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
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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
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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.
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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
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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.
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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
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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
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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
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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
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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
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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.
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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,
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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
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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
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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
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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
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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:
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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
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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;
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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
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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
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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
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