University Of Calgary Department of Mathematics and Statistics

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1 University Of Calgary Department of Mathematics and Statistics Hawkes Seminar May 23, / 46 Some Problems in Insurance and Reinsurance Mohamed Badaoui Department of Electrical Engineering National Polytechnic Institute

2 2 / 46 Risk Process with Investment

3 Introduction 3 / 46 The risk theory in general and ruin probability in particular are considered as part of insurance mathematics, dealing with stochastic models of an insurance business. Good references are Asmussen (2000), Rolski et al (1999) and Embrechts et al (1999).

4 4 / 46 In 1903 Lundberg (Swedish mathematician) introduced in his famous Uppsala thesis a simple model which is capable of describing the basic dynamic of an insurance portfolio. "Lundberg realized that the Poisson processes lie in the heart of non-life insurance... which is similar to the recognition by Bachelier in 1900 that the Brownian motion is the key building block for financial models. Embrechts et al (1999)."

5 4 / 46 In 1903 Lundberg (Swedish mathematician) introduced in his famous Uppsala thesis a simple model which is capable of describing the basic dynamic of an insurance portfolio. "Lundberg realized that the Poisson processes lie in the heart of non-life insurance... which is similar to the recognition by Bachelier in 1900 that the Brownian motion is the key building block for financial models. Embrechts et al (1999)."

6 Introducción Let (Ω, F, P) be a complete probability space. The Cramér-Lundberg process or classical risk process X, is defined as follows: where x is the initial capital c is the premium rate N t X t = x + ct i=1 Y i {N t } is a Poisson process with rate λ The claims {Y i } are iid rvs independent of {N t }, having common distribution function G with G(0) = 0, and ν = E[Y ]. 5 / 46

7 Introducción Let (Ω, F, P) be a complete probability space. The Cramér-Lundberg process or classical risk process X, is defined as follows: where x is the initial capital c is the premium rate N t X t = x + ct i=1 Y i {N t } is a Poisson process with rate λ The claims {Y i } are iid rvs independent of {N t }, having common distribution function G with G(0) = 0, and ν = E[Y ]. 5 / 46

8 Introducción Let (Ω, F, P) be a complete probability space. The Cramér-Lundberg process or classical risk process X, is defined as follows: where x is the initial capital c is the premium rate N t X t = x + ct i=1 Y i {N t } is a Poisson process with rate λ The claims {Y i } are iid rvs independent of {N t }, having common distribution function G with G(0) = 0, and ν = E[Y ]. 5 / 46

9 Introducción Let (Ω, F, P) be a complete probability space. The Cramér-Lundberg process or classical risk process X, is defined as follows: where x is the initial capital c is the premium rate N t X t = x + ct i=1 Y i {N t } is a Poisson process with rate λ The claims {Y i } are iid rvs independent of {N t }, having common distribution function G with G(0) = 0, and ν = E[Y ]. 5 / 46

10 6 / 46 By M Y (r) = E[e ry 1] we denote the moment generating function.

11 7 / 46 A classical risk process {X t } as defined above, is a model for the time evolution of the reserves of an insurance company. A possible measure of risk is the ruin probability. The event that X t ever falls below zero is called Ruin Ruin = {X t < 0, for some t > 0} The ruin time is the time when the process falls below zero for the first time: τ = inf {t, X t < 0} and by convention inf =

12 7 / 46 A classical risk process {X t } as defined above, is a model for the time evolution of the reserves of an insurance company. A possible measure of risk is the ruin probability. The event that X t ever falls below zero is called Ruin Ruin = {X t < 0, for some t > 0} The ruin time is the time when the process falls below zero for the first time: τ = inf {t, X t < 0} and by convention inf =

13 8 / 46 The ruin probability is defined as ψ(x) = P[Ruin X 0 = x] = P[τ < ] It is sometimes convenient to use the survival probability for practical purposes δ(x) = 1 ψ(x) = P[τ = ]

14 The main purpose is to avoid the ruin with probability 1. Remark If c < λν the ruin is unavoidable, that is ψ(x) = 1. If c = λν some deep theory of random walks (see, Mikosch 2000), ruin occurs with probability 1. c > λν is then the obvious condition towards solvency which implies the premium rate: where ρ is the safety loading. c = (1 + ρ)λν 9 / 46

15 The main purpose is to avoid the ruin with probability 1. Remark If c < λν the ruin is unavoidable, that is ψ(x) = 1. If c = λν some deep theory of random walks (see, Mikosch 2000), ruin occurs with probability 1. c > λν is then the obvious condition towards solvency which implies the premium rate: where ρ is the safety loading. c = (1 + ρ)λν 9 / 46

16 The main purpose is to avoid the ruin with probability 1. Remark If c < λν the ruin is unavoidable, that is ψ(x) = 1. If c = λν some deep theory of random walks (see, Mikosch 2000), ruin occurs with probability 1. c > λν is then the obvious condition towards solvency which implies the premium rate: where ρ is the safety loading. c = (1 + ρ)λν 9 / 46

17 10 / 46 The condition c > λν is also called the net profit condition, which means that, per unit of time the premium income exceeds the expected aggregate claim amount.

18 11 / 46 Theorem The survival probability δ(x) is continuous and differentiable everywhere except for the countable set, where G is not continuous. Moreover δ(x) satisfies the following integro-differential equation: [ x ] cδ (x) = λ δ(x) δ(x y) dg(y) 0 (1)

19 Remark Integrating (1) from 0 to z we get: δ(z) = δ(0) + λ c z 0 δ(z y)(1 G(y)) dy Since δ(0) and ψ(0) are still unknown, then from the above equation we get: δ(0) = 1 λν c λν and ψ(0) = c (2) 12 / 46

20 Remark Integrating (1) from 0 to z we get: δ(z) = δ(0) + λ c z 0 δ(z y)(1 G(y)) dy Since δ(0) and ψ(0) are still unknown, then from the above equation we get: δ(0) = 1 λν c λν and ψ(0) = c (2) 12 / 46

21 13 / 46 Example (Lundberg, 1903) Let the claims be Exp(α) distributed. Then by equation (1) we have: x ] cδ (x) = λ [δ(x) e αx δ(y)αe αy dy (3) Differentiating (3) yields δ (x) = and the solution is given by: 0 ( ) λ c α δ (x) δ(x) = A + Be ( λ c α)x

22 14 / 46 Then: δ(x) = 1 λ αc e( λ c α)x and ψ(x) = λ αc e( λ c α)x

23 Remark In general (1) cannot be solved analytically. We can compute the survival probability in (1) numerically 15 / 46

24 Remark In general (1) cannot be solved analytically. We can compute the survival probability in (1) numerically 15 / 46

25 16 / 46 The fact that in general equation (1) doesn t have an explicit solution has motivated seeking upper and lower bounds for the ruin probability. To obtain this upper bound we begin our study by introducing a number R > 0, called The adjustment coefficient or the Lundberg exponent. For the classical risk process, R is defined to be the unique positive root of h(r) = 0 where h(r) = λ (M Y (r) 1) cr

26 17 / 46 Assumption For some quantity 0 < γ, M Y (r) is finite for all r < γ with lim r γ M Y (r) = The assumption above is a technical condition which we require to have a turning point.

27 The following theorem is considered as a connection between the adjustment coefficient and the ruin probability, which gives an upper bound for the ruin probability. This upper bound is attributed to Lundberg. Theorem (Lundberg inequality) ψ(x) e Rx, x 0 where R is the adjustment coefficient. Remark If we start with large capitals the ruin probability is very small, but also we observe that the ruin depends on the adjustment coefficient. 18 / 46

28 19 / 46 For r R such that M Y (r) <,the stochastic process { e rx t h(r)t } t 0 is a martingale.

29 20 / 46 For iid exponential Exp(α) claim sizes ( ) ψ(x) e α λ x c

30 Remark The lower bound for the ruin probability is given by ψ(x) Ce Rx, for more details, (see Asmussen, 2000), where: C = inf z 1 E[e R(Y z) Y > z] 21 / 46

31 22 / 46 The next theorem gives an asymptotic behavior for the ruin probability. Theorem (Cramér, 1930) Assume that the adjustment coefficient R exists and that λ c 0 xe Rx (1 G(x)) dx < Then: lim x ψ(x)erx = c λν λm Y (R) c (4)

32 23 / 46 Even though the Cramér-Lundberg model does not fit the data, it is still used as a skeleton for more general models. If a problem cannot be solved for the Cramér-Lunberg model, then it is very difficult to do it for more realistic models.

33 24 / 46 Some generalizations of the Cramér-Lunberg model were proposed in the following papers: Estimates for the ruin probability of ruin with special emphasis on the probability of large claims, by Embrechts and Veraverbeke Ruin probabilities and overshoots for general Lévy insurance risk processes, by Klüppelberg, Kyprianou and Maller, Risk model based on general compound Hawkes process, by Swishchuk 2017.

34 25 / 46 In this section we assume that the insurance company is allowed to take a reinsurance with a retention level b t also called reinsurance strategy. If a claim is happening at time t then the insurer pays b t Y and the reinsurer pays (1 b t )Y. The premium rate paid to the reinsurer as a compensation for the risk is c c(b). The surplus process satisfies the following equation: t N t Xt b = x + c(b s ) ds b Ti Y i 0 where {T i } are the claim times i=1

35 26 / 46 The ruin time The survival probability { } τ b = inf t, Xt b < 0 δ b (x) = P[τ b = ] The purpose is to find a strategy that maximizes the following objective: δ(x) = sup δ b (x) b [0,1]

36 27 / 46 Let b = inf {b [0, 1] : c(b) > 0}, then δ(x) satisfies the following Hamilton-Jacobi-Bellman equation: [ ] x δ λ b (x) = inf δ(x) δ(x by) dg(y), (5) b ( b,1] c(b) 0 For more details we refer to Schmidli, 2008.

37 28 / 46 By defining the adjustment coefficient as the solution to: inf {λ(m Y (br) 1) c(b)r} = 0 b [0,1] and following the same approach described in previous section we get lower bound, upper bound and an asymptotic approximation for the ruin probability.

38 29 / 46 Gerber (1970) added a Brownian motion W t to the classical risk model to: N t X t = x + ct Y i + σw t Let R the solution to: i=1 λ(m Y (r) 1) cr + σ2 2 r 2 = 0 and the Cramer-Lunberg approximation becomes: lim x ψ(x)erx = c λν λm Y (R) c + σ2 R

39 30 / 46 In this section the insurer has the possibility to invest in a risky asset described by a geometric Brownian motion ds t = S t (µdt + σdw t ), (6) µ, σ R are fixed constant. W is a standard Brownian motion assumed to be N t independent of the process Y i. i=1 F = (F t ) t 0 is the filtration generated by the processes N t Y i and S t. i=1

40 31 / 46 We will denote by K = {K t, t 0} the investment strategy of the insurer in each time period t in the risky asset, and by K the set of all admissible adapted strategies, i.e., K t is a non-anticipative function and satisfies, for any T, T 0 K 2 t dt <, a.s. (7)

41 32 / 46 Let Xt K := X(t, x, K ) be the wealth of the insurer at time t if he chooses the admissible strategy K to invest in the risky asset, then the process X K satisfies the following: X K t = x + t 0 t (c + µk s ) ds + 0 N t σk s dw s Y i (8) i=1

42 The time of ruin is defined as { } τ(x, K ) = inf t : X K < 0, The ruin probability is given by ψ(x, K ) = P [τ(x, K ) < ]. The problem of the insurer consists in minimizing the ruin probability, that is solving ψ(x) := inf ψ(x, K ), K K and finding an optimal strategy K K such that ψ(x) = ψ(x, K ). 33 / 46

43 34 / 46 Hipp and Plum 2000 Obtained the Hamilton-Jacobi-Bellman equation for the optimal ruin probability: { } 1 λe[ψ(x Y ) ψ(x)] + inf K R 2 σ2 K 2 ψ (x)+(c+µk )ψ (x) = 0. They proved a verification theorem as well as existence theorem!

44 35 / 46 The optimal strategy of investment is given by: K t = K (X t ) = µ σ 2 ψ (X t ) ψ (X t ). (9)

45 36 / 46 Theorem (Gaier, Grandits and Schachermayer, 2003) The minimal ruin probability ψ(x) of an insurer investing in a risky asset can be bounded from above by ψ(x) e ˆrx, where 0 < ˆr < γ is the positive solution of: λ (M Y (r) 1) = cr + µ2 2σ 2.

46 Definition Let 0 < r < γ be given. We say that Y has a uniform moment in the tail distribution for r if the following condition holds true sup E[e r(z Y ) Y > z] <. z 0 37 / 46

47 38 / 46 Theorem (Gaier, Grandits and Schachermayer, 2003) Assume that Y has a uniform exponential in the tail distribution for ˆr. Then for every K K: ψ(x, K ) Ce ˆrx, where C = inf y 0 y y dg(u). e ˆr(y z) dg(z)

48 Theorem (Hipp and Schmidli, 2004) There exists a constant ξ (0, ) such that lim x ψ(x)eˆrx = ξ. 39 / 46

49 40 / 46 Schmidli 2001 considered the risk process with investment and reinsurance are allowed, the wealth process behaves as follows: X K t = x + t 0 t (c(b s ) + µk s ) ds + 0 N t σk s dw s b Ti Y i i=1

50 41 / 46 The ruin time The survival probability { } τ K,b = inf t, Xt b < 0 δ K,b (x) = P[τ K,b = ] The purpose is to find and strategy that maximize the following objective: δ(x) = sup δ K,b (x) K,b

51 42 / 46 The optimal survival probability satisfies the following HJB equation: { } 1 λe[δ(x by ) δ(x)]+ sup sup 2 σ2 K 2 δ (x)+(c+µk )δ (x) = 0. b [0,1] K R

52 43 / 46 By the defining the adjustment coefficient as the solution to: {λ(m Y (br) 1) (c(b) + µk )r + 12 } σ2 r 2 K 2 = 0 inf b [0,1],K R and following the same approach described in the previous section we get lower bound, upper bound and an asymptotic approximation for the ruin probability.

53 Bibliography 44 / 46 Gaier, J., Grandits, P. & Schachermeyer, W. (2003). Asymptotic ruin probabilities and optimal investment. Ann. Appl. Probab. 13, Gerber, H.U. (1970). An extention of the renewal equation and its application in the collective theory of risk. Skand. Actuar Tidskr. 53, Hipp, C. & Plum, M (2000). Optimal investment for insurers. Insurance: Math. Econom. 27, Hipp, C. & Schmidli, H (2004). Asysmptotics of ruin probabilities for controlled risk processes in the small claims case. Scand. Actuarial J. 27,

54 Bibliography 45 / 46 Schmidli, H. (2002). On minimizing te ruin probability by investment and reinsurance. The annals of applied probability. 12 3, Schmidli, H. (2008). Stochastic control in insurance. Springer

55 46 / 46 THANK YOU FOR YOUR ATTENTION!