Calculating the Probability of a Negative Return Under a Normal Model
Summary
The document distinguishes an ex ante return estimate based on assumed parameters from an ex post calculation using realized mean and risk. It gives a simple normal-distribution method for estimating the probability that a return falls below zero: standardize zero relative to the return mean and standard deviation, then evaluate the standard normal cumulative probability.
With a zero mean, a symmetric normal distribution assigns equal probability to positive and negative returns. With the example’s positive mean equal to one standard deviation, the standardized threshold for a loss is one standard deviation below the mean, yielding a probability of about 15.87%. This calculation is conditional on the normal model and the stated mean and standard deviation; it does not establish that actual returns are normally distributed or that historical averages will persist.
Key ideas
- A zero-mean normal return has equal probabilities of being above and below zero.
- To estimate loss probability, standardize the zero-return threshold using the mean and standard deviation.
- A mean one standard deviation above zero implies a negative-return probability of about 15.87% under normality.
- The result depends on the normal-distribution assumption and the chosen return parameters.
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# Calculating ex ante returns & probability of a negative return over some horizon
# Calculating ex ante returns & probability of a negative return over some horizon
One way to go on about this is to parametrically calculate the returns, i.e. hold the exposure constant and backtest against the factor changes over that horizon. This is not forward looking per se but it's using assumption based model to arrive at that distribution (so no longer assuming a normal distribution). The other way is to simulate returns through a Black Scholes Model.
Would the following statement be correct: in an ex ante framework there is a 50% chance of returns being either negative or position, assuming normal distribution and a mean of zero. Whereas in an ex-post framework where we have a realised mean return of 3% per year and run approximately 3% of risk then there is a probability of 16% of being negative over a one year period. Does this make any sense at all to you? How would you back out that 16%?
## Answer by Gordon (score 1)
https://quant.stackexchange.com/a/25954
For a normal random variable $\xi$ with mean 0, then \begin{align*} P(\xi < 0) = P(\xi > 0) = 50\,\%. \end{align*}
For a normal random variable $\eta$ with mean (i.e., realized mean return) $\mu=3\,\%$ and risk (i.e., standard deviation) $\sigma = 3\,\%$, then \begin{align*} P(\eta < 0) &= P\left(\frac{\eta - \mu}{\sigma} < \frac{ - \mu}{\sigma} \right)\\ &= P\left(\frac{\eta - \mu}{\sigma} < -1 \right)\\ &\approx 15.87\,\%. \end{align*}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.