BIBLIOGRAPHY AND REFERENCES
5. Net Debt
4.2 BASIC STATISTICAL METHODS USED FOR FORECASTING
As the world becomes more and more unpredictable, forecasting practice is one of the priorities for a financial modeller. Accurate forecasts play an important role in helping businesses operate in an efficient and effective manner. In this section we will try to describe some basic statistical methods frequently used in financial statement forecasting.
The objective of a forecast is to predict or estimate a future activity level such as demand, sales volume, capital expenditure, inventory levels, and so on based on the analysis of historical data.
John Tennent and Graham Friend, in their book Guide to Business Modelling2, describe 3 different approaches to forecasting which can be classified as follows:
▪Extrapolation techniques, such as time series analysis, implicitly assume that the past will be a reasonable predictor of the future. This assumption may be valid for mature and stable businesses, such as water and gas utilities. However, many industry sectors are experiencing increasing levels of structural change. The use of extrapolative techniques for these sectors may generate poor results.
▪Causative techniques, such as regression analysis, attempt to understand the funda-mental relationships that determine the dynamics of a market. This understanding, combined with a set of assumptions about the future, provides the basis for the forecast.
Because the underlying relationships are often estimated from historical data, these techniques are useful when only small, incremental changes in assumptions are expected in the future.
▪Judgemental techniques: modellers may often be asked to produce a forecast for a new product or market where there is no available historic data. In these cases, forecasting can become judgemental and highly subjective. Although the forecasts can be refined through studying the results of market research and by examining the experiences of similar or related products in other markets and countries, the task of forecasting becomes more like an art than a science.
In practice, most modellers rely on a mixture of all 3 techniques. They may establish the current market trends through time series analysis, and attempt to understand market dynamics through regression techniques. We will give a short description of time series analysis and its various forms (e.g. exponential smoothing and moving averages – simple and weighted) and we will then elaborate on linear regression as a tool to derive forecasts based on historical data.
A time series is a set of data observed over equally spaced time intervals like the monthly sales volume of SteelCo. Time series analysis seeks to discern whether there is some pattern in the data collected to date and assumes that this pattern will continue in the future. A time series can be broken down into 4 components: the trend (long-term direction), the seasonality (related movements over the year), the cyclical (changes due to cyclical movements – recession or expansion – of the economy), and the irregular (unsystematic short-term fluctuations). The development of a time series model requires a moderate amount of statistical knowledge that is beyond the scope of this book. Neverthe-less the moving average and weighted moving average smoothing methods that are used
to average out the irregular component of the time series are relatively simple, especially when a trend is present and there is no clear evidence of seasonality or cycles. The moving average method consists of computing an average of the most recent n data values for the series and using this average to forecast the value of the time series for the next period.
Moving averages are useful if one can assume that the item to be forecast will stay fairly steady over time. In the weighted moving averages, older data are usually less important and the more recent observations are typically given more weight than older observations.
In any case the weights are based on intuition. Moving averages and weighted moving averages are effective in smoothing out sudden fluctuations in a trend pattern in order to provide stable estimates but require maintaining extensive records of past data. On the other hand exponential smoothing requires little record keeping of past data. It is a form of weighted moving average where the weights decline exponentially as we move from the most recent data. That is, the most recent data have the most weight. A brief presentation on the various forms of time series analysis, entitled “Time Series: The Art of Forecasting”, can be found on the website of Sacramento University provided by Professor Emeritus Bob Hopfe.3
For the rest of this section we will elaborate on regression analysis. We have already used a linear trend function in Chapter 3 to forecast the growth rate in sales volume based on historic data. The function we used in Chapter 3 is the simplest form of regression analysis.
Regression analysis is a statistical technique that finds a mathematical expression that best describes a set of historic data set (X,Y).Then this expression may be used to forecast future data values of Ys given future data values of Xs. The latter is called extrapolation and is the process by which we use the regression line to predict a value of the Y variable for a value of the X variable that is outside the range of the data set. The opposite is called interpolation and is the process by which we use the regression line to predict a value of the Y variable for a value of the X variable that is not one of the data points but is within the range of the data set.
Let us see how we forecast in Excel the growth rate of 2014 equal to −11.3% based on the growth rates of 2012 and 2013 which were −14.8% and −13.0% respectively. The linear trend function fits a straight line to an array of known Ys and known Xs and then returns the y-values along that line for any new x’s that you specify. How does it do this? You will recall from your introductory maths class that you can always draw a straight line between any 2 points. If these 2 points are (Y1,X1) and (Y2,X2) then the straight line is defined as:
y = m x + b,
where m is the so-called slope of the line or gradient and equals to:
m Y Y
X X
= −
−
( )
( )
2 1
2 1
and b is the intercept, determines the point at which the line crosses the y-axis, and equals to:
b = y − m x= − − * *
− = − −
Y Y Y −
X X X Y Y Y
X X X
1 2 1
2 1
1 2 2 1
2 1
2
( )
( )
( )
( )
As far as SteelCo’s case is concerned, if (Y1, X1) = (−14.8, 1) and (Y2,X2) = ( −13.0, 2) and we want to find (Y3, X3) where X3= 3 then:
m= − − −
−
( . ( . ))
( )
13 0 14 8
2 1 = 1.77% and
b= −14 8. %−1 77. %*1= −16 59. % Thus
Y3= 1.77% *x − 16.59% = 1.77%*3 −16.59% = −11.27% (see Exhibit 3.7).
Instead of doing all the above calculations back in Chapter 3, we simply selected the 2 cells containing the values of 14.8% and -13.0% respectively and we dragged them to the right.
Now imagine we have a set of data which consists of the N observed data points (x1, y1), (x2, y2), ..., (xN, yN) and we want to fit a straight line in the form y = m x + b described above.
This straight line, or best-fitting line as it is called, can be obtained by using the method of least squares. The idea behind the least squares method is that we can find m and b by mini-mizing the sum of the squares of the deviations (least square error) of the actual values of yi from the line’s calculated value of y. The formulae for m and b are:
m
xy x y
n
x x
n
= −
−
Σ Σ Σ
Σ Σ
( )( ) ( )
2 2
and b= −Y mX,
where Y Y
= Σn and
X X
= Σn
In all equations, the summation sign is assumed to be from 1 to n.
The easiest way to get the best fit of a data set is to plot it in Excel and add a linear trend-line. Let us take the 7 data points of the annual revenues of SteelCo for the years 2011 to 2017 as presented in Exhibit 3.11 and plot them using a line chart (see Exhibit 4.14). By right click-ing on the line and selectclick-ing the Add Trendline at the window that pops up you get another window like the one in Exhibit 4.15.
Please note that the window popup of the Trendline has by default ticked the Linear trendline button. Please also note that at the bottom of this window we have ticked the Display Equation on chart and Display R-square value on chart boxes respectively.
EXHIBIT 4.14 Plotting and adding a trendline to SteelCo’s revenues
On the left hand side of Exhibit 4.15 we see the line chart with the best fit (least squares equation):
y = −14952x + 18654
and another value, that of R2 which equals R2= 0.925. R2 is a statistic that gives information about the goodness of fit of a linear trendline. In regression, the R2 coefficient is a statistical EXHIBIT 4.15 Using a linear trendline to model SteelCo’s revenues
measure of how well the regression line approximates the real data points. An R2 of 1 indicates that the regression line perfectly fits the data whereas an R2 of 0 indicates that there is no fit between the regression line and the data points.
Apart from the linear trendline you see in Exhibit 4.15 many other options (e.g. nonlinear trendlines) that can also be inserted if appropriate. Describing these options is definitely out-side the scope of this book but we will make another attempt and try to fit a nonlinear trendline in order to get a better R2 than the 0.925 of the linear trendline. If we choose the polynomial trendline with order 2 then we get the graph shown in Exhibit 4.16. You see that now R2= 0.997 this is an almost perfect fit. Even without the R2 statistic you can assess visually that this is a much better fit than that of the linear trendline.
Polynomial trendlines are useful when there is reason to believe that the relationship between the Y (dependent) and the X (independent) variables is curvilinear. Curvilinear relation-ships are presented either by a maximum or a minimum in the (X,Y) curve. In a polynomial regression model of order n the relationship between the dependent and independent variables is modelled as a nth order polynomial function:
y=anxn+an−1xn−1+...+a2x2+a1x+a0
Polynomial models are also useful in approximating functions to unknown and possibly very complex nonlinear relationships. The most frequently used polynomial model is the quadratic one (second-order model) like the one we have used to model SteelCo’s revenues in Exhibit 4.16.
The interested reader can add other regression trendlines to the chart with the revenue data of SteelCo and see how well they fit. This example, as well as the whole model we are building, is provided in an Excel spreadsheet so that the user can experiment him- or herself with various regression options.
Now it is time to use the analysis provided above in order to forecast the sales revenues of SteelCo for 2018.
185,500
154,684
132,500
y = 2,406. 7x2− 34,206. 3x + 215,424. 0 R2 = 0,997
140,000 160,000 180,000 200,000
106,395 98,171
92,322 60,000
80,000 100,000 120,000
20,000 -40,000
3
1 2 4 5 6 7
117,564
EXHIBIT 4.16 Using a quadratic polynomial trendline to model SteelCo’s revenues