Risk Management
EABH & Swiss Re
Mathematics at Swiss Re in the 1960s
Hans Bühlmann
June 2013
1. Mathematics in insurance / Actuaries
Actuaries of the first kind
~ 18th century
Life insurance
Individual Risk Theory
each policy
bottom up
Cash Flow Accounting:
EQUIVALENCE PRINCIPLE
Present Value Matching
interest rates
mortality
All this: In a Deterministic Model !
Law of Large Numbers
1. Mathematics in insurance / Actuaries
Actuaries of the second kind
~ middle of 20th century
All Branches of Insurance + Total of Insurance Company
Collective Risk Theory
total
topdown
Swiss Re decision in 1961
Witness of this pioneering period
Risk management without calling it so
2 concrete examples to get insight into the then activity
2. Rating of Excess of Loss Reinsurance Treaties
(XL-Treaties)
History of XL-Treaty last five years
For covered portfolio ceding company has earned premium T = 10mio
Total claim
Ceding company
120 000
100 000
20 000
80 000
80 000
-
235 000
100 000
135 000
97 000
97 000
-
….
….
….
XL Treaty with limit 100 000
(retention)
Reinsurer
550 000
100 000
450 000
320 000
100 000
220 000
165 000
100 000
65 000
2 350 000
1 015 000
1 335 000
B
2. Rating of Excess of Loss Reinsurance Treaties
(XL-Treaties)
B
B%
is called Burning Cost
Burning Cost in percent : 13.35%
of total premium T
Burning Cost Rate
What premium should we change for next years?
First Idea
Premium = B% . T next year
Underwriter
Premium =
Actuary
Premium = (B% + α F%) T next year
100
75
______________________
B% . T next year
2. Rating of Excess of Loss Reinsurance Treaties
(XL-Treaties)
F%
~ fluctuation measure of data %
e.g. standard deviation
α
~ chosen such that total of all XL-treaties is producing a loss with small probability
Generally:
Actuary produces class of Premium Calculation Principles.
Choice by consideration of total business of reinsurer => topdown
3. How to Choose the Retention Limit
Also Swiss Re cannot run arbitrarily high engagements
SR buys reinsurance on reinsurance business
retrocession
How should Swiss Re fix its retention under retrocession treaties?
3. How to Choose the Retention Limit
Actuarial method (Bruno De Finetti 1940)
a) Relative Retentions
(Retention) Branchi ί =
Cα
ί
αί
~ Profitability Branch ί
Limit in XL
MPL in proportional reinsurance
Historical remark: H. Markowitz, 1952
3. How to Choose the Retention Limit
b) Absolute retention
Fix the constant C in a)
ί) Choose operative capital
buffer to absorb losses
buffer which you are willing to loose
buffer ?
- such that the company can survive
- such that the company can digest it on its balance sheet
- full surplus
3. How to Choose the Retention Limit
ίί) Choose acceptable probability of loosing the buffer
- long term => Cramér-Lundberg formula: probability of ruin
- short term
: probability of default
operative capital
Ψ (u) = e-ku
prob. ruin
depends on riskportfolio
retention limits
on the constant C
Conclusion
•
Probability of ruin enters into formula for C on logarithmic scale
•
Operative capital and profitability are the main drivers: C depends linearly on k and u
Interesting for today
•
Questions of solvency
•
Attitude towards risk: - Security
- Profitability