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 |
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| 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.
