Lessons

Probability of One Portfolio Outperforming Another Portfolio

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


	Mean	Standard Deviation	Correlation
A	$\mu\_{A}=10%$	$\sigma\_{A}=20%$	$\rho\_{AB}=30%$
B	$\mu\_{B}=12%$	$\sigma\_{B}=26%$

We have a total of $10 million to invest. Our objective is to reach a target return of $5 million. Let us look at the following three options and find out the probability of reaching our target in each case:

Entire $10 million in Asset A
Entire $10 million in Asset B
$5 million in A and $5 million in B

Assuming r as the return from each portfolio, our objective can be expressed as follows:

P (10million*r > 5million)

Or

P (r > 0.5)

We know that if the returns of an asset are normally distributed, it can be expressed as a function of standard normal distribution. We can associate the return distribution to a standard normal distribution, which has a zero mean and a standard deviation of one.

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