Estimating Return Probabilities from a Normal Distribution

Summary

The document explains a basic probability calculation for a stock return using its historical average and standard deviation. It assumes returns follow a normal distribution and asks for the probability of a return below a threshold that lies one standard deviation below the mean. The stated solution uses the familiar normal-curve areas: half of outcomes are above the mean, and roughly 34% lie between the mean and one standard deviation below it. Subtracting both regions from the total gives the lower-tail probability presented in the answer.

This is an illustration of reading probabilities from a normal distribution, rather than a test that historical returns actually are normal. The calculation depends on that assumption, and the document gives no evidence that the stock's returns follow a normal distribution or that historical mean and volatility reliably describe future outcomes. It also does not discuss estimation uncertainty, changing volatility, skewness, or fat tails, all of which can affect return probabilities in practice.

Key ideas

• The calculation assumes returns are normally distributed.
• Half of a normal distribution lies above its mean.
• About 34% of outcomes lie between the mean and one standard deviation from the mean on either side.
• A threshold one standard deviation below the mean leaves the lower-tail probability after subtracting the upper half and the intervening region.
• The result depends on the normality assumption and the use of historical estimates.

Tags

# Probability of a return from historical average and standard deviation


# Probability of a return from historical average and standard deviation












I have a question from a sample exam paper that I'm having some trouble figuring out.

The question is: Bavarian Sausage stock has an average historical return of 16.3% and a standard deviation of 5.3%. What is the probability that the return on Bavarian Sausage will be less than 11%?

ANS: 16% 1-.5-.34 = .16

The answer provided shows the upper and lower values to be .5 and .34 (50% and 34%). How do you arrive at these figures?

## Answer by Nikos (score 2)

https://quant.stackexchange.com/a/9551

The answer assumes a normal distribution.

As you can see in this graph, in the normal distribution:

- 50% of the outcomes are in the right hand part of the distribution (i.e. higher than the mean)

- 34% of the outcomes are between the mean and 1 standard deviation

The question wants you to determine the probability that your stock returns less than 11%. If you notice that the 11% are exactly 1 standard deviation away from the mean (11% = 16.3%-5.3%) you know that you can compute the probability by doing:

1 (all the outcomes) - 0.5 (all the outcomes above the mean) - 0.34 (outcomes between mean and standard deviation, below the mean).

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.