Bonus-malus systems and Markov chains

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1 Bonus-malus systems and Markov chains Dutch car insurance bonus-malus system class % increase new class after # claims > New insurants enter at stage 100%. The second column represents the percentage increase / decrease of the basic premium (based on rating factors like price of the car, horsepower of the car, etc.). Definition and basic properties of Markov chains Definition: A stochastic process X = {X t, t } in discrete time is called a (first order) Markov chain if P (X t+1 = j X t = i, X t 1 = i t 1,..., X 0 = i 0 ) = P (X t+1 = j X t = i) for all j, i, i t 1,..., i 0 E, where E is the state space of the Markov chain. The probability p ij (t) = P (X t+1 = j X t = i) is called (one step) transition probability. The matrix P (t) = (p ij (t)) 1

2 is called transition matrix or matrix of transition probabilities. It is a stochastic matrix, since all rows sum to one. If p ij (t) p ij (and therefore P (t) P ), the Markov chain is called homogeneous. The probabilities p (h) ij (t) = P (X t+h = j X t = i) are called h-step transition probabilities, in particular, p ij (t) p ij (t, 1). Similarly, the matrix is called h-step transition matrix. P (h) (t) = (p (h) ij (t)) The distribution of X t+1 is conditionally independent from the past, given the current value X t. The above definition is only valid for discrete valued random variables, i.e. if the state space E is countable. We will only consider discrete Markov chains. Markov chains can be generalised to continuous time and are then called (discrete) Markov processes. Distribution of a Markov chain: Given the transition probabilities p ij and a starting distribution p (0) i of a Markov chain is uniquely determined. = P (X 0 = i), the distribution The h-step transition probabilities are determined by the Chapman-Kolmogorov equations: and therefore P (h) = P h = P... P. P (h+l) = P (h) P (l) For the transition probabilities the Chapman-Kolmogorov equations can be expressed as p (h+l) ij = k E p (h) ik p(l) kj. Based on the h-step transition probabilities, we obtain P (X tn = i n, X tn 1 = i n 1,..., X t1 = i 1, X 0 = i 0 ) = p (0) i 0 p (t 1) i 0 i 1 p (t 2 t 1 ) i 1 i 2... p (tn t n 1) i n 1 i n Example: Bonus-Malus system Bonus-Malus systems can be considered as homogeneous Markov chains with a finite state space E of bonus malus classes. Suppose that the l classes are ordered such that the corresponding premiums are decreasing, i.e. π 1 π 2... π l. The first class is sometimes called super malus class and the last class super bonus class. 2

3 Frequently a Poisson distribution is used to model the transition probabilities within a bonusmalus system. To be more specific, the Poisson distribution describes the number of claims for an individual and the transition probabilities are determined from this claim frequency distribution. class German bonus-malus system (up to 2002) premium new class after # claims rate >

4 4 where 1.. = 0 Transition matrix for the German bonus-malus system (up to 2002) i/j {0}..... {1}... {2}. {3}.... {4, 5,...} 2 {0}..... {1}... {2}. {3}.... {4, 5,...} 3. {0}..... {1}.. {2}. {3}.... {4, 5,...} 4.. {0}..... {1}. {2}. {3}.... {4, 5,...} 5... {0}..... {1}. {2} {3}.... {4, 5,...} {0}..... {1}. {2} {3}... {4, 5,...} {0}.... {1}. {2} {3}... {4, 5,...} {0}... {1}. {2} {3}... {4, 5,...} {0}.. {1}. {2} {3}... {4, 5,...} {0}.. {1} {2} {3}... {4, 5,...} {0}.. {1} {2}. {3}. {4, 5,...} {0}. {1} {2}. {3}. {4, 5,...} {0}. {1}. {2} {3} {4, 5,...} {0}.. {1} {2} {3, 4,...} {0}.. {1} {2} {3, 4,...} {0}... {1} {2, 3,...} {0}.... {1, 2,...} {0}.... {1, 2,...} 2. {k} = p k = λk k! e λ. 3. {k, k + 1,...} = i=k p i Supplementary material (Risk Theory) Summer term 2007

5 1. The classes with premium rate 100% are called bonus classes, the classes with premium rate > 100% are malus classes. 2. In the German bonus-malus system, the classes are called Schadenfreiheitsklassen. 3. Since 2003 Germany has a new bonus-malus system with 23 bonus classes and 3 malus classes (and reversed order as in the Dutch example from above). Stationary distribution of a Markov chain Under regularity conditions on the transition matrix P, the linear systems of equations µ = µp has a unique, strictly non-zero solution µ, where µ = (µ j, j E). This solution µ is called the stationary distribution of the Markov chain, since µ j = lim p (t) ij, t and for any starting distribution. lim t p(0) P t = p (0) P = µ 1. If p (0) = µ, then P p (0) = µ, i.e. the state probabilities are time-constant. This is the reason why µ is called stationary distribution. 2. The stationary distribution can be obtained as µ = (I P + Q) 1 where = (1,..., 1), I is the identity matrix and Q = In a bonus-malus system, the stationary distribution represents the percentage of drivers in the different classes after the bonus-malus system has been run for a long time. 5

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