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This study tackled portfolio selection problem for an insurer as well as a reinsurer aiming at maximizing the probability of survival of the Insurer and the Reinsurer, to assess the impact of proportional reinsurance on the survival of insurance companies as well as to determine the condition that would warrant reinsurance according to the optimal reinsurance proportion chosen by the insurer. It was assumed the insurer’s and the reinsurer’s surplus processes were approximated by Brownian motion with drift and the insurer could purchase proportional reinsurance from the reinsurer and their risk reserves followed Brownian motion with drift. Obtained were Hamilton-Jacobi-Bellman (HJB) equations which solutions gave the optimized values of the insurer’s and the reinsurer’s optimal investments in the risky asset and the value of the discount rate that would warrant reinsurance as a ratio of their portfolio weights in the risky asset.

The first study on optimal reinsurance was done by Bruno de Finetti as pointed out by Centeno and Simões [

Schimidli [

[

Taksar and Markussen [

Hipp and Plum [

Liu and Yang [

Castillo and Parrocha [

Irgens and Paulsen [

Paulsen et al. [

Paulsen [

Meng and Zhang [

Kasumo [

Mata [

In this study we consider the risk reserve of an insurer and a reinsurer to follow Brownian motion with drift and tackled their portfolio optimization problem. The optimized values of the insurer and the reinsurer are calculated. Also calculated are the insurer’s and the reinsurer’s optimal investment in the risky asset and then the discount value,

To make for clear understanding of this work, we defined the following few terms;

Insurance: Insurance is an arrangement by which a company gives customers financial protection against loss or harm such as theft or illness in return for a payment called a premium (Encarta World English Dictionary [

Reinsurance: This is the transfer of risk from a direct insurer (the cedent) to a second insurance carrier (the reinsurer). It may also be defined as insurance for insurers. It serves the purpose of offering protection to cedents against very large individual claims or fluctuations in their aggregate portfolio of risks, as well as diversifying the financial losses caused by it [

Risk: This is the probability of loss to an insurer or the amount that an insurer is in danger of losing [

Optimal portfolio: An optimal portfolio is a portfolio in which the risk-reward combination is such that it yields the maximumreturns (provides the highest utility) possible under the current and anticipated circumstances. Its mathematical formulation was provided the University of California's noble laureate economist Harry Markowitz (born 1927) in 1952.

Portfolio reinsurance: The practice whereby an insurer transferssome or all of the risk attached to a portfolio to another insurer, or reinsurer. Insurers use portfolio reinsurance to reducethe risk of having to pay large claims in the event of significant losses to the value of the portfolio.

Suppose the claim process

where a and b are positive constant and

with safety loading (security risk premium)

Using equation (1), the surplus process of the insurer is given by;

The insurance company has the permission to purchase proportional reinsurance to reduce her risk and pays reinsurance premium continuously at the rate of

The surplus of the insurance company is then given as;

for the insurer, and

for the reinsurer, (Danping et al. [

Assuming that the insurer and reinsurer invest their surplus in the same market consisting of two assets: a risky asset (stock) and a riskless asset (bond) which rate of return is a linear function of time, let the prices the riskless and be risky assets

and

(Osu and Ihedioha, [

Both the insurer and there insurer hold the risky asset as long as.

Let

and for the reinsurer the strategy

Assume that

For the corresponding admissible strategies,

for the insurer, and

for the reinsurer (Wokiyi, [

Substituting the expressions for,

for the insurer and;

for the insurer.

The quadratic variations of the wealth processes of the insurer and the reinsurer are;

Suppose the investor has a power utility function, the Arrow-Pratt measure of relative risk aversion (RRA) or coefficient of relative risk aversion is defined as;

where w is the wealth level of an investor. The special case being considered is where the utility function is of the form,

which has a constant relative risk aversion parameter

where

and

Subject to:

for the insurer and;

for the reinsurer.

The theorem that follows gives the optimization of the insurer’s wealth;

Theorem 1: The optimal policy that maximizes the expected power utility at terminal time T is to invest at each time

with optimal proportion reinsured,

and value function;

where

Proof:

We derive the Hamilton-Jacobi-Bellman (HJB) partial differential equation starting with the Bellman equation:

where

Rewriting Equation (25) as,

and dividing both sides of the equation by

Ito’s lemma (Miao, [

For the insurer, substituting in the Ito’s lemma for

Equation (28) simplifies to;

Applying (29) to the Bellman Equation (26) and taking expectation, we get the HJB equation;

where,

satisfying terminal condition,

Observing the homogeneity of the objective function, the restriction and the terminal condition, we conjecture that the value function V must be linear to

Let

be such a value function, such that at the terminal date, T

then

Substituting Equation (35) into Equation (30), we obtain; the new H-J-B equation,

To obtain the optimal value

This simplifies to;

This is the insurance company’s optimal investments in the risky asset, stock, that is both horizon and wealth dependent.

Also, differentiating Equation (36) with respect to

This reduces to;

The solution of the HJB Equation (36) is thus; replacing

From which we obtain,

Since t is the dominating variable, as implied in our choice of

where

The integral of the differential equation of the function g is obtained as;

where

That is;

Applying the terminal condition,

This implies that the horizon dependent solution to the insurance company’s investment problem is:

This is the maximized expected power utility value at time t under optimal investment policy.

For the reinsurer, we state the following theorem 2.

Theorem 2: The optimal policy to maximize the expected power utility at T is to invest at each time

with optimal proportion reinsured,

and value function;

where

Proof:

Adopting Equations (25) to (35) and replacing

which reduces to;

To obtain the optimal investment in the risky asset, Equation (53) is differentiated with respect to

Solving for

The differentiation of (53) with respect to

From equation (55) which when reduced gives;

where,

the definite integration within

Applying the terminal condition,

This implies that the horizon dependent solution to the insurance company’s investment problem is:

Here we find the condition under which the proportion reinsured by the Insurer equals the amount accepted to be insured by the Reinsurer.

Therefore, we equate the values of

That is;

This reduces to;

where;

Therefore,

where, w_{r} and w_{i} are the Reinsurer’s and the Insurer’s portfolio weights in the risky asset, respectively.

Clearly, the optimal policies that maximize the expected power utility and the value functions for both the insurer and the Reinsurer are horizon dependent.

In this study, we consider the optimal investment problem for both an insurer and a reinsurer. The basic claim process is assumed to follow a Brownian motion with drift and the Insurer could purchase proportional reinsurance from the Reinsurer.

The Reinsurer and the Insurer were allowed to invest in a risky and a risk-free assets and expressions for their optimal portfolios obtained solving the corresponding HJB equations. The discount value,

Silas A. Ihedioha,Bright O. Osu, (2015) Optimal Portfolios of an Insurer and a Reinsurer under Proportional Reinsurance and Power Utility Preference. Open Access Library Journal,02,1-11. doi: 10.4236/oalib.1102033