Optimal Dividend Policy of A Large Insurance Company with Solvency - PowerPoint PPT Presentation

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Dividend Policy of A Large Insurance Company with Solvency Constraints Zongxia Liang Department of Mathematical Sciences Tsinghua University, Beijing 100084, China zliang@math.tsinghua.edu.cn Joint work with Jicheng Yao Hsu 100 Conference, July 5-7, 2010, Peking University

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Outline Mathematical Models 1 Optimal Control Problem without solvency constraints 2 Optimal Control Problem with solvency constraints 3 Economic and financial explanation 4 8 steps to get solution 5 References 6

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Methods for Making Maximal Profit · · · The insurance company generally takes the following means to earn maximal profit, reduce its risk exposure and improve its security: Proportional reinsurance Controlling dividends payout Controlling bankrupt probability(or solvency) and so on

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Cramér-Lundberg model of cash flows The classical model with no reinsurance, dividend pay-outs

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Cramér-Lundberg model of cash flows The classical model with no reinsurance, dividend pay-outs

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Cramér-Lundberg model of cash flows The classical model with no reinsurance, dividend pay-outs The cash flow (reserve process) r t of the insurance company follows N t � r t = r 0 + pt − U i , i = 1

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Cramér-Lundberg model of cash flows The classical model with no reinsurance, dividend pay-outs The cash flow (reserve process) r t of the insurance company follows N t � r t = r 0 + pt − U i , i = 1 where claims arrive according to a Poisson process N t with intensity ν on (Ω , F , {F t } t ≥ 0 , P ) .

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Cramér-Lundberg model of reserve process U i denotes the size of each claim. Random variables U i are i.i.d. and independent of the Poisson process N t with finite first and second moments given by µ 1 and µ 2 . p = ( 1 + η ) νµ 1 = ( 1 + η ) ν E { U i } is the premium rate and η > 0 denotes the safety loading .

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Diffusion approximation of Cramér-Lundberg model By the central limit theorem, as ν → ∞ , d r t ≈ r 0 + BM ( ηνµ 1 t , νµ 2 t ) .

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Diffusion approximation of Cramér-Lundberg model By the central limit theorem, as ν → ∞ , d r t ≈ r 0 + BM ( ηνµ 1 t , νµ 2 t ) . So we can assume that the cash flow { R t , t ≥ 0 } of insurance company is given by the following diffusion process dR t = µ dt + σ dW t , where the first term " µ t " is the income from insureds and the second term " σ W t " means the company’s risk exposure at any time t .

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making Proportional reinsurance to reduce risk The insurance company gives fraction λ ( 1 − a ( t )) of its income to reinsurance company

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making Proportional reinsurance to reduce risk The insurance company gives fraction λ ( 1 − a ( t )) of its income to reinsurance company As a return, the reinsurance share with the insurance company’s risk exposure σ W t by paying money ( 1 − a ( t )) σ W t to insureds.

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making Proportional reinsurance to reduce risk The insurance company gives fraction λ ( 1 − a ( t )) of its income to reinsurance company As a return, the reinsurance share with the insurance company’s risk exposure σ W t by paying money ( 1 − a ( t )) σ W t to insureds. The cash flow { R t , t ≥ 0 } of the insurance company then becomes dR t = ( µ − ( 1 − a ( t )) λ ) dt + σ a ( t ) dW t , R 0 = x . We generally assume that λ ≥ µ based on real market.

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making dividends payout for the company’s shareholders

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making dividends payout for the company’s shareholders If L t denotes cumulative amount of dividends paid out to the shareholders up to time t ,

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making dividends payout for the company’s shareholders If L t denotes cumulative amount of dividends paid out to the shareholders up to time t ,then the cash flow { R t , t ≥ 0 } of the company is given by dR t = ( µ − ( 1 − a ( t )) λ ) dt + σ a ( t ) dW t − dL t , R 0 = x , (1)

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Making dividends payout for the company’s shareholders If L t denotes cumulative amount of dividends paid out to the shareholders up to time t ,then the cash flow { R t , t ≥ 0 } of the company is given by dR t = ( µ − ( 1 − a ( t )) λ ) dt + σ a ( t ) dW t − dL t , R 0 = x , (1) where 1 − a ( t ) is called the reinsurance fraction at time t , the R 0 = x means that the initial capital is x , the constants µ and λ can be regarded as the safety loadings of the insurer and reinsurer, respectively.

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Control Problem for the model (1) Notations:

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Control Problem for the model (1) Notations: A policy π = { a π ( t ) , L π t } is a pair of non-negative càdlàg F t -adapted processes defined on a filtered probability space (Ω , F , {F t } t ≥ 0 , P )

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Control Problem for the model (1) Notations: A policy π = { a π ( t ) , L π t } is a pair of non-negative càdlàg F t -adapted processes defined on a filtered probability space (Ω , F , {F t } t ≥ 0 , P ) A pair of F t adapted processes π = { a π ( t ) , L π t } is called a admissible policy if 0 ≤ a π ( t ) ≤ 1 and L π is a nonnegative, t non-decreasing, right-continuous with left limits.

Mathematical Models Optimal Control Problem without solvency constraints Optimal Control Problem with solvency constraints Optimal Control Problem for the model (1) Notations: A policy π = { a π ( t ) , L π t } is a pair of non-negative càdlàg F t -adapted processes defined on a filtered probability space (Ω , F , {F t } t ≥ 0 , P ) A pair of F t adapted processes π = { a π ( t ) , L π t } is called a admissible policy if 0 ≤ a π ( t ) ≤ 1 and L π is a nonnegative, t non-decreasing, right-continuous with left limits. Π denotes the whole set of admissible policies.