## Guaranteed investment contracts: distributed and undistributed excess return.(English)Zbl 1092.91053

The paper deals with a model consisting an investment/saving plan or contract between two parties called the insurer and the customer. At date zero the customer deposits an amount $$X$$ into an account $$A$$, which is invested by the insurer for a period of $$T$$ years. The insurer promises the customer an annual rate of return on the account $$A$$ in the year $$i$$ equal to $$g_{i}+\alpha(\delta_{i}-g_{i})^{+}$$, where the constant $$g_{i}$$ is a specified minimum rate of return guarantee in the year $$i$$; $$\delta_{i}$$ is the random rate of return of the specified benchmark portfolio in the year $$i$$; and $$\alpha$$ is the fraction of the positive excess rate of return which is credited to the customer’s account. In return for the minimum rate of return guarantee the insurer receives a fraction $$\beta$$ of the excess rate of return, i.e. the return $$\beta(\delta_{i}-g_{i})^{+}$$ is credited to the insurer’s account, denoted by $$C$$. In addition, the model includes a surplus distribution mechanism working through the bonus account $$B$$, which is managed by the insurer. The case of Gaussian returns on the benchmark portfolio and deterministic short term interest rates are considered. A closed-form solution for the value of the customer’s account is derived. The value of the bonus account is solved by Monte Carlo simulations. Corresponding values of annual minimum rate of return guarantees and the fractions of the excess return distributed to the customer’s account and the bonus account are plotted for fair contracts.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 91B28 Finance etc. (MSC2000) 62P05 Applications of statistics to actuarial sciences and financial mathematics 60J65 Brownian motion 62E10 Characterization and structure theory of statistical distributions
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### References:

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