Question

If the daily returns on the stock market are normally distributed with a mean of .05% and a standard deviation of 1%, the probability that the stock market would have a return of -23% or worse on one particular day (as it did on Black Monday) is approximately __________.

Answer 1

The probability that the stock market would have a return of -23% or worse on one particular day (as it did on Black Monday) is approximately 0%.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

If the daily returns on the stock market are normally distributed with a mean of .05% and a standard deviation of 1%

This means that
`\mu = 0.05, \sigma = 1`

The probability that the stock market would have a return of -23% or worse on one particular day (as it did on Black Monday) is approximately

This is the p-value of Z when X = -23. So

`Z = (X - \mu)/(\sigma)`

`Z = (-23 - 0.05)/(1)`

`Z = -23.05`

`Z = -23.05` has a p-value of approximately 0. So

The probability that the stock market would have a return of -23% or worse on one particular day (as it did on Black Monday) is approximately 0%.