Nonlinear Feedback Loops using @RISK: Adding uncertainty to a dynamic model of oil prices William...

Nonlinear Feedback Loops using @RISK: Adding uncertainty to a dynamic model of oil prices

William StraussFutureMetrics, LLC

Presented at thePalisade User ConferenceOctober 25, 2007

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.

© FutureMetrics, LLC

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.


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.


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.


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.).


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


Dynamically linking the @RISK generated data to the complex non-linear model provides a unique method of generating simulation based forecasts for oil prices.

The architecture of the model is as follows:


The foundation is the flow of oil



Next we add the demand structure that drives consumption. Don’t worry, prices will arrive later.


Here is the beginning of how prices are determined.


A difference between actual and desired inventories drive the oil futures prices. Spot prices adjust in response.


Prices influence exploration and the development of new sources.


The stock of proven reserves influences extraction productivity.


Prices also influence the rate of production.


Prices also influence energy efficiency which influences consumption.


Oil Price also drives the demand relationship


Finally, new discoveries add to the proven reserves.

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.


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.


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.

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.


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).



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 Now for some results!results!


Extraction Intensity forecasts “peak oil”.


Blue is median Red are 10th and 90th Percentiles


•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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Mean of Forecasted PriceConf BandsMean of Simulated PriceConf Bands


Assuming Normal Distribution.

(Oil price has no chance of going below zero so this is bogus!)

Assuming Lognormal Distribution

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Mean of Simulated PricesMean of Forecasted PricesConf BandConf Band

90% Confidence Intervals


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.


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!


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.

100 iterations

Blue is medianRed are 10th and 90th percentiles

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.


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.


Nonlinear Feedback Loops using @RISK: Adding uncertainty to a dynamic model of oil prices William...

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