Finding the Probability a Lognormal Return Falls Below Its Mean
Summary
The question works through the probability that a stock’s one-period gross return is below its expected value, assuming that gross return is lognormally distributed. Taking logarithms turns the comparison into a normal probability: compare the log return with the logarithm of the expected gross return, standardize using the normal distribution’s mean and standard deviation, and evaluate the normal cumulative distribution function.
The accepted explanation confirms the probability calculation and clarifies a terminology issue: the task asks for a probability, not a quantile. A quantile is the value of a random variable associated with a specified cumulative probability, such as the median. The discussion supplies no broader model validation or market evidence; its conclusion depends on the stated lognormal assumption and parameters.
Key ideas
- A lognormal gross return becomes normally distributed after taking its logarithm.
- The probability of falling below the expected gross return can be evaluated by standardizing its logarithm and using the normal cumulative distribution function.
- A probability and a quantile are different quantities.
- The calculation depends on the assumed lognormal distribution and its parameters.
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# Clarification on this author's solution for this problem on lognormal stock distribution
# Clarification on this author's solution for this problem on lognormal stock distribution
I am self-studying from a manual on financial economics, and I am trying to completely wrap my head around this solution:
I'm trying to fill in the in-between steps of this solution based on first principles, so please tell me if my understanding is correct (I'll be referencing the paragraph from the textbook shown below my solution):
The rate of return over a one year period is $S_1/S_0$. The expected rate of return is $\textbf{E}[S_1/S_0].$
The problem is therefore asking to find $\text{Pr}[S_1/S_0 < \textbf{E}[S_1/S_0]]$.
Now $S_1/S_0$ is lognormally distributed, so we have:
$\text{Pr}[\ln(S_1/S_0) < \ln(\textbf{E}[S_1/S_0])] = \text{Pr}[\ln(S_1/S_0) < \alpha]$.
We have that $\ln(S_1/S_0)$ is normally distributed with parameters $m = 0.1 - 0.5(0.30)^2 = 0.055$ and $v = 0.30$.
Hence $\text{Pr}[\ln(S_1/S_0) < \alpha] = \text{Pr}[z < (\alpha - m)/v] = N(\frac{0.1 - 0.055}{0.3}) = N(0.15) = 0.55962$.
My question is:
- Is the logic of the in-between steps correct? Please correct any detail that I don't have quite right.
- What does the author mean by the quantile of the expected return, and why is that helpful?
## Answer by Quantuple (score 2, accepted)
https://quant.stackexchange.com/a/28381
- Yes, your steps are valid
- This is a wrong use of the term "quantile". Here you need to compute a probability (through the normal cdf) and not a quantile (i.e. the value of a random variable corresponding to a given level of the cdf, e.g. the quantile 0.5 (or percentile 50%) is the median)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.