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., $P(r_{1} > r_{2})$

This can be expressed as:

$P(r_{1} - r_{2} > 0)$

We can write this as:

$P(0.5A + 0.5B - 0.4A - 0.6B > 0)$

or

$P(0.1A - 0.1B) > 0$

0.1A - 0.1B is normally distributed.

Therefore, it's mean and standard deviation will be given as follows:

Mean, $E(0.1A-0.1B) = 0.1\mu_{A} - 0.1\mu_{B} = 0.1(10\%)-0.1(12\%) = -0.2\%$

Standard Deviation, $\sigma(0.1A-0.1B) = \sqrt{0.1^{2}\sigma_{A}^{2}+0.1^{2}\sigma_{B}^{2}-2(0.1)(0.1)\sigma_{A}\sigma_{B}\rho}$

$= 2.76\%$

We can write our probability as follows:

$P(0.1A-0.1B>0) = P\left ( \frac{0.1A-0.1B+0.002}{0.0276} > \frac{0 + 0.002}{0.0276} \right )$

$= P(Z>0.0725)$

where Z is the standard normal variable.

$P(Z>0.0725) = 47.1\%$, using 1-NORMSDIST(0.0725) in excel.

Therefore, the 50-50 portfolio has a 47.1% chance of beating the benchmark portfolio of 40-60.

This probability of beating the benchmar depends on the correlation between the assets. With high correlation, the probability will decrease and vice verse.