# The probability distribution of the bioethanol crush margin

A chain is no stronger than its weakest link.

### Introduction

Producing bioethanol is a high risk endeavor with adverse price development and crumbling margins.

In the following we will illustrate some of the risks the bioethanol producer is facing using corn as feedstock. However, these risks will persist regardless of the feedstock and production process chosen. The elements in the discussion below can therefore be applied to any and all types of bioethanol production:

1. What average yield (kg ethanol per kg feedstock) can we expect? And what is the shape of the yield distribution?

2. What will the future price ratio of feedstock to ethanol be? And what volatility can we expect?

The crush margin^{1} measures the difference between the sales proceeds of finished bioethanol and its feedstock^{2}.

With current technology, one bushel of corn can be converted into approx. 2.75 gallons of corn and 17 pounds of DDG (distillers’ dried grains). The crush margin (or gross processing margin) is then:

**1. Crush margin = 0.0085 x DDG price + 2.8 x ethanol price – corn price**

Since from 65 % to 75 % of the variable cost in bioethanol production is cost of corn, the crush margin is an important metric especially since the margin in addition shall cover all other expenses like energy, electricity, interest, transportation, labor etc. and – in the long term the facility’s fixed costs.

The following graph taken from the CME report: Trading the corn for ethanol crush, (CME, 2010) gives the margin development in 2009 and the first months of 2010:

This graph gives a good picture of the uncertainties that faces the bioethanol producers, and can be a helpful tool when hedging purchases of corn and sale of the products^{3}.

### The Crush Spread, Crush Profit Margin and Crush Ratio

There are a number of other ways to formulate the crush risk (CME, July 11. 2011):

The CBOT defines the “Crush Spread” as the Estimated Gross Margin per Bushel of Corn. It is calculated as follows:

**2. Crush Spread = (Ethanol price per gallon X 2.8) – Corn price per bushel, or as**

** 3. Crush Profit margin = Ethanol price – (Corn price/2.8).**

Understanding these relationships is invaluable in trading ethanol stocks^{4}.

By rearranging the crush spread equation, we can express the spread as its ratio to the product price (simplifying by keeping bi-products like DDG etc. out of the equation):

**4. Crush ratio = Crush spread/Ethanol price = y – p, **

Where: y = EtOH Yield (gal)/ bushel corn and p = Corn price/Ethanol price.

We will in the following look at the stochastic nature of y and p and thus the uncertainty in forecasting the crush ratio.

The crush spread and thus the crush ratio is calculated using data from the same period. They therefore give the result of an unhedged operation. Even if the production period is short – two to three days – it will be possible to hedge both the corn and ethanol prices. But to do that in a consistent and effective way we have to look into the inherent volatility in the operations.

### Ethanol yield

The ethanol yield is usually set to 2.682 gal/bushel corn, assuming 15.5 % humidity. The yield is however a stochastic variable contributing to the uncertainty in the crush ratio forecasts. As only starch in corn can be converted to ethanol we need to know the content of extractable starch in a standard bushel of corn – corrected for normal loss and moisture. In the following we will lean heavily on the article: “A Statistical Analysis of the Theoretical Yield of Ethanol from Corn Starch”, by Tad W. Patzek (Patzek, 2006) which fits our purpose perfectly. All relevant references can be found in the article.

The aim of his article was to establish the mean extractable starch in hybrid corn and the mean highest possible yield of ethanol from starch. We however are also interested in the probability distributions for these variables – since no production company will ever experience the mean values (ensembles) and since the average return over time always will be less than the return using ensemble means^{5} (Peters, 2010).

The purpose of this exercise is after all to establish a model that can be used as support for decision making in regard to investment and hedging in the bioethanol industry over time.

From (Patzek, 2006) we have that the extractable starch (%) can be described as approx. having a normal distribution with mean 66.18 % and standard deviation of 1.13:

The nominal grain loss due to dirt etc. can also be described as approx. having a normal distribution with mean 3 % and a standard deviation of 0.7:

The probability distribution for the theoretical ethanol yield (kg/kg corn) can then be found by Monte Carlo simulation^{6} as:

– having an approx. normal distribution with mean 0.364 kg EtHO/kg of dry grain and standard deviation of 0.007. On average we will need 2.75 kg of clean dry grain to produce one kilo or 1.74 liter of ethanol^{7}.

Since we now have a distribution for ethanol yield (y) as kilo of ethanol per kilo of corn we will in the following use price per kilo both for ethanol and corn, adjusting for the moisture (natural logarithm of moisture in %) in corn:

We can also use this to find the EtHO yield starting with wet corn and using gal/bushel corn as unit (Patzek, 2006):

giving as theoretical value a mean of 2.64 gal/wet bushel with a standard deviation of 0.05 – which is significantly lower than the “official” figure of 2.8 gal/wet bushel used in the CBOT calculations. More important to us however is the fact that we easily can get yields much lower than expected and thus a real risk of lower earnings than expected. Have in mind that to get a yield above 2.64 gallons of ethanol per bushel of corn all steps in the process must continuously be at or close to their maximum efficiency – which with high probability never will happen.

### Corn and ethanol prices

Looking at the price developments since 2005 it is obvious that both the corn and ethanol prices have a large variability ($/kg and dry corn):

The long term trends show a disturbing development with decreasing ethanol price, increasing corn prices and thus an increasing price ratio:

“Risk is like fire: If controlled, it will help you; if uncontrolled,it will rise up and destroy you.”

Theodore Roosevelt

### The unhedged crush ratio

Since the crush ratio on average is:

Crush ratio = 0.364 – p, where:

0.364 = Average EtOH Yield (kg EtHO/kg of dry grain) and

p = Corn price/Ethanol price

The price ratio (p) has to be less than 0.364 for the crush ratio in the outset to be positive. As of January 2011 the price ratios has overstepped that threshold and have for the first months of 2011 stayed above that.

To get a picture of the risk an unhedged bioethanol producer faces only from normal variation in yield and forecasted variation in the price ratio we will make a simple forecast for April 2011 using the historic time series information on trend and seasonal factors:

The forecasted probability distribution for the April price ratio is given in the frequency graph below:

This represents the price risk the producer will face. We find that the mean value for the price ratio will be 0.323 with a standard deviation of 0.043. By using this and the distribution for ethanol yield we can by Monte Carlo simulation forecast the April distribution for the crush ratio:

As we see, will negative values for the crush ratio be well inside the field of possible outcomes:

The actual value of the average price ratio for April turned out to be 0.376 with a daily maximum of 0.384 and minimum of 0.363. This implies that the April crush ratio with 90 % probability would have been between -0.005 and -0.199, with only the income from DDGs to cover the deficit and all other costs.

### Hedging the crush ratio

The distribution for the price ratio forecast above clearly points out the necessity of price ratio hedging (Johnson, 1960) and (Stein, 1961).

The time series chart above shows both a negative trend development and seasonal variations in the price ratio. In the short run there is nothing much to do about the trend development, but in the longer run will probably other feedstock and better processes change the trend development (Shapouri et al., 2002).

However, what immediately stand out are the possibilities to exploit the seasonal fluctuations in both markets:

Ideally, raw material is purchased in the months seasonal factors are low and ethanol sold the months seasonal factor are high. In practice, this is not possible, restrictions on manufacturing; warehousing, market presence, liquidity, working capital and costs set limits to the producer’s degrees of freedom (Dalgran, 2009).

Fortunately, there are a number of tools in both the physical and financial markets available to manage price risks; forwards and futures contracts, options, swaps, cash-forward, and index and basis contracts. All are available for the producers who understand financial hedging instruments and are willing to participate in this market. See: (Duffie, 1989), (Hull, 2003) and (Bjørk, 2009).

The objective is to change the margin distributions shape (red) from having a large part of its left tail on the negative part of the margin axis to one resembling the green curve below where the negative part have been removed, but most of the upside (right tail) has been preserved, that is to: eliminate negative margins, reduce variability, maintain the upside potential and thus reduce the probability of operating at a net loss:

Even if the ideal solution does not exist, large number of solutions through combinations of instruments can provide satisfactory results. In principle, it does not matter where these instruments exist, since both the commodity and financial markets are interconnected to each other. From a strategic standpoint, the purpose is to exploit fluctuations in the market to capture opportunities while mitigating unwanted risks (Mallory, et al., 2010).

### Strategic Risk Management

To manage price risk in commodity markets is a complex topic. There are many strategic, economic and technical factors that must be understood before a hedging program can be implemented.

Since all the hedging instruments have a cost and since only future outcomes ranges and not exact prices, can be forecasted in the individual markets, costs and effectiveness is uncertain.

In addition, the degrees of desired protection have to be determined. Are we seeking to ensure only a positive margin, or a positive EBITDA, or a positive EBIT? With what probability and to what cost?

A systematic risk management process is required to tailor an integrated risk management program for each individual bioethanol plant:

*The choice of instruments will define different strategies that will affect company liquidity and working capital and ultimately company value. Since the effect of each of these strategies will be of stochastic nature it will only be possible to distinguish between them using the concept of stochastic dominance. (s*electing strategy*)*

Models that can describe the business operations and underlying risk can be a starting point, to such an understanding. Linked to balance simulation they will provide invaluable support to decisions on the scope and timing of hedging programs.

It is only when the various hedging strategies are simulated through the balance so that the effect on equity value can be considered that the best strategy with respect to costs and security level can be determined – and it is with this that S@R can help.

### References

Bjørk, T.,(2009). Arbitrage Theory in Continuous Time. Oxford University Press, Oxford.

CME Group., (2010).Trading the corn for ethanol crush,

http://www.cmegroup.com/trading/agricultural/corn-for-ethanol-crush.html

CME Group., (July 11. 2011). Ethanol Outlook Report, , http://cmegroup.barchart.com/ethanol/

Dalgran, R.,A., (2009) Inventory and Transformation Hedging Effectiveness in Corn Crushing. Journal of Agricultural and Resource Economics 34 (1): 154-171.

Duffie, D., (1989). Futures Markets. Prentice Hall, Englewood Cliffs, NJ.

Hull, J. (2003). Options, Futures, and Other Derivatives (5th edn). Prentice Hall, Englewood Cliffs, N.J.

Johnson, L., L., (1960). The Theory of Hedging and Speculation in Commodity Futures, Review of Economic Studies , XXVII, pp. 139-151.

Mallory, M., L., Hayes, D., J., & Irwin, S., H. (2010). How Market Efficiency and the Theory of Storage Link Corn and Ethanol Markets. Center for Agricultural and Rural Development Iowa State University Working Paper 10-WP 517.

Patzek, T., W., (2004). Sustainability of the Corn-Ethanol Biofuel Cycle, Department of Civil and Environmental Engineering, U.C. Berkeley, Berkeley, CA.

Patzek, T., W., (2006). A Statistical Analysis of the Theoretical Yield of Ethanol from Corn Starch, Natural Resources Research, Vol. 15, No. 3.

Peters, O. (2010). Optimal leverage from non-ergodicity. Quantitative Finance, doi:10.1080/14697688.2010.513338.

Shapouri,H., Duffield,J.,A., & Wang, M., (2002). The Energy Balance of Corn Ethanol: An Update. U.S. Department of Agriculture, Office of the Chief Economist, Office of Energy Policy and New Uses. Agricultural Economic Report No. 814.

Stein, J.L. (1961). The Simultaneous Determination of Spot and Futures Prices. American Economic Review, vol. 51, p.p. 1012-1025.

### Footnotes

- The relationship between prices in the cash market is commonly referred to as the Gross Production Margin. [↩]
- It can also be considered as the productions throughput; the rate at which the system converts raw materials to money. Throughput is net sales less variable cost, generally the cost of the most important raw materials. (see: Throughput Accounting [↩]
- The historical chart going back to APR 2005 is available at the CBOT web site [↩]
- We will return to this in a later post. [↩]
- We will return to this in a later post [↩]
- See formula #3 in (Patzek, 2006 [↩]
- With a specific density of 0.787 kg/l [↩]