## The Mathematics of a Performance Based Contract (PBC) – Part 2

In the previous article (see The Mathematics of a Performance Based Contract (PBC) – Part 1) I described how a standard Australian Department of Defence Non-Linear PBC Payment Curve could be described by 2 equations for straights lines of the form:

y = mx + b

Accordingly, from a PBC perspective this equation can be rewritten as:

APS = m (Achieved Performance) + Minimum APS Value

The hardest calculation is to determine ‘m’, the slope of the curve.  However, we can simply think of it as “if my Achieved Performance increases by say 1 percentage, how much does the APS go up by?”  In mathematics this is typically written up as: where:

• Rise = change in the APS over the length of the line
• Run = change in the Achieved Performance over the length of the line

Again, if you are a little rusty, you may want to have a look at the following website (see https://www.mathsisfun.com/gradient.html).

Consider the example in Figure 2. Figure 2 – Example Non-Linear PBC Payment Curve

For Performance Band C the gradient would be: That is, for each 1% increase in the Achieved Performance score from 85% to 90%, the APS would increase by 16%, from 0% to 80%.

Alternatively, for Performance Band B the gradient would be: That is, for each 1% increase in the Achieved Performance score from 90% to 95%, the APS would increase by 4%, from 80% to 100%.

So why have I done all this maths?  What is the point?

In operation, say the seller achieved an Achieved Performance score of 87%, what would be the corresponding APS?  In this case, using the equation we derived above for Performance Band C we can simply put in the value of 87% as follows:

APS (Performance Band C) = 16 x (APS – 85%) + 0%

= 16 x ( 87% – 85%) + 0%

= 16 x (3%) + 0% = 48%

Therefore, an Achieved Performance score of 87% is equal to an APS of 48%.

Importantly, you will notice an additional term in the equation above, specifically (APS – 85%). The reason for this is to have the same ‘range’, or distance, that the ‘run’ had in calculating the gradient; that is between 85% and 90%.  Therefore, we need to subtract 85% from the Achieved Performance score before multiplying by the gradient.

Alternatively, say the seller achieved an Achieved Performance score of 94%, what would be the corresponding APS?  In this case, using the equation we derived above for Performance Band B we can simply put in the value of 87% as follows:

APS (Performance Band B) = 4 x (APS – 90%) + 80%

= 4 x ( 94% – 90%) + 80%

= 4 x (4%) + 80%

= 16% + 80% = 96%

Therefore, an Achieved Performance score of 94% is equal to an APS of 96%.

Again, you will notice the additional term in the equation above, specifically (APS – 90%) giving the same ‘range’, or distance, that the ‘run’ had in calculating the gradient; that is between 90% and 95%.  Therefore, we need to subtract 90% from the Achieved Performance score before multiplying by the gradient.  However, in this circumstance, the APS had a starting value of 80% and therefore we had to add 80% to the final score.

By being able to do this maths both buyer and seller can determine the APS for each performance measure, and in some cases, determine the payment.  Accordingly, it is important for PBC practitioners, regardless of whether buyer or seller, to be able to understand and complete these calculations.

In the next article we’ll look at an alternate option to using this maths.

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