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December 2011 – Strategy @ Risk

Strategy @ Risk

Month: December 2011

M&A: When two plus two is five or three or …

When two plus two is five (Orwell, 1949)

Introduction

Mergers & Acquisitions (M&A) is a way for companies to expand rapidly and much faster than organic growth – that is coming from existing businesses – would have allowed. M&A’s have for decades been a trillion-dollar business, but empirical studies reports that a significant proportion must be considered as failures.

The conventional wisdom – is that the majority of deals fail to add shareholder value to the acquiring company. According to this research, only 30-50% of deals are considered to be successful (See Bruner, 2002).

If most deals fail, why do companies keep doing them? Is it because they think the odds won’t apply to them, or are executives more concerned with extending its influence and company growth (empire building) and not with increasing their shareholder (s) value?

Many writers argue that these are the main reasons driving the M&A activities, with the implication that executives are basically greedy (because their compensation is often tied to the size of the company) – or incompetent.

To be able to create shareholder value the M&A must give rise to some forms of synergy. Synergy is the ability of the merged companies to generate higher shareholder value (wealth) than the standalone entities. That is; that the whole will be greater than the sum it’s of parts.

For many of the observed M&A’s however, the opposite have been the truth – value have been destroyed; the whole have turned out to be less than the sum of its parts (dysergy).

“When asked to name just one big merger that had lived up to expectations, Leon Cooperman, former co-chairman of Goldman Sachs’ Investment Policy Committee, answered: I’m sure there are success stories out there, but at this moment I draw a blank.” (Sirower, 1997)

The “apparent” M&A failures have also been attributed to both methodological and measurement problems, stating that evidence – as cost saving or revenue enhancement brought by the M&A is difficult to obtain after the fact. This might also apply to some of the success stories.

What is surprising in most (all?) of the studies of M&A success and failures is the lack understanding of the stochastic nature of business activities. For any company it is impossible to estimate with certainty its equity value, the best we can do is to estimate a range of values and the probability that the true value will fall inside this range. The merger two companies amplify this, and the discussion of possible synergies or dysergies can only be understood in the context of randomness (stochasticity) ((See: the IFA.com – Probability Machine, Galton Board, Randomness and Fair Price Simulator, Quincunx at http://www.youtube.com/watch?v=AUSKTk9ENzg)).

[tube] http://www.youtube.com/watch?v=AUSKTk9ENzg, 400,300 [/tube]

The M&A cases

Let’s assume that we have two companies A and B that are proposed merged. We have the distribution for each company’s equity value (shareholders value) for both companies and we can calculate the equity distribution for the merged company. Company A’s value is estimated to be in the range of 0 to 150M with expected value 90M. Company B’s value is estimated to be in the range of -40 to 200M with expected value 140M. (See figure below)

If we merge the two companies assuming no synergy or dysergy we get the value (shareholder) distribution shown by the green curve in the figure. The merged company will have a value in the range of 65 to 321M, with an expected value of 230M. Since there is no synergy/dysergy no value have been created or destroyed by the merger.

For company B no value would be added in the merger if A was bought at a price equal to or higher than the expected value of the company. If it was bought at a price less than expected value, then there is a probability that the wealth of the shareholders of company B will increase. But even then it is not with certainty. All increase of wealth to the shareholders of company B will be at the expenses of the shareholders of company A and vice versa.

Case 1

If we assume that there is a “connection” between the companies, such that an increase in one of the company’s revenues also will increase the revenues in the other, we will have a synergy that can be exploited.

This situation is depicted in the figure below. The green curve gives the case with no synergy and the blue the case described above. The difference between them is the synergies created by the merger. The synergy at the dotted line is the synergy we can expect, but it might turn out to be higher if revenues is high and even negative (dysergy) when revenues is low.

If we produce a frequency diagram of the sizes of the possible synergies it will look as the diagram below. Have in mind that the average synergy value is not the value we would expect to find, but the average of all possible synergy values.

Case 2

If we assume that the “connection” between the companies is such that a reduction in one of the company’s revenues streams will reduce the total production costs, we again have a synergy that can be exploited.
This situation is depicted in the figure below. The green curve gives the case with no synergy and the red the case described above. The difference between them is again the synergies created by the merger. The synergy at the dotted line is the synergy we can expect, but it might turn out to be higher if revenues is lower and even negative (dysergy) when revenues is high.

In this case, the merger acts as a hedge against revenue losses at the cost of parts of the upside created by the merger. This should not deter the participants from a merger since there is only a 30 % probability that this will happen.

The graph above again gives the frequency diagram for the sizes of the possible synergies. Have in mind that the average synergy value is not the value we would expect to find, but the average of all possible synergy values.

Conclusion

The elusiveness of synergies in many M&A cases can be explained by the natural randomness in business activities. The fact that a merger can give rise to large synergies does not guarantee that it will occur, only that there is a probability that it will occur. Spread sheet exercises in valuation can lead to disaster if the stochastic nature of the involved companies is not taken into account. AND basing the pricing of the M&A candidate on expected synergies is pure foolishness.

References

Bruner, Robert F. (2002), Does M&A Pay? A Survey of Evidence for the Decision-Maker. Journal of Applied Finance, Vol. 12, No. 1. Available at SSRN: http://ssrn.com/abstract=485884

Orwell, George (1949). Nineteen Eighty-Four. A novel. London: Secker & Warburg.

The whole is more than the sum of its parts. Aristotle, Metaphysica

Sirower, M. (1997) The Synergy Trap: How Companies Lose the Acquisition Game. New York. The Free Press.

14/12/2011
Introduction to Simulation Models
This entry is part 4 of 6 in the series Balance simulation

Simulation models sets out to mimic real life company operations, that is describing the transformation of raw materials and labor to finished products in such a way that it can be used as support for strategic decision making.

A full simulation model will usually consist of two separate models:
1. an EBITDA model that describes the particular firm’s operations and
2. an generic P&L and Balance simulation model (PL&B).
The EBITDA model ladder

Both the deterministic and stochastic balance simulation can be approached as a ladder with two steps, where the first is especially well suited as an introduction to risk simulation and the second gives a full blown risk analysis. In these successive steps the EBITDA calculations will be based on:
1. financial information only, by using coefficients of fabrications and unit prices (e.g. kg flour per 1000 bread and cost of flour per kg, etc.) as direct input to the balance model – the direct method and
2. EBITDA models to give a detailed technical description of the company’s operations.
The first step uses coefficients of fabrications and their variations give a low effort (cost) alternative, usually using the internal accounting as basis. In many cases, this will often 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.

This step is especially well suited for introduction to risk simulation and the art of communicating risk and uncertainty throughout the firm. It can also profitably be used in cases where time and data is limited and where one wishes to limit efforts in an initial stage. Data and assumptions can later be augmented to much more sophisticated analyses within the same framework. This way the analysis can be successively built in the direction the previous studies suggested.

The second step implies setting up a dedicated EBITDA subroutine to the balance model. This can then 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 EBITDA model can both be a stand-alone model and a subroutine to the PL&B model. The stand-alone EBITDA model can be used to in detail study the firm’s operations and how different operational strategies will or can affect EBITDA outcomes and distribution.

When connected to the PL&B model it will act as a subroutine giving the necessary information to produce the P&L and ultimately the Balance and the – outcomes distributions.

This gives great flexibility in model formulations and the opportunity to fit models to different industries and accommodate for the data available.

P&L and Balance simulation

The generic PL&B model – based on the IFRS standard – can be used for a wide range of business activities both:
1. describes the firm’s financial environment (taxes, interest rates, currency etc.) and
2. acts as a testing bed for financial strategies (hedging, translation risk, etc.)
Since S@R has set out to create models that can give answers to both deterministic and stochastic questions thus the PL&B model is a real balance simulation model – not a simple cash flow forecast model.

All runs in the simulation produces a complete P&L and Balance it enables uncertainty curves (distributions) for any financial metric like ‘Yearly result’, ‘free cash flow’, economic profit’, ‘equity value’, ‘IRR’ or’ translation gain/loss’ etc. to be produced.
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.

The results from these analyses can be presented in different forms from detailed traditional financial reports to graphs describing the range of possible outcomes for all items in the P&L and Balance (+ much more) looking at the coming one to five (short term) or five to fifteen years (long term) and showing the impacts to e.g. equity value, company value, operating income etc.

The goal is to find the distribution for the firm’s equity value which will incorporate all uncertainty facing the firm.

This uncertainty gives both shape and location of the equity value distribution, and this is what we – if possible – are aiming to change:
1. reducing downside risk by reducing the left tail (blue curve)
2. increasing expected company value by moving the curve to the right (green curve)
3. increasing the upside potential by increasing the right tail (red curve) etc.
The Data

To be able to simulate the operations we need to put into the model all variables that will affect the firm’s net yearly result. Most of these will be collected by S@R from outside sources like central banks, local authorities and others, but some will have to be collected from the firm.

The production and firm specific variables are related to every day’s activities in the firm. Their historic values can be collected from internal accounts or from production reports. Someone in the procurement-, production- or sales department will have their records and most always the controllers. The rest will be variables inside the domain of the CEO and the company treasurer.

The variables fall in five groups:

i.      general variables describing the firm’s financial environment ,
ii.      variables describing the firms strategy,
iii.      general variables used for forecasting purposes,
iv.      direct problem related variables and
v.      the firm specific:

a. production coefficients and

b. cost of raw materials and labor related variables.

The first group will contain – for all countries either delivering raw materials or buying the finished product (s) – variables like: taxes, spot exchange rates etc. For the firm’s domestic country it will in addition contain variables like: Vat rates, taxes on investments and dividend income, depreciation rates and method, initial tax allowances, overdraft interest rates etc.

The second group will contain variables like: minimum cash levels, debt distribution on short and long term loans and currencies, hedge ratios, targeted leverage, economic depreciation etc.

The third group will contain variables needed for forecasting purposes: yield curves, inflation forecasts, GDP forecasts etc. The expected values and their 5 % and 95 % probability limits will be used to forecast exchange rates, interest rates, demand etc. They will be collected by S@R.

The fourth group will contain variables related to sales forecasts: yearly air temperature profiles (and variation) for forecasting beer sales and yearly water temperature profiles (and variation) for forecasting increase in biomass in fish farming.

The fifth group will contain variables that specify the production and costs of production. They will vary according to the type of operations e.g.: operating rate (%), max days of production, tools maintenance (h per 10.000 units) , error rate (errors per 1000 units), waste (% of weight of prod unit), cycle time (units per min), number of machines per shift (#), concession density (kg per m³), feed rates (%), mortality rates (%) etc., etc.. This variable specifies the production and will they be stochastic in the sense that they are not constant but will vary inside a given – theoretical or historic – range.

To simulate costs of production we use the coefficients of fabrication and their unit costs. Both the coefficients and their unit costs will always be of stochastic nature and they can vary with capacity utilization: energy per unit produced (kwh/unit) and energy price (cost per Kwh), malt use (kg per hectoliter), malt price (per kg), maximum takeoff weight (ton), takeoff charge (per ton), specific consumption of wood, (m³/Adt), cost of round wood (per m³), etc., etc.

The uncertainty (and risk) stemming from all groups of variables will be propagated through the P&L and down to the Balance, ending up as volatility in the equity distribution.

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

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.

Most companies have some sort of model describing the company’s operations. They are used mostly for budgeting, but in some cases also for forecasting cash flow and other important performance measures.

If the client already has spread sheet models describing the operations, we can build on this. There is no reason to reinvent what has already been done – thus saving time and resources that can be better utilized in other phases of the project.

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.

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.

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 first approach can also be considered as an introduction and stepping-stone to a more complete EBITDA model and detailed simulations.

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, key people from the controller group will be adequate for this work and if they do not have the direct knowledge, they usually know whom 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 – depending on the scope of the project.

For S&R, the period 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 second alternative for more complex models. The total time will also depend on the number of analyses that needs to be run and the type of reports that has to be delivered.

The team’s participation in the project also makes communication of the results up or down in the system simpler. Since the input data is collected by templates this gives the responsible departments and persons, ownership to assumptions, data and results. These templates thus visualize the flow of data thru the organization and the interdependence between the departments – facilitating the communication of risk and the different strategies both reviewed and selected.

No knowledge or expertize on uncertainty calculations or statistical methods is required from the clients side. The team will thru ‘osmosis’ acquires the necessary knowledge. Usually the team finds this as an exciting experience.
09/12/2011
Corn and ethanol futures hedge ratios
This entry is part 2 of 2 in the series The Bio-ethanol crush margin

A large amount of literature has been published discussing hedging techniques and a number of different hedging models and statistical refinements to the OLS model that we will use in the following. For a comprehensive review see “Futures hedge ratios: a review,” (Chen et al., 2003).

We are here looking for hedge models and hedge ratio estimations techniques that are “good enough” and that can fit into valuation models using Monte Carlo simulation.

The ultimately purpose is to study hedging strategies using P&L and Balance simulation to forecast the probability distribution for the company’s equity value. By comparing the distributions for the different strategies, we will be able to select the hedging strategy that best fits the boards risk appetite /risk aversion and that at the same time “maximizes” the company value.

Everything should be made as simple as possible, but not simpler. – Einstein, Reader’s Digest. Oct. 1977.

To use futures contracts for hedging we have to understand the objective: a futures contract serves as a price-fixing mechanism. In their simplest form, futures prices are prices set today to be paid in the future for goods. If properly designed and implemented, hedge profits will offset the loss from an adverse price moves. In a like fashion, hedge losses will also eliminate effects of a favorable price change. Ultimately, the success of any hedge program rests on the implementation of a correctly sized futures position.

The minimum variation hedge

This is often referred to as – the volatility-minimizing hedge for one unit of exposure. It can be found by minimizing the variance of the hedge payoff at maturity.

For an ideal hedge, we would like the change in the futures price () to match as exactly as possible the change in the value of the asset () we wish to hedge, i.e.:

The expected payoff from the hedge will be equal to the value of the cash position at maturity plus the payoff of the hedge (Johnson, 1960) or:

$E(H) = X_S delim{[} {E (S2)-S1} {]} + X_F delim{[} {E (F2)-F1 }{]}$

With spot position X_S, a short futures market holding X_F, spot price S₁and expected spot price at maturity E (S₂), current future contract price F₁andexpected future price E (F₂) – excluding transaction costs.

What we want is to find the value of the futures position that reduces the variability of price changes to the lowest possible level.

The minimum-variance hedge ratio is then defined as the number of futures per unit of the spot asset that will minimize the variance of the hedged portfolio returns.

The variance of the portfolio return is: ((The variance of the un-hedged position is: )):

Where is the variance in the future price change, is the variance of the change in the spot price and the covariance between the spot and future price changes. Letting represent the proportion of the spot position hedged, minimum value of Var (H) can then be found ((by minimizing Var (H) as a function of h)) as:

Where is the correlation between the spot and future price changes while assuming that X_S is exogenous determined or fixed.

Estimating the hedge coefficient

It is also possible to estimate the optimal hedge (h*) using regression analysis. The basic equation is:

with as the change in spot price not explained by the regression model. Since the basic OLS regression for this equation estimates the value of h* as:

we can use this regression to find the solution that minimizes the objective function E(H). This is one of the reasons that use of the objective function E (H) is so appealing. ((Note that other and very different objective functions could have chosen.))

We can then use the coefficient of determination, or , as an estimate of the percentage reduction in the variability of changes in the value of the cash position from holding the hedged position – the hedge effectiveness. (Ederington, 1979) ((Not taking into account variation margins etc.)).

The basis

Basis is defined as the difference between the spot price (S) and the futures price (F). When the expected change in the future contract price is equal to the expected change in the spot price, the optimal variance minimizing strategy is to set h*=1. However, for most future contract markets the future price does not perfectly parallel the spot price, causing an element of basis risk to directly affect the hedging decision.

A negative basis is called contango and a positive basis backwardation:
1. When the spot price increases by more than the futures price, the basis increases and is said to “strengthen the basis” (when unexpected, this is favorable for a short hedge and unfavorable for a long hedge).
2. When the futures price increases by more than the spot price, the basis declines and this is said to “weaken the basis” (when unexpected, this is favorable for a long hedge and unfavorable for a short hedge).
There will usually be a different basis for each contract.

The number of futures contracts

The variance minimizing number of futures contracts N* will be:

N*=h* ${ X_S}/{Q_F}$

Where is the size of one futures market contract. Since futures contracts are marked to market every day, the daily losses are debited and daily gains credited the parties accounts – settlement variations – i.e. the contracts are closed every day. The account will have to be replenished if the account falls below the maintenance margin (margin call). If the account is above the initial margin withdrawals can be made from the account.

Ignoring the incremental income effects from investing variation margin gains (or borrowing to cover variation margin losses), we want the hedge to generate h*. Appreciating that there is an incremental effect, we want to accrue interest on a “tailed” hedge such that (Kawaller, 1997):

h*h or
ĥ = h* or h* if time to maturity is less than one year.

Where:
r = interest rate and
n = number of days remaining to maturity of the futures contract.

This amounts to adjusting the hedge by a present value factor. Tailing converts the futures position into a forward position. It negates the effect of daily resettlement, in which profits and losses are realized before the day the hedge is lifted.

For constant interest rates the tailed hedge (for h* < 1.) rises over time to reach the exposure at the maturity of the hedge. Un-tailed the hedge will over-hedge the exposure and increase the hedger’s risk. Tailing the hedge is especially of importance when the interest rate is high and the time to maturity long.

An appropriate interest rate would be one that reflects the average of the firm’s cost of capital (WACC) and the rate it would earn on its investments (ROIC) both which will be stochastic variable in the simulation. The first would be relevant in cases when the futures contracts generate losses, while the second when the futures contracts generate gains. In practice some average of these rates are used. ((See FAS 133 and later amendments))
There are traditionally two approaches to tailing:
1. Re-balance the tail each day. In this case the tailed hedge ratio is adjusted each day to maturity of the futures contract. In this approach the adjustment declines each day, until at expiration there is no adjustment.
2. Use a constant tail (average): ĥ= h* where N is the original number of days remaining to maturity. In this shortcut, the adjustment is made at the time the hedge is put on, and not changed. The hedge will start with being too big and ends with being too small, but will on average be correct.
For investors where trading is active, the first approach is more convenient, for inactive traders, the second is often used.

Since our models always incorporate stochastic interest rates, hedges discounted with the appropriate rates are calculated. This amounts to solving the set of stochastic simultaneous equations created by the hedge and the WACC/ROIC calculations since the hedges will change their probability distributions. Note that the tailed hedge ratio will be a stochastic variable, and that minimizing the variance of the hedge will not necessarily maximize the company value. The value of – ĥ – that maximizes company value can only be found by simulation given the board’s risk appetite / risk aversion.

The Spot and Futures Price movements

At any time there are a number of futures contracts for the same commodity simultaneously being priced. The only difference between them is the delivery month. A continuous contract takes the individual contracts in the futures market and splices them together. The resulting continuous series ((The simplest method of splicing is to tack successive delivery months onto each other. Although the prices in the history are real, the chart will also preserve the price gaps that are present between expiring deliveries and those that replace them.)) allows us to study the price history in the market from a single chart. The following graphs show the price movements ((To avoid price gap problems, many prefer to base analysis on adjusted contracts that eliminate roll-over gaps. There are two basic ways to adjust a series.
Forward-adjusting works by beginning with the true price for the first delivery and then adjusting each successive set of prices up or down depending on whether the roll-over gap is positive or negative.
Back-adjusting reverses the process. Current price are always real but historical prices are adjusted up or down. This is the often preferred method, since the series always will show the latest actual price. However, there is no perfect method producing a continuous price series satisfying all needs.)) for the spliced corn contracts C-2010U to 2011N and the spliced ethanol contracts EH-2010U to 2011Q.

In the graphs the spot price is given by the blue line and the corresponding futures price by the red line.

For the corn futures, we can see that there is a difference between the spot and the futures price – the basis ((The reasons for the price difference are transportation costs between delivery locations, storage costs and availability, and variations between local and worldwide supply and demand of a given commodity. In any event, this difference in price plays an important part in what is being actually pay for the commodity when you hedge.)) – but that the price movements of the futures follow the spot price closely or – vice versa.

The spliced contracts for bioethanol are a little different from the corn contracts. The delivery location is the same and the curves are juxtaposed very close to each other. Here are however other differences.

The regression – the futures assay

The selected futures contracts give us five parallel samples for the relation between the corn spot and futures price, and six for the relation between the ethanol spot and ethanol futures price. For every day in the period 8/2/2010 to 7/14/2011 we have from one to five observations of the corn relation (five replications) and from 8/5/2010 to 8/3/2011 we have one to twelve observations of the ethanol relation. Since we follow a set of contracts, the number of daily observations of the corn futures prices starts with five (twelve for the ethanol futures) and ends with only one as the contracts matures. We could of course also have selected a sample giving an equal number of observations every day.

There are three likely models which could be fit:
1. Simple regression on the individual data points,
2. Simple regression on the daily means,and
3. Weighted regression on the daily means using the number of observations as the weight.
When the number of daily observations is equal all three models will have the same parameter estimates. The weighted and individual regressions will always have the same parameter estimates, but when the sample sizes are unequal these will be different from the unweighted means regression. Whether the weighted or unweighted model should be used when the number of daily observations is unequal will depend on the situation.

Since we now have replications of the relation between spot and the futures price we have the opportunity to test for lack of fit from the straight line model.

In our case using this approach have a small drawback. We are looking for the regression of the spot price changes against the price changes in the futures contract. This model however will give us the inverse: the regression of the price changes in the futures contract against the changes in spot price. The inverse of the slope of this regression, which is what we are looking for, will in general not give the correct answer (Thonnard, 2006). So we will use this approach (model#3) to test for linearity and then model #1 with all data for estimation of the slope.

Ideally we would like to find stable (efficient) hedge ratios in the sense that they can be used for more than one hedge and over a longer period of time, thus greatly simplifying the workload for ethanol producing companies.

All prices, both spot and futures in the following, have been converted from $/gallon (ethanol) or $/bushel (corn) to $/kg.

The Corn hedge ratio

The analysis of variance table (ANOVA) for the weighted regression of the changes in the corn futures prices on the changes in corn spot prices (model#3):

The analysis of variance cautions us that the lack of fit to a linear model for all contracts is significant. However the sum of squares due to this is very small compared to the sum of squares due to linearity – so we will regard the changes in the futures prices to have been generated by a linear function of the changes in the spot prices and the hedge ratios found as efficient. In figure below the circles gives the daily means of the contracts and the line the weighted regression on these means:

Nevertheless, this linear model will have to be monitored closely as further data becomes available.

The result from the parameter estimation using simple regression (model#1) is given in the table below:

The relation is:

Giving the un-tailed corn hedge ratio h* = 1.0073

First, since the adjusted R-square value (0.9838) is an estimate of the percentage reduction in the variability of changes in the value of the cash position from holding the hedged position, a hedge based on this regression coefficient (slope) should be highly effective.

The ratio of the variance of the hedged position and the un-hedged position is equal to 1-R². The variance of a hedged position based on this hedge ratio will be 12.7 % of the unhedged position.

We have thus eliminated 87.3 % of the variance of the unhedged position. For a simple model like this this can be considered as a good result.

In the figure the thick black line gives the 95% confidence limits and the yellow area the 95% prediction limits. As we can see, the relationship between the daily price changes is quite tight thus promising the possibility of effective hedges.

Second, due to the differencing the basis caused by the difference in delivery location have disappeared, and even if the constant term is significant, it is so small that it with little loss can be considered zero.

The R-square values would have been higher for the regressions on the means than for the regression above. This is because the total variability in the data would have been reduced by using means (note that the total degrees of freedom is reduced for the regressions on means). A regression on the means will thus always suggest greater predictive ability than a regression on individual data because it predicts mean values, not individual values.

The Ethanol hedge ratio

The analysis of variance table (ANOVA) for the weighted regression of the changes in the ethanol futures prices on the changes in ethanol spot prices (model#3):

The analysis of variance again cautions us that the lack of fit to a linear model for all contracts is significant. In this case it is approximately ten times higher than for the corn contracts.

However the sum of squares due to this is small small compared to the sum of squares due to linearity – so we will regard the changes in the futures prices to have been generated by a close to linear function of the changes in the spot prices and the hedge ratios found as “good enough”. In figure below the circles gives the daily means of the contracts and the line the weighted regression on these means:

In this graph we can clearly see the deviation from a strictly linear model. The assumption of a linear model for the changes in ethanol spot and futures prices will have to be monitored very closely as further data becomes available.

The result from the parameter estimation using simple regression (model#1) is given in the table below:

The relation is:

Giving the un-tailed corn hedge ratio h* = 1.0135

The adjusted R-square value (0.8105) estimating the percentage reduction in the variability of changes in the value of the cash position from holding the hedged position is high even with the “lack of linearity”. A hedge based on this regression coefficient (slope) should then still be highly effective.

The variance of a hedged position based on this hedge ratio will be 43.7 % of the unhedged position. It is not as good as for the corn contracts, but will still give a healthy reduction in the ethanol price risk facing the company.

As this turned out, we can use both of these estimation methods for the hedge ratio as basis for strategy simulations, but one question remains unanswered: will this minimize the variance of the crush ratio?

References

Understanding Basis, Chicago Board of Trade, 2004. http://www.gofutures.com/pdfs/Understanding-Basis.pdf

http://www.cmegroup.com/trading/agricultural/files/AC-406_DDG_CornCrush_042010.pdf

Bond, Gary E. (1984). “The Effects of Supply and Interest Rate Shocks in Commodity Futures Markets,” American Journal of Agricultural Economics, 66, pp. 294-301.

Chen, S. Lee, C.F. and Shrestha, K (2003) “Futures hedge ratios: a review,” The Quarterly Review of Economics and Finance, 43 pp. 433–465

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Endnotes
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