Nonlinear Feedback Loops using @RISK:
Adding uncertainty to a dynamic model of oil prices
Presented at the
Palisade User Conference
October 25, 2007
William Strauss
FutureMetrics, LLC
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Nonlinear feedback loops describe many on the
processes that make up the real world. The
resulting rising and falling patterns with
overshoot (feast or famine) are observed in many
area of study in the natural world.
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The oil market’s many positive and negative
feedbacks and the time delayed responses of the
supply side and demand side to price signals and
shocks suggest a complex dynamic model.
Long-term trends are accompanied by cyclic
price fluctuations. The many interacting
feedback loops make the system very complex
and sensitive to small changes.
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The model that follows is a stylized construction of
the oil market. FutureMetrics is not expert in the oil
sector. However:
The model is calibrated to start with the median
estimates of current global consumption and
proven reserves in the correct ratio (these initial
conditions, along with many other variables in the
model, are allowed to vary based on their
uncertainty). Many of the other relationships are
based on observed economic data.
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For the oil price forecasting model we use a tool
that is specific to developing very complex
nonlinear systems by using visual maps to create
the structure. However this tool (STELLA from
ISSE Systems) does not provide a robust method
of incorporating variable uncertainty into the
model’s dynamics.
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Many of the driving variables are uncertain.
For example: what are the expected proven
reserves over time and how rapidly will supply
respond to price signals (this includes
extraction productivity, discovery rates and
amounts, price elasticity of supply, etc.); how
rapidly will demand respond to price signals
(efficiency of transportation, power
generation, price elasticity of demand, etc.);
how rapidly will demand grow (the developing
world, etc.).
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In order to bring uncertainty into the model,
Palisade Corporation’s Excel based risk modeling
product @RISK is used to generate the
appropriate probability distributions and then
sample these distributions over many simulation
runs.
@RISK is also used to co-vary variables that are
expected to move in correlation to each other.
@RISK provides orderly (following correlation
rules) random inputs
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Dynamically linking the @RISK generated data
to the complex non-linear model provides a
unique method of generating simulation based
forecasts for oil prices.
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The architecture of the model is as follows:
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The foundation is the flow of oil
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Next we add the demand structure that drives
consumption. Don’t worry, prices will arrive later.
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Here is the beginning of how prices are determined.
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A difference between actual and desired
inventories drive the oil futures prices.
Spot prices adjust in response.
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Prices influence exploration and the development of new sources.
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The stock of proven reserves influences extraction productivity.
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Prices also influence the rate of production.
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Prices also
influence energy
efficiency which
influences
consumption.
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Oil Price also
drives the
demand
relationship
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Finally, new
discoveries add to the
proven reserves.
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13 of the key inputs to the model have stochastic
elements. Within 7 of those inputs there are
multiple and complex relationships.
For example, the demand schedule has 24 nodes
relating oil price to demand. Each node can
randomly vary around a base value but each
node is correlated to both past and expected
future demand and past and expected oil futures
prices.
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The ability to create a demand schedule and a
correlation matrix (48x48) in Excel and then use
@RISK to drive the randomly (with rules)
generated cell values for the demand schedule is
the key to estimating multiple demand curves for
simulation (next slide for an aside).
This technique is repeated for many of the inputs
that vary non-linearly with other inputs and
variables.
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Here is the @RISK generated
raw output from the 24 nodes
in the demand schedule with a
fitted second order polynomial
regression line.
Everyone knows how to get the
fit equation on the chart using
the “trendline” option. But
what about getting those
parameters in cells so you can
recreate the trendline and send
the smoothed data to the
simulation?
y = 0.0028x2 - 0.0207x + 0.8608
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The documentation for “LINEST” in Excel does not tell you about the function’s ability to
take array inputs. Thus, by highlighting three cells in a row and then typing in
{=LINEST(Yvalues,Xvalues^{1,2})} and entering using Crtl-Shft-Enter to create an array
function (and getting the function encased in { }), the parameters for X2, X, and the intercept
will be in the three highlighted cells. The parameters update for each @RISK iteration and
send a new smoothed demand schedule to each simulation run.
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The demand schedule therefore changes
randomly but within certain rules based on the
correlation matrix (using RiskCorrmat) that
assures that demand becomes more elastic
(flatter) over time if oil prices remain high.
Many other key variables follow similar rules.
The way in which new discovery is modeled is
perhaps the weakest part of the model; not in
mathematical terms but in terms of
understanding the oil discovery process. The next
slide shows how that was done for this model.
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The initial proven reserves have a mean value that
calibrates the simulation to a real-world ratio of
consumption to proven reserves (note that this mean
value also varies stochastically based on the range of
estimates for current proven reserves.
New discovery require that an “event” happens. Using
@RISK to define a discrete distribution, the event has a
probability of occurring of 0.1. There is the possibility of
an “event” happening every 16 months or 15 times in 20
years).
If the “event” happens then a quantity of new oil is
moved into the proven reserves. Another discrete
distribution provides the amount that is moved
(screenshot from @RISK on the next slide).
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So in any given iteration, the total added to the proven reserves
could be zero or, potentially, a value that is significantly higher
than the initial condition. In one iteration, new discoveries
totaled more than 2.5 times initial conditions. The median total
new discovery was about .43 initial conditions.
Now for some results!
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Extraction Intensity
forecasts “peak oil”.
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Blue is median
Red are 10th and 90th Percentiles
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•We took the data from the simulation (almost 24,000 data points) and used
econometric methods to estimate a set of model parameters and generate
predicted oil prices and a vector of standard errors (gory details on request) .
•As the previous charts suggest and testing confirms, the data is not normally
distributed and thus calculating standard confidence intervals based on an
assumption of normality would produce erroneous results. To find the
probability distribution that best describes the simulation data we used
Palisade's BestFit. The monthly simulation data is very close to a lognormal
distribution. Thus the upper and lower confidence intervals are not
symmetrical.
•We then used @RISK to generate a lognormal distribution for each of the
240 months’ data across all 100 simulations and calculate the values that
define the upper and lower confidence intervals (at 90%) using the 240
standard errors generated by the regression.
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$500
$400
Mean of Forecasted Price
Conf Bands
Mean of Simulated Price
Conf Bands
Assuming Normal Distribution.
(Oil price has no chance of going
below zero so this is bogus!)
$300
$200
Assuming Lognormal Distribution
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$100
$0
2010
2015
2020
2025
$500
Mean of Simulated Prices
Mean of Forecasted Prices
Conf Band
Conf Band
$400
$300
90% Confidence Intervals
$200
$100
$0
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2010
2015
2020
2025
From month 12 to month 24 delivery of oil is at 40% of pre-shock capacity
In this single run, oil price
TRIPLES – And this is assuming
that alternative extraction
sources will be developed.
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If there are limited source
alternatives, in this single run
the price quadruples and there
is no lower price overshoot.
Since inventory cannot recover,
prices remain volatile!
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The war starts at month 48
and continues for a period
that has a mean of 18
months and a standard
deviation of 6 months.
Blue is median
Red are 10th and 90th
percentiles
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100 iterations
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By integrating the powerful capabilities of
@RISK for generating correlated random
variables that change values for each iteration
into a complex non-linear feedback loop model,
we have been able to forecast an array of future
time paths of oil prices.
We also relied on @RISK’s ability to fit and
generate non-normal distributions to find the
confidence bands for our simulated oil prices.
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This model has illustrated potential oil price
dynamics under a future of both consistent (but
stochastic) supply and demand relationships and
a future with a severe supply disruption.
Many other scenarios and model refinements are
possible (suggestions welcomed).
Finally, as has been noted, the parameters used
in the model could be refined by oil sector
experts to bring more confidence to the
forecasts.
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