1. Portfolio Management using CAT Modeling Software: An Reinsurer’s perspective Jim Maher, FCAS, MAAA CAS Ratemaking Seminar Las Vegas, March 2001

2. Use of CAT Modeling Software Initially used primarily as a Pricing Tool Post-event loss reserving -US CAT events -International CAT events Increasingly used as a Portfolio Management Tool -managing aggregates on a per event basis -modeling the portfolio’s loss distribution

3. CAT Portfolio Management Goal: Optimize portfolio of CAT risk What would an optimal portfolio look like? - High returns, low risk Concepts from investment portfolio theory - Efficient frontier - minimize std dev of return for given expected return

4. Efficient Frontier

5. Portfolio Optimization Your (re)insurer’s current portfolio is as follows:

6. CAT Model Parameters Where the above loss cost rates have been determined by using the following catastrophe rating model:

7. Loss cost rates E[Loss] =  E[F]*E[S], (sum over event ids) k(a) = f(1) L(1,a) + f(2) L(2,a) = = 40%*5% +20%*30%= 8% k(b) = f(1) L(1,b) + f(2) L(2,b) = = 40%*10% + 20%*20% = 8%

8. Portfolio Optimization Your CEO wants your recommendation on how to best optimize the above portfolio. His idea is as follows:

9. Risk vs. reward To evaluate the CEO’s proposal, return to idea of risk vs. reward Minimize variance of return for a given expected return E[Return] = Premium – E[Loss] E[Return] = r(a)T(a) + r(b) T(b) where r(a) = p(a) – k(a), r(b) = p(b)-k(b)

10. Risk vs. Reward, ctd. Var[Return]= Var[Prem-Loss]=Var[Loss] Var[Loss] =  {E[F] Var[S] + E[S] 2 Var[F]} (sum over event ids) = f(1)[v(1,a)T(a) 2 + v(1,b)T(b) 2 ] + [L(1,a)T(a)+L(1,b)T(b)] 2 w(1) + f(2)[v(2,a)T(a) 2 + v(2,b)T(b) 2 ] + [L(2,a)T(a)+L(2,b)T(b)] 2 w(2)

11. Risk vs. Reward, ctd. Var[Loss] = h(a) T(a) 2 + h(a,b)T(a)T(b) + h(b) T(b) 2 where, h(a) = f(1) v(1,a) + f(2) v(2,a) + w(1) L(1,a) 2 + w(2) L(2,a) 2 h(b) = f(1) v(1,b) + f(2) v(2,b) + w(1) L(1,b) 2 + w(2) L(2,b) 2 h(a,b) = 2w(1)L(1,a)L(1,b) + 2w(2)L(2,a)L(2,b)

12. Risk vs. Reward, ctd. Then we have: E[Return] = r(a)T(a) + r(b)T(b) = $90,000 Var[Return] = h(a)T(a) 2 + h(a,b)T(a)T(b) +h(b)T(b) 2 Want to find the value of T(a) that minimizes Var[Return]

13. Risk vs. Reward- solution Solution: T(a) = E[Return]/ r(a) * [ h(b) r(a) 2 – ½ h(a,b) r(a)r(b) ] [h(b)r(a) 2 – h(a,b)r(a)r(b) + h(a)r(b) 2 ] = $1.175 MM

14. Minimizing Standard Deviation

15. Comparison of Portfolios The 3 portfolios compare as follows: (Surplus has been allocated proportional to std dev.)

16. Alternative Approaches Other measures of risk: -Expected Downside (EPD) - 100 year Downside Optimize Portfolio based on minimizing these - requires full distribution of results

17. Minimum EPD

18. Minimum Downside

19. Comparison of Portfolios

20. Portfolio Optimization summary No one correct answer Depends on how risk and reward are defined Need direction from senior management - corporate utility function