Risk Analysis and Valuation of Life Insurance Contracts
Combining Actuarial and Financial Approaches
Stefan Graf, Alexander Kling, Jochen Russ
Agenda
• Traditional life insurance products
• Model and Methodology
• Conclusion
Traditional life insurance products
• Often equipped with minimum interest rate guarantees
• Types of guarantee
– point-to-point guarantee  only relevant at maturity
– cliquet guarantee  relevant on a year-by-year basis
• Plus some additional surplus participation
– regular and terminal surplus participation
The paper‘s title
• Valuation:
– use risk-neutral valuation
• Risk Analysis:
– investigate the (real-world-) risk exposure
• Combination:
– [Barbarin and Devolder, 2005] propose a methodology of
combining above paradigms
What’s new?
• Extension to cliquet guarantees
• Discuss methodology proposed by [Barbarin and Devolder, 2005]
– when does it work in practice?
• Derivation of risk-minimizing asset allocations
– what is a ‘good’ risk-measure?
Agenda
• Traditional life insurance products
• Model and Methodology
• Conclusion
Model and Methodology
• Financial market:
– Gaussian framework modeling stochastic interest rate and stock
markets
• Insurer’s assets:
– Stocks, bonds and cash
Model and Methodology
• Liabilities:
– point-to-point guarantee
– cliquet guarantee
• minimum surplus participation according to German
regulatory framework
• model mirroring actually applied surplus policy
Model and Methodology
• Proposed methodology works fine in arbitrage-free market
• Yields following pricing approach:
– Determine risk-minimizing asset allocation
– Adjust terminal bonus participation rate to ensure fair contract
Model and Methodology
• Determine risk-minimizing asset allocation
– using a heuristic search algorithm based on Evolution Strategies
Pitfalls
• Which risk measure to choose?
Optimal Asset Allocation - Risk measure: Shortfall probability
100%
100%
Money market
Bond
Stock
Shortfall Probability
10%
0%
0%
6.
50
6.
20
5.
90
5.
60
5.
30
5.
00
4.
70
4.
40
4.
10
3.
50
3.
20
2.
90
2.
60
2.
30
2.
00
1.
70
3.
80
Guaranteed interest rate
%
10%
%
20%
%
20%
%
30%
%
30%
%
40%
%
40%
%
50%
%
50%
%
60%
%
60%
%
70%
%
70%
%
80%
%
80%
%
90%
%
90%
Pitfalls
• Which risk measure to choose?
Optimal Asset Allocation - Risk measure: Expected Shortfall
100%
100%
Money market
Bond
Stock
Relative Expected Shortfall
10%
0%
0%
6.
50
6.
20
5.
90
5.
60
5.
30
5.
00
4.
70
4.
40
4.
10
3.
50
3.
20
2.
90
2.
60
2.
30
2.
00
1.
70
3.
80
Guaranteed interest rate
%
10%
%
20%
%
20%
%
30%
%
30%
%
40%
%
40%
%
50%
%
50%
%
60%
%
60%
%
70%
%
70%
%
80%
%
80%
%
90%
%
90%
Agenda
• Traditional life insurance products
• Model and Methodology
• Conclusion
Conclusion
• Proposed methodology allows for extension to cliquet guarantees
and works fine given absence of arbitrage
• Optimal asset allocations vary dramatically using different risk
measure
– wrong incentives applying inappropriate risk-measure?
Thanks for your attention
• Contact details
Stefan Graf
s.graf@ifa-ulm.de
Alexander Kling
a.kling@ifa-ulm.de
Jochen Russ
j.russ@ifa-ulm.de