1. Fair Valuation of Guaranteed Contracts: the Interaction Between Assets and Liabilities Erwin Charlier Tilburg University and ABN AMRO Bank Joint work with Ruud Kleynen Maastricht University and Kleynen Consultants

2. Overview 1. Introduction 2. General theoretical framework 3. Modelling the assets and the short-rate 4. Data and parameter estimation 5. Results 6. Conclusions 7. Further research

3. Introduction  Balance sheet: book value accounting  fair or market value of assets and liabilities  Market value of assets:  Market prices for publicly traded assets (stocks, bonds)  Valuation models for less liquid assets like real estate  Market value of liabilities:  Very little traded liabilities  Optionalities

4. Introduction In this presentation: Simple insurer:  Assets: investments in stocks and bonds  Liabilities and equity:  Single guaranteed return contract (policy)  Equity Policy characteristics:  Guaranteed return, r offered  Bonus: if the return on equity exceeds r offered then fraction of surplus to policyholder

5. General theoretical framework t=0: t=T: AssetsLiabilities A0A0 L 0 = αA 0 E 0 = (1-α)A 0

6. General theoretical framework 0<=t<=T: t=0: no cross-subsidizing Note: prices under risk-neutral measure

7. Modelling the assets and the instantaneous short-rate Instantaneous short-rate: stochastic, Vasicek LN gross asset returns: normal Geometric Brownian motions correlated Under risk-neutral measure: analytic formulae for price of put and call Real-world measure used to describe economy at time t, also input for prices

8. Data and parameter estimation Parameters in process for instantaneous short-rate:  Cross-section of FR bond prices (Feb 28, 2002)  Time-series of 1-month FIBOR rates Also used to derive instantaneous short-rate series Parameters in process for assets:  Assume two investment categories: stocks and bonds (monthly, Nov 1990-Feb 2002)  Use weights to construct time-series of portfolio returns  But: high mean  used Dimson(2002) Correlation: use imputed instantaneous short-rate and portfolio returns

9. Results alpha=0.95, delta=0.91, r offered =0.04, T=10

10. Results alpha=0.8, delta=0.72, r offered =0.04, T=10

11. Conclusions  Model allows for stochastic interest rates that can be correlated with process for assets.  Parameters in the model estimated from data instead of choosing some value.  Using both risk-neutral and real-world measure we can derive risk-return profiles for both policyholders and equityholders.  Different specifications of the debt-equity ratio and the contract did not lead to satisfying return profiles for both policyholders and equityholders.  Best results for equityholder occur with low debt- equity ratios, conflicting practice.

12. Further research  Further investigate causes of unsatisfactory risk- return profiles.  Extend to more complicated balance sheet (more than one product, different maturities for the policies, etc.).  Consider balance sheet at intermediate times with rule for regulator to interfere.  Use more advanced models to describe the instantaneous short-rate and the assets, while keeping closed-form solutions for the options.  Drop the requirement of no cross-subsidizing.