## The Mathematics of a Performance Based Contract (PBC) – Part 1

Over the 15 years of applying and teaching Performance Based Contracting (PBC) I have always had trouble trying to describe the mathematics of PBCs, especially to those who

Reminded by my family’s current viewing of the US TV Series, Numbers, I am inspired to try and make PBC mathematics more accessible in order for non-mathy people to better understand, and hopefully use, these techniques.  So where is maths used in a PBC?  Interestingly, it is used throughout a PBC, from the performance levels to the percentage weighting of each of the say three Key Performance Indicators (KPIs) (e.g. 50% for KPI-1, 30% for KPI-2 and 20% for KPI-3) through to adjustment of the performance fee / At-Risk Amount.  However, for the majority of circumstances, the math in these instances is fairly simple and doesn’t tend to worry most people.  But there are two areas where this isn’t the case:

1. working out the Adjusted Performance Score (APS) from an Achieved Performance score for a non-linear payment curve; and
2. using historical data to set a performance level.

In these instances while the maths is fairly simple, they tend to confuse people leading to either avoidance by using an alternate approach which doesn’t require the same level of mathematical ability or removal altogether.  So let’s look at each in turn starting with the Adjusted Performance Score calculation.

Calculating the Adjusted Performance Score (APS) in a Performance Based Contract

In a series of previous articles (see Payment Curves Part 1, Part 2 and Part 3) I highlight the various methods for describing how to turn an Achieved Performance score (i.e. the raw score of the performance measure such as number of days late from a milestone, percentage of satisfied deliveries, number of outages per 1,000 operating hours, etc.) into an APS, which is always a percentage.  While there are a number of ways of doing this, a common method is to use a straight line between the minimum performance level (at which APS = 0%) and the required performance level (at which APS = 100%).  This can be a simple straight line between these 2 points, or as in the Australian Department of Defence approach, this line is broken into 2 segments with an inflection / elbow point making it two lines. However, the issue remains the same, how do I calculate the APS from the Achieved Performance score based on the straight line?

Consider the generic non-linear Payment Curve described in Figure 1 which has four “bands” of performance.

Figure 1 – Generic Non-Linear PBC Payment Curve

Here, all but Band D has a requirement to do some maths in order to determine the APS from the Achieved Performance level using the equation for a straight line:

y = mx + b

where:

• y = Adjusted Performance Score (APS), or the vertical axis of the payment curve in percentage
• m = gradient of slope of the line – the lower the value of m the less steep the line, while a higher value of m results in a steeper line
• x = Achieved Performance score, or the horizontal axis, in whatever scale (e.g. percentage of satisfied deliveries)
• b = starting or ‘offset’ point which is the value for y when x = 0 (e.g. for Band B the minimum APS is 80%, therefore z = 80%)

If you are a bit rusty you can get more information from the following website (see https://www.mathsisfun.com/equation_of_line.html).

In the next article I will look at how we apply this to a PBC using an example from the standard Australian Department of Defence Non-Linear PBC Payment Curve.

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