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Vectorial Risk Model (VRM)Please refer to: Livvarcin, O. and Fikes, L.T., A New Mathematics Based Model Proposal in Risk Management: Vectorial Risk Model (VRM), Submitted to Journal of the School of Business Administration, Istanbul University, 2009 Vectorial Risk Model (VRM) As a natural variable of business world [22], risk is not only hard to measure but also hard to understand [23]. On the other hand, reliable measurement of risk and its appropriate perception by decision makers is very crucial for successful risk management. The purpose of this study is to support both business world and risk researchers by developing a valid, trustable, and simple mathematical model. This research assesses risk as a measurable variable and in addition to several mathematical calculations it also provides a visual expression for simplicity and intelligiblity. The proposed Vectorial Risk Model (VRM) as coined by the authors may be applied to all ventures of a company as well as a particular organizational activity. Simplification of a multidimentioanl and complex concept such as risk has always been the main focus of many researchers working on risk management [24]. Comprehensive approach in models that are to be used in risk management is very important from the perspectives of validity and applicability [25]. In the proposed model, risk is expressed as a vector on a two dimensional coordinate system with gain and loss axis as illustrated in Figure 1. This vector is coined as risk vector. The projection of risk vector on horizontal (loss) axis is called as �loss rate� and abbreviated with letter �l�. Similarly the projection of risk vector on vertical gain axis is called as �gain rate� and abbreviated with letter �g�. Loss rate (l) is the multiplication of maximum loss with the probability of its occurrence whereas gain rate (g) is the multiplication of maximum gain with the probability of the occurrence of gain.
Figure 1 Vectorial Risk Model, balance angle and risk vector The proposed model may also provide solutions in scenarios where multible gain and/or loss alternatives are the case. Generic equations for multible gain and loss occasions are provided below:
g : Gain rate l : Loss rate G : Maximum possible gain from an occasion L : Minimum possible loss from an occasion N : Number of unfavorable occasions M : Number of favorable occasions O : Bir durumun oluşma ihtimali The direction and the magnitude of the risk vector are dependent on the expected gain, potential loss and their probabilities to occur. Gain and loss rates in (for example financially) low level organizational activities will also be low and cause the magnitude of risk vector to be small. The angle between the risk vector and the gain axis is symbolized as shown as θ and named as �Balance Angle�. Balance angle expresses the overall risk in a certain organizational activity and calculated by the followinf formula:
Although it does not have to be linear there is always a positive correlation between risk and balance angle. Balance angle basically represent the ratio between gain rate (g) and loss rate (l). When loss rate increases, balance angle increases as well. Increase in loss rate (l) also indicates augmentation in risk. Two sample scenarios are illustrated in Figure 2. In the first scenario demonstrated in Figure 2-(1), loss rate (l) has relatively high value with respect to gain rate (g), points out to a high risk situation with big balance angle. On the contrary, in the second scenario shown in Figure 2-(2) gain rate (g) has higher value then loss rate (l). Thus, the second situation embraces less risk and represents smaller balance angle.
Figure 2 Wide and narrow balance angles To apply VRM to an organizational activity involving risk certain actions need to be taken. First, variables including possible maximum gain, possible maximum loss, and their probabilities, need to be determined by measurements, calculations or if none is possible by expert supported estimations. Second gain and loss rates are to be calculated and plotted on the two dimentional graph of VRM. Risk vector, as the third step, requires to be drawn by cross connecting the lines perpendicular to gain and loss rates. The vector starting from the origin of the coordinate system and ending at the intersection of perpendicular lines is called as the risk vector as demonstrated in Figure 1 and Figure 2. VRM might be used by various organizations for various scenarios. Term gain rate (g) and loss rate (l) coined in this study does not necessarily indicate only financial variables but also non-financial benefits and detriments. Although only the basic concepts of the proposed model are introduced in this paper, VRM may be applied on all organizations. It only needs relatively minor efforts for the adjustment of the case including the determination of gain/loss dimensions and probabilities. 1.1 Sample ScenarioA lottery might be used as a simple but sufficient example both to explain the basic concepts VRM and to demonstrate its applicability. Since parameters such as probability of winning or loosing, prizes, ticket prices are known, lottery was a fit sample. It eliminates ambiguity and establishes common understanding. As illustrated in Figure 3, three different lottery scenarios are assessed.
Figure 3 Sample scenarios (1) Gain and loss rates are equal (2) Gain rate is less than loss rate (3) Loss rate is less than gain rate In the first scenario, there are total of 100 tickets and each is sold for 1 TL. There is only one winning prize of 100 TL. Thus the possible maksimum loss (L) is 1 TL with the possibility of 99 %. Similarly the possibility of winning possible maksimum gain (G), 99 TL (1 TL ticket price is subtracted from the 100 TL winnign price) is 1%. By using these G and L values; gain rate (g), loss rate (l), and balance angle is calculated as follows:
In contrast to the huge difference between the maximum gain (99 TL) and loss (1TL), the values of gain rate (g) and loss rate (l) are calculated to be equal and 0.99. The reason is the high ratio between propabilities of loss and gain. The chances to win are only 1% whereas the probability of loss is 99%. As a consequence of this equality between gain rate (g) and loss rate (l), the balance angle is caluculated to be 45�. In the second scenario the prize of the lottery is reduced to 30 TL where ticket prices and total number of tickets are kept unchanged. Since maximum loss (1 TL) and its propobility did not change with respect to the first scenario, the loss rate (l) did not change either. The probability of winning the lottery prize did not change either however due to the change in the prize both gain rate (g) and balance angle changed as calculated below. The decrease in the gain rate increased the ratio between loss rate (l) and gain rate (g) and correspondingly the balance angle.
In the last scenario illustrated in Figure 3, the lottery company reduced the ticket price in half as a campaign. Additionaly the company announced one 50 TL and five 10 TL prizes instead of a single 100 TL winning prize. With reference to these new regulations, gain rate (g), loss rate (l) and balance angle is calculated as follows:
As seen in this sample, balance angle has the highest value in the second occasion. Further analysis will indicate that the scenario with the highest value of risk is also involved in this occasion. On the contrary, the risk is least in the third occasion where the balance angle also has the smallest value. The relationship signifies positive correlation between the amount of risk and the balance angle. Either profit oriented or non-profit, all kinds of organizations that have to face risk, may use VRM as an effective tool for the clearly manifestation and comprehensive assessment of risk, by adopting VRM to their risk involving activities. The most difficult part for this alignment might be the determination of risk according to the criteria of VRM. However, adequate understanding of the basic concept the proposed model will ease this process. 1.2 Change of Risk in TimeOne of the most important dimensions of risk management is coping with uncertainty which is also a responsibility that managers have to confront [26]. The responsibility of the success or failure of managers or organizations depends on how well this responsibility is managed [27]. There is no doubt that, the right actions bring the success to the manager or the organization while the unsuccessful ones can easily turn into a nightmare [28]. The consequences of wrong decisions or not deciding in time may be wider and more serious [2]. The Balance Angle concept can be used as a tool to overcome this problem by the managers of the organizations those who make the decisions. The board or the top management of an organization can set a predefined level for the balance angle in order to be able to empower the subordinates. The major issue of risk management is the time dependant change of risk and balance angle. Since, this change affects the decision making process while coping with risk. Risk simply shows three different behaviours in relationship with time. Alternatives are shown in Figure 4. The risk or balance angle may increase, decrease or stay constant in time. The change of risk does not have to be linear nor has to show the same pattern. In this paper, the change of risk will be considered to be linear for simplification while demonstrating the model. More complex forms of risk, in which risk may change parabolically, exponentially or some other way, are not included in this research, since the main purpose of this paper is to demonstrate the conceptual model. It is inevitable to deal with more complex forms of risk behaviours in future applications, where real time conditions apply.
Figure 4 Change of risk over time (1) Constant risk (2) Decreasing risk (3) Increasing risk The most critical issue in risk management is to monitor the change of risk and tailoring the decisions in accordance with the changing conditions. Managers often lose sight of risks. Their perception of risk can be influenced and their capabilities to evaluate opportunities may vary [29]. The frequency of reconsideration and deciding is directly affected because of the change pattern of risk. The more the risk ascending, the more frequent should be reevaluation and deciding. The opposite is true for descending risk situations. In order to be able to better understand the change of risk over time and evaluation periods, it is appropriate to use close to reality scenarios. As an example, three simplified activities of an international transportation company are analyzed for the explenation of the model. The first scenario expresses a situation where the company operates certain number of ships between Turkey and India periodically. The ships might be seized by pirates when passing through Somali coasts. As illustrated in Figure 5, due to existence of NATO forces in the area for protecting merchant ships the balance valu was at low, θ1 level. After a while NATO withdraw its forces due to various political reasons and risk caused by pirates increased. Accordingly the balance angle increased to a higher, θ2 level.
Figure 5 Constant risk and balance angle The areas indicated with numbers in Figure 5 are named as Decision Coverage Zones (A) and formulized as follows:
Δt, is the proposed time period between two comprehensive evaluation cycles. According to its own strategies and policies the company might have considered the period Δt adequately for the first condition. At the first condition, due to constant risk, the balance angle was also constant at θ1 level and tolerating Δt period of time between critical decision was enough. However after disappearance of NATO protection, due to the increase in the balance angle, previously determined Δt period will not be sufficient for an effective risk management. The company has to increase the frequency of his comprehensive evaluation cycles to be able to cope with increased risk in the area. For an effectual risk management, Decision Coverage Zones (A) are proposed to be kept constants even when the risk changes. As illustrated in Figure 5 in correspondence to the increased value of balance angle the company increased to frequency of evaluation and decision cycles to keep Decision Coverage Zones (A) the same. The areas of numbered zones in Figure 5 have all the same value. Since balance angle doubled the tolerated time period decreased to half. A scenario where risk decreases in time is illustrated in Figure 6. This scenario explains the situation where new sailors are hired. At the beginning, since there would very limited knowledge about the performance of the new employees, personnel related risk will be very high. Evantually while personnels experience increases the uncertainties related to their performance will decrease.
Figure 6 Decreasing risk and balance angle In such a scenario, organization needs to establish initially frequent and eventually rarer evaluation and decision cycles. For an effective risk management areas 1, 2, and 3 demonstrated in Figure 6 shall be equal. This preposition may be expressed mathematically as follows:
In real business applications, newly employed workers are generally inspected soon after they start working. If no problem is observed in the first inspection, managers will wait longer for the next inspection. This pattern will continue while the ambiguities about the employees vanish. This common practice proves the validity of the proposed model. It also proves the basic concepts of the proposed model have already been in use for long. Third and the last scenario, represents a situation where risk increases linearly in time. According to this scenario, as illustrated in Figure 7, while ships of the organization become older, the balance angle increases in correspondence to the probability of ship based faults and failures.
Figure 7 Increasing risk and balance angle Whenever risk increases linearly in time, the balance angle will also increase in parallel. At the beginning when balance angle is low, the organization has longer period of time for decision making. Since ships are relatively new at this period, the probability of any ship related failures is low. When time passes and ships become older, ship based failures will intensify. This increase in risk shortens the allocated decision periods for the organization. The organization needs to review its decisions and align itself to the conditions of new situation more often than before. For example, organization has to increase the frequency of ship inspections. Increasing the frequency of overviews or decreasing decision periods may continue until the balance angle reaches the predetermined threshold of the organization. This threshold is the maximum level that is considered as the limit by the organization in advance. When balance angle reaches to the threshold the organization has to take an action aither to reduce the balance angle or stop the activity. In this scenario, for example, the organization reduced the balance angle at threshold level to tolerable levels by acquiring a new ship. This meddling to risk illustrated in Figure 8 might be expressed as interference to risk.
Figure 8 Interference to risk when risk is increasing After interference, risk parameters including balance angle will change even if not the pattern of the risk. The changed balance angle will require new assessment of risk and new attitudes in risk management. In the sample scenario, for example, acquiring a same type new ship will significantly decrease the probability ship based failures and overall risk and correspondingly the balance angle. Since new ship is similar to the previous one, the risk patterns caused by ships� failures, as illustrated in Figure 8, would be similar too. After the acquisition the risk and the balance will again start to increase in time. Buying a new ship is not the only alternative to modify risk or balance angle. The overall maintenance of the ship would also decrease the probabilities of failures but not as much as the acquisition. Thus neither balance angle nor the pattern of risk would be the same. This situation illustrated in Figure 9 might be addressed as limited interference.
Figure 9 Limited interference to risk when risk is increasing Interference to risk might be the case also when risk is decreasing or constant in time. For example in our first scenario NATO�s decision not to protect Somali region is akind of interference to the originally constant risk. The risk will increas after the leave of NATO forces however, afterwards, it will again stay constant. Hiring new personnel for ships may be given as a sample for interference in decreasing risk scenarios. Uncertainties on new personnel will increase the risk to a certain point as illustrated in Figure 10. After the interference the ambiguities on the new personnel will diminish and the risk will continue to decrease in time.
Figure 10 Interference to risk when risk is decreasing
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