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Operations – Strategy @ Risk

Strategy @ Risk

Category: Operations

Inventory management – Stochastic supply
This entry is part 4 of 4 in the series Predictive Analytics

Introduction

We will now return to the newsvendor who was facing a onetime purchasing decision; where to set the inventory level to maximize expected profit – given his knowledge of the demand distribution. It turned out that even if we did not know the closed form (( In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain “well-known” functions.)) of the demand distribution, we could find the inventory level that maximized profit and how this affected the vendor’s risk – assuming that his supply with certainty could be fixed to that level. But what if that is not the case? What if the supply his supply is uncertain? Can we still optimize his inventory level?

We will look at to slightly different cases:
1. one where supply is uniformly distributed, with actual delivery from 80% to 100% of his ordered volume and
2. the other where the supply have a triangular distribution, with actual delivery from 80% to 105% of his ordered volume, but with most likely delivery at 100%.
The demand distribution is as shown below (as before):

Maximizing profit – uniformly distributed supply

The figure below indicates what happens as we change the inventory level – given fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profit will increase to a point of maximum.

If we let the actual delivery follow the uniform distribution described above, and successively changes the order point expected profit will follow the red line in the graph below. We can see that the new order point is to the right and further out on the inventory axis (order point). The vendor is forced to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2300 to 2900 units with a mean close to the inventory level giving maximum profit for the fixed supply case:

The realized profits are as shown in the frequency graph below:

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.

Maximizing profit – triangular distributed supply

Again we compare the expected profit with delivery following the triangular distribution as described above (red line) with the expected profit created by known and fixed supply (blue line). We can see as we successively move to higher inventory levels (from left to right on the x-axis) that expected profits will increase to a point of maximum. However the order point for the stochastic supply is to the right and further out on the inventory axis than for the non-stochastic case:

The uncertain supply again forces the vendor to order more newspapers to ‘outweigh’ the supply uncertainty:

At the point of maximum profit the actual deliveries spans from 2250 to 2900 units with a mean again close to the inventory level giving maximum profit for the fixed supply case ((This is not necessarily true for other combinations of demand and supply distributions.)) .

The realized profits are as shown in the frequency graph below:

Average profit has somewhat been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation this variability has increased by 10%, and this is again mainly caused by an increased negative skewness – again have the down side been raised.

The introduction of uncertain supply has shown that profit can still be maximized however the profit will be reduced by increased costs both in lost sales and in excess inventory. But most important, profit variability will increase raising issues of possible other strategies.

Summary

We have shown through Monte-Carlo simulations, that the ‘order point’ when the actual delivered amount is uncertain can be calculated without knowing the closed form of the demand distribution. We actually do not need the closed form for the distribution describing delivery, only historic data for the supplier’s performance (reliability).

Since we do not need the closed form of the demand distribution or supply, we are not limited to such distributions, but can use historic data to describe the uncertainty as frequency distributions. Expanding the scope of analysis to include supply disruptions, localization of inventory etc. is thus a natural extension of this method.

This opens for use of robust and efficient methods and techniques for solving problems in inventory management unrestricted by the form of the demand distribution and best of all, the results given as graphs will be more easily communicated to all parties than pure mathematical descriptions of the solutions.

Average profit has to some extent been reduced compared with the non-stochastic supply case, but more important is the increase in profit variability. Measured by the quartile variation, this variability has increased by almost 13%, and this is mainly caused by an increased negative skewness – the down side has been raised.
23/09/2012
Inventory management – Some effects of risk pooling

This entry is part 3 of 4 in the series Predictive Analytics

Introduction

The newsvendor described in the previous post has decided to branch out having news boys placed at strategic corners in the neighborhood. He will first consider three locations, but have six in his sights.

The question to be pondered is how many of the newspaper he should order for these three locations and the possible effects on profit and risk (Eppen, 1979) and (Chang & Lin, 1991).

He assumes that the demand distribution he experienced at the first location also will apply for the two others and that all locations (point of sales) can be served from a centralized inventory. For the sake of simplicity he further assumes that all points of sales can be restocked instantly (i.e. zero lead time) at zero cost, if necessary or advantageous by shipment from one of the other locations and that the demand at the different locations will be uncorrelated. The individual point of sales will initially have a working stock, but will have no need of safety stock.

In short is this equivalent to having one inventory serve newspaper sales generated by three (or six) copies of the original demand distribution:

The aggregated demand distribution for the three locations is still positively skewed (0.32) but much less than the original (0.78) and has a lower coefficient of variation – 27% – against 45% for the original ((The quartile variation has been reduced by 37%.)):

The demand variability has thus been substantially reduced by this risk pooling ((We distinguish between ten main types of risk pooling that may reduce total demand and/or lead time variability (uncertainty): capacity pooling, central ordering, component commonality, inventory pooling, order splitting, postponement, product pooling, product substitution, transshipments, and virtual pooling. (Oeser, 2011))) and the question now is how this will influence the vendor’s profit.

Profit and Inventory level with Risk Pooling

As in the previous post we have calculated profit and loss as:

Profit = sales less production costs of both sold and unsold items
Loss = value of lost sales (stock-out) and the cost of having produced and stocked more than can be expected to be sold

The figure below indicates what will happen as we change the inventory level. We can see as we successively move to higher levels (from left to right on the x-axis) that expected profit (blue line) will increase to a point of maximum – ¤16541 at a level of 7149 units:

Compared to the point of maximum profit for a single warehouse (profit ¤4963 at a level of 2729 units, see previous post), have this risk pooling increased the vendors profit by 11.1% while reducing his inventory by 12.7%. Centralization of the three inventories has thus been a successful operational hedge ((Risk pooling can be considered as a form of operational hedging. Operational hedging is risk mitigation using operational instruments.)) for our newsvendor by mitigating some, but not all, of the demand uncertainty.

Since this risk mitigation was a success the newsvendor wants to calculate the possible benefits from serving six newsboys at different locations from the same inventory.

Under the same assumptions, it turns out that this gives an even better result, with an increase in profit of almost 16% and at the same time reducing the inventory by 15%:

The inventory ‘centralization’ has then both increased profit and reduced inventory level compared to a strategy with inventories held at each location.

Centralizing inventory (inventory pooling) in a two-echelon supply chain may thus reduce costs and increase profits for the newsvendor carrying the inventory, but the individual newsboys may lose profits due to the pooling. On the other hand, the newsvendor will certainly lose profit if he allows the newsboys to decide the level of their own inventory and the centralized inventory.

One of the reasons behind this conflict of interests is that each of the newsvendor and newsboys will benefit one-sidedly from shifting the demand risk to another party even though the performance may suffer as a result (Kemahloğlu-Ziya, 2004) and (Anupindi and Bassok 1999).

In real life, the actual risk pooling effects would depend on the correlations between each locations demand. A positive correlation would reduce the effect while a negative correlation would increase the effects. If all locations were perfectly correlated (positive) the effect would be zero and a correlation coefficient of minus one would maximize the effects.

The third effect

The third direct effect of risk pooling is the reduced variability of expected profit. If we plot the profit variability, measured by its coefficient of variation (( The coefficient of variation is defined as the ratio of the standard deviation to the mean – also known as unitized risk.)) (CV) for the three sets of strategies discussed above; one single inventory (warehouse), three single inventories versus all three inventories centralized and six single inventories versus all six centralized.

The graph below depicts the situation. The three curves show the CV for corporate profit given the three alternatives and the vertical lines the point of profit for each alternative.

The angle of inclination for each curve shows the profits sensitivity for changes in the inventory level and the location each strategies impact on the predictability of realized profit.

A single warehouse strategy (blue) gives clearly a much less ability to predict future profit than the ‘six centralized warehouse’ (purple) while the ‘three centralized warehouse’ (green) fall somewhere in between:

So in addition to reduced costs and increased profits centralization, also gives a more predictable result, and lower sensitivity to inventory level and hence a greater leeway in the practical application of different policies for inventory planning.

Summary

We have thus shown through Monte-Carlo simulations, that the benefits of pooling will increase with the number of locations and that the benefits of risk pooling can be calculated without knowing the closed form ((In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain “well-known” functions.)) of the demand distribution.

Since we do not need the closed form of the demand distribution, we are not limited to low demand variability or the possibility of negative demand (Normal distributions etc.). Expanding the scope of analysis to include stochastic supply, supply disruptions, information sharing, localization of inventory etc. is natural extensions of this method ((We will return to some of these issues in later posts.)).

This opens for use of robust and efficient methods and techniques for solving problems in inventory management unrestricted by the form of the demand distribution and best of all, the results given as graphs will be more easily communicated to all parties than pure mathematical descriptions of the solutions.

References

Anupindi, R. & Bassok, Y. (1999). Centralization of stocks: Retailers vs. manufacturer. Management Science 45(2), 178-191. doi: 10.1287/mnsc.45.2.178, accessed 09/12/2012.

Chang, Pao-Long & Lin, C.-T. (1991). Centralized Effect on Expected Costs in a Multi-Location Newsboy Problem. Journal of the Operational Research Society of Japan, 34(1), 87–92.

Eppen,G.D. (1979). Effects of centralization on expected costs in a multi-location newsboy problem. Management Science, 25(5), 498–501.

Kemahlioğlu-Ziya, E. (2004). Formal methods of value sharing in supply chains. PhD thesis, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, July 2004. http://smartech.gatech.edu/bitstream/1853/4965/1/kemahlioglu ziya_eda_200407_phd.pdf, accessed 09/12/2012.

OESER, G. (2011). Methods of Risk Pooling in Business Logistics and Their Application. Europa-Universität Viadrina Frankfurt (Oder). URL: http://opus.kobv.de/euv/volltexte/2011/45, accessed 09/12/2012.

Endnotes

13/09/2012
Inventory Management: Is profit maximization right for you?
This entry is part 2 of 4 in the series Predictive Analytics

Introduction

In the following we will exemplify how sales forecasts can be used to set inventory levels in single or in multilevel warehousing. By inventory we will mean a stock or store of goods; finished goods, raw materials, purchased parts, and retail items. Since the problem discussed is the same for both production and inventory, the two terms will be used interchangeably.

Good inventory management is essential to the successful operation for most organizations both because of the amount of money the inventory represents and the impact that inventories have on the daily operations.

An inventory can have many purposes among them the ability:
1. to support independence of operations,
2. to meet both anticipated and variation in demand,
3. to decouple components of production and allow flexibility in production scheduling and
4. to hedge against price increases, or to take advantage of quantity discounts.
The many advantages of stock keeping must however be weighted against the costs of keeping the inventory. This can best be described as the “too much/too little problem”; order too much and inventory is left over or order too little and sales are lost.

This can be as a single-period (a onetime purchasing decision) or a multi-period problem, involving a single warehouse or multilevel warehousing geographically dispersed. The task can then be to minimize the organizations total cost, maximize the level of customer service, minimize ‘loss’ or maximize profit etc.

Whatever the purpose, the calculation will have to be based on knowledge of the sales distribution. In addition, sales will usually have a seasonal variance creating a balance act between production, logistic and warehousing costs. In the example given below the sales forecasts will have to be viewed as a periodic forecast (month, quarter, etc.).

We have intentionally selected a ‘simple problem’ to highlight the optimization process and the properties of the optimal solution. The last is seldom described in the standard texts.

The News-vendor problem

The news-vendor is facing a onetime purchasing decision; to maximize expected profit so that the expected loss on the Qth unit equals the expected gain on the Qth unit:

I. , where

Co = The cost of ordering one more unit than what would have been ordered if demand had been known – or the increase in profit enjoyed by having ordered one fewer unit,

Cu = The cost of ordering one fewer unit than what would have been ordered if demand had been known – or the increase in profit enjoyed by having ordered one more unit, and

F(Q) = Demand Probability for q<= Q. By rearranging terms in the above equation we find:

II.

This ratio is often called the critical ratio (CR). The usual way of solving this is to assume that the demand is normal distributed giving Q as:

III. , where: , is normal distributed with zero mean and variance equal one.

Demand unfortunately, rarely haves a normal distribution and to make things worse we usually don’t know the exact distribution at all. We can only ‘find’ it by Monte Carlo simulation and thus have to find the Q satisfying the equation (I) by numerical methods.

For the news-vendor the inventory level should be set to maximize profit given the sales distribution. This implies that the cost of lost sales will have to be weighed against the cost of adding more to the stock.

If we for the moment assume that all these costs can be regarded as fixed and independent of the inventory level, then the product markup (% of cost) will determine the optimal inventory level:

IV.

In the example given here the critical ratio is approx. 0.8. The question then is if the inventory levels indicated by that critical ratio always will be the best for the organization.

Expected demand

The following graph indicates the news-vendors demand distribution. Expected demand is 2096 units ((Median demand is 1819 units and the demand lies most typically in the range of 1500 to 2000 units)), but the distribution is heavily skewed to the right ((The demand distribution has a skewness of 0.78., with a coefficient of variation of 0.45, a lower quartile of 1432 units and an upper quartile of 2720 units.)) so there is a possibility of demand exceeding the expected demand:

By setting the product markup – in the example below it is set to 300% – we can calculate profit and loss based on the demand forecast.

Profit and Loss (of opportunity)

In the following we have calculated profit and loss as:

Profit = sales less production costs of both sold and unsold items
Loss = value of lost sales (stock-out) and the cost of having produced and stocked more than can be expected to be sold

The figure below indicates what will happen as we change the inventory level. We can see as we successively move to higher levels (from left to right on the x-axis) that expected profit (blue line) will increase to a point of maximum ¤4963 at a level of 2729 units:

At that point we can expect to have some excess stock and in some cases also lost sales. But regardless, it is at this point that expected profit is maximized, so this gives the optimal stock level.

Since we include both costs of sold and unsold items, the point giving expected maximum profit will be below the point minimizing expected loss –¤1460 at a production level of 2910 units.

Given the optimal inventory level (2729 units) we find the actual sales frequency distribution as shown in the graph below. At this level we expect an average sale of 1920 units – ranging from 262 to 2729 units ((Having a lower quartile of 1430 units and an upper quartile of 2714 units.)).

The graph shows that the distribution possesses two different modes ((The most common value in a set of observations.)) or two local maxima. This bimodality is created by the fact that the demand distribution is heavily skewed to the right so that demand exceeding 2729 units will imply 2729 units sold with the rest as lost sales.

This bimodality will of course be reflected in the distribution of realized profits. Have in mind that the line (blue) giving maximum profit is an average of all realized profits during the Monte Carlo simulation given the demand distribution and the selected inventory level. We can therefore expect realized profit both below and above this average (¤4963) – as shown in the frequency graph below:

Expected (average) profit is ¤4963, with a minimum of –¤1681 and a maximum of ¤8186, the range of realized profits is therefore very large ((Having a lower quartile of ¤2991 and an upper quartile of ¤8129.)) ¤9867.

So even if we maximize profit we can expect a large variation in realized profits, there is no way that the original uncertainty in the demand distribution can be reduced or removed.

Risk and Reward

Increased profit comes at a price: increased risk. The graph below describes the situation; the blue curve shows how expected profit increases with the production or inventory (service) level. The spread between the green and red curves indicates the band where actual profit with 80% probability will fall. As is clear from the graph, this band increases in width as we move to the right indicating an increased upside (area up to the green line) but also an increased probability for a substantial downside (area down to the red line:

For some companies – depending on the shape of the demand distribution – other concerns than profit maximization might therefore be of more importance – like predictability of results (profit). The act of setting inventory or production levels should accordingly be viewed as an element for the boards risk assessments.

On the other hand will the uncertainty band around loss as the service level increases decrease. This of course lies in the fact that loss due to lost sales diminishes as the service level increases and the that the high markup easily covers the cost of over-production.

Thus a strategy of ‘loss’ minimization will falsely give a sense of ‘risk minimization’, while it in reality increases the uncertainty of future realized profit.

Product markup

The optimal stock or production level will be a function of the product markup. A high markup will give room for a higher level of unsold items while a low level will necessitate a focus on cost reduction and the acceptance of stock-out:

The relation between markup (%) and the production level is quadratic ((Markup (%) = 757.5 – 0.78*production level + 0.00023*production level²)) implying that markup will have to be increasingly higher, the further out on the right tail we fix the production level.

The Optimal inventory (production) level

If we put it all together we get the chart below. In this the green curve is the accumulated sales giving the probability of the level of sales and the brown curve the optimal stock or production level given the markup.

The optimal stock level is then found by drawing a line from the right markup axis (right y-axis) to the curve (red) for optimal stock level, and down to the x-axis giving the stock level. By continuing the line from the markup axis to the probability axis (left y-axis) we find the probability level for stock-out (1-the cumulative probability) and the probability for having a stock level in excess of demand:

By using the sales distribution we can find the optimal stock/production level given the markup and this would not have been possible with single point sales forecasts – that could have ended up almost anywhere on the curve for forecasted sales.

Even if a single point forecast managed to find expected sales – as mean, mode or median – it would have given wrong answers about the optimal stock/production level, since the shape of the sales distribution would have been unknown.

In this case with the sales distribution having a right tail the level would have been to low – or with low markup, to high. With a left skewed sales distribution the result would have been the other way around: The level would have been too high and with low markup probably too low.

In the case of multilevel warehousing, the above analyses have to be performed on all levels and solved as a simultaneous system.

The state of affairs at the point of maximum

To have the full picture of the state of affairs at the point of maximum we have to take a look at what we can expect of over- and under-production. At the level giving maximum expected profit we will on

average have an underproduction of 168 units, ranging from zero to nearly 3000 ((Having a coefficient of variation of almost 250%)). On the face of it this could easily be interpreted as having set the level to low, but as we shall see that is not the case.

Since we have a high markup, lost sales will weigh heavily in the profit maximization and as a result of this we can expect to have unsold items in our stock at the end of the period. On average we will have a little over 800 units left in stock, ranging from zero to nearly 2500. The lower quartile is 14 units and the upper is 1300 units so in 75% of the cases we will have an overproduction of less than 1300 units. However in 25% of the cases the overproduction will be in the range from 1300 to 2500 units.

Even with the possibility of ending up at the end of the period with a large number of unsold units, the strategy of profit maximization will on average give the highest profit. However, as we have seen, with a very high level of uncertainty about the actual profit being realized.

Now, since a lower inventory level in this case only will reduce profit by a small amount but lower the confidence limit by a substantial amount, other strategies giving more predictability for the actual result should be considered.
22/08/2012
You only live once

This entry is part 4 of 4 in the series The fallacies of scenario analysis

You only live once, but if you do it right, once is enough.
— Mae West

Let’s say that you are considering new investment opportunities for your company and that the sales department has guesstimated that the market for one of your products will most likely grow by a little less than 5 % per year. You then observe that the product already has a substantial market and that this in fifteen years’ time nearly will be doubled:

Building a new plant to accommodate this market growth will be a large investment so you find that more information about the probability distribution for the products future sales is needed. Your sales department then “estimates” the market yearly growth to have a mean close to zero and a lower quartile of minus 5 % and an upper quartile of plus 7 %.

Even with no market growth the investment is a tempting one since the market already is substantial and there is always a probability of increased market shares.

As quartiles are given, you rightly calculate that there will be a 25 % probability that the growth will be above 7 %, but also that there will be a 25 % probability that it can be below minus 5 %. At the face of it, and with you being not too risk averse, this looks as a gamble worth taking.

Then you are informed that the distribution will be heavily left skewed – opening for considerable downside risk. In fact it turns out that it looks like this:

A little alarmed you order the sales department to come up with a Monte Carlo simulation giving a better view of the future possible paths of the market development.

The return with the graph below giving the paths for the first ten runs in the simulation with the blue line giving average value and the green and red the 90 % and 10 % limits of the one thousand simulated outcomes:

The blue line is the yearly ensemble averages ((A set of multiple predictions that is all valid at the same time. The term “ensemble” is often used in physics and physics-influenced literature. In probability theory literature the term probability space is more prevalent.

An ensemble provides reliable information on forecast uncertainties (e.g., probabilities) from the spread (diversity) amongst ensemble members.

Also see: Ensemble forecasting; a numerical prediction method that is used to attempt to generate a representative sample of the possible future states of dynamic systems. Ensemble forecasting is a form of Monte Carlo analysis: multiple numerical predictions are conducted using slightly different initial conditions that are all plausible given the past and current set of observations. Often used in weather forecasting.)); that is the time series of average of outcomes. The series shows a small decline in market size but not at an alarming rate. The sales department’s advice is to go for the investment and try to conquer market shares.

You then note that the ensemble average implies that you are able jump from path to path and since each is a different realization of the future that will not be possible – you only live once!

You again call the sales department asking them to calculate each paths average growth rate (over time) – using their geometric mean – and report the average of these averages to you. When you plot both the ensemble and the time averages you find quite a large difference between them:

The time average shows a much larger market decline than the ensemble average.

It can be shown that the ensemble average always will overestimate (Peters, 2010) the growth and thus can falsely lead to wrong conclusions about the market development.

If we look at the distribution of path end values we find that the lower quartile is 64 and the upper quartile is 118 with a median of 89:

It thus turns out that the process behind the market development is non-ergodic ((The term ergodic is used to describe dynamical systems which have the same behavior averaged over time as averaged over space.)) or non-stationary ((Stationarity is a necessary, but not sufficient, condition for ergodicity. )). In the ergodic case both the ensemble and time averages would have been equal and the problem above would not have appeared.

The investment decision that at first glance looked a simple one is now more complicated and can (should) not be decided based on market development alone.

Since uncertainty increases the further we look into the future, we should never assume that we have ergodic situations. The implication is that in valuation or M&A analysis we should never use an “ensemble average” in the calculations, but always do a full simulation following each time path!

References

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

Endnotes

02/08/2011
The probability distribution of the bioethanol crush margin

This entry is part 1 of 2 in the series The Bio-ethanol 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 ((The relationship between prices in the cash market is commonly referred to as the Gross Production Margin.)) measures the difference between the sales proceeds of finished bioethanol and its feedstock ((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)).

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 ((The historical chart going back to APR 2005 is available at the CBOT web site)).

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 ((We will return to this in a later post.)).

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 ((We will return to this in a later post)) (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 ((See formula #3 in (Patzek, 2006)) 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 ((With a specific density of 0.787 kg/l)).

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. (selecting 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

24/07/2011
Uncertainty modeling
This entry is part 2 of 3 in the series What We Do

Prediction is very difficult, especially about the future.
Niels Bohr. Danish physicist (1885 – 1962)

Strategy @ Risks models provide the possibility to study risk and uncertainties related to operational activities; cost, prices, suppliers, markets, sales channels etc. financial issues like; interest rates risk, exchange rates risks, translation risk , taxes etc., strategic issues like investments in new or existing activities, valuation and M&As’ etc and for a wide range of budgeting purposes.

All economic activities have an inherent volatility that is an integrated part of its operations. This means that whatever you do some uncertainty will always remain.

The aim is to estimate the economic impact that such critical uncertainty may have on corporate earnings at risk. This will add a third dimension – probability – to all forecasts, give new insight: the ability to deal with uncertainties in an informed way and thus benefits above ordinary spread-sheet exercises.

The results from these analyzes can be presented in form of B/S and P&L looking at the coming one to five (short term) or five to fifteen years (long term); showing the impacts to e.g. equity value, company value, operating income etc. With the purpose of:
- Improve predictability in operating earnings and its’ expected volatility
- Improve budgeting processes, predicting budget deviations and its’ probabilities
- Evaluate alternative strategic investment options at risk
- Identify and benchmark investment portfolios and their uncertainty
- Identify and benchmark individual business units’ risk profiles
- Evaluate equity values and enterprise values and their uncertainty in M&A processes, etc.
Methods

To be able to add uncertainty to financial models, we also have to add more complexity. This complexity is inevitable, but in our case, it is desirable and it will be well managed inside our models.

People say they want models that are simple, but what they really want is models with the necessary features – that are easy to use. If something is complex but well designed, it will be easy to use – and this holds for our models.

Most companies have some sort of model describing the company’s operations. They are mostly used for budgeting, but in some cases also for forecasting cash flow and other important performance measures. Almost all are deterministic models based on expected or average values of input data; sales, cost, interest and currency rates etc.

We know however that forecasts based on average values are on average wrong. In addition will deterministic models miss the important uncertainty dimension that gives both the different risks facing the company and the opportunities they bring forth.

S@R has set out to create models that can give answers to both deterministic and stochastic questions, by linking dedicated Ebitda models to holistic balance simulation taking into account all important factors describing the company. The basis is a real balance simulation model – not a simple cash flow forecast model.

Both the deterministic and stochastic balance simulation can be set about in two different alternatives:
1. by a using a EBITDA model to describe the companies operations or
2. by using coefficients of fabrications (e.g. kg flour pr 1000 bread etc.) as direct input to the balance model – the ‘short cut’ method.
The first approach implies setting up a dedicated Ebitda subroutine to the balance model. This will give detailed answers to a broad range of questions about markets, capacity driven investments, operational performance and uncertainty, but entails a higher degree of effort from both the company and S@R. This is a tool for long term planning and strategy development.

The second (‘the short cut’) uses coefficients of fabrications and their variations, and is a low effort (cost) alternative, usually using the internal accounting as basis. This will in many cases give a ‘good enough’ description of the company – its risks and opportunities. It can be based on existing investment and market plans. The data needed for the company’s economic environment (taxes, interest rates etc) will be the same in both alternatives:

The ‘short cut’ approach is especially suited for quick appraisals of M&A cases where time and data is limited and where one wishes to limit efforts in an initial stage. Later the data and assumptions can be augmented to much more sophisticated analysis within the same ‘short cut’ framework. In this way analysis can be successively built in the direction the previous studies suggested.

This also makes it a good tool for short-term (3-5 years) analysis and even for budget assessment. Since it will use a limited number of variables – usually less than twenty – describing the operations, it is easy to maintain and operate. The variables describing financial strategy and the economic environment come in addition, but will be easy to obtain.

Used in budgeting it will give the opportunity to evaluate budget targets, their probable deviation from expected result and the probable upside or down side given the budget target (Upside/downside ratio).

Done this way analysis can be run for subsidiaries across countries translating the P&L and Balance to any currency for benchmarking, investment appraisals, risk and opportunity assessments etc. The final uncertainty distributions can then be “aggregated’ to show global risk for the mother company.

An interesting feature is the models ability to start simulations with an empty opening balance. This can be used to assess divisions that do not have an independent balance since the model will call for equity/debt etc. based on a target ratio, according to the simulated production and sales and the necessary investments. Questions about further investment in divisions or product lines can be studied this way.

Since all runs (500 to 1000) in the simulation produces a complete P&L and Balance the uncertainty curve (distribution) for any financial metric like ‘Yearly result’, ‘free cash flow’, economic profit’, ‘equity value’, ‘IRR’ or’ translation gain/loss’ etc. can be produced.

In some cases we have used both approaches for the same client, using the last approach for smaller daughter companies with production structures differing from the main companies.
The second approach can also be considered as an introduction and stepping stone to a more holistic Ebitda model.

Time and effort

The work load for the client is usually limited to a small team of people ( 1 to 3 persons) acting as project leaders and principal contacts, assuring that all necessary information, describing value and risks for the clients’ operations can be collected as basis for modeling and calculations. However the type of data will have to be agreed upon depending on the scope of analysis.

Very often will key people from the controller group be adequate for this work and if they don’t have the direct knowledge they usually know who to ask. The work for this team, depending on the scope and choice of method (see above) can vary in effective time from a few days to a couple of weeks, but this can be stretched from three to four weeks to the same number of months.

For S&R the time frame will depend on the availability of key personnel from the client and the availability of data. For the second alternative it can take from one to three weeks of normal work to three to six months for the first alternative for more complex models. The total time will also depend on the number of analysis that needs to be run and the type of reports that has to be delivered.

Selecting strategy

Models like this are excellent for selection and assessment of strategies. Since we can find the probability distribution for equity value, changes in this brought by different strategies will form a basis for selection or adjustment of current strategy. Models including real option strategies are a natural extension of these simulation models:

If there is a strategy with a curve to the right and under all other feasible strategies this will be the stochastic dominant one. If the curves crosses further calculations needs to be done before a stochastic dominant or preferable strategy can be found:

Types of problems we aim to address:

The effects of uncertainties on the P&L and Balance and the effects of the Boards strategies (market, hedging etc.) on future P&L and Balance sheets evaluating:
- Market position and potential for growth
- Effects of tax and capital cost
- Strategies
- Business units, country units or product lines – capital allocation – compare risk, opportunity and expected profitability
- Valuations, capital cost and debt requirements, individually and effect on company
- The future cash-flow volatility of company and the individual BU’s
- Investments, M&A actions, their individual value, necessary commitments and impact on company
- Etc.
The aim regardless of approach is to quantify not only the company’s single and aggregated risks, but also the potential, thus making the company capable to perform detailed planning and of executing earlier and more apt actions against uncertain factors.

Used in budgeting, this will improve budget stability through higher insight in cost side risks and income-side potentials. This is achieved by an active budget-forecast process; the control-adjustment cycle will teach the company to better target realistic budgets – with better stability and increased company value as a result.

This is most clearly seen when effort is put into correctly evaluating strategies-projects and investments effects on the enterprise. The best way to do this is by comparing and Choosing strategies by analyzing the individual strategies risks and potential – and select the alternative that is dominant (stochastic) given the company’s chosen risk-profile.

A severe depression like that of 1920-1921 is outside the range of probability. –The Harvard Economic Society, 16 November 1929
24/11/2010