Which Insurance Policy Is Mean-Variance Efficient with Dependent Background Risk?

何種保險單於存在相依背景風險下具有均異效率性？

汪青萍(Ching-Ping Wang)；黃鴻禧( Hung-Hsi Huang)

背景風險 ； 均異效率前緣 ； 比率保險 ； 自付額保險 ； 上限型保險 ； Background risk ； Mean-variance frontier ； Proportional coinsurance ； Deductible insurance ； Upper-limit insurance

經濟論文

43卷1期（2015 / 03 / 01）

115 - 152

英文

本研究測試當存在相依背景風險下，比率保險、自負額保險、上限型保險等三種普遍存在之保險契約形式的均異效率性。本研究考處被保險人同時面臨隨機的損失與所得，其中所得線性相關於損失。首先，在給定相同保險費的支付下，本研究理論地比較這三種保單的變異數差異。其次，在損失金額服從對數常態分配的假設下，本研究將此三種保險的均異效率前緣繪於月一平面圖；結果顯示，除上限型保險外，自負額保險與比率共保皆可能最有均異效率。其次，若相依背景風險與所得呈負相關，則自付額保險較其餘兩種保險具有均異效率優勢。尤其，當存在正相依背景風險時，適當地組合此三種保單可以提高均異效率。

Thisstudyexaminesthemean-varianceefficienciesofthreecommoninsurancepolicies,i.e.proportionalcoinsurance,deductible,andupper-limit,inthepresenceofdependentbackgroundrisk.Thisstudyconsidersaninsuredwhofacessimultaneouslystochasticlossandincome,wherethelatterislinearlyrelatedtotheformer.First,giventhesamepremium,thisstudytheoreticallycomparesthedifferencesinvarianceamongthethreeinsurancepolicies.Additionally,themean-variancefrontiercurvesforthethreeinsurancesaredepictedonthesamegraph,assumingthelossobeysalognormaldistribution.Thisexampleshowsthat,althoughpossiblydeductibleorcoinsurance,theefficientinsurancepoliciesarenotupper-limit.Additionally,inthecasewherethedependentbackgroundriskisnegativelyrelatedtotheincome,thedeductibleinsurancealwaysdominatestheothertwo.Particularly,ifthereexistsapositivelydependentbackgroundrisk,anadequateportfolioofthethreecommoninsurancesmayimprovethemean-varianceefficiency.

1. Brazauskas, V.,Kleefeld, A.(2011).Folded and Log-Folded-t Distributions as Models for Insurance Loss Data.Scandinavian Actuarial Journal,2011,59-74.
2. Cavallo, A.,Rosenthal, B.,Wang, X.(2012).Treatment of the Data Collection Threshold in Operational Risk: A Case Study Using the Lognormal Distribution.Journal of Operational Risk,7,3-38.
3. Chiu, M. C.,Wong, H. Y.(2012).Mean-Variance Asset-Liability Management: Cointegrated Assets and Insurance Liability.European Journal of Operational Research,223,785-793.
4. Cox, S. H.,Lin, Y.,Tian, R.(2013).Morality Portfolio Risk Management.Journal of Risk and Insurance,80,853-890.
5. Doherty, N. A.,Schlesinger, H.(1983).Optimal Insurance in Incomplete Markets.Journal of Political Economy,91,1045-1054.
6. Doherty, N. A.,Schlesinger, H.(1983).The Optimal Deductible for an Insurance Policy When Initial Wealth Is Random.Journal of Business,56,555-565.
7. Gollier, C.(1996).Optimal Insurance of Approximate Losses.Journal of Risk and Insurance,63,369-380.
8. Gollier, C.,Schlesinger, H.(1996).Arrow's Theorem on the Optimality of Deductibles: A Stochastic Dominance Approach.Economic Theory,7,359-363.
9. Guiso, L.,Jappelli, T.(1998).Background Uncertainty and the Demand for Insurance Against Insurable Risks.The Geneva Papers on Risk and Insurance Theory,23,7-27.
10. Han, Z.,Gau, W. C.(2008).Estimation of Loss Reserves with Lognormal Development Factors.Insurance Mathematics & Economics,42,389-395.
11. He, L.,Liang, Z.(2013).Optimal Investment Strategy for the DC Plan with the Return of Premiums Clauses in a Mean-Variance Framework.Insurance Mathematics & Economics,53,643-649.
12. Huang, H. H.(2006).Optimal Insurance Contract under a Value-at-Risk Constraint.The Geneva Risk and Insurance Review,31,91-110.
13. Huang, H. H.,Shiu, Y. M.,Wang, C. P.(2013).Optimal Insurance Contract with Stochastic Background Wealth.Scandinavian Actuarial Journal,2013,119-139.
14. Jeleva, M.(2000).Background Risk, Demand for Insurance, and Choquet Expected Utility Preferences.The Geneva Papers on Risk and Insurance Theory,25,7-28.
15. Jorion, P.(2001).Value at Risk: The New Benchmark for Managing Financial Risk.NewYork:McGraw-Hill.
16. Kaluszka, M.(2001).Optimal Reinsurance under Mean-Variance Premium Principles.Insurance Mathematics & Economics,28,61-67.
17. Lee,W. C.(2012).Fitting the Generalized Pareto Distribution to Commercial Fire Loss Severity: Evidence from Taiwan.Journal of Risk,14,63-80.
18. Luciano, E.,Kast, R.(2001).A Value at Risk to Background Risk.The Geneva Papers on Risk and Insurance Theory,26,91-115.
19. Mahul, O.(2000).Optimal Insurance Design with Random Initial Wealth.Economics Letters,69,353-358.
20. Markowitz, H.(1952).Portfolio Selection.Journal of Finance,7,77-91.
21. Nadarajah, S.,Bakar, S. A. A.(2014).New Composite Models for the Danish Fire Insurance Data.Scandinavian Actuarial Journal,2014,180-187.
22. Raviv, A.(1979).The Design of an Optimal Insurance Policy.American Economic Review,69,84-96.
23. Schlesinger, H.(1997).Insurance Demand without the Expected-Utility Paradigm.Journal of Risk and Insurance,64,19-39.
24. Schlesinger, H.,Doherty, N. A.(1985).Incomplete Markets for Insurance: An Overview.Journal of Risk and Insurance,52,402-423.
25. Vercammen, J.(2001).Optimal Insurance with Nonseparable Background Risk.Journal of Risk and Insurance,68,437-448.
26. Verlaak, R.,Beirlant, J.(2003).Optimal Reinsurance Programs: An Optimal Combination of Several Reinsurance Protections on a Heterogeneous Insurance Portfolio.Insurance Mathematics & Economics,33,381-403.
27. Wang, C. P.,Huang, H. H.(2012).Optimal Insurance Contract and Coverage Levels under Loss Aversion Utility Preference.Quantitative Finance,12,1615-1628.
28. Wang, C. P.,Shyu, D.,Huang, H. H.(2005).Optimal Insurance Design under a Value-at-Risk Framework.The Geneva Risk and Insurance Review,30,161-179.
29. Zeng, Y.,Li, Z.(2011).Optimal Time-Consistent Investment and Reinsurance Policies for Mean-Variance Insurers.Insurance Mathematics & Economics,49,145-154.