The following illustration shows how the consensus is calculated for a range of broker forecasts of earnings per share. Each forecast is date-weighted over 180 days, giving maximum emphasis to the most recent forecast, and reducing progressively to zero emphasis for a forecast six months old.

(NOTE: When calculating the date weighting, the reference point used is midway between first publication date and the latest reconfirmation date.)

First, a date-weighted average is established to determine the standard deviation of the eligible forecasts. This, in turn, enables any outlying forecasts to be identified, and excluded. Finally, a date-weighted average of the remaining forecasts is taken to be the consensus.

A total of nine brokers are providing forecasts in this example.

Date-weighted average:
The first stage is to calculate a date-weighted average of all the forecasts as follows, giving a result of 35.5p. The working figures are shown beneath:

• Step 1: Date-weight each forecast by its age in days, subtracted from 180.
  
• Step 2: Add together the weighted EPS forecasts.
  
• Step 3: Divide the sum of the weighted EPS forecasts, 38442.5, by the sum of the weights, 1085, to find the weighted average EPS, 35.5p.

This can be summarised as follows:

   Sum of the weighted
      EPS forecasts                weighted average
   -------------------      =            EPS (p)
   Sum of the weights

Working figures:

                 38442.5
Date weighted average EPS   =   -------   =   35.5P
                                  1085

Note that the forecast from Broker ‘A’ gets zero weighting on account of its age, being more than 180 days old, and is therefore excluded.

Standard deviation:
Next, the standard deviation is calculated so as outlying forecasts can be identified. The calculation gives a result of 1.33p, and involves four steps to establish the root mean square of the deviations from the weighted average, as follows:

• Step 1: Calculate the deviation of each forecast from the weighted average
  
• Step 2: Calculate the square of each deviation
  
• Step 3: Find the mean of the squared deviations
  
• Step 4: Calculate the square root of the mean to give the standard deviation

The following table illustrates how the standard deviation is derived from the data in the example. The number of brokers is now eight (one having been excluded by age of forecast) and the weighted average EPS is 35.5p:

Total of the squares of each deviation        = 14.07P

Mean of the squarred deviations
(14.07 divided by 8)                          = 1.76P

Standard deviation (squareroot of 1.76)       = 1.33P

Note that in order to lie within one standard deviation of the weighted average, the forecasts in the above example must remain within the range 35.5p + or - 1.3p, namely between 34.2p and 36.8p.

Brokers ‘D’ and ‘G’, forecasting 38.8p and 34.0p respectively, are therefore regarded as outlying forecasts, and are excluded from the final consensus along with Broker ‘A’, whose forecast, being more than 180 days old, is excluded already.

Consensus:
The consensus is taken to be the date-weighted average of the remaining forecasts, having excluded those which are old or outlying. The final calculation is therefore as follows, giving a consensus EPS of 35.1p. The working figures are shown beneath:

• Step 1: Date-weight each qualifying forecast by its age in days, subtracted from 180.
  
• Step 2: Add the together the weighted EPS forecasts (27546.1p)
  
• Step 3: Divide the sum of the weighted EPS forecasts, 27546.1, by the sum of the weights, 784, to give weighted average EPS, 35.1p

This can be summarised as follows:

   Sum of the weighted
      EPS forecasts                weighted average
   -------------------      =            EPS (p)
   Sum of the weights

Working figures:

                     27546.1
Consensus EPS    =  --------     =    35.1p
                         784

(Note: the illustration above describes the calculation of an earning per share consensus value. The calculation for dividend per share is exactly the same, but please see Foreign income dividends for further explanation.)