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Probability of Attaining a Return Goal

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  \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 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)


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.

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