Understanding Reinsurance: Benefits and Modelling

Reinsurance Introduction I. Why does an insurer buy Reinsurance? (For example) in case of exceptional event, i.e. accumulation of many small events or one extreme event that exterminate the time diversification of the Insurer s portfolio (ex. Earthquake) What is reinsurance? Formally, it s a contract between an Insurer and a Reinsurer, upon which the Reinsurer commits to pay his share of a claim against a premium Risk wise, it s a way for the insurer to optimize and homogenize its portfolio of risks. Insurance mathematics VI. lecture

Reinsurance Introduction II. Functions of reinsurance: Risk transfer Insurer can assume greater individual risks than its assets allows Income smoothing predictable by absorption of large losses and reduction of capital needed Surplus relief Insurer s underwritings being limited by solvency margin, reinsurance allows insurer to keep underwriting without increasing its capital Reinsurer s expertise Insurer desires to benefit from the expertise and rating ability of the reinsurer Creating a manageable and profitable portfolio Insurer improves balance and homogeneity of its portfolio by getting rid of peak exposure and reducing volatility Managing cost of capital Reinsurance cost is less than capital cost and more convenient results are more Insurer s Insurance mathematics VI. lecture

Reinsurance Modelling I. Let X the variate of claim payment. Let P the premium of direct insurance, F(x) the distribution function of direct insurance. Then the reinsurance contract can be modelled with ?1reinsurance premium and T measurable function: if the claim payment is x then the direct insurer will pay T(x), the reinsurance partner will pay x-T(x). The name of T(x) is own section or own part. In this chapter we will sign ?0? = ? ?(?) < ? ,?1? = ? ? ?(?) < ? ,?0= ? ?1. The risk of direct insurer will be characterized with ??,??? and the risk of reinsurer will be characterized with ??,??? pair, pair. Insurance mathematics VI. lecture

Reinsurance Modelling II. If we collect premium due exactly the risk then we get: ??? ? = ? ???0? = ???? ? =?0 ???1? = (? ??)?? ? =?1 + Rational assumption is that: 0 ? ? ? ? ?0 It follows: 0 ?0 ? Insurance mathematics VI. lecture

Reinsurance Special cases + then there is no any reinsurance. If ? ? = ? ? ?0 +then the direct insurer does not keep on any risk and If ? ? = 0 ? ?0 of course the whole premium will be transferred to the reinsurer. The direct insurer will get commission from the reinsurer because of treaty administration. The name of this type is fronting . The reason of fronting can be among others when the reinsurance partner does not want to be a direct insurance company but they would like to sell one (or more) special product. Then they are searching a company via which they can realize it. Insurance mathematics VI. lecture

Reinsurance Basic distinctions I. The reinsurance contract can be proportional or non-proportional. Under proportional reinsurance, one or more reinsurers take a stated percentage share of each policy that is covered by the reinsurance agreement. This means that the reinsurer will receive that stated percentage of the premiums and will pay the same percentage of claims. In addition, the reinsurer will pay a "ceding commission" to the insurer to cover the costs incurred by the insurer (marketing, underwriting, claims etc.). Under non-proportional reinsurance the reinsurer only pays out if the total claims suffered by the insurer in a given period exceed a stated amount, which is called the "retention" or "priority". Insurance mathematics VI. lecture

Reinsurance Basic distinctions II. The reinsurance contract can be treaty or facultative. According to treaty reinsurance the reinsurer covers the specified share of more than one insurance policy issued by the ceding company which come within the scope of that contract. Facultative reinsurance is normally purchased by ceding companies for individual risks not covered, or insufficiently covered, by their reinsurance treaties, for amounts in excess of the monetary limits of their reinsurance treaties and for unusual risks. Insurance mathematics VI. lecture

Reinsurance Proportional contracts I. 1. Quota share The direct insurer transmits the 1-q part of either business line or product, and the q-th part remains at direct insurer. Of course reinsurance partner will reimburse the 1-q-th part of claims to the direct insurer (and some commission for costs of direct insurer). At first let P is the net premium of direct business. It means that ? ? = ? ? Then: ?1= ???1? = ? ?? ?? ? = (1 ?) ??? ? = 1 ? ? Insurance mathematics VI. lecture

Reinsurance Proportional contracts II. It means that regarding net premium the ceded premium is fair. But there are costs (and profit rates) also. Let the total cost loading (need) of direct insurer is ?1, the total cost loading (need) of reinsurer is ?2(?1>?2). Then the gross premium of direct insurance will be as follows: ? = 1 ?1 The ceded premium will be the next: ? 1= (1 ?) 1 ?1 Insurance mathematics VI. lecture

Reinsurance Proportional contracts III. But the ceded premium with cost need of reinsurer would be as follows: ? = (1 ?) 1 1 ?2 It means that the fair reinsurance commission would be as next: ?3= 1 1 = 1 1 ?1 1 1 ?2 Advantages of quota share treaty: - decreasing the fluctuation of profit/loss - decreasing the capital need (direct insurer) - simple administration Disadvantage of quota share treaty: - direct insurer can not select between good and bad risks Insurance mathematics VI. lecture

Reinsurance Proportional contracts IV. 2. Surplus Surplus treaty is similar to quota share, but in this case direct insurer can decide q own part per risk. There are two limits about own part: the reinsurer takes over maximum the c-fold of own part (take over c layer), i.e. 1 ? 1 + ? In the other side ? ? ? where R the maximum of own part, S the sum insured (probable maximum loss). It follows: ? ? 1 + ? 1 ? (1 + ?) ? Insurance mathematics VI. lecture

Reinsurance Proportional contracts V. Example: We suppose that there are 9 layers and the maximum own part is 100. 1000 reinsurers part free choice 200 100 10 1000 100 200 own part Insurance mathematics VI. lecture

Reinsurance Proportional contracts VI. Advantage of surplus treaty: direct insurer can choose more q related to good risks reinsurer can give back premium refund based on its result Disadvantage of surplus treaty: more complicated administration (because of individual register of q-s) Insurance mathematics VI. lecture

Reinsurance Proportional contracts VII. Statement: Let X is a risk and ?1 that the variance of ceded portfolio is ?1 Then there is a q for which ? ? = ? ? is optimal, i.e. ?2?(? ) is minimal. 2> 0 for which 0 ?1 2 ?2(?). We suppose 2, i.e. ?2? ?(?) = ?1 2. Proof: 1 2. Then ?2? ?(?) = ?1 2 ?1 Let ? = 1 ( We sign ?1? = ? ? ? ? 2= (1 ?)2 ?2(?) ?2(?)) If ?1 2= ?2? then with q=0 the statement is true. Insurance mathematics VI. lecture

Reinsurance Proportional contracts VIII. Proof (continued): If ?1 It follows: 2< ?2? then ? ? ? = ? ? ? ?1(?) ?2? ?(?) = ?2? ? ? ?1(?) = 2?? ? = = ? ?? ?1? ? ? ?? ?1? = (((? ? ? ) (1 ?) (?1? ? ?1? )2?? ? = 2 2(1 ?) ? ? ? (?1? ? ?1? 2)?? ? = = ((1 ?)2? ? ? + (?1? ? ?1? = 1 ?2?2? 2 1 ? ? ? ? (?1? ? ?1? ?? ? + ?2(?1? ) Insurance mathematics VI. lecture

Reinsurance Proportional contracts IX. Proof (continued): We saw earlier: ?2? ?(?) = ?1 It follows: ? ? ? (?1? ? ?1? 2= (1 ?)2 ?2(?) 1 2(1 ?)?2(?1? ) ?? ? = For ?1? = ? ? ? ?, it follows: 2 ?2?(?) = ? ? ? ? ? ?? ? = ?? + ?1? ?(?? + ?1(?)2?? ? = + (?1? ? ?1? )2?? ? = = (? ? ? ? 2+ 2? ? ? ? = (?2? ? ? + (?1? ? ?1? )2?? ? = (?1? ? ?1? = ?2?2? + 2? ? ? ? (?1? ? ?1? ?? ? + ?2(?1? ) Insurance mathematics VI. lecture

Reinsurance Proportional contracts X. Proof (continued): From above equations we get: 1 ?2?(?) = ?2?2? + 1 ??2(?1? ) We know that 0 ? < 1 and ?2(?1? ) 0, it follows: ?2(? ? ) ?2?2? If ? ? ?? then the equation will be valid. Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts I. Excess of Loss (XL) 1. In this treaty ? ? = min ?,? ,i.e. ? ? = ?,?? ? ? ?,?? ? > ?, ? 0 Then the reinsurance premium will be as follows: ? ?1= ? ? ? ?? ? = ? ? ? ?? ? + ? ? ? ?? ? = 0 ? ? = ? ? ?? ? + ? ? ?? ? = ??? ? ??? ? 0 ? ? ? If F(x) is absolutely continuous with f(x) probability density function we get: ?1= ? ?(?)?? ? ? ? ?? ? ? Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts II. Example 1.: Let X is continuous uniform distribution on (0,100) and we suppose that exactly 1 claim will happen. (We will calculate just the net premium.) Then the direct premium will be as next: ? = ? ? =0 + 100 If the direct insurer want to keep maximum 80 of the claim then it has to be transferred 20% in case of quota-share treaty. The premium of quota-share treaty as follows: ?1,?= 1 ? ? ? = 0,2 50 = 10 But if the direct insurer will buy XL treaty then the premium will be the next: 100 ? 100?? 80 80 = 50 2 100 100 ?2 2 80? 1 1 ?1,??= 100?? = = 2 100 80 80 It means that XL treaties are relative cheap. Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts III. Example 2.: If there is no limit of direct insurer then it can be worthy to buy XL treaty. Let X is exponential distribution with a parameter. (We will calculate just the net premium.) Then the direct premium will be as next: ? = ? ? =1 ? We suppose that the own part is? .Then the reinsurance premium will be: ? ??? ???? ? ?? ???? =1 ? = ?? ?? ( ?? 1)? ? ? ?? ?1,??= ? ? ? ? ? ? ? ? = 1 ?? ? k 1 ? =1 ? ? ?? ? ? Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts IV. Example 2. (continued): It means that for example k=2 than the reinsurance premium will be just about 14% of net premium. Advantage of XL treaty: Disadvantages of XL treaty: - simple administration - difficult to calculate reinsurance premium; (usually the risk has no known distribution) - protect just against big claims, does not protect against more small losses. Catastrophe XL treaty: There is a special XL treaty: reinsurer will pay when because of one huge insurance event the total claim excess a pre-defined limit. Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts V. 2. Stop Loss The reinsurer will pay if the total claim of one pre-defined period (typically one year) will excess a pre-defined limit or a pre- defined percentage of premium. Let ??(1 ? ?) the claim payment of i-th risk in the pre- defined period, than the contract can be modelled as follows: ? ? ??,?? ??< ? ? ?=1 ?=1 ? ?( ??) = ?=1 ?,?? ?? ? ?=1 Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts VI. or ? ? ??,?? ??< ? ? ? ?=1 ?=1 ? ?( ??) = ?=1 ? ?,?? ?? ? ? ?=1 q signs a claim ratio, D signs a premium income. Statement: If the reinsurance premium is fixed then ?0 0 for which ?0? = ?,?? ? ?0 ?,?? ? > ?0 is optimal, i.e. in case of any ? transformation ?2?(?) ?2?0(?) Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts VII. Proof: We know that ?1= in case of XL treaty. ? ? ?? ? ? Generally ?1(?) is continuous in M and strictly monotone decreasing. If M=0 then ?1= ? and lim ? ?1= 0. Then because of Bolzano theorem ?0that for pre-defined ?1 Than we will get as follows: ? ?0?? ? ?1= ?0 2 2?? ? = ?2?(?) = ? ? ? ? ? ?? ? = ( ? ? ?0) + (?0 ? ? ? 0 0 2)?? ? = 2+2 ? ? ?0(?0 ? ? ? ) + (?0 ? ? ? = ( ? ? ?0 0 Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts VIII. Proof (continued): 2= 2??? + 2( = ? ? ?0 ? ? ?? ? ?0) (?0 ? ? ? + (?0 ? ? ? 0 0 2+ (?0 ? ? ? 2= 2??? 2(?0 ? ? ? = ? ? ?0 0 ?0 2= 2??? (?0 ? ? ? 2??? + = ? ? ?0 ? ? ?0 0 0 2 2??? (?0 ? ? ? + ? ? ?0 ?0 In the first integral ? ? < ? < ?0 (? ? ?0)2 (? ?0)2 The second integral is non-negative that is why: ?0 2 ?2?(?) 2??? (?0 ? ? ? ? ?0 0 Insurance mathematics VI. lecture

Reinsurance Non-proportional contracts IX. Proof (continued): Whereas because of definition ?0? ?,?? ? ?0 we will get the next: ?0 ?0 2??? = 2??? ?0? ?0 ? ?0 0 0 and ?0? ?0,?? ? > ?0, it means: 2??? =0 ?0? ?0 ?0 2 ?0? ?0 It follows: ?2?0(?) = 0 2??? (?0 ? ? ? It means: ?2?(?) ?2?0(?) Insurance mathematics VI. lecture

Reinsurance Comparison Surplus QS XL Cat XL Stop Loss Type proportional proportional non- non- non- proportional proportional proportional Administration easy difficult easy depend on the definition easy Selection no possible no no no Is it a part of reinsurer in each claim yes yes no no no Prem. calc. easy easy difficult subjective difficult Premium proportional proportional cheap relative cheap relative expensive Saving against big claim freq. no no no no yes Saving against cumulated cl. no no no yes uninterested Saving against bad loss ratio no no no no yes Insurance mathematics VI. lecture