Abstract
We investigate the dividend and equity issuance problem in the presence of interest rate. The evolution of the financial reserves of an insurance company, where management payout dividends and issue new equity, is described by a stochastic differential equation. The work of Lokka and Zervos [1] is extended by including the interest rate component into the model in order to make the model more realistic. The aim is to maximise the expected discounted dividends pay-out until the time of bankruptcy. In order to investigate this problem, the stochastic control theory for diffusion processes will be used. In order to handle the problem, the Hamilton-Jacobi-Bellman equation (HJB) is derived and solved. The second order ordinary differential equation associated with the problem turns out to belong to the class of Kummer’s confluent hyper-geometric differential equations. This category of equations is not easy to solve. The equation is non-dimensionalised and change of variables is effected in two different stages. The results show that interest rate affected the rate at which the value function and threshold level change.
Highlights
The field of stochastic control has evolved a long way from the 1970s, in its application to finance
We investigate the dividend and equity issuance problem in the presence of interest rate
The results show that interest rate affected the rate at which the value function and threshold level change
Summary
The field of stochastic control has evolved a long way from the 1970s, in its application to finance. Taksar and Zhou [22] considered the case in which the drift and the volatility of the liquid assets are related since higher risk implies higher expected return Lin He and Zongxia Liang [23] examined an insurance company in which the dividends payout, equity issuance and the risk exposure are controlled by the management. The study by Chevalier, Vath and Scotti [26] examined the problem of determining the optimal control on the dividend and investment policy of a firm They reviewed the fact that the firm carries a debt obligation in its balance sheet. They pointed out that the manager of a corporate must balance the retention of earnings and the distribution of dividends They applied dynamic-programming approach, where the state variables are the prevailing levels of cash reserves and of the stochastic short-rate, in addition to time. The solution of the optimal control problem is established that is the optimal value function V ( x) and the corresponding optimal control u* are derived
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