Calculation of optimum premium values

Introduction

Here we consider the subject of the calculus of variations by analogy with the classical topic of maximization and minimization in calculus. Richard Bellmann (1957) [1] in his book of dynamic programming states that an optimal policy as the property that, whatever the initial state or initial decision are, the remaining decision must constitute optimal policy with regard to the outcome resulting from the first decision. Gelfand and Fomin (1963) [5] , Young (1969) [19] , and Leitmann (1981) [12] have given rigorous treatments of the subject. The problem of the calculus of variations is that of determining a function that maximizes a given functional, the objective function. An analogous problem in calculus is that of determining a point (using the first-order condition for a maximum) at which a specific function, the objective function, is maximum. A similar procedure will be employed to derive the first-order condition for the variational problem. The analogy with classical optimization extends also to the second-order maximization condition of calculus. For more rigorous derivations of maximum principle by dynamic programming approach we refer Pontryagin et al. (1962) [17] , Berkovitz (1961) [2] , Halkin (1967) [6] , Hartberger (1973) [7] . The dynamic programming tool is a problem in which a random element enters into the movement of the system; the optimal control must be stated in feedback form, in terms of the system, rather than in terms of time alone (because the state that has to be obtained cannot be known in advance, due to the stochastic disturbance). Karatzas and Shreve (1997) [11] cover most of stochastic optimal control problems and methods (see also Fleming and Rishel, 1975; Fleming-Zariphopoulou, 1991; Merton, 1992; Hipp and Plum, 2000; Oksendal, 2003; Hipp and Plum, 2003 and Li, 2017) [3,4,8,15,16,9,13] . Sethi and Thompson (2005) [18] discussed many concepts related optimal control theory such as HJB equation, Pontryagin's principle etc. In the present paper we optimize the maximum premium values using stochastic optimal control theory of dynamic programming problems such as Hamiltonian-Jacobi-Bellaman (HJB) equation. Finally we obtain the solution, using the relationship between the maximum principle of optimal control theory and the necessary conditions of the calculus of variations (Sethi and Thompson, 2005) [18] .

Computation of optimum premium values for Quadratic utility function:

The maximum premium (P max ) that an insured is willing to pay can be determined using utility theory. Kapoor and Jain (2011) [10] determined P max by considering different forms of the utility functions assuming the loss random variable to follow different forms of continuous statistical distributions. We determine optimum premium values using HJB equation for some discretized distributions.

The first two moments of some discretized distributions are given in the following table. See Mallappa and Talawar (2020) [14] for maximum premiums of various discrete loss distributions with different utility functions.

Table 1: First two moments of some discretized distributions

Discrete Distribution

The maximum premium (𝑃 𝑚𝑎𝑥 ) for quadratic utility function.

We optimize the above equation subject to

The HJB equation is given by

Where 𝜆 is Hamiltonian constant and its value is obtained using the following condition.

This is the second-order condition for the maximization of the Hamiltonian. Hence optimal premium equation by HJB is given in the following form

We compute and plot the optimum premium values for different values of quadratic utility function i.e., 𝑢(𝑤) = -16, -36, -64, -100, -144 𝑎𝑛𝑑 249001 against the different parametric values of discretized probability distributions. Following are the plots of optimum premiums using quadratic utility function assuming loss follows some descritized probabilty distributions.

Computation of optimum premium values for Fractional power utility function:

The maximum premium equation for fractional power utility function

We optimize the above equation subject to

Thus the HJB equation is given by

Where 𝜆 is Hamiltonian constant and its value is obtained using the following condition,

and hence the second-order condition for the maximization of the Hamiltonian is

Following are the plots of optimum premiums using fractional power utility function assuming loss follows some probabilty distributions. Thus the quadratic utility function is more beneficial to the insured because the optimum premium values are high compared to maximum premium values (Mallappa and Talawar, 2020) [14] .

And in the case of fractional power utility function, we observe the following  For discrete Exponential, Gamma, Weibull, Pareto and Burr distributions the optimum premium value increase as the parameter 𝜆 increase and the values are higher slightly than the maximum premium values.  It is also observed that even for the smaller values of utility function, the optimum premium values are very high.

Hence fractional power utility function is more beneficial to insurer.

References

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2. Fleming WH, Zariphopoulou T. An optimal investment/consumption model with borrowing. Mathematics of Operations Research. 1991;16(4):802-22.
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4. Hipp C, Plum M. Optimal investment for insurers. Insurance: Mathematics and Economics. 2000;27(2):21528.
5. Hipp C, Plum M. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics. 2003;7(3):299-321.
6. Kapoor HPS, Jain K. Premium Evaluation For Different Loss Distributions Using Utility Theory, Discussiones Mathematicae, Probability and Statistics. 2011;31(1-2):4158. doi:10.7151/dmps.1137
7. Mallappa, Talawar AS. Premium calculation for different loss discrete analogues of continuous distributions using utility theory. International Journal of Agricultural & Statistical Sciences. 2020;16(1):61-72.