Question

Many science-based projects have a few assumptions that are based on future events and their results. Thus, the concept of probability is used to estimate the future outcomes of the events.

a) Value at Risk, or VaR for short, is a statistical concept that is usually based on properties of normal distribution and signifies what level of financial risk a company is potentially exposed to. Explore this concept and explain how normal distribution is used to calculate the VaR.

b) Draw an argument explaining why means of randomly selected samples are normally distributed around the population mean. Use the central limits theorem in your explanation. Provide some original examples of this concept.

c) Use the central limits theorem to solve this problem and draw conclusions from it: A population of first year undergraduate students working part time has a mean salary of £15,572 with a standard deviation of £3,150. If a sample of 50 students is taken, what is the probability that the mean of their salaries will be less than £15,000? How would this probability change if the sample had only 5 students?

Answer 1

a) VaR tells us the maximum loss that a company can make at a given confidence level. A 95% VaR means the maximum loss possible in 5 out of 100 times/days.

Since returns over an extended period can be expected to be normally distributed, to calculate 95% VaR, we find out the point on the normal curve such that the total probability to the left of it is 5%.

b) As per Central Limit Theorem, average of a large number of independent random variables tend to normal distribution, with mean equal to the population mean. Hence, if we consider randomly selected sample with > 30 data points in each sample, their means will be found to be normally distributed with mean nearly same as the mean of the population the samples are drawn from.

An example is coin flips experiment, where if we flip the coin a number of times, the probability of getting a given no. of heads is normally distributed, with mean equal to half the number of flips.

c) Given the sample size is more than 30, we have a large sample here and hence the mean of the sample (i.e. mean salary) is normally distributed with mean same as the population is is drawn from.

Hence, P(X < 15000) = P( Z < (15000-15572 / 3150 ) (using the corresponding Z-score)

= P( Z < -0.1816 )

= 0.429

A sample with only 5 students will not have its mean normally distributed. This is expected to have a Student's T-distribution with thicker tails, hence a higher probability of the probability for mean salary less than $ 15000