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:

Mean | Standard Deviation | Correlation | |

A | |||

B |

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:

Suppose the benchmark has the following returns:

We need to find that probability that our portfolio will beat the benchmark index, i.e.,

This can be expressed as:

We can write this as:

or

0.1A – 0.1B is normally distributed.

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

Mean,

Standard Deviation,

We can write our probability as follows:

where Z is the standard normal variable.

, 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.

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