# P&L attribution methodology

P&L Attribution analysis and reporting provides users with a coherent breakdown of the drivers of P&L movements between two points in time with reference to a select number of easily understandable pricing factors.

P&L Attribution can be calculated in two ways, either the Sensitivities method or the Scenario based method.

Sensitivities method

The Sensitivities Method involves the calculation of a trade’s sensitivities (also known as the Greeks) and then using them to predict the expected change in P&L from one period to the next by using the actual market changes in the factors driving the transaction price over the same period and the transaction’s sensitivity to those factors. For example, let’s assume the value, V, of a trade is affected by three variables, time τ; underlying price p and another generic factor, x:

If we call the trade value V’s sensitivity to a change in the underlying asset price ‘Delta’ or ‘δ’ and the observed underlying asset price p at time t is ‘pt‘ then the expected P&L change due only to the change in underlying asset price can be approximated as follows:

Impact of underlying price on value V ≈ δ.(pt-1 – pt-2)

To calculate the total expected impact on the trade value we simply repeat this approach for the other factors which drive the value of the transaction and then aggregate these to give a final expected value change. This predicted value change can then compared to the observed change in the transaction’s value.

Impact of the change in Time = V(τt – 1,pt – 2,xt – 2) – V(τt – 2,pt – 2,xt – 2)

Impact of the change in Time = V(τt – 1,pt – 2,xt – 2) – V(τt – 2,pt – 2,xt – 2)

Scenario method

The Scenario or Revaluation method executes a series of valuation scenarios, one for each factor driving the transaction value, whereby each scenario changes one factors’ value at a time by an amount equal to that observed between the two time points in question. P&L attribution is then calculated by aggregating the impact of these valuation scenarios and not on fixed sensitivities. Therefore,

Impact of the change in Time = V(τt – 1,pt – 2,xt – 2) – V(τt – 2,pt – 2,xt – 2)

Impact of the change in Underlying Prices = V(τt – 2,pt – 1,xt – 2) – V(τt – 2,pt – 2,xt – 2)*

The Scenario method will tend to be more onerous on computing time and resource than the Sensitivities Method but it has the advantage of more accurately taking into account second order effects, such as gamma.

For each method the expected P&L may be different from the actual P&L, this will then give a ratio of explained to unexplained P&L.

* *strictly speaking, pt – 2, should actually be the forward value of pt – 2 at time t – 1.*