduced linearly between d 1 and d 2. The payment function is thus given by: h DD(d1,d 2 )(x) = 0, x d 1, d 2 (x d 1 ) d 2 d 1, d 1 < x d 2, x, otherwise.

17 Pure risk premiums under deductibles 16 Disappearing deductible cont. The following formula shows the premium under the disappearing deductible in terms of the premium in the case of no deductible and the corresponding limited expected value function P DD(d1,d 2 ) = P + d 1 d 2 d 1 E(X, d 2 ) d 2 d 2 d 1 E(X, d 1 ).

18 Pure risk premiums under deductibles 17 Figure 5: The payment function under the disappearing deductible (solid line) and no deductible (dashed line). STFded05.xpl

19 Pure risk premiums under deductibles 18 Lognormal distribution of loss Consider a random variable Z which has the normal distribution. Let X = e Z. Then the distribution of X is called a lognormal distribution. The distribution function is given by ( ) { ln t µ t 1 F (t) = Φ = exp 1 ( ) } 2 ln y µ dy, σ 2πσy 2 σ 0 where t, σ > 0, µ R and Φ(.) is the standard normal distribution function. We will illustrate the formulae for premium under deductibles using the catastrophe data example. As the example we consider the lognormal loss distribution with parameters µ = and σ =

20 Pure risk premiums under deductibles 19 Figure 6: The premium under the franchise deductible (thick line) and fixed amount deductible (thin line). STFded06.xpl

21 Pure risk premiums under deductibles 20 Figure 7: The thick solid line - the premium for c = 0.2 and m 1 = 10 million, the solid - for c = 0.4 and m 1 = 10 million, the dotted - for c = 0.2 and m 1 = 50 million, and the dashed - for c = 0.4 and m 1 = 50 million. STFded07.xpl

22 Pure risk premiums under deductibles 21 Figure 8: The thick line represents the premium for d 1 = USD 10 million and the thin line the premium for d 1 = USD 50 million. STFded08.xpl

23 Pure risk premiums under deductibles 22 Pareto distribution of loss The Pareto distribution is defined by the formula: F (t) = 1 ( λ λ + t ) α, where t, α, λ > 0. The expectation of the Pareto distribution exist only for α > 1. As the example we consider the PCS data. The analysis showed that the catastrophe-linked losses can be well modelled by the Pareto distribution with parameters α = and λ =

24 Pure risk premiums under deductibles 23 Figure 9: The premium under the franchise deductible (thick line) and fixed amount deductible (thin line). STFded09.xpl

25 Pure risk premiums under deductibles 24 Figure 10: The thick solid line - for c = 0.2 and m 1 = 10 million, the solid - for c = 0.4 and m 1 = 10 million, the dotted - for c = 0.2 and m 1 = 50 million, and the dashed - for c = 0.4 and m 1 = 50 million. STFded10.xpl

26 Pure risk premiums under deductibles 25 Figure 11: The thick line represents the premium for d 1 = USD 10 million and the thin line the premium for d 1 = USD 50 million. STFded11.xpl

27 Pure risk premiums under deductibles 26 Burr distribution of loss Experience has shown that the Pareto formula is often an appropriate model for the claim size distribution, particularly where exceptionally large claims may occur. However, there is sometimes a need to find heavy tailed distributions which offer greater flexibility than the Pareto law. Such flexibility is provided by the Burr distribution which distribution function is given by F (t) = 1 ( λ λ + t τ ) α, where t, α, λ, τ > 0. Its mean exists only for ατ > 1.

28 Pure risk premiums under deductibles 27 For the Burr distribution with ατ > 1 the following formulae hold: (a) franchise deductible premium P F D(a) = λ 1 τ Γ (α 1/τ) Γ (1 + 1/τ) Γ(α) { 1 B (1 + 1τ, α 1τ, a τ λ + a τ )}, (b) fixed amount deductible premium P F AD(b) = λ 1 τ Γ (α 1/τ) Γ (1 + 1/τ) Γ(α) { 1 B (1 + 1τ, α 1τ, b τ λ + b τ )} b ( λ λ + b τ ) α,

29 Pure risk premiums under deductibles 28 (c) proportional deductible premium P P D(c) = (1 c) λ 1 τ Γ(α 1/τ)Γ(1 + 1/τ), Γ(α) (d) limited proportional deductible premium { P LP D(c,m1,m 2 ) = λ 1 τ Γ (α 1/τ) Γ (1 + 1/τ) Γ(α) ( 1 B τ, α 1 τ, m τ ) 1 λ + m τ + cb (1 + 1τ, α 1τ, (m 1 /c) τ ) 1 λ + (m 1 /c) τ cb (1 + 1τ, α 1τ, (m 2 /c) τ ) } λ + (m 2 /c) τ ( ) α ( ) α ( ) α λ λ λ m 1 λ + m τ + m 1 1 λ + (m 1 /c) τ m 2 λ + (m 2 /c) τ,

30 Pure risk premiums under deductibles 29 (e) disappearing deductible premium P DD(d1,d 2 ) = λ 1 τ Γ (α 1/τ) Γ (1 + 1/τ) Γ(α) { } d 2 d 1 + d 1 B 1 + 1/τ, α 1/τ, d τ 2/(λ + d τ 2) d 2 d 1 { } d 2 B 1 + 1/τ, α 1/τ, d τ 1/(λ + d τ 1) d 2 d 1 + λ 1 τ Γ (α 1/τ) Γ (1 + 1/τ) Γ(α) d 2 d 1 d 2 d 1 {( λ λ + d τ 2 ) α ( λ λ + d τ 1 ) α }, where the functions Γ( ) and B(,, ) are defined as: Γ(a) = y a 1 e y dy and B(a, b, x) = Γ(a+b) x 0 Γ(a)Γ(b) 0 ya 1 (1 y) b 1 dy.

31 Pure risk premiums under deductibles 30 Weibull distribution of loss Another frequently used analytic claim size distribution is the Weibull distribution which is defined by { ( ) α } t F (t) = 1 exp, β where t, α, β > 0. For the Weibull distribution the following formulae hold: (a) franchise deductible premium ( P F D(a) = βγ α ) [ 1 Γ { ( ) α }] a α,, β (b) fixed amount deductible premium ( P F AD(b) = βγ ) [ { 1 Γ ( ) α }] b α α, β b exp { ( ) α } b, β

32 Pure risk premiums under deductibles 31 (c) proportional deductible premium P P D(c) = (1 c)βγ ( ), α (d) limited proportional deductible premium ( P LP D(c,m1,m 2 ) = βγ ) [ 1 Γ α + βγ ( ) [ { c Γ ( α α, m1 cβ m 1 exp { ( ) α } m1 β + m 1 exp { ( α, m1 β ) α } Γ { ( ) α } m1 cβ ) α }] { ( α, m2 β m 2 exp ) α }] { ( ) α } m2, cβ

33 Pure risk premiums under deductibles 32 (e) disappearing deductible premium )[ P DD(d1,d 2 ) = βγ ( α d 2 d 1 d 2 Γ d 2 d 1 + d 1 Γ { ( ) α } ] α, d1 + d 1d 2 cβ d 2 d 1 where the function Γ(, ) is defined as { ( ) α } α, d2 β [ { ( ) α } d2 exp β exp { ( ) α }] d1, β Γ(a, x) = 1 Γ(a) x 0 y a 1 e y dy.

34 Pure risk premiums under deductibles 33 Gamma distribution of loss The gamma distribution given by F (t) = F (t, α, β) = t 0 1 Γ(α)β α yα 1 e y β dy, for t, α, β > 0 does not have these drawbacks. For the gamma distribution following formulae hold: (a) franchise deductible premium P F D(a) = αβ {1 F (a, α + 1, β)}, (b) fixed amount deductible premium P F AD(b) = αβ {1 F (b, α + 1, β)} b {1 F (b, α, β)},

35 Pure risk premiums under deductibles 34 (c) proportional deductible premium P P D(c) = (1 c)αβ, (d) limited proportional deductible premium + cαβ P LP D(c,m1,m 2 ) = αβ {1 F (m 1, α + 1, β} { F F ( m1 c, α + 1, β ) + m 1 {F (m 1, α, β) F ( m2 c, α + 1, β )} ( m1 c, α, β )} m 2 {1 F ( m2 c, α, β )},

36 Pure risk premiums under deductibles 35 (e) disappearing deductible premium P DD(d1,d 2 ) = αβ d 2 d 1 [ d2 {1 F (d 1, α + 1, β)} d 1 {1 F (d 2, α + 1, β)} ] + d 1d 2 d 2 d 1 {F (d 1, α, β) F (d 2, α, β)}.