Forecasting sales and forecasting uncertainty

This entry is part 1 of 4 in the series Predictive Analytics



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 chart1 picturing the uncertainty in their estimates.

This fan chart2  shows as blue colored bands the uncertainty around the WEO baseline forecast with 50, 70, and 90 percent confidence intervals3 :

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:


























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 results4 , 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.


Gardner, D & Tetlock, P., (2011), Overcoming Our Aversion to Acknowledging Our Ignorance,

World Economic Outlook Database, April 2012 Edition;




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  1.   The Inflation Report Projections: Understanding the Fan Chart By Erik Britton, Paul Fisher and John Whitley, BoE Quarterly Bulletin, February 1998, pages 30-37. []
  2. Figure 1.12. from:, World Economic Outlook (April 2012), International Monetary Fund, Isbn  9781616352462 []
  3. 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. []
  4. in this case the probability distribution of future net sales []


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S@R develops models for support of decision making under uncertainty. Taking advantage of recognized financial and economic theory, we customize simulation models to fit specific industries, situations and needs.

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