Risk management - Breakdown of variables
Breakdown of variables
Sometimes it is difficult to assess the outcome of a variable without breaking it down into smaller parts.
In the example below we are planting trees and want to know how many weeks it will take us to plant 600.
Our experts breakdown the task on a weekly basis and assume for week one the possibilities are:
| Trees planted | Probability |
|---|---|
| 100 | 0.3 |
| 200 | 0.5 |
| 300 | 0.2 |
At the end of week 2 we could have a minimum of 200 or a maximum of 600.
At the end of week 3 we could have a minimum of 300 or a maximum of 900.
The calculations for the probability are:
| Week end | Trees planted (100%) | Calculation | Impact | |||
|---|---|---|---|---|---|---|
| 0 | 0 | |||||
| 1 | 1 | 0.3 x 1 | 0.30 | |||
| 2 | 0.5 x 1 | 0.50 | ||||
| 3 | 0.2 x 1 | 0.20 | ||||
| 2 | 2 | 0.3 x 0.3 | 0.09 | |||
| 3 | 0.3 x 0.5 + | 0.5 x 0.3 | 0.30 | |||
| 4 | 0.3 x 0.2 + | 0.5 x 0.5 + | 0.2 x 0.3 | 0.37 | ||
| 5 | 0.5 x 0.2 + | 0.2 x 0.5 | 0.20 | |||
| 6 | 0.2 x 0.2 | 0.04 | ||||
| 3 | 3 | 0.3 x 0.09 | 0.03 | |||
| 4 | 0.3 x 0.3 + | 0.5 x 0.09 | 0.14 | |||
| 5 | 0.3 x 0.37 + | 0.5 x 0.3 + | 0.2 x 0.09 | 0.28 | ||
| 6 | 0.3 x 0..2 + | 0.5 x 0.37 + | 0.2 x 0.3 | 0.31 | ||
| 7 | 0.3 x 0.04 + | 0.5 x 0.2 + | 0.2 x 0.37 | 0.19 | ||
| 8 | 0.5 x 0.04 + | 0.2 x 0.2 | 0.06 | |||
| 9 | 0.2 x 0.04 | 0.01 | ||||
The reasoning behind the calculations is a bit tricky.
If we look at the example of planting 300 in week 2 we have:
We know that we can plant 300 by either planting 100 in week 1 and 200 in week 2, or 200 in week 1 and 100 in week 2.
We know we must plant at least 100 each week.
Hence, the probability is:
100 in week 1 and 200 in week 2 = 0.3 x 0.5 = 0.15
200 in week 1 and 100 in week 2 = 0.5 x 0.3 = 0.15
And we need to add these together to get = 0.3
The rest are calculated in a similar fashion of potential combinations.
The contribution after 1 week is:
100 x 0.3 = 30
200 x 0.5 = 100
300 x 0.2 = 60
Total = 190
Therefore after 3 weeks would expect 3 x 190 = 570 trees planted.
We can also see from the calculations that in week 3 the probability that 300 to 600 are planted is:
0.03 + 0.14 + 0.28 + 0.31 = 0.75 (2 decimal places) or 75%.
In other words, there is a 25% chance of planting 700 to 900 trees by the end of week 3.