Optimal control of the surplus in an insurance policy

Journal of the Korean Statistical Society 39 (2010) 431–437

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Optimal control of the surplus in an insurance policy Mi Ock Jeong a , Eui Yong Lee b,∗ a

Research Planning Division, Statistics Research Institute, Daejeon, 302-280, Republic of Korea

b

Department of Statistics, Sookmyung Women’s University, Seoul, 140-742, Republic of Korea

article

info

Article history: Received 28 May 2009 Accepted 29 September 2009 Available online 8 October 2009 AMS 2000 subject classifications: Primary 90B05 Secondary 60J25

abstract A classical continuous time surplus process is modified by adding two actions. If the level of the surplus goes below τ ≥ 0, we increase the level of the surplus up to initial level u > τ by injecting capital to the surplus. Meanwhile, the excess amount of the surplus over V > u is invested continuously to other business. After assigning several costs related to managing the surplus, we obtain the long-run average cost per unit time and illustrate a numerical example to show how to find an optimal investment policy minimizing the cost. © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

Keywords: Continuous time surplus process Long-run average cost Optimal investment policy

1. Introduction In this paper, we introduce a generalized continuous time surplus process where two actions, investment and injection, are added. The level of the surplus is initially at u > 0, thereafter, increases at constant rate c > 0 by incoming premiums. The level of the surplus also decreases jumpwise due to the claims occurring according to a Poisson process of rate λ > 0. The amounts of the claims are assumed to be independent and identically distributed with general distribution function G. If the level of the surplus reaches V > u, the amount over V is continuously invested to other business until the next claim arrives and decreases the level of the surplus to the level below V . Meanwhile, if the level of the surplus goes below τ (0 ≤ τ < u), we inject capital to the surplus and immediately increase the level of the surplus up to u. Here, V can be considered as the enough level of the surplus at which an investment is possible and τ can be considered as the minimum level of the surplus which is required by the regulator. The continuous time surplus process and its properties have been studied by many researchers. The results on the probability of the ruin that the level of the surplus ever goes below 0 in an infinite horizon are well summarized in Klugman, Panjer, and Willmot (2004). Recently, Cai (2004) and Tang (2005) extended the earlier results of the ruin probability to the cases with stochastic and constant interest rates, respectively. Ng, Hailiang, and Zhang (2004) also obtained the ruin probability of the surplus process which had a random discount factor. The first passage time to a certain level in the continuous time surplus process was introduced by Gerber (1990). Thereafter, Gerber and Shiu (1997) obtained the joint distribution of the time to the ruin, the level of the surplus before the ruin and the deficit at the ruin. Wang and Politis (2002) studied the conditional hitting time of the surplus process to an upper barrier under the condition that the ruin had not occurred. Dickson and Willmot (2005) calculated the density of the time to the ruin by an inversion of its Laplace transform. Recently, Oh, Jeong, and Lee (2007) applied the martingale optional sampling theorem to the continuous time surplus process with exponential claims and obtained the expected time from the origin to the point where the level of the surplus

∗

Corresponding author. Tel.: +82 2 710 9436; fax: +82 2 710 9283. E-mail address: [email protected] (E.Y. Lee).

1226-3192/$ – see front matter © 2009 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jkss.2009.09.002

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reaches either 0 or V . They also derived the expected total amount of the surplus during this period. Jeong, Lim, and Lee (2009) made use of the results of Oh et al. (2007) to obtain the long-run average cost per unit time after assigning several management costs to the process and to show how to find the optimal investment policy minimizing the long-run average cost. In this paper, we extend the earlier analyses by assuming that the amount of the surplus over V is continuously invested to other business and that an injection of capital to the surplus is made when the level of the surplus runs short. We also assume that the claims are independent and generally distributed with common distribution function. We assign to the process the cost of handling claims, the cost of the injection, the reward of the investment and the opportunity cost of holding the surplus. In Section 2, we derive some interesting characteristics, such as the expected length of a period between two successive points of injection, the expected total surplus during the period, and the expected level of the surplus prior to an injection. We obtain, in Section 3, the long-run average cost per unit time by making use of the interesting characteristics obtained in Section 2 and the strong Markov property of the surplus process. In Section 4, we finally show through a numerical example how we minimize the long-run average cost. 2. Interesting characteristics Let {U (t ), t ≥ 0} be the ordinary continuous time surplus process without any actions, then U (t ) = U (0) + ct −

N (t ) −

Xi ,

(2.1)

i=1

where c is the premium rate, {N (t ), t ≥ 0} is a Poisson process of rate λ > 0, and Xi denotes the size of a claim whose distribution function is G with mean µ > 0. Let T (x) be the time to the point where U (t ), starting at x(τ < x < V ), either reaches V or goes below τ . Jeong et al. (2009) showed that E [T (y)] = E [T (y + τ )], 0 < y < V − τ , is given by E [T (y)] = E [T (0)]W (y) −

y c

∗ W (y)

(2.2)

with

 V −τ E [T (0)] = where W (y) =

0

(V − τ − t )dW (t ) , cW (V − τ )

n =0

K ∗n (y), K (y) =

λµ

y

Ge (y) and Ge (y) = µ1 0 (1 − G(t ))dt. Here, ∗ denotes the Stieltjes convolution and K ∗n denotes the n-fold recursive Stieltjes convolution of K with K ∗0 being the Heaviside function. Let P (l, x) = Pr {U (T (x)) < τ − l|U (0) = x}, for l > 0, be the probability that the level of the surplus at T (x) is less than τ − l. Jeong et al. (2009) showed that P (l, y) = P (l, y + τ ), 0 < y < V − τ , is given by

∑∞

P (l, y) = P (l, 0)W (y) −

1 c

c

 G ∗ W (y)

(2.3)

with

 V −τ P (l, 0) =

0

{ G(V − τ − t )}dW (t ) , cW (V − τ )

y

where  G(y) = λ 0 {1 − G(t + l)}dt. Let {Z (t ), t ≥ 0} be our generalized continuous time surplus process where two actions, investment and injection, are added. A sample path of {Z (t ), t ≥ 0} is shown in Fig. 1. In the following subsections, we obtain several characteristics of {Z (t ), t ≥ 0} by making use of E [T (x)] and P (l, x), which are interesting by themselves but also needed later to calculate the long-run average cost per unit time. 2.1. Period between two successive points of injection Let T ∗ (x) be the time to the next point of injection, where x is the current level of the surplus. We define as a cycle the period between two successive points of injection, then T ∗ (u) is the length of a cycle. We, in this subsection, obtain E [T ∗ (u)], the expected length of a cycle. Observe that the surplus process eventually either reaches V or goes below τ . Notice that P (0, u) is the probability that the surplus process goes below τ before reaching V and 1 − P (0, u) is the probability that the surplus process reaches V before going below τ . By the strong Markov property of our surplus process, we can have the following relation: E [T ∗ (u)] = E [T (u)] +

[

1

λ

] + E [T ∗ (V − Y )] [1 − P (0, u)] .

(2.4)

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433

Fig. 1. A sample path of surplus process Z (t ).

In other words, E [T ∗ (u)] is at least E [T (u)] and λ1 + E [T ∗ (V − Y )] is added if the surplus process reaches V instead of going below τ . Here, λ1 is the expected time to the next claim after the surplus process reaching V and Y is the amount of the claim having distribution function G. Applying again the strong Markov property to E [T ∗ (V − Y )] and conditioning on Y we can have another relation given by E [T (V − Y )] = ∗

∫



V −τ

E [T (V − y)] +

0

[

1

λ

]



+ E [T (V − Y )] [1 − P (0, V − y)] dG(y). ∗

Solving the above equation for E [T ∗ (V − Y )] gives

 V −τ E [T ∗ (V − Y )] =

1−

0 V −τ 0



E [T (V − y)]dG(y)

[1 − P (0, V − y)]dG(y)

 V −τ

[1 − P (0, V − y)]dG(y) .  V −τ 1 − 0 [1 − P (0, V − y)]dG(y) 1

+

λ

0

(2.5)

Inserting Eq. (2.5) into Eq. (2.4), we obtain the formula of E [T ∗ (u)], where E [T (u)], E [T (V − y)], P (0, u) and P (0, V − y) can be derived from Eq. (2.2) and (2.3) by noting that E [T (x)] = E [T (x − τ )] and P (l, x) = P (l, x − τ ), for τ ≤ x ≤ V . Remark 1. When the amount of a claim is exponentially distributed with mean µ, that is, G(x) = 1 − e−x/µ , for x > 0, we can show that E [T ∗ (u)] is given by E [T ∗ (u)] = E [T (u)] +

[

1

λ

] + E [T ∗ (V − Y )] [1 − P (0, u)] ,

where

µλeθ(V −τ ) (µ + u − τ ) − µλeθ(u−τ ) (µ + V − τ ) + c (V − u) , (µλ − c )(µλeθ(V −τ ) − c ) λ(µλ − c )(V − τ )eθ(V −τ ) + c 2 (1 − eθ (V −τ ) ) E [T ∗ (V − Y )] = and λ(µλ − c )2 eθ(V −τ ) µλeθ(V −τ ) − µλeθ(u−τ ) P (0, u) = µλeθ(V −τ ) − c E [T (u)] =

with θ =

µλ−c . µc

2.2. Total surplus during a cycle In this subsection, we obtain the expected total surplus between two successive injection points. Let M (x) be the expected total area under the surplus process {U (t ), t ≥ 0} during T (x), that is, M (x) = E

T (x)

[∫ 0

]

U (t )dt |U (0) = x .

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By establishing a backward differential equation for M (x), Jeong, Lim and Lee showed that M (y) = M (y +τ ), 0 < y < V −τ , is given by y2 + 2τ y

M (y) = M (0)W (y) −

2c

∗ W (y)

(2.6)

with

 V −τ M (0) =

0

(V − τ − t )2 + 2τ (V − τ − t )dW (t ) . 2cW (V − τ )

Now, let K (u) be the total surplus during a cycle, the period between two successive points of injection. That is, K (u) is the total area under our surplus process {Z (t ), t ≥ 0} in a cycle. By the strong Markov property of {Z (t ), t ≥ 0}, we can show that E [K (u)] satisfies E [K (u)] = M (u) +

[

]

V

+ E [K (V − Y )] [1 − P (0, u)] .

λ

(2.7)

Eq. (2.7) means that E [K (u)] is at least M (u) and Vλ + E [K (V − Y )] is added if the surplus process reaches V before going below τ . Here, Vλ is the expected total surplus to the next claim after the surplus process reaching V and E [K (V − Y )] is the expected total surplus until the next injection starting at V − Y . Applying the strong Markov property again to E [K (V − Y )] and conditioning on Y give E [K (V − Y )] =

V −τ

∫

M (V − y)dG(y) +

V −τ

∫

0

[

V

λ

0

]

+ E [K (V − Y )] [1 − P (0, V − y)]dG(y).

If we solve the above equation for E [K (V − Y )], we have

 V −τ E [K (V − Y )] =

0

1−

 V −τ 0

M (V − y)dG(y)

[1 − P (0, V − y)]dG(y)

+

 V −τ

[1 − P (0, V − y)]dG(y) 0 .  V −τ 1 − 0 [1 − P (0, V − y)]dG(y) V

λ

(2.8)

Inserting Eq. (2.8) into Eq. (2.7), we obtain the formula of E [K (u)], where M (u) and M (V − y) can be driven from Eq. (2.6). Note again that M (x) = M (x − τ ), for τ ≤ x ≤ V . Remark 2. When the amount of a claim is exponentially distributed with mean µ, we can show that E [K (u)] is given by E [K (u)] = M (u) +

[

] + E [K (V − Y )] [1 − P (0, u)] ,

V

λ

where M ( u) =



1

µλ − c

E [K (V − Y )] =

A(µλe 1



1−R

θ(u−τ )

Q +

− c ) − B(e

θ(u−τ )

− 1) − D(u − τ ) −

(u − τ )2 2

 ,

c  R ,

λ

with Q =



  c + e−(V −τ )/µ µ2 − B + B − µD µλ − c µλ µλ  (V − τ )2 2 − Ac + B − D(V − τ − µ) − + µ(V − τ ) − µ , 1

ceθ(V −τ )



A−

B



2

c eθ(V −τ ) − 1



R =

A =



µλeθ(V −τ ) − c 1

µλeθ(V −τ ) − c

,  

B e

θ(V −τ )

µ2 λ(µc + µλτ − c τ ) , (µλ − c )2 µ2 λ D = + τ. µλ − c B =



−1 −



  µ2 λ (V − τ )2 , + τ (V − τ ) − µλ − c 2

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2.3. Level of the surplus prior to an injection We, in this subsection, obtain the expected level of the surplus just before an injection is made, denoted by E [Z ∗ (u)]. Conditioning on whether our surplus process goes below τ before reaching V or reaches V before going below τ , we have the following relation: E [Z ∗ (u)] = P {U (T (u)) < τ } {τ − E [τ − U (T (u))|U (T (u)) < τ ]} + P {U (T (u)) = V }

× {τ − EY [E [τ − U (T (V − Y ))|U (T (V − Y )) < τ ]]}  ∞  ∫ ∞ P (l, V − Y )dl 0 P (l, u)dl − [1 − P (0, u)]EY =τ− , P (0, V − Y ) 0 where we again use the strong Markov property of our surplus process. If we condition on the values of Y , the above equation becomes E [Z (u)] = τ − ∗

∞

∫

P (l, u)dl − [1 − P (0, u)]

∫

=τ−

∞

P (l, u)dl − [1 − P (0, u)]

 y−(V −τ ) 0

V −τ

0

∫

∞

∫

1

1dl

dG(y) +

∞

y − (V − τ )dG(y) +

∞ 0

V −τ

∫

∞ 0

0

P (l, V − y)dl

P (0, V − y)

0

V −τ

0

V −τ

∫

P (l, V − y)dl

P (0, V − y)

 dG(y)

 dG(y) .

(2.9)

Remark 3. When the amount of a claim is exponentially distributed with mean µ, we can show that E [Z ∗ (u)] is given by E [Z ∗ (u)] = τ − µ. The above result can also be driven directly from the fact that the exponential claim has a memoryless property. 3. Long-run average cost per unit time We, in this section, obtain the long-run average cost per unit time after assigning to the surplus process several costs related to managing the surplus. Let c1 be the cost of handling a claim, c2 be the cost of injecting a unit amount to the surplus, c3 be the reward of investing a unit amount of the surplus to the other business, and c4 be the cost of holding a unit amount of the surplus per unit time. Here, the holding cost is a kind of an opportunity cost. Let N be the number of investments during a cycle, the period between two successive points of injection. Applying the strong Markov property of our surplus process and also conditioning on Y , we have the following relations: P {N = 0} = P (0, u), P {N = n} = [1 − P (0, u)] p∗ (1 − p∗ )n−1 , with p∗ =



V −τ 0

n = 1, 2, . . .  P (0, V − y)dG(y) + G(V − τ ) . Here, p∗ is the probability that the surplus process, starting at V − Y , goes

below τ before reaching V again. Hence, the expected value of N is given by E (N ) =

1 − P (0, u) p∗

.

Let S denote the amount of an investment, then the expected value is given by λ

E (S ) = E [E (S |E )] = E

  2  λ c E

2

=

c

λ2

,

where E λ is the exponential random variable of rate λ, since the time to the next claim after the surplus reaching V follows an exponential distribution with mean 1/λ due to the memoryless property. Hence, the expected total amount of investment during a cycle is E (N )E (S ). Let R be the amount of an injection during a cycle, then the expected value is given by E (R) = u − E [Z ∗ (u)]. The foregoings enable us to find the long-run average cost per unit time. By the renewal reward theorem of Ross (1983, p. 78), we finally obtain that C (V ) = c1 λ +

c2 E (R) − c3 E (N )E (S ) + c4 E [K (u)] E [T ∗ (u)]

.

(3.1)

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M.O. Jeong, E.Y. Lee / Journal of the Korean Statistical Society 39 (2010) 431–437

0.4

Long–Run Average Cost

0.2 0 –0.2 –0.4 –0.6 –0.8 –1 –1.2 –1.4 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 V Fig. 2. C (V ) with respect to V . Table 1 Amounts of claims in a medical insurance policy (10,000 wons, per month). Amounts of claims

Average numbers of occurrence

0–10 10–20 20–40 40–60 60–80 80–100 100–200

29.8 22.0 28.3 8.3 7.0 5.5 4.3

Total

105.2

Table 2 Optimal values V ∗ minimizing C (V ). c

V∗

C (V ∗ )

100 110 120 130 140 150

5019 2574 1802 1378 1158 1024

12.4124 1.4081 −1.2885 −4.4738 −7.4934 −10.4377

4. Numerical example We, in this section, illustrate how our model can be applied to real data and how we can obtain the target level of the investment minimizing the long-run average cost per unit time. Table 1 shows the amounts of claims which are grouped into 7 categories and their average numbers of occurrence per month in a medical insurance policy of a commercial insurance company in Korea during year 2008. As a result of the goodness-of-fit test at the level of significance α = 0.05, we can show that the amounts of claims follow an exponential distribution with mean 32.1(p-value = 0.655). The average number of claims per day is calculated to be λ = 3.50. We now assign actual values to the costs related to managing the surplus with consideration of interest rates as follows. The cost of handling a claim, c1 , is assumed to be 0.02, both the cost of injecting a unit, c2 , and the cost of investing a unit, c3 , are assumed to be 1, and the cost of holding a unit per day, c4 , is assumed to be 0.00001. The initial surplus, u, is put to be 1000 and the minimum level of the surplus, τ , is set to be 0. In Fig. 2, C (V ) is illustrated as a function of V when the premium rate, c, is 120. From Fig. 2, we can see that C (V ) is minimized when V is around 1800. The optimal values of V minimizing the long-run average cost per unit time are calculated for various values of c and shown in Table 2. The negative value of C (V ∗ ) means that we make a profit in our policy. From Table 2, we can see that we make a profit only when premium rate c is larger than µλ, the average amount of claims per unit time. We can also see that as c increases we can make more profit in our policy. Acknowledgements This work was supported by the Korea Science and Engineering Foundation grant funded by the Korea government (MOST) (NO. R01-2007-000-20045-0) and by the Sookmyung Women’s University research grants 2009.

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