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 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)
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.
Therefore, our returns can be expressed as follows:
Our probability can be expressed as follows:
Let us now evaluate each of the three portfolios.
1. Entire $10 million in Asset A
That is, P(Z > 2) = 2.28%
2. Entire $10 million in Asset B
That is, P(Z > 1.46) = 7.19%
3. $5 million in A and $5 million in B
First we need to calculate the expected return and standard deviation of the portfolio.
That is, P(Z > 2.09) = 1.8%
You can observe that the portfolio with the highest risk has the greatest chance of reaching the target, while the diversified portfolio has the least chance of reaching the target.