The monthly return of your portfolio is 3%. What is the probability of realizing a 1% return on a given month
The correct answer and explanation is:
To find the probability of realizing a 1% return on a given month when the expected monthly return of your portfolio is 3%, more information about the return distribution is needed. Typically, financial returns are modeled as normally distributed random variables. The expected return (mean) is 3%, but the probability of getting exactly 1% depends on the standard deviation (volatility) of returns.
If the returns follow a normal distribution with mean μ=3%\mu = 3\% and standard deviation σ\sigma, the probability of exactly 1% return is zero because the probability of any single point in a continuous distribution is zero. Instead, we consider the probability of returns falling within an interval around 1%, for example between 0.5% and 1.5%.
To calculate this, we standardize the return using the Z-score formula: Z=X−μσZ = \frac{X – \mu}{\sigma}
where XX is the target return (1%). Using this, the probability of observing a return near 1% is the area under the normal curve between the Z-scores corresponding to the chosen interval.
Without the standard deviation, the exact probability cannot be computed. If the standard deviation is known, the steps are:
- Calculate the Z-score for 1%.
- Use a normal distribution table or calculator to find the cumulative probability up to that Z.
- Find the probability of an interval around 1% if needed.
In practice, portfolio returns fluctuate around the mean 3% with some variability. A 1% return is less than the average return, so the probability will depend on how spread out returns are. A low standard deviation means returns cluster near 3%, making a 1% return less likely. A high standard deviation means returns vary widely, making a 1% return more likely.
In summary, the probability of realizing exactly 1% return in a month cannot be stated without knowing the return volatility or distribution shape. With a normal distribution assumption and known standard deviation, the probability can be found by standardizing the target return and looking up cumulative probabilities