“ Formation of the optimal portfolio of insurer ’ s services of the voluntary types of insurance ”

The article studies the possibility of using optimization modelling to form the optimal structure of insurance services’ portfolio of insurance companies. Based on the data of net insurance payments and profitability of the voluntary types of insurance in 2005-2015, the authors conducted their analysis according to the possibility to be included in the general insurance portfolio of the insurance company. The optimization model is based on the approach developed by G. Markowitz. The formation of insurance services portfolio is conducted by solving the optimization problem to maximize the portfolios’ profitability or to minimize the portfolio’s risks. The obtained results can be used in making strategic decisions by the management regarding the development of insurance companies.


Introduction 1
The activity of insurance companies in the current conditions could be described as a rethinking of the ways of the future development. Uncertainty of the course of economic development, financial market's stagnation and significant problems of banking institutions do not contribute to the formation of a clear development strategy and the insurance market, especially when it comes to the sales of insurance products. Deterioration of financial performance of individual insurers, as well as the new possibilities that have to appear along with the gradual recovery of the insurance market, bring to the fore the issue of choosing a new development strategy, a key element of which should be the choice of the optimal set of insurance services that will ensure maximization of profits based on acceptable risk levels. Insurance companies need to define a list of optimization tasks according to which insurance services should be given priority.
Significant contribution to the study of the nature of insurance portfolio and its impact on the financial reliability of insurers was made by the following domestic scientists: V. Bazylevych (1997) optimal portfolio of insurance services in the contemporary Ukrainian realities of the insurance market's development.
This problem will be solved by analyzing the performance indicators of insurance companies in Ukraine in terms of the provided insurance services and the use of the principles of G. Markowitz optimization model for the formation of the optimal portfolio of insurance services.

Literature review
The study of the insurance market and the role of adequate levels of insurance management is done in the works of many scientists. Each of them noted key aspects of quality management of insurance companies. For example, Bomhard (2005) notes that particularly relevant in conditions of financial instability is to harmonize the requirements of state regulations of the insurance industry, to use separate management of risks of certain types of insurance, paying special attention to the placement of insurance reserves. Krenn & Oschischnig (2003) identify a number of key factors that must be taken into account by insurance companies in the management of insurance portfolios. They include underwriting risk, market risk, credit risk, liquidity risk, operational and other risks.
Ungureanu, M.A., Gruiescu, M., Ioan , C., Morega, D. (2010) proposed an approach that makes it possible to assess the individual risk of separate assets or assets portfolio taking into account the specifics of activities of insurance companies. Their method is based on the use of Value at Risk (VaR) methods.
Fwu-Ranq Chang (2008) shows how the connection between the property insurance and the portfolio choice of the insurer is realized, emphasizing their interdependence. The researcher notes that the size of the premium for the contract of property insurance plays a key role in making investment decisions, as its increase leads to this type of insurance becoming dominant in the portfolio, while the general approach of the insurer to the choice of insurance types becomes more conservative.
N. Dolhosheya (2010) emphasizes that the analysis of insurance portfolio of the insurer should start with determining the number of insured objects, the total insurance amount, and the size of collected insurance payments. The very structure of the portfolio determines the financial stability of the insurance company, and the study of unprofitableness of the insurance amount should solve the problem of identifying the actual level of unprofitableness and its comparison with the level of insurance rates used by the insurance company.
Thus, the contemporary works, which investigate insurance services portfolios, work out in detail various impact factors and their peculiarities. Nevertheless, the problems of optimizing insurance portfolios in modern conditions require further research. This paper offers one way to solve them.

Methodology
At the present stage of development of the insurance market in Ukraine, insurers have accumulated sufficient information on the volumes of insurance activities, which to a certain extent, continue to grow. Therefore, insurance companies can make forecasts of their activities and assess their potential.
The logic of the study involves the solving of practical problems of building an optimized portfolio of insurance services by the insurer based on the profitability of their provision. The general methodology requires the following steps: to make a selection of those types of insurance, which have a significant specific weight in insurance payments; exclude from the study those types of insurance that do not give significant amounts of insurance premiums; to calculate the profitability of the types of insurance using statistical data of the National Commission for the State Regulation of Financial Services Markets; to calculate statistical key indicators for portfolio optimization; to carry out portfolio optimization based on the approach of G. Markowitz; to make conclusions about the results of the study.
This approach is built on the principles, which take into account the current market importance of certain types of insurance. Calculations are made by using the general statistical information of the insurance market in Ukraine. They serve as the basis for the extension of this approach to the specific activities of a particular insurer.
As for the Markowitz optimization model, which is traditionally used to optimize the structure of the securities portfolio, it can also be applied to insurance. It is not only a matter of the insurance company's investment activity, but also of its insurance activities, such as optimization of the insurance services' portfolio. Indicators used by Markowitz model to calculate the optimal investment portfolio, have their counterparts in insurance business. The profitability of securities is similar to insurance profitability. The risk of securities in the model is calculated as a standard deviation of profitability. It is possible to calculate such indicator for a particular type of insurance.
The Markowitz model, which is based on the fact that the data for the previous periods used to calculate profitability and the risk of securities, fully reflects future values of profitability, is also acceptable for insurance, because the development of the insurance market, in general, is characterized by certain stability.
The extent and the nature of the relationship between the securities, which is expressed by the coefficients of linear correlation, can also be calculated for the types of insurance.
The Markowitz model is based on the fact that indicators of the profitability of securities are interdependent, which means that one can observe an increase in profitability of one type of securities with a simultaneous increase for other securities, while the third one remain the same, and the fourth one is generally reduced. This situation is inherent to the insurance market, where the profits of insurance products may grow in some categories and decrease in others.
Therefore, in our opinion, the Markowitz model in its basic approach is fully acceptable for selecting the optimal portfolio of insurance services.

Results
For the analysis, we collected statistical data on the voluntary types of insurance, which were provided by insurance companies in the period 2005-2015.
Indicators of insurance companies engaged in life insurance were not included in the analysis due to the nature of their activities, limited types of their insurance, which are present in statistical reporting of the relevant insurers. The preliminary analysis of state insurance and compulsory insurance has shown that these types are not connected to significant insurance claims and provide very high profitability. The paper does not contain relevant statistical data. It can be viewed on the website of the National Commission for the State Regulation of Financial Services Markets (Doff, 2011).
Based on the above-mentioned arguments, the paper will focus on the voluntary types of risk insurance. In order to assess their profitability, we will use the data of net income of insurance premiums ( Table 1, see Appendix) and net insurance payments ( Table 2, see Appendix). The preference is given to them, not to the gross indicators, because the net values of indicators are adjusted for the amounts, which are actually withdrawn from the insurance company through operations with a partial transfer of risks to other insurers as a result of reinsurance.
The types of voluntary insurance have undergone a procedure of additional selection by excluding the types of insurance with zero (or actually zero) indicators of the obtained net insurance premiums and insurance payments. As a threshold indicator of the selection we have taken 1% of the total amount of premiums and payments. This makes it possible to avoid working with such insurance services, which actually do not have volumes or the market of their consumption is limited to a small number of customers. These types of insurance are shown in Tables 1, 2 in gray color  (see Appendix).
What is left includes 11 types of voluntary insurance. For each of them, we have made calculations for the indicators of profitability in the period 2005-2015 (Table 3, see Appendix).
Calculations of profitability of a specific type of insurance are made according to formula (1) and the obtained results are presented in Table 3 (see  Appendix).
where -profitability of the i-th type of insurance; -net insurance payments according to the i-th type of insurance, UAH; -net income of insurance premiums according to the i-th type of insurance, UAH.
Indicators of profitability of the selected types of insurance were used in the Markowitz model (1, 2), which is formalized by formula 2.
where w i -the share of the i-th type of insurance in the portfolio of insurance services; r i -profitability of the i-th type of insurance; N -number of the types of insurance; cov ab -coefficient of covariance between the pairs of specific types of insurance; req -the maximum allowable risk of the insurance services portfolio.
Other options for formulating the problem are also possible, namely, the achievement of minimal risks at a certain level of profitability. However, in this article, we confine ourselves to maximization of profitability that can be achieved at different values of the general risk of insurance services portfolio.
For each type of insurance, we calculated indicators that are necessary for the formalized model, namely the average value of profitability, variance and standard deviation, which are actually a normal risk for each type of insurance (Table 4). These calculations are made on the basis of the profitability data of a particular type of insurance in the period 2005-2015 presented in Table 3 (see Appendix).  For the final use of the model, it is necessary to obtain the values of pairwise covariance coefficients for all types of insurance services, which was done by using the standard functions of MS Excel 2010.
The obtained results were presented in matrix form (Table 5), where the numbers from 1 to 11 are the symbols of the corresponding types of insurance from Table 4. The search of the optimal portfolio of insurance services is done by using the functions of MS Excel 2010 "Search of Solutions" from the packet "Analysis" (Markovitz, 1952). In doing so, restrictions were imposed in accordance with model (2). The values of the specific weight of a particular type of insurance were limited to 15%, which, in our opinion, will make it possible to include the maximum number of selected insurance services. The search of a decision was conducted for a class of nonlinear problems by using the method of generalized consolidated gradient. Two variants of optimized portfolio were proposed: according to the criterion of profitability maximization (3) and the criterion of risk minimization (4). The obtained results are presented in Table 6. (4)  The obtained results demonstrate that the indicators of profitability and riskiness of portfolios are almost identical for the portfolio of profit maximization and the portfolio of risk minimization. A characteristic feature of these portfolios is the exclusion of such types of insurance as insurance of financial risks and ground transport insurance.
They are risky and do not give high levels of profitability. Within the existing restrictions, the portfolio includes such types of insurance as accident insurance, cargo and baggage insurance, insurance from fire risks and risks of natural disasters, property insurance and third-party liability insurance.

Conclusion and recommendations
One could argue that the obtained results, like the used methodology, can be used by the managers of insurance companies or developing their strategies. The choice of the voluntary types of insurance gives insurers sufficient flexibility not to conclude highrisk insurance contracts or, in some cases, an opportunity to transfer reinsurance contracts. It is clear that the realities of specific regions, the existing developments, customer base and additional risks, which are inherent to a particular type of insurance, may affect the final choice, and the final version of insurance services portfolio may differ from the one obtained in this article, but certain guidelines could prove to be quite useful.