Financial Mathematics in Derivative Securities and Risk Reduction

1. University of Economics, Faculty of Informatics Dolnozemská cesta 1, 852 35 Bratislava Slovak Republic Financial Mathematics in Derivative Securities and Risk Reduction Insurance and Risk Reduction, Financial Layering Ass. Prof. Ľudovít Pinda, CSc. Department of Mathematics, Tel.:++421 2 67295 813, ++421 2 67295 711 Fax:++421 2 62412195 e-mail: pinda@dec.euba.sk

2. Sylabus of the lectures ·        Preloss financing: - Retention funding. ·        Composite financing strategies: - Full insurance and partial insurence. - Insurance and risk reduction : - Risk reduction with coinsurance. - Risk reduction with deductibles. ·        Financial layering.

3. Identify Risk, Management Events: Measure Capital Costs Estimate Effects on Corporate Earnings and Cost of Capital Immediate Investment Decision Contingent Investment Decision Loss Reduction No Loss Reduction Postloss Reinvestment Postloss Abandonment Financing Financing Preloss Financing Postloss Financing Internal funds Dept Equity Dept Internal funds Insurance Funding Equity Contingent Loans Tab. 1 Decision Framework for financial Risk Management

4. Retention funds when alternative sources of finance have no transaction costs

8. The value of the firm: •   The commercial non-risk management activities. •   The value of retention fund. •   The value of the firms loss exposure. Retention funds when the alternative sources of finance have transaction costs

9. The value of the firm: • The value from commercial non-risk management activities. • The value or cost from establishing a fund and investing its assets minus the cost of raising capital to • finance the fund. • The ( negative ) value contributed by the loss exposure. • The ( negative ) contribution to value arising from the prospect of incurring transaction costs for • unfunded loses.

10. Composite financing strategies • Full insurance and partial insurance Fig. 1 Loss distribution and Proportionale Coinsurance

11. Risk Reduction with Coinsurance

15. Example 1 Solution Tab.1

16. Tab. 2

17. Tab. 3

18. Fig. 2 Fig. 3

19. E(E) cov(X,L)=20 cov(X,L)=0 17,8 17,7 17,6 17,5 17,4 17,3 s 8,5 9 9,5 10 10,5 11 ( E ) a a =1 a =0,75 a =0,5 a =0,25 a =0 Fig. 4

20. Settlement under policy with 20 000 deductibles Per loss deductible Cumulative deductible Loss Payment Cumulative loss Payment 16 000 0 173 000 173 000-20 000=153 000 33 000 33 000-20 000=13 000 124 000 124 000-20 000=104 000 Total settlement 117 000 153 000 Risk reduction with deductibles Tab. 4

21. Fig. 5 Distribution of retained loss Distribution of policy payment

23. l 0 100 1000 10 000 20 000 p(l) 0.6 0.2 0.1 0.04 0.06 Example 2 The loss distribution by Tab. 5 Tab. 5

25. E(E) 13400 13200 13000 12800 12600 12400 12200 1000 2000 3000 4000 5000 6000 Tab. 6 Fig. 6

26. Financial Layering Tab. 7

27. C C=L C L 0 C C=L C C=L L 0 L 0 Insurance New Issue Fig. 7 Fig. 8 Internal Liquid Resources Fig. 9

28. Two-layer financing methods Fig. 10

29. Three-layer financing method Tab. 8 Fig. 11