Your SlideShare is downloading. ×
  • Like
Nancy Beneda 1
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×

Now you can save presentations on your phone or tablet

Available for both IPhone and Android

Text the download link to your phone

Standard text messaging rates apply
Published

 

  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
294
On SlideShare
0
From Embeds
0
Number of Embeds
0

Actions

Shares
Downloads
2
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. MANAGING AN ASSET MANAGEMENT FIRM’S RISK PORTFOLIO Nancy Beneda1 Vaaler Insurance Fellow Associate Professor, Department of Finance, University of North Dakota, Box 7096, Grand Forks, ND 58202-7096, USA 1 University of North Dakota Box 7096 Grand Forks, ND 58201 (701) 777-4690 fax (701) 777-5099 nancy.beneda@und.nodak.edu MANAGING AN ASSET MANAGEMENT FIRM’S RISK PORTFOLIO 1
  • 2. Abstract: This paper presents a simplified model for quantifiably measuring and managing various types of risk, as a portfolio of risks. An asset management firm may face a variety of risks due to the broad nature of various investments. The technique utilizes computerized simulation and optimization modeling. The software used to administer the simulations is Crystal Ball. The use of simulation allows risk managers to combine various categories of risk, a firm faces, into one risk portfolio. These techniques will enable risk managers to have the information needed to achieve the desired level of overall firm risk and the expected cost of managing risk. The firm’s overall risk metric selected for use in this paper is the standard deviation of after-tax operating earnings. 1. INTRODUCTION A primary objective of risk management is to preserve the operating effectiveness of the organization. The focus is to ensure that the organization is not pre- vented from achieving its objectives of earning a profit and maximizing the wealth of the 2
  • 3. stockholders. In recent years, there has been considerable discussion about the potential shift toward enterprise risk management, which would bring together the management of all risks: financial, pure (traditionally insured hazards), operational, and strategic risks into a single risk portfolio. The use of enterprise risk management may be especially useful to asset management firms. Innovation and growth are common characteristics of asset management firms. Further, asset management firms may be involved in a broad range of investing activities in various economic and business segments. Firms, which are expanding either into new markets or new product areas may have a higher degree of operational and strategic risks. Further these firms will want to evaluate the effect of new investment projects on overall firm risk. Being able to more accurately measure the total risk, which a firm faces, will result in a better understanding of the extent to which the firm will be able to handle new speculative projects. Further, if a firm is able to lessen the current risk it faces, it may have greater latitude in the speculative risks it can undertake. This paper presents a simplified model for quantifiably measuring and managing the overall risk of an asset management firm by using computerized simulation and optimization modeling. The firm’s overall risk metric selected for use in this paper is the standard deviation of after-tax operating earnings. The software used to administer the simulations is Crystal Ball. The use of simulation allows risk managers to combine the various categories of risk, a firm faces, into one risk portfolio. These techniques will enable risk managers to have the information needed to achieve the desired level of overall firm risk and the expected cost of managing risk. 3
  • 4. The rest of the paper is organized as follows. Section 2 includes a description of the methodology. Section 3 provides a description of the hypothetical situation and includes a description of the types of risk that the hypothetical asset management firm faces. Section 4 provides the results of the simulation and optimization modeling and reports the after-tax operating earnings and standard deviation of operating earnings under various risk management decision scenarios. Section 5 provides concluding remarks. 2. METHODOLOGY Several techniques and concepts which are currently included in various literature sources are combined in this paper, to develop a methodology of measuring a firm’s overall risk. Some of these techniques include 1) risk categorization (i.e. dividing firm risks into various components such as financial, pure, strategic, and operational), 2) simulation modeling, 3) value-at-risk, and 4) optimization modeling and portfolio theory. In this paper the standard deviation of after-tax operating earnings is the firm’s overall risk metric. Rather than calculating a unique value for after-tax operating earnings, simulation modeling is used to create a probability distribution. Assumptions regarding each of the four categories of risks (i.e. financial, pure, strategic, and operational) are developed and incorporated into the model. Assumptions about model inputs include type of probability distribution, range of possible occurrences, and/or volatility of possible occurrences. The assumptions are used in the simulation to create the possible outcomes of after-tax operating earnings and the probability distribution of 4
  • 5. outcomes. Simulation modeling is simply an advanced form of sensitivity analysis in which a probability is attached to each possible outcome. Risk Categorization and Components of Overall Risk Generally the major risks a firm faces can be categorized into one of the following risk categories (Vaughan and Vaughan, 2003; Harrington and Niehaus, 2004): 1. Financial risk – controllable 2. Pure risk - controllable 3. Operational – uncontrollable 4. Strategic – uncontrollable Generally, financial risks and pure risks are considered to be manageable in the sense that loss financing techniques can be used to mitigate them. Examples of financial risk include interest rate risk, commodities risk, and foreign currency exchange risk and generally are managed by using futures or options contracts. Pure risk generally includes loss of property or a required payment of cash due to different types of liabilities. These types of hazards are generally managed through the purchase of insurance. Risk reduction techniques may also be used to manage financial and pure risks. For example sprinkler systems might be installed to reduce the severity of damage caused by fire. Another example is the installation of safety regulations to prevent worker injuries. Examples of strategic risks include product obsolescence and increased competition. Examples of operational risks include increasing cost of operations or input shortages. Generally loss financing techniques, such as futures, options, and insurance are not commonly used for managing strategic and operational risks. Operational and strategic risk reduction may be achieved from making appropriate choices about which products to produce and which markets to enter. 5
  • 6. In this paper it is illustrated how overall firm risk can be reduced by using loss financing techniques to manage financial risk and pure risk. An asset management firm with a high degree of operational and strategic risk will want to reduce financial risk and pure risk to a greater degree than one with low operating and strategic risk. The techniques presented in this paper illustrate how risk managers can quantifiably measure overall risk and risk reduction from loss financing. Simulation modeling and probability distributions Simulation is the process of building a mathematical or logical model of a system or a decision problem, and experimenting with the model to assist in solving the decision problem (Powell and Baker, 2004; Evans and Olson, 2002). Simulation modeling is an alternative to deterministic modeling. With deterministic models, input and output variables are fixed. No uncertainty can be built into a deterministic model. If a risk analyst is able to make assumptions concerning the shape of the distributions, of the revenue and expense items, which are affected by various types of risks, computer simulation can be employed to estimate the probability distribution of total operating earnings. Under such assumptions the model outputs will not have a unique value, but rather will be characterized by a probability distribution. Knowing the probability distribution of outputs provides insights into the risks involved in making decisions about purchasing futures contracts or purchasing insurance. The technique of simulation modeling is especially useful when the probability distributions of the input variables are not “normal.” Many input variables do not follow a normal distribution. For example, a Poisson distribution is used in many cases to represent the distribution of expected frequency of losses during a given period. The 6
  • 7. Poisson distribution describes the number of times an event occurs in a given interval, such as the number of telephone calls per minute or the number of errors per page in a document (Powell and Baker, 2004). The Poisson distribution is used in this paper to represent frequency of customer liability lawsuits. A lognormal distribution is used to represent the distribution of expected average loss severity. The lognormal distribution is widely used in situations where values are positively skewed or where most of the values occur near the minimum value (Powell and Baker, 2004). This type of distribution is common for security valuation or in estimating accident severity, in which the value cannot fall below ‘zero.’ In this paper the lognormal distribution is used to describe the possible outcomes for lawsuit severity. The ability to make model assumptions about the probability distribution of an uncertain input variable (i.e. unit price or expected foreign currency exchange rate) is the essence of simulation modeling. Unless a large number of exposures are present, the true distribution of total losses will likely exhibit positive skewness and the difference from the “normal” distribution could be substantial. In this case the normal distribution will tend to understate the probability of large losses. As a consequence the firm will underestimate the likelihood of large potentially disruptive losses. The software used to administer the simulations is Crystal Ball. The type of simulation modeling used is Monte Carlo simulation, which is a sampling experiment whose purpose is to estimate the distribution of an outcome variable that depends on several probabilistic input variables. Thus, it is not necessary to rely on the assumption that total earnings are normally distributed. Value-at-Risk 7
  • 8. It is useful to apply the concept of value-at-risk, when evaluating the overall risk of an enterprise (Vaughan and Vaughan, 2003; Harrington and Niehaus, 2004). Value-at-risk is simply an alternative technique used to describe a probability distribution for the value or earnings (losses) of a firm (or portfolio). Value-at-risk is used in addition to developing an average for an outcome and the associated probability distribution. A risk manager might be interested in determining the probability of certain critical events occurring, such as the probability of negative profits. For example, suppose a probability distribution exists for a random variable, such as a firm’s after-tax operating earnings. One might describe the probability distribution as “the value-at-risk for this year’s earnings is $10 million at the 5 percent level.” This statement could be interpreted to mean that the probability that the firm will have a loss greater than $10 million is 5 percent”. Simulation modeling is used in this paper to determine values-at -risk for different risk levels. This information is also helpful to risk managers in making decisions about how much risk financing to incur. Optimization Modeling and Portfolio Theory Optimization is a technique which attempts to maximize or minimize an objective function, by changing the values of the decision variables, which are subject to one or more constraints. The technique is similar to achieving an optimal portfolio (Bodie, et al. 2002). Portfolio theory suggests that optimal allocations of a pool of money exist which maximizes the targeted expected return, given a specified portfolio variance. In the example selected for this paper, the objective is to maximize the after-tax operating 8
  • 9. income given a specified level of standard deviation of after-tax operating income, for a hypothetical enterprise. Obtaining optimal values generally requires that you search in an iterative or ad hoc fashion. This involves running a simulation for an initial set of values, analyzing the results, changing one or more values, re-running the simulation, and repeating the process until you find a satisfactory (optimal) solution. This process can be very tedious and time consuming and it is often not clear how to adjust the values from one simulation to the next. Computerized optimization overcomes the limitations of the ad hoc and the enumerative methods by intelligently searching for optimal solutions to optimization problems. Once an optimization problem is described (by selecting decision variables, identifying the objective, and imposing constraints and requirements), the computer- generated optimization and simulation software is invoked to evaluate the simulation model for different sets of decision variable values. This is an iterative process that successively generates new sets of decision variable values, until an optimal solution is found. 3. HYPOTHETICAL SITUATION A hypothetical asset management firm is used to illustrate the procedure. In this example, the overall after-tax operating income is used for measuring the value- at-risk for all of the various investments. In this simplified illustration it is assumed that the firm has two investment activities and has risk in all four of the specified categories: financial, operational, strategic and pure risks. The assumptions (i.e. type of distribution, range, and/or standard deviation) for each of the uncertain 9
  • 10. variables in the model are presented in Schedule 1. The month of January, 2004 is arbitrarily selected as the reporting month. It is further assumed that the firm is located in the US, reports all of its revenues and expenses in US dollars, and pays only US taxes at a tax rate of 35%. Investment Activity A - Financial Risk (Foreign Currency Exchange Risk) and operating risk The financial risk the hypothetical company faces is foreign exchange risk. The expected annual sales for the upcoming month are 20,000 units, which are sold to a company located in Canada. All the units sold to the Canadian firm are contracted in Canadian dollars on the transaction date of December 15, 2003. However the Canadian dollars will not be received by the hypothetical asset management firm until January 31, 2004. In other words the amount of Canadian dollars is determined on December 15, 2003 and deliverable on January 31. Foreign exchange risk results because it is not known what the exchange rate will be on January 31, 2004. The company may however choose to hedge the foreign currency risk with futures contracts. There are two uncertain variables which describe the foreign currency risk this company faces. If the company does not hedge, the uncertain variable is the expected price per Canadian dollar on January 31, 2004. If the company hedges with futures contracts, the uncertain variable of concern is the expected basis on January 31, 2004. Schedule 1, Panel A presents the assumptions concerning expected price per franc and expected basis. These assumptions were developed based on an analysis of foreign exchange rates and bases, obtained from the Wall Street Journal, over the period, January 10
  • 11. 1990 through December 2003. In this example a March 2004 futures contract is entered on December 15, 2003 and closed out two months early on January 31, 2004. Thus the historical data to evaluate basis risk, included the futures price and exchange rate, which occur two months prior to each of the four contract maturities of March, June, September, and December. Schedule 1 also shows the computation of the price per Canadian dollar when futures contracts are used. This calculation incorporates the futures basis. Another source of risk related to Investment activity A is operating risk, which reflects the volatility of its Cost of Goods Sold (COGS) and operating expenses. See Schedule 1, Panel A. It is assumed that the probability distribution which best represents the outcomes for operating costs is the normal distribution with a standard deviation of $192,500. Investment Activity B Pure Risk (Customer Liability Lawsuits) and Strategic Risk Another major risk the company faces is potential customer liability lawsuits. Generally, in a situation in which there is a significant number of losses per period, the expected loss can be calculated as: 1) Expected loss = expected frequency of losses * expected average loss severity In this formula, as long as the distributions of both expected frequency of losses and expected average loss severity are ‘normal,’ the formula works accurately in estimating an expected loss in a static scenario. However, typically neither one of the distributions for these variables exhibit characteristics of a normal distribution. Further it is difficult to estimate a probability distribution of the expected losses without simulation modeling. 11
  • 12. In this paper a Poisson distribution is used to represent the distribution of expected frequency of losses and a lognormal distribution is used to represent expected average loss severity. Historical data on frequency and severity can help a company estimate future losses. In this hypothetical example, the expected frequency of customer liability lawsuits is two and the expected severity per lawsuit is $320,000. Thus, using a deterministic model, the total expected losses from customer liability lawsuits for January 2004 is $640,000. However, as will be shown, simulation modeling produces quite different results. See Schedule 1, Panel B for a description of these uncertain variables. The other two sources of risk for Investment Activity B are strategic and operating risk. The strategic risk represents the volatility of selling price and the Uniform distribution is used is used to represent the probability of occurrences. See Schedule 1, Panel B. The operating risk reflects the volatility of operating expenses, including COGS. It is assumed that the probability distribution which best represents the outcomes for operating costs is the normal distribution with a standard deviation of $127,500 4. RESULTS Deterministic Model A model in which the inputs are fixed is referred to as a deterministic model. Deterministic modeling precludes simulation modeling and the development of probability distributions. Schedule 2 presents a schedule of the deterministic computation of the expected after-tax operating income for Investment activities A and B, and overall. In addition to being deterministic, the computation of after-tax operating income also assumes that no loss financing is being used to manage risk. In other words, 12
  • 13. in Schedule 2, it is assumed that no insurance is purchased and no futures contracts are used to hedge the foreign currency risk. Investment Activity A includes foreign revenues in Canadian dollars and operating expenses. The current price of the Canadian dollar on December 15, 2003 is $0.8121. Thus, the Canadian dollars (CD) to be received on January 31, 2004 from the December transaction are fixed in the amount of 2,462,750.89 CD. This is calculated as revenues in dollars divided by the current exchange rate: $2,000,000 / $0.8121 = 2,462,750.89 CD Since the expected exchange rate, on January 31, 2004, is 0.8139, if the position is left open the foreign revenues are projected to be $2,004,433. This is calculated as the fixed number of Canadian dollars times the expected exchange rate: 2,462,750.89 CD * $0.8139 = $2,004,432.95 Also included in Investment Activity A is expected cost of goods sold (COGS) and other operating expenses of $1,925,000. Investment Activity B includes local revenues in US dollars, operating expenses and losses from customer lawsuits. Operating expenses for Investment B are estimated to be $1,275,000. The expected losses from liability lawsuits is $640,000 (deterministic outcome), which is the expected frequency (two) times the expected severity per lawsuit ($320,000). Simulation Modeling Simulation Modeling is used to produce a more accurate reflection of after-tax operating income. The uncertain parameter information and probability distributions (presented in Schedule 1) are incorporated into the simulated model. The uncertain input 13
  • 14. variables include price per unit in US dollars, expected Canadian exchange rate, expected futures basis, operating expenses, expected frequency of lawsuits, and expected severity of lawsuits. The computerized simulation program then creates a probability distribution for the output variable, the after-tax operating income (see Schedule 3). The results presented in Schedule 3 represent a quantifiable measure of risk, assuming the hypothetical firm uses no loss financing (i.e. no futures contracts and no insurance). The probability distribution for after-tax operating income can be presented in several ways. Schedule 3 presents the statistics (Panel A), the frequency chart (Panel B), the percentile ranges (Panel C), and several percent levels for values-at risk (Panel D). Schedule 3, Panel A presents the mean, median, standard deviation and skewness. The standard deviation of $859,983 is quite large, almost 6 times the expected after-tax operating earnings of $146,575. Notice that the simulation results in an expected after- tax operating income which is different from the result obtained from the deterministic model. This results from the variation of probability distributions of the input variables. Panel B of Schedule 3 presents a frequency chart, which is simply a picture of the frequency of the outcomes. Panel C of Schedule 3 presents the ranges of outcomes by quartile. Panel D presents various values at risk. For example the value at risk at the 5% level is $1,400,000. In other words there is a 5% probability that the firm will have a loss greater than $1,400,000. Simulation Modeling and the affects of Loss Financing A simulation similar to that reported in Schedule 3 was run, in which the hypothetical losses are completely financed. See Schedule 4. In this simulation the foreign currency risk is completely hedged using futures contracts and the customer 14
  • 15. liability lawsuits are fully insured. In this scenario basis risk replaces foreign currency risk. Using futures contracts the expected price of the Canadian Dollar on January 31, 2004 is $0.8125, calculated as shown in Schedule 1 Panel A. The results of this simulation indicated a much lower standard deviation of $306,721. However the lower risk is not without cost. The expected after-tax operating income was lower as well, $93,196 under this scenario. The ranges also tightened up substantially. The maximum loss drops from $11.4 million to $4.6 million. Further, under this scenario, the value-at risk at the 5% level is only $370,000 versus $1,400,000 under no loss financing.. Simulation Modeling and Optimal Risk Financing Schedule 5 illustrates the results of utilizing optimization software (i.e. Crystal Ball OptQuest) to determine how much hedging and insurance should be utilized to achieve a specified standard deviation. If a firm has an idea of how much risk can be incurred by the firm, a risk level can be specified. Assume that management feels that a standard deviation of after-tax operating income of $400,000 could be tolerated if profits were sufficient. The firm would like to know what level of after-tax operating income could be achieved given a standard deviation equal to $400,000. The decision variables are how many Canadian dollars to hedge and how much insurance to purchase. At the specified level of risk (i.e. standard deviation = $400,000), Crystal Ball OptQuest identifies the optimal number of futures contracts as eleven. This results in 1,375,000 Canadian dollars being hedged, calculated as 11 * 125,000 (contract size). The optimal amount of insurance identified by 15
  • 16. OptQuest is $580,162. Running a new simulation under these assumptions yields the results as shown in Schedule 5. The after-tax operating income, of $125,234 is only a little higher than that when full loss financing is utilized, but less than when none is used. The maximum loss of $6.9 million, and the value-at risk at 5%, of $0.9 million, are also mid-range values. 5. CONCLUSION This paper presents a simplified model for quantifiably measuring and managing the overall risk of a firm as a risk portfolio, using computerized simulation and optimization modeling. The software used to administer the simulations is Crystal Ball. The use of simulation allows risk managers to analyze the impact of risk management decisions on overall firm risk. These techniques will enable risk managers to have the information needed to achieve the desired level of overall firm risk and the expected cost of managing risk. Enterprise risk management brings together the management of all risks: financial, pure (traditionally insured hazards), operational, and strategic risks into a single risk portfolio. The use of enterprise risk management is especially useful to firms which are highly innovative. Firms, which are expanding either into new markets or new product areas may have a higher degree of operational and strategic risks. If a firm is able to lessen the current risk it faces, it may have greater latitude in the speculative risks it can undertake. REFERENCES 16
  • 17. Bodie, Zvi, Alex Kane, and Alan Marcus; 2002, Investments, McGraw Hill, New York, New York. Evans, James R. and David L. Olson; 2002, Simulation and Risk Analysis, Prentice Hall, Upper Saddles River, New Jersey. Harrington, Scott E. and Gregory R. Niehaus; 2004, Risk Management and Insurance, McGraw Hill Irwin, New York, New York. Powell, Stephen G. and Kenneth R. Baker; 2004, The Art of Modeling with Spreadsheets, John Wiley & Sons, Inc., New York, New York. Vaughan, Emmett J. and Therese Vaughan; 2003, Fundamentasl of Risk and Insurance, John Wiley & Sons, Inc., New York, New York. Schedule 1 Information about four types of risk: 17
  • 18. Panel A. Investment Activity A Financial Risk (foreign exchange risk) uncertain variables Expected basis -0.0024 Uniform distribution; max = -0.004; min = -0.008 Expect price per CD 1/31/2004 0.8139 Normal distribution; standard deviation = 0.13139 Computation of expected price of CD under futures contracts March futures price on Dec 15, 2003 0.8149 Known Expected basis (St –Ft) on January 31, 2004 -0.0024 Uncertain Expected price of CD under futures contract 0.8125 N/A Operational Risk (Volatility of operating costs and expenses) Other operating expenses $1.925mil Normal distribution; std dev = $192,500 Panel B. Investment Activity B Strategic Risk (expected volatility of unit price) Unit price of product $100 Uniform distribution; max = $110; min = $90 Pure Risk (customer liability lawsuits) Expected frequency of lawsuits 2 Poisson distribution; std dev = 1; min = 0 Expected severity per lawsuit $320,00 Lognormal distribution; standard dev = $700,000 0 Operational Risk (Volatility of operating costs and expenses) Other operating expenses $1.275mil Normal distribution; std dev = $127,500 Schedule 2 Deterministic modeling of expected after-tax operating income 18
  • 19. Investment Activity A Price per unit in US dollars $100.001 Units sold to Canadian firm 20,000 Current (December 15) price of unit in CD $0.8121 Fixed number of Canadian dollars to be received January 31, 2004 2,462,750.89 Expected exchange rate (January 31) 0.81391 Revenues (position open) $2,004,432.95 COGS and Other Operating Expenses -$1,925,000.001 Operating Income (A) $79,432.95 Investment Activity B Price per unit in US dollars $100.00 Units sold locally 20,000 Revenues $2,000,000.00 COGS and Operating Expenses -$1,275,000.001 Expected frequency of lawsuits next month 21 Expected severity per lawsuit -$320,000.001 Expected Losses from Customer Lawsuits -$640,000.00 Operating Income (B) $85,000.00 Total Operating Income (A and B) Operating Income $164,432.95 Taxes -$57,551.53 After-tax Operating Income $106,881.42 1 Variables in model which are uncertain Schedule 3 Simulation modeling of after-tax operating income; 19
  • 20. no risk controls in place Forecast: After-tax Operating Income Statistic Panel A s Trials 500 Mean $146,575 Median $287,534 Standard Deviation $859,983 Skewness -2.79 Panel B Frequency Chart Forecast: After-tax Operating Income 500 Trials Frequency Chart 481 Displayed Panel C Percentiles for after-tax operating income . 044 22 . 033 16.5 Percentile Range . 022 11 0% to 25% ($11,440,002) to ($42,788) 25% to 50% . 011 ($42,788) to $287,533 5.5 50% to 75% . 000 $287,533 to $596,187 0 75% to 100% ($1,768,605.88) ($953,652 00) . ($138,698. 12) $676, 255.76 $596,187 to 1,596,476 $1,491, 20 63 9. dollars Panel D Values at Risk Probability Value at risk1 5% ($1,400,000) 10% ($500,000) 15% ($260,000) 1 indicates that the after-tax operating loss will be greater than the indicated amount at the indicated probability 20
  • 21. Schedule 4 Simulation modeling of after-tax operating income Hedging in place and full insurance coverage Panel A Statistics Trials 500 Mean $93,196 Median $102,480 Standard Deviation $306,721 Skewness -2.03 Panel B Frequency Chart Forecast: After-tax Operating Income 500 Trials Frequency Chart 496 Displayed . 030 15 . 023 11.25 . 015 7.5 . 008 3.75 . 000 0 ($644,959. 70) ($300,947 98) . $43, 063.75 $387, 075.48 $731, 087.21 dollars Panel C Percentiles for after-tax operating income Percentile Range 0% to 25% ($4,581,327) to ($108,459) 25% to 50% ($105,900) to $102,480 50% to 75% $102,480 to $287,743 75% to 100% $287,743 to $972,952 Panel D Values at Risk Probability Value at risk1 5% ($370,000) 10% ($270,000) 15% ($200,000) 1 indicates that the after-tax operating loss will be greater than the indicated amount at the indicated probability 21
  • 22. Schedule 5 Simulation modeling of after-tax operating income Optimal hedging in place and optimal insurance purchased and SD <= $400,000 Panel A Statistics Trials 500 Mean $125,234 Median $135,859 Standard Deviation $374,293 Skewness -5.10 Panel B Percentiles for after-tax operating income Percentile dollars 0% to 25% ($6,921,728) to ($77,099) 25% to 50% ($77,099 to $135,859 50% to 75% $135,859 to 343,257 75% to 100% $343,257 to $1,134,372 Panel D Values at Risk Probability Value at risk1 5% ($905,000) 10% ($380,000) 15% ($230,000) 1 indicates that the after-tax operating loss will be greater than the indicated amount at the indicated probability 22