Earlier we looked at calculating the probability of beating a fixed target. Now we will look at calculating the probability of beating a benchmark which is itself stochastic.

Let us consider two assets A and B with the following details:

We have a total of $10 million to invest. Our objective is to beat a benchmark.

Let us take the 50-50 portfolio, which has the following returns:

$r\_{1} = 0.5A + 0.5B$

Suppose the benchmark has the following returns:

$r\_{2} = 0.4A + 0.6B$

We need to find that probability that our portfolio will beat the benchmark index, i.e., $P(r\_{1} > r\_{2})$