Insurance Abstract
A method for producing an insurance model includes the steps of
calculating insurance payouts, calculating daily call payouts, calculating
statistics of insurance payout obligations, selecting an insurance
premium, and selecting a hedge amount. One aspect of the present
invention is to determine a combination of insurance premium to
charge and a hedge amount to place using daily calls, or other such
financial instruments, such that some measure of return of the total
portfolio of the insurance policy meets some maximum or minimum
criteria.
Insurance Claims
What is claimed is:
1. A method for generating an insurance premium and hedge amount
for event risk insurance for a commodity, the method comprising
the steps of: receiving a price scenario input; receiving an event
risk scenario input; and generating an insurance premium and a hedge
amount in response to receiving the price scenario input and the
event risk scenario input.
2. The method of claim 1, further comprising the steps of: calculating
insurance payouts; calculating daily call payouts; and calculating
statistics of insurance payout obligations.
3. The method of claim 2, wherein the step of calculating insurance
payouts further comprises the steps of: calculating a total loss
for a first event risk; and calculating a potential insurance payout
by subtracting an event deductible from the total loss for the first
event risk.
4. The method of claim 3, wherein the step of calculating insurance
payouts further comprises the step of: calculating an insurance
payout by limiting the potential insurance payout to a maximum insurance
payout.
5. The method of claim 3, wherein the step of calculating insurance
payouts further comprises the steps of: calculating the total loss
for a subsequent event risk, wherein one or more prior event risks
have occurred; calculating a subsequent event risk deductible by
subtracting the total of deductibles paid for prior event risks
during a period from a maximum period deductible; calculating a
potential insurance payout by subtracting the subsequent event deductible
from the total loss for the subsequent event risk; and calculating
an insurance payout by limiting the potential insurance payout to
a maximum insurance payout.
6. The method of claim 2, wherein the step of calculating insurance
payouts incorporates dollar and megawatt deductibles and maximum
payouts.
7. The method of claim 2, wherein the step of calculating statistics
of insurance payout obligations is performed by examining the statistics
of: P=I+x.times.CA16.times.N.times.x.times.C.sub.p.
8. The method of claim 7, wherein the step of selecting an insurance
premium is performed using the statistics examined in the step of
calculating statistics of insurance payout obligations.
9. The method of claim 2, wherein the step of selecting an insurance
premium comprises the steps of: picking a measure of desired performance;
and calculating a minimum premium that will achieve a desired ratio.
10. A method for generating an insurance premium and hedge amount
for forced outage insurance for electric power, the method comprising
the steps of: receiving a price scenario input; receiving a unit
outage scenario input; and generating an insurance premium and a
hedge amount in response to receiving the price scenario input and
the unit outage scenario input.
11. The method of claim 10, further comprising the steps of: calculating
insurance payouts; calculating daily call payouts; and calculating
statistics of insurance payout obligations.
12. The method of claim 11, wherein the step of calculating insurance
payouts further comprises the steps of: calculating a total loss
for a first forced outage event; and calculating a potential insurance
payout by subtracting an event deductible from the total loss for
the first forced outage event.
13. The method of claim 12, wherein the step of calculating insurance
payouts further comprises the step of: calculating an insurance
payout by limiting the potential insurance payout to a maximum insurance
payout.
14. The method of claim 12, wherein the step of calculating insurance
payouts further comprises the steps of: calculating the total loss
for a subsequent forced outage event, wherein one or more prior
forced outage events have occurred; calculating a subsequent forced
outage event deductible by subtracting the total of deductibles
paid for prior forced outage events during a period from a maximum
period deductible; calculating a potential insurance payout by subtracting
the subsequent event deductible from the total loss for the subsequent
forced outage event; and calculating an insurance payout by limiting
the potential insurance payout to a maximum insurance payout.
15. The method of claim 11, wherein the step of calculating insurance
payouts incorporates dollar and megawatt deductibles and maximum
payouts.
16. The method of claim 11, wherein the step of calculating statistics
of insurance payout obligations is performed by examining the statistics
of: P=I+x.times.CA16.times.N.times.x.times.C.sub.p.
17. The method of claim 16, wherein the step of selecting an insurance
premium is performed using the statistics examined in the step of
calculating statistics of insurance payout obligations.
18. The method of claim 11, wherein the step of selecting an insurance
premium comprises the steps of: picking a measure of desired performance;
and calculating a minimum premium that will achieve a desired ratio.
19. A forced outage insurance model, comprising: a price scenario
input; a unit outage scenario input; and in response to receiving
price scenarios and unit outage scenarios, being operative to: generate
an insurance premium; and generate a hedge amount.
Insurance Description
TECHNICAL FIELD
[0001] The present invention relates to commodity industries and,
more particularly, to an approach to insure event risks while also
calculating an appropriate hedge amount.
BACKGROUND OF THE INVENTION
[0002] One of the most technological advancements that occurred
during the 20.sup.th century was the widespread use of electric
power. As power distribution moved from locally based generators
to a massive, intertwined and interconnected grid that spanned the
entire U.S. continent, industrial plants were set free from the
constraints of having to be established in close proximity to power
sources. During this time, electric power migrated from luxury,
to necessity and today, is currently traded on the open market as
a commodity by pioneers such as Mirant.
[0003] One risk that electric power producers face is a forced
outage of one of the producer's generation facilities. The main
risk of a forced outage is that the forced outage will occur while
power prices are high and the power plants will not be able to generate
electricity. This prevents the power producer from collecting the
associated profit from the power during this period of outage. In
addition, to supplement the loss of electricity, the producer must
seek an alternate source and pay a price determined by the market
at the time of the forced outage. Therefore, many power producers
are interested in purchasing insurance that will reimburse them
when their power plants experience a forced outage at the same time
that power prices are above a certain price. Insurance companies
are not able to issue insurance to cover a risk unless a model is
created that will predict the liability associated with an outage
of a power plant.
[0004] Therefore, there is a need in the art for an approach to
insure event risks while also calculating an appropriate hedge amount.
SUMMARY OF THE INVENTION
[0005] The present invention is directed towards solving the aforementioned
needs in the art, as well as other needs in the art, by providing
an approach to insure event risks while also calculating an appropriate
hedge amount using the underlying commodity, derivative of the underlying
commodity, or related commodity.
[0006] One aspect of the present invention is a method for determining
a combination of insurance premiums to charge and a hedge amount
to place using, commodity call options or such instruments, such
that some measure of return of the total portfolio of the insurance
policy meets some minimum or maximum criteria. The unique aspect
of this invention id finding the optimal hedge for a given commodity
such that there is a superior measure of return on the insurance
portfolio compared to other hedged or unhedged portfolios. The
present invention provides a method for valuating insurance policies
comprising (1) generation of price paths (for onpeak power), including
dependence of prices and outages, either from a fundamental asset
model, or from a combination hybrid lognormal/stackbased model;
(2) calculating insurance payouts, incorporating deductibles and
maximum payouts; (3) calculating daily call payouts; (4) calculating
various statistics of a composite portfolio consisting of insurance
payout obligations and daily call options; and (5) selecting an
insurance premium and a hedge amount.
[0007] These aspects and embodiments, as well as other aspects
and embodiments of the present invention are more clearly described
in the specification, figures and claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] FIG. 1 is a block diagram illustrating a typical system
architecture for a system for implementing an exemplary embodiment
of the present invention.
[0009] FIG. 2 is a flow diagram illustrating a method of producing
a forced outage insurance model according to an exemplary embodiment
of the present invention.
[0010] FIG. 3 is a flow diagram illustrating the calculation of
a forced outage insurance payout according to an exemplary embodiment
of the present invention in which deductibles and maximum payouts
are used.
[0011] FIG. 4 is a graph illustrating the process of identifying
the optimal premium  hedge combination that satisfies a given Sharpe
ratio Criteria.
[0012] FIG. 5 is an example of forward price scenarios (price paths)
that are used by a forced outage insurance model.
[0013] FIGS. 6a and 6b are an example of the unit outage simulation
scenarios (outage paths) that are used by a forced outage insurance
model.
DETAILED DESCRIPTION
[0014] Turning now to the figures in which like numerals represent
like elements throughout the several views, several exemplary embodiments
of the present invention are described.
[0015] This invention can be applied to any commodity market exposed
to event risks where hedging instruments are available in the commodity
or in a related commodity. Commodities that are covered include
but are not limited to electric power, natural gas, grains, agricultural
products, metals, petrochemicals, energy products such as heating
oil, jet fuel, crude oil and distillate products thereof, pulp and
paper, plastics, integrated circuit chips such as Dynamic Random
Access Memory (DRAM), and other traded commodities. The electric
power industry is used as an exemplary embodiment of the present
invention.
[0016] The present invention predicts the costs associated with
a forced outage and models insurance products designed to protect
power producers from the losses associated with a forced outage.
Typically, a forced outage is any outage or partial outage that
is not voluntary. A forced outage may be caused by, but is not limited
to an outage caused by, mechanical failure, environmental shutdown,
weather events, natural disasters, or lack of necessary resources.
In accordance with alternative embodiments of the present invention,
a company offering forced outage insurance may limit or expand the
definition of a forced outage and the operation of the present invention
is not limited to any one definition.
[0017] FIG. 1 is a block diagram illustrating an exemplary embodiment
of the present invention. The forced outage insurance model 110
interfaces with a stack model 105. The details of the operation
of the stack model 105 may vary among various embodiments of the
present invention. However, in an exemplary embodiment, the stack
model 105 receives forward market prices 115 and unit outage rates
120 and outputs price scenarios 125 and unit outage scenarios 130.
The price scenarios 125 and the unit outage rates 130 of the stack
model 105 are communicated to the forced outage insurance model
110 for processing. The forced outage insurance model 110 also incorporates
contract terms 135 and generates a suggested insurance premium 140
and hedge amount 145. Contract terms 135 refer to the terms of the
insurance contract. Contract terms 135 may include, but are not
limited to, policy limits, deductible amounts, definitions of forced
outages, and other terms generals specified in an insurance contract.
The insurance premium 140 refers to the price paid to obtain an
insurance policy for a specified insurance period. The hedge amount
145 refers to the megawatt (MW) quantity of daily calls that are
purchased in the power market to insure access to power at a given
price.
[0018] FIG. 2 is a flow diagram illustrating the operation of a
forced outage insurance model according to an exemplary embodiment
of the present invention. In an exemplary embodiment of the present
invention, the first step in determining the insurance premium 140
to be charged for forced outage insurance is the calculation of
the expected payoffs of the forced outage insurance by generating
price paths 205. Without deductibles and caps, the daily payoff
to the owner of forced outage insurance is:
16xCxmax(0, P.sup.onK)
[0019] where
[0020] P.sup.on is the average on peak power price for that day
[0021] K is the forced outage insurance strike price, and
[0022] C is the MW capacity lost to a forced outage or derating
of the power plant.
[0023] In a simplified exemplary embodiment of the present invention,
the owner of the forced outage insurance will receive the above
payoffs everyday during the duration of the insurance. In practice
however, and in accordance with an exemplary embodiment of the present
invention, one or more deductibles are usually written into the
forced outage insurance contract, which cause the insurance owner
to receive only the payoff in excess of a dollar denominated deductible,
and for the lost megawatts (MW) in excess of a MW deductible. The
owner of the forced outage insurance will receive only the payoff
in excess of the per event deductible, every time the plant experiences
a forced outage, until a period deductible limit has been reached.
Once the period deductible limit has been reached, the owner will
receive the full daily payoffs due to future forced outages. Period
deductible limits limit the total deductible paid for multiple events
within an insurance period. In an exemplary embodiment of the present
invention, the period is one year and the period deductible resets
at the end of the year. Alternatively, any length period may be
used. Megawatt deductibles are defined on an aggregate or franchise
basis. In the presence of an aggregate MW deductible, only lost
megawatts above the deductible are covered. If a MW deductible is
of the franchise type, all lost megawatts are covered if the losses
exceed the amount of the deductible. Finally, there is usually a
maximum amount of money that the owner of the forced outage insurance
can receive. Once the owner receives payoffs that sum to the forced
outage insurance maximum (period payout cap), the owner will receive
no more payoffs. In an exemplary embodiment of the present invention,
the period payout cap resets each insurance period (i.e. at the
end of each year). Alternatively, a lifetime cap may be used or
any other period length may be used to limit the total insurance
payout.
[0024] In an exemplary embodiment of the present invention, the
forced outage insurance does not take effect unless there is a forced
outage and the price of onpeak power is above a predetermined strike
price. In an exemplary embodiment of the present invention, the
strike price is the price of power above which the insurance policy
is in effect. Accordingly, if a power plant experiences a forced
outage while the price of power is below the strike price, no claim
may be made against the forced outage insurance policy. The strike
price of the forced outage insurance will typically be set around
150% to 200% of the onpeak forward power price. In an exemplary
embodiment of the present invention, forced outage insurance is
intended to insure against outages only when the price of power
is particularly high. Thus, the inclusion of a strike price more
accurately corresponds to the desired insurance. Since the present
invention deals with contracts for which insurable losses are set
above some high strike value, it is assumed that the insured unit
is needed for operation when the power demand is high enough to
cause the price to reach the strike value and, thus, in operation
when an insurable outage occurs. Thus, for the purposes of insurance
valuation, dispatch issues of whether a plant would have been running
when an outage occurred is ignored.
[0025] Those familiar with insurance contracts will understand
that the deductibles described herein are examples of typical deductible
formulation. Alternatively, any, or no, deductible schedule may
be used in accordance with the present invention.
[0026] In accordance with an exemplary embodiment of the present
invention, the calculation of the insurance payout 210 proceeds
as follows. First, power prices and unit outages over the insurance
term are simulated using a stack model and imported into the forced
outage insurance model. (See FIGS. 5 and 6ab). The number of simulation
runs (paths) is selected so that the resulting sampling error is
below a certain dollar threshold. For each particular simulation
path, for each day d, the following loss function is calculated:
L.sub.d=16xmax(0, CMWDeductable)xmax(0, P.sub.d.sup.onK)
[0027] where again C denotes the megawatts of lost unit capacity,
K is the insurance strike, and P.sub.d.sup.on is the average onpeak
power price on day d.
[0028] Next, the insurable events are determined. An insurable
event is defined to be a group of consecutive days that the power
generation unit is down or derated beyond the MW deductible, and
which results in a total loss (the sum of the losses over each day
in the event) that is greater than the current deductible. The current
dollar deductible is equal to the event deductible up to the value
of the period deductible. There can be a different deductible for
each unit in the portfolio. In an exemplary embodiment of the present
invention, the outages of each unit are simulated independently.
In such an embodiment, the cases in which two or more units are
down due to the same proximate cause are not considered. When two
or more units are down due to the same proximate cause, it is considered
one event from the perspective of the insurance contract. The current
dollar deductible must be distinguished from the event dollar deductible.
For each simulated path, a cumulative deductible is computed and
is equal to the sum of all deductibles paid up to the current time,
including losses below the current deductible. If the cumulative
deductible has not exceeded the maximum period deductible, then
the current deductible is either the per event deductible, or, if
smaller, the difference between the maximum period deductible and
the cumulative deductible. If the maximum period deductible has
been exceeded for that path, then the current deductible is zero.
[0029] For example, suppose there is an event deductible of $1,000,000,
and a maximum period deductible of $3,000,000. Then, suppose there
are three events, each of which is above the per event deductible.
For the next event, no deductible will be applied, since the maximum
period deductible has been satisfied. In other words, each subsequent
loss (for which the price is above the strike) will be entirely
covered.
[0030] For each outage event along a simulation path, there is
a corresponding current deductible. The potential insurance payout
is the difference between the loss due the event and the current
deductible, if the loss exceeds the latter. The word "potential"
is used, because if there is a cap or a maximum payout, then the
insurance payout just computed must be modified. For example, suppose
there is a cap of $100,000 and the amount that has already been
paid out in insurance losses is $95,000. Suppose an insurable loss
of $10,000 is incurred. In this case, the insurance company would
only pay $5,000 because paying the entire $10,000 would exceed the
cap. So for each insurable loss, the insurance company must check
Oust as in this example) how much must be paid out to remain within
the cap. Once the cap has been attained, then subsequent losses
are not insured.
[0031] FIG. 3 is a flow diagram illustrating the calculation of
a forced outage insurance payout according to an exemplary embodiment
of the present invention in which deductibles and maximum payouts
are used. Based on the terms defined above, the entire process may
be outlined as follows:
[0032] For each path,
[0033] 1. Starting with the first day of the coverage period, find
the first event, and calculate the total loss for this event 305.
[0034] 2. If the capacity forced out is less than the MW deductible
307, there is no insurance payout 309. If the capacity forced out
is greater than the MW deductible 307, determine if the total loss
is greater than the current deductible 310.
[0035] 3. If the total loss is greater than the current deductible
310, calculate the potential insurance payout by subtracting this
deductible from the total loss 315. Next, increment the cumulative
deductible (initialized to zero) by the current deductible, and
set the current deductible to MIN (per event deductible, maximum
cumulative deductiblecumulative deductible) 325. For example,
if the cumulative deductible is $2,500,000, the maximum cumulative
deductible is $3,000,000, and the per event deductible is $1,000,000,
the current deductible (the deductible for the next loss) would
be $500,000. If the total loss is less than the current deductible
310, there is no potential insurance payout, but the cumulative
deductible is incremented by the total loss 320.
[0036] 4. If the total insured losses have not been exceeded 335,
and the potential payout is less than the difference between the
maximum coverage and the current total payouts 330, make the entire
payout 345; otherwise, pay out the difference 350. If the total
insured losses have been exceeded 335, there is no insurance payout
340.
[0037] This procedure takes place for each path, and the total
expected payout is simply the average over all paths.
[0038] In an exemplary embodiment of the present invention, exposure
to loss is reduced by hedging. Hedging may be performed by purchasing
daily calls of power to assure access to a certain quantity of power
at a predetermined price. Assume that an insurance company sells
500 MW of forced outage insurance. Since this will leave them exposed
to some very large losses, a hedging strategy has to be constructed
to minimize the maximum loss. Forced outage insurance can be viewed
as a call option on the daily average onpeak power price, with
the additional condition that the insured power plant must be experiencing
a forced outage. Therefore, a straightforward way to hedge a short
position in forced outage insurance is to buy call options on the
daily average onpeak power price (daily calls). However, purchasing
500 MW of daily calls would not be an appropriate hedge because
the daily calls do not have the additional condition that the power
plant must experience a forced outage. Therefore, some hedge amount
less than 500 MW must be purchased so that the payoffs of the call
option more accurately match the payoffs of the forced outage insurance.
[0039] In an exemplary embodiment of the present invention, the
daily call payout is calculated 215 as follows: for each onpeak
day along a path, if the onpeak power price is above the call strike,
the payout is the price minus the strike, times 16 (onpeak) hours.
Since the number of hours is already incorporated and the total
payout is a running sum along onpeak days, the units of this output
is $/MW.
[0040] In an exemplary embodiment of the present invention, statistics
are examined to determine appropriate insurance hedges and premiums
220. Once the insurance and call payouts have been calculated for
each path, the next step is to examine the statistics of the following
quantity (the daily profit or loss of our total position):
P=I+x.times.CA16.times.N.times.x.times.C.sub.p
[0041] where
[0042] I is the insurance loss ($)
[0043] C is the call payout ($/MW)
[0044] x is the hedge amount (MW)
[0045] A is the insurance premium ($)
[0046] N is the number of onpeak days
[0047] C.sub.p is the price paid for the daily call ($/MWh)
[0048] Calculation of I involves parameters determined by the forced
outage insurance contract (e.g., insurance strikes) and the probability
of forced outage (determined from historical data), and C.sub.p
is a quoted price. Therefore, the only remaining free variables
are A, the daily forced outage insurance premium, and x, the size
of the hedge. (Note that the insurance and call payouts are summed,
for each path, over all the insured periods, e.g., the summer months
of some year.) The following statistics may be determined, as a
function of the hedge amount x and the insurance premium A:
[0049] Expected value of P
[0050] Standard deviation of P
[0051] Maximum loss (negative of the minimum over all paths of
P)
[0052] Expected loss (average over the negative part of each P)
[0053] "Sharpe" ratio: ratio of the expected value of
P to its standard deviation
[0054] Conditional Expected Profit to Conditional Expected Loss
[0055] Payoff to 95%: ratio of the expected value of P to the 95
percentile loss
[0056] The next step is to use these statistics to choose a particular
set of hedges and premiums 225, 230.
[0057] In an exemplary embodiment of the present invention, insurance
premiums and call hedges are selected after examination of the statistics
225, 230. One problem to be solved is how to determine an insurance
premium to charge, and how much of the exposure to hedge through
purchasing daily call options. One solution, according to an exemplary
embodiment of the present invention, may be illustrated as follows.
The insurance company picks a measure of desired performance, based
on the statistically derived ratios discussed above, and finds the
minimum premium that may be charged while still being able to hedge
and achieve that desired ratio. This objective is displayed in FIG.
4 for sample data involving the Sharpe ratio.
[0058] In FIG. 4, it is assumed that a portfolio with a Sharpe
ratio (ratio of expected payoff to standard deviation) of 0.6 is
desired. In this graph, by increasing the premium charged (A, in
millions), ranges of x (the amount to hedge) that will result in
a portfolio with this desired Sharpe ratio may be produced. However,
by pricing the premium too high, there is a risk of making the contract
nonmarketable. Therefore the smallest premium possible to satisfy
the desired return parameter(s) is sought. For the example under
consideration, the graph of FIG. 4 indicates that a (total) premium
of about 4.75 million dollars should be charged and about 100 MW
of daily calls should be bought.
[0059] This idea can be extended to any of the ratios computed
for the portfolio (or others deemed to be of importance). The rationale
behind these ratios can be briefly summarized:
[0060] "Sharpe" ratio: The Sharpe ratio is a classic
measure of portfolio performance which indicates the expected rate
of return per unit of risk associated with achieving that return.
According to an exemplary embodiment of the present invention, an
expected payoff is used, not a rate of return. Thus, this is not,
strictly speaking, a "pure" Sharpe ratio.
[0061] Payoff to 95%: Motivated by the need for captive capital
requirements, equal to 95% of the maximum loss. This ratio reflects
the rate of return on capital and is often used by hedge funds.
[0062] Conditional payoff to the conditional loss: Similar to Payoff
to 95% ratio.
[0063] Another aspect of an exemplary embodiment of the present
invention allows the partnership of two insuring parties to allocate
losses, and thus share premiums. In such an embodiment, there may
be some cutoff of losses, below which the first company is responsible
for insuring some high percentage of losses (i.e. 90100%), and
above which the first company is responsible for covering some smaller
percentage of losses (i.e. 25%). The second insuring company would
cover the remaining losses. However, this feature, while incorporated
into the model, does not affect things qualitatively, as the invention
still deals with an insurance contract and an obligation to cover
some losses. The only thing that changes is quantitative, the size
of the losses covered (and hence the size of the premiums that can
be charged).
[0064] While this invention has been described in detail with particular
reference to preferred embodiments thereof, it will be understood
that variations and modifications can be effected within the scope
of the invention as defined in the appended claims.
