# Joi, Inc. stock has an annual return mean and standard deviation of 11% and 32%, respectively....

## Question:

Joi, Inc. stock has an annual return mean and standard deviation of 11% and 32%, respectively. What is the smallest expected loss, in percentages, in the coming month with a probability of 2.5%? (Note: Use 2 as the multiple in your probability statement)

Also, is the below similar?

You find a particular stock has an annual standard deviation of 25%. What is the standard deviation for a 2-month period?

## Value at risk (VAR)

Value at Risk (VAR) is a statistical measure to evaluate how much an asset or investment is risky for investment. VAR is is calculated as the maximum possible loss for a predefined time horizon within a confidence level.

Taking an example, if a stock has a one-day 2% VAR of $100 , it can be assumed that if there is no trading. there is a 2% probability that the stock can fall down more than $100 over a one-day period.

## Answer and Explanation:

*Joi, Inc. stock has an annual return mean and standard deviation of 11% and 32%, respectively. What is the smallest expected loss, in percentages, in the coming month with a probability of 2.5%? (Note: Use 2 as the multiple in your probability statement) *

{eq}\mu= 11\% {/eq}

Annual standard deviation = {eq}32\% {/eq}

Variance {eq}= 0.32 ^2 = 0.1024 {/eq}

Monthly Variance {eq}= 0.1024/12 = 0.008533333 {/eq}

Monthly standard deviation {eq}(\sigma) = \sqrt {0.008533333} = 0.0924 {/eq}

From the z score table we get, the z score of 2.5% = 1.9 = 2 (approx.)

So smallest expected loss {eq}= \mu - 2*\sigma {/eq}

{eq}=11\% - 2*0.0924 {/eq}

{eq}= -7.48\% {/eq}

**You find a particular stock has an annual standard deviation of 25%. What is the standard deviation for a 2-month period?**

Annual variance {eq}= 0.25^2 = 0.0625 {/eq}

So, 2-month variance {eq}= 0.0625/2 = 0.03125 {/eq}

Standard deviation for a 2-month period {eq}= \sqrt {0.03125} = 17.68\% {/eq}

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