1.4 normal distribution

Question 1

The normal distribution

Answer 1

popular for quantitative work

–> partly because of the central limit theorem

can be completely described by two parameters – mean (μ) and variance (σ2)

It is symmetric (skewness = 0) and has a kurtosis of 3

The mean, median, and mode are all the same

Also, a linear combination of two or more normal random variables is normally distributed

Question 2

Multivariate distributions

Answer 2

specify the probabilities for a group of related random variables

They are common in investment work

Question 3

A multivariate normal distribution for n random variables is defined by which parameters?

Answer 3

n means

n variances

n*(n−1)/2 distinct correlations

For example, a multivariate normal distribution with 3 random variables will have:

3 means
3 variances
3(2)/2 = 3 distinct correlations

–> So, this multivariate distribution has 3 + 3 + 3 = 9 parameters in total.

Question 4

The normal distribution is used to model many security returns. However, the fit is not good for which type of distributions?

Answer 4

distributions with fat tails (i.e., high kurtosis) or asymmetry

Question 5

The normal distribution is not appropriate for which asset prices?

Answer 5

with a floor of zero

Question 6

explain the amount of observations within a certain amount of standard deviations away from the mean

Answer 6

50% of observations are within ±(2/3)σ

68% of observations are within ±σ

95% of observations are within ±2σ

99% of observations are within ±3σ

Question 7

how do we standardize the normal random variable X?

Answer 7

by subtracting the mean and dividing by the standard deviation

This results in a standard normal random variable Z that is normally distributed with a mean of 0 and a standard deviation of 1

Question 8

formula for Z

Answer 8

z = (X - μ) / σ

μ = mean return

X = random variable

Question 9

mean-variance analysis

Answer 9

focuses on symmetric risk

Question 10

The focus with safety first

Answer 10

the probability the portfolio return, RP, falls below the threshold level, RL

If the returns are normally distributed, P(RP < RL) can be minimized by maximizing the safety-first ratio (SFRatio)

–> (also known as Roy’s safety-first ratio)

Question 11

SFRatio formula

Answer 11

(E(RP) - RL) / σP

The SFRatio is just the Sharpe ratio with RL
substituted for RF

Question 12

The probability of RP falling below RL is:

Answer 12

P(RP < RL) = N*(−SFRatio)

Question 13

stress testing

Answer 13

measuring risk at financial institutions

uses very unfavorable events or scenarios to manage the risk

Question 14

Which of the following is most likely one of the parameters required to completely describe a multivariate normal distribution?

A
Kurtosis

B
Skewness

C
Correlation

Answer 14

C
Correlation

Question 15

Safety-first rules are most likely to be used by investors who:

A
10%
consider risk symmetrically.

B
88%
are concerned about shortfall risk.

C
do not consider the correlations of returns on assets within a portfolio.