Quantitative Methods - 4.2 - Normal Distributions

Question 1

what are the key properties of a normal distribution?

Answer 1

• completely described by mean and variance
• symmetric about the mean (skewness=0)
• kurtosis=3
• A linear combination of normally distributed random variables is also normally distributed (e.g. if asset returns are normally distributed then the portfolio returns are normally distributed)
• posibilities of outcomes decrease further from the mean but tails go on forever

Question 2

what is the difference between a univariate and multivariate distribution?

Answer 2

• univariate: single random variable
• multivariate: more than one random variable (to analyse we need the means, variances, asset weightings and correlation coefficients)

Question 3

what is a confidence interval?

Answer 3

a range of values within which a random variable is expected to be a certain percentage of the time

Question 4

what are the three confidence intervals we need to know?

Answer 4

• The 90% confidence interval for X is e(x) − 1.65s to e(x) + 1.65s.
• The 95% confidence interval for X is e(x) − 1.96s to e(x) + 1.96s.
• The 99% confidence interval for X is e(x) − 2.58s to e(x) + 2.58s.

(s=standard deviations)

Question 5

what is a standard normal distribution?

Answer 5

a normal distribution where mean=0 and standard deviation=1

Question 6

what is a z-value/score?

Answer 6

the number of standard deviations a variable lies from the mean

Question 7

what do the values in z-tables show?

Answer 7

the probabilities of observing a z-value that is less than a given value, z [i.e., P(Z < z)]

Question 8

what is shortfall risk?

Answer 8

the probability that a portfolio return or value will be below a target return or value

Question 9

what is Roy’s saftey-first ratio?

Answer 9

the number of standard deviations that you need to move before you hit the target expected return/value

Question 10

in Roy’s saftey-first ratio, the more standard deviations you need to move before you hit the target/threshold, the…..?

Answer 10

the smaller the probability of breaching the threshold level i.e. the safest

Question 11

what is the process of standardisation?

Answer 11

the process of converting an observed value for a random variable to its z-value

Question 12

what is the formula for converting negative z-values on a positive z-values table?

Answer 12

F(–Z) = 1 − F(Z)