Tag: Simple regression

Forecasting sales and forecasting uncertainty
This entry is part 1 of 4 in the series Predictive Analytics

Introduction

There are a large number of methods used for forecasting ranging from judgmental (expert forecasting etc.) thru expert systems and time series to causal methods (regression analysis etc.).

Most are used to give single point forecast or at most single point forecasts for a limited number of scenarios. We will in the following take a look at the un-usefulness of such single point forecasts.

As example we will use a simple forecast ‘model’ for net sales for a large multinational company. It turns out that there is a good linear relation between the company’s yearly net sales in million euro and growth rates (%) in world GDP:

with a correlation coefficient R= 0.995. The relation thus accounts for almost 99% of the variation in the sales data. The observed data is given as green dots in the graph below, and the regression as the green line. The ‘model’ explains expected sales as constant equal 1638M and with 53M in increased or decreased sales per percent increase or decrease in world GDP:

The International Monetary Fund (IMF) that kindly provided the historical GDP growth rates also gives forecasts for expected future change in the World GDP growth rate (WEO, April 2012) – for the next five years. When we put these forecasts into the ‘model’ we ends up with forecasts for net sales for 2012 to 2016 as depicted by the yellow dots in the graph above.

So mission accomplished! … Or is it really?

We know that the probability for getting a single-point forecast right is zero even when assuming that the forecast of the GDP growth rate is correct – so the forecasts we so far have will certainly be wrong, but how wrong?

“Some even persist in using forecasts that are manifestly unreliable, an attitude encountered by the future Nobel laureate Kenneth Arrow when he was a young statistician during the Second World War. When Arrow discovered that month-long weather forecasts used by the army were worthless, he warned his superiors against using them. He was rebuffed. “The Commanding General is well aware the forecasts are no good,” he was told. “However, he needs them for planning purposes.” (Gardner & Tetlock, 2011)

Maybe we should take a closer look at possible forecast errors, input data and the final forecast.

The prediction band

Given the regression we can calculate a forecast band for future observations of sales given forecasts of the future GDP growth rate. That is the region where we with a certain probability will expect new values of net sales to fall. In the graph below the green area give the 95% forecast band:

Since the variance of the predictions increases the further new forecasts for the GDP growth rate lies from the mean of the sample values (used to compute the regression), the band will widen as we move to either side of this mean. The band will also widen with decreasing correlation (R) and sample size (the number of observations the regression is based on).

So even if the fit to the data is good, our regression is based on a very small sample giving plenty of room for prediction errors. In fact a 95% confidence interval for 2012, with an expected GDP growth rate of 3.5%, is net sales 1824M plus/minus 82M. Even so the interval is still only approx. 9% of the expected value.

Now we have shown that the model gives good forecasts, calculated the confidence interval(s) and shown that the expected relative error(s) with high probability will be small!

So the mission is finally accomplished! … Or is it really?

The forecasts we have made is based on forecasts of future world GDP growth rates, but how certain are they?

The GDP forecasts

Forecasting the future growth in GDP for any country is at best difficult and much more so for the GDP growth for the entire world. The IMF has therefore supplied the baseline forecasts with a fan chart (( The Inflation Report Projections: Understanding the Fan Chart By Erik Britton, Paul Fisher and John Whitley, BoE Quarterly Bulletin, February 1998, pages 30-37.)) picturing the uncertainty in their estimates.

This fan chart ((Figure 1.12. from:, World Economic Outlook (April 2012), International Monetary Fund, Isbn 9781616352462)) shows as blue colored bands the uncertainty around the WEO baseline forecast with 50, 70, and 90 percent confidence intervals ((As shown, the 70 percent confidence interval includes the 50 percent interval, and the 90 percent confidence interval includes the 50 and 70 percent intervals. See Appendix 1.2 in the April 2009 World Economic Outlook for details.)) :

There is also another band on the chart, implied but un-seen, indicating a 10% chance of something “unpredictable”. The fan chart thus covers only 90% of the IMF’s estimates of the future probable growth rates.

The table below shows the actual figures for the forecasted GDP growth (%) and the limits of the confidence intervals:

Lower

Baseline

Upper

90%

70%

50%

50%

70%

90%

2012

2.5

2.9

3.1

.5

3.8

4.0

4.3

2013

2.1

2.8

3.3

4.1

4.8

5.2

5.9

The IMF has the following comments to the figures:

“Risks around the WEO projections have diminished, consistent with market indicators, but they remain large and tilted to the downside. The various indicators do not point in a consistent direction. Inflation and oil price indicators suggest downside risks to growth. The term spread and S&P 500 options prices, however, point to upside risks.”

Our approximation of the distribution that can have produced the fan chart for 2012 as given in the World Economic Outlook for April 2012 is shown below:

This distribution has: mean 3.43%, standard deviation 0.54, minimum 1.22 and maximum 4.70 – it is skewed with a left tail. The distribution thus also encompasses the implied but un-seen band in the chart.

Now we are ready for serious forecasting!

The final sales forecasts

By employing the same technique that we used to calculate the forecast band we can by Monte Carlo simulation compute the 2012 distribution of net sales forecasts, given the distribution of GDP growth rates and by using the expected variance for the differences between forecasts using the regression and new observations. The figure below describes the forecast process:

We however are not only using the 90% interval for The GDP growth rate or the 95% forecast band, but the full range of the distributions. The final forecasts of net sales are given as a histogram in the graph below:

This distribution of forecasted net sales has: mean sales 1820M, standard deviation 81, minimum sales 1590M and maximum sales 2055M – and it is slightly skewed with a left tail.

So what added information have we got from the added effort?

Well, we now know that there is only a 20% probability for net sales to be lower than 1755 or above 1890. The interval from 1755M to 1890M in net sales will then with 60% probability contain the actual sales in 2012 – see graph below giving the cumulative sales distribution:

We also know that we with 90% probability will see actual net sales in 2012 between 1720M and 1955M.But most important is that we have visualized the uncertainty in the sales forecasts and that contingency planning for both low and high sales should be performed.

An uncertain past

The Bank of England’s fan chart from 2008 showed a wide range of possible futures, but it also showed the uncertainty about where we were then – see that the black line showing National Statistics data for the past has probability bands around it:

This indicates that the values for past GDP growth rates are uncertain (stochastic) or contains measurement errors. This of course also holds for the IMF historic growth rates, but they are not supplying this type of information.

If the growth rates can be considered stochastic the results above will still hold, if the conditional distribution for net sales given the GDP growth rate still fulfills the standard assumptions for using regression methods. If not other methods of estimation must be considered.

Black Swans

But all this uncertainty was still not enough to contain what was to become reality – shown by the red line in the graph above.

How wrong can we be? Often more wrong than we like to think. This is good – as in useful – to know.

“As Donald Rumsfeld once said: it’s not only what we don’t know – the known unknowns – it’s what we don’t know we don’t know.”

While statistic methods may lead us to a reasonably understanding of some phenomenon that does not always translate into an accurate practical prediction capability. When that is the case, we find ourselves talking about risk, the likelihood that some unfavorable or favorable event will take place. Risk assessment is then necessitated and we are left only with probabilities.

A final word

Sales forecast models are an integrated part of our enterprise simulation models – as parts of the models predictive analytics. Predictive analytics can be described as statistic modeling enabling the prediction of future events or results ((in this case the probability distribution of future net sales)) , using present and past information and data.

In today’s fast moving and highly uncertain markets, forecasting have become the single most important element of the management process. The ability to quickly and accurately detect changes in key external and internal variables and adjust tactics accordingly can make all the difference between success and failure:
1. Forecasts must integrate both external and internal drivers of business and the financial results.
2. Absolute forecast accuracy (i.e. small confidence intervals) is less important than the insight about how current decisions and likely future events will interact to form the result.
3. Detail does not equal accuracy with respect to forecasts.
4. The forecast is often less important than the assumptions and variables that underpin it – those are the things that should be traced to provide advance warning.
5. Never relay on single point or scenario forecasting.
The forecasts are usually done in three stages, first by forecasting the market for that particular product(s), then the firm’s market share(s) ending up with a sales forecast. If the firm has activities in different geographic markets then the exercise has to be repeated in each market, having in mind the correlation between markets:
1. All uncertainty about the different market sizes, market shares and their correlation will finally end up contributing to the uncertainty in the forecast for the firm’s total sales.
2. This uncertainty combined with the uncertainty from other forecasted variables like interest rates, exchange rates, taxes etc. will eventually be manifested in the probability distribution for the firm’s equity value.
The ‘model’ we have been using in the example have never been tested out of sample. Its usefulness as a forecast model is therefore still debatable.

References

Gardner, D & Tetlock, P., (2011), Overcoming Our Aversion to Acknowledging Our Ignorance, http://www.cato-unbound.org/2011/07/11/dan-gardner-and-philip-tetlock/overcoming-our-aversion-to-acknowledging-our-ignorance/

World Economic Outlook Database, April 2012 Edition; http://www.imf.org/external/pubs/ft/weo/2012/01/weodata/index.aspx

Endnotes
27/04/2012
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

Ederington, Louis H. (1979). “The Hedging Performance of the New Futures Markets,” Journal of Finance, 34, pp. 157-70

Einstein, Albert (1923). Sidelights on Relativity (Geometry and Experience). P. Dutton., Co.

Figlewski, S., Lanskroner, Y. and Silber, W. L. (1991) “Tailing the Hedge: Why and How,” Journal of Futures Markets, 11: pp. 201-212.

Johnson, Leland L. (1960). ” The Theory of Hedging and Speculation in Commodity Futures,” Review of Economic Studies, 27, pp. 139-51.

Kawaller, I. G. (1997 ) ”Tailing Futures Hedges/Tailing Spreads,” The Journal of Derivatives, Vol. 5, No. 2, pp. 62-70.

Li, A. and Lien, D. D. (2003) “Futures Hedging Under Mark-to-Market Risk,” Journal of Futures Markets, Vol. 23, No. 4.

Myers Robert J. and Thompson Stanley R. (1989) “Generalized Optimal Hedge Ratio Estimation,” American Journal of Agricultural Economics, Vol. 71, No. 4, pp. 858-868.

Thonnard, M., (2006), Confidence Intervals in Inverse Regression. Diss. Technische Universiteit Eindhoven, Department of Mathematics and Computer Science, Web. 5 Apr. 2013. <http://alexandria.tue.nl/extra1/afstversl/wsk-i/thonnard2006.pdf>.

Stein, Jerome L. (1961). “The Simultaneous Determination of Spot and Futures Prices,” American Economic Review, 51, pp. 1012-25.

Endnotes
01/12/2011