Portfolio Management using CAT Modeling Software:

An Reinsurer’s perspective

Jim Maher, FCAS, MAAACAS Ratemaking SeminarLas Vegas, March 2001

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

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

Efficient Frontier

01234567

0 200 400 600

expected return

std

dev

retu

rn

Portfolio Optimization

Your (re)insurer’s current portfolio is as follows:

ZoneTotal sum insured

loss cost rate

premium rate

a T(a) k(a) p(a)b T(b) k(b) p(b)

ZoneTotal sum insured

loss cost rate

premium rate Premium

Expected Return

a 1,000,000 8% 13% 130,000 50,000 b 1,000,000 8% 12% 120,000 40,000

Tot 2,000,000 250,000 90,000

CAT Model Parameters

Mean Loss Ratios Variance of Loss Ratios

Event idAnnual

Frequencyvariance of frequency Zone a Zone b Zone a Zone b

1 f(1) w(1) L(1,a) L(1,b) v(1,a) v(1,b)2 f(2) w(2) L(2,a) L(2,b) v(2,a) v(2,b)

Mean Loss Ratios Variance of Loss Ratios

Event idAnnual

Frequencyvariance of frequency Zone a Zone b Zone a Zone b

1 40% 24% 5% 10% 0.16% 0.00%2 20% 16% 30% 20% 0.00% 1.21%

Where the above loss cost rates have been determined by using thefollowing catastrophe rating model:

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%

Portfolio Optimization

• Your CEO wants your recommendation on how to best optimize the above portfolio.

• His idea is as follows:

ZoneTotal sum insured

loss cost rate

premium rate Premium

Expected Return

a 1,800,000 8% 13% 234,000 90,000 b - 8% 12% - -

Tot 1,800,000 234,000 90,000

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)

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)

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)

Risk vs. Reward, ctd.

Then we have:E[Return] = r(a)T(a) + r(b)T(b) = $90,000Var[Return] = h(a)T(a)2 + h(a,b)T(a)T(b) +h(b)T(b)2

r(a) 5.0000%r(b) 4.0000%h(a) 1.5640%

h(a,b) 2.1600%h(b) 1.1220%

Want to find the value of T(a) that minimizes Var[Return]

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

ZoneTotal sum insured

loss cost rate

market premium

rate PremiumExpected

Returna 1,175,815 8% 13% 152,856 58,791 b 780,231 8% 12% 93,628 31,209

Tot 1,956,046 246,484 90,000

Minimizing Standard Deviation

215,000

220,000

225,000

230,000

235,000

240,000

0 500,000 1,000,000 1,500,000 2,000,000

T(a) [Sum insured in Zone a]

St D

ev R

etur

n

Comparison of PortfoliosThe 3 portfolios compare as follows:

(Surplus has been allocated proportional to std dev.)

Zone a Zone b PremiumExpected Return

Std Dev Return ROP Surplus ROE

1,000,000 1,000,000 250,000 90,000 220,136 36.0% 500,000 22.00%1,800,000 - 234,000 90,000 225,108 38.5% 511,292 21.60%1,175,815 780,231 246,484 90,000 219,703 36.5% 499,015 22.04%

Sum Insured

Alternative Approaches

• Other measures of risk:-Expected Downside (EPD)- 100 year Downside

• Optimize Portfolio based on minimizing these- requires full distribution of results

Minimum EPD

Scenario probability

Aggregate Loss Ratio

Zone a

Aggregate Loss Ratio

Zone b Return1 48% 0.0% 0.0% 263,494 T(a) 325,301 2 16% 9.0% 10.0% 49,880 T(b) 1,843,373 3 16% 1.0% 10.0% 75,904 Premium 263,494 4 6% 30.0% 31.0% (405,542) stdev 229,637 5 6% 30.0% 9.0% 0 EPD (56,602) 6 2% 39.0% 41.0% (619,157) Downside (619,157) 7 2% 31.0% 41.0% (593,133) 8 2% 39.0% 19.0% (213,614) 9 2% 31.0% 19.0% (187,590)

Total 100% 8.0% 8.0% 90,000

Minimum Downside

Scenario probability

Aggregate Loss Ratio

Zone a

Aggregate Loss Ratio

Zone b Return1 48% 0.0% 0.0% 234,000 T(a) 1,800,000 2 16% 9.0% 10.0% 72,000 T(b) 0 3 16% 1.0% 10.0% 216,000 Premium 234,000 4 6% 30.0% 31.0% (306,000) stdev 225,108 5 6% 30.0% 9.0% (306,000) EPD (68,400) 6 2% 39.0% 41.0% (468,000) Downside (468,000) 7 2% 31.0% 41.0% (324,000) 8 2% 39.0% 19.0% (468,000) 9 2% 31.0% 19.0% (324,000)

Total 100% 8.0% 8.0% 90,000

Comparison of Portfolios

Zone a Zone b PremiumExpected Return

Std Dev Return EPD

50 year downside

1,000,000 1,000,000 250,000 90,000 220,136 (62,000) (550,000) 1,800,000 0 234,000 90,000 225,108 (68,400) (468,000) 1,175,815 780,231 246,484 90,000 219,703 (63,407) (531,979)

325,301 1,843,373 263,494 90,000 229,637 (56,602) (619,157)

Sum Insured

Portfolio Optimization summary

• No one correct answer• Depends on how risk and reward are

defined• Need direction from senior management

- corporate utility function