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This chapter is from the book

This chapter is from the book

Fibonacci Risk

The Fibonacci series is a sequence of numbers in which every item in the series equals the sum of the previous two, with the exception that the first two values are defined as 1.


This recursive definition yields the series of 1, 1, 2, 3, 5, 8, 13, ….

The ratio between two (sufficiently large) consecutive Fibonacci numbers is an irrational number known as phi (the Greek letter φ), whose value is 1.618..., and the series is expressed as:

  • Fibi = φ *Fibi-1

Since ancient times, φ has been known as the golden ratio. It is observed throughout nature and human enterprises alike. Two famous (and quite disparate) examples based on the golden ratio are the way the invertebrate nautilus’s shell spirals and the way markets retrace their former price levels.

Notice that the weights in Table 10-1 are similar to the beginning values of the Fibonacci series. As an alternative to Table 10-1, you can choose any four consecutive members from the Fibonacci series (such as [89, 144, 233, 377]) as weights. Regardless of your choice, when you use them to evaluate the network in Figure 10-4, the risk will always be 0.64 because the weights maintain the ratio of φ. If WG is the weight of the green activities, the other weights are:


and the criticality risk formula can be written as:

Since WG appears in all elements of the numerator and the denominator, the equation can be simplified:


Approximating the value of φ, the formula is reduced to:

I call this risk model the Fibonacci risk model.

Fibonacci Risk Values

The maximum value that the Fibonacci risk formula can reach is 1.0 in an all-critical network. The minimum value that it can reach is 0.24 (1/4.24), slightly less than the minimum criticality risk model value of 0.25 (when using the set [1, 2, 3, 4] for weights). This supports the notion that risk has a natural lower limit of about 0.25.

Activity Risk

The criticality risk model uses broad risk categories. For example, if you define float greater than 25 days as green, then two activities—one with 30 days of float and the other with 60 days of float—will be placed in the same green bin and will have the same risk value. To better account for the risk contribution of each individual activity, I created the activity risk model. This model is a far more discrete than the criticality risk model.

The activity risk formula is:


  • Fi is the float of activity i.

  • N is the number of activities in the project.

  • M is the maximum float of any activity in the project or Max(F1, F2, …, FN).

As with the criticality risk, you should exclude activities of zero duration (milestones and dummies) from this analysis.

Applying the activity risk formula to the network in Figure 10-4 yields:

Activity Risk Values

The activity risk model is undefined when all activities are critical. However, at the limit, given a large network (large N) that includes only one noncritical activity with float M, the model approaches 1.0:

The minimum value of the activity risk is 0 when all activities in the network have the same level of float, M:

While activity risk can in theory reach zero, in practice it is unlikely that you will encounter such a project because all projects always have some non-zero amount of risk.

Calculation Pitfall

The activity risk model works well only when the floats of the projects are more or less uniformly spread between the smallest float and the largest float in the network. An outlier float value that is significantly higher than all other floats will skew the calculation, producing an incorrectly high-risk value. For example, consider a one-year project that has a single week-long activity that can take place anywhere between the beginning and the end of the project. Such an activity will have almost a year’s worth of float, as illustrated in the network in Figure 10-5.


FIGURE 10-5 Network with outlier high float activity

Figure 10-5 shows the critical path (bold black) and many activities with some color-coded level of float (Fi) below. The activity shown above the critical path itself is short but has an enormous amount of float M.

Since M is much larger than any other Fi, the activity risk formula yields a number approaching 1:

The next chapter demonstrates this situation and provides an easy and effective way of detecting and adjusting the float outliers.

The activity risk also produces an incorrectly low activity risk value when the project does not have many activities and the floats of the noncritical activities are all of similar or even have identical value. However, except for these rare, somewhat contrived examples, the activity risk model measures the risk correctly.

Criticality Versus Activity Risk

For decent-size real-life projects, the criticality and activity risk models yield very similar results. Each model has pros and cons. In general, criticality risk reflects human intuition better, while activity risk is more attuned to the differences between individual activities. Criticality risk modeling often requires calibration or judgment calls, but it is indifferent to how uniformly the floats are spread. Activity risk is sensitive to the presence of large outlier floats, but it is easy to calculate and does not require much calibration. You can even automate the adjustment of float outliers.

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