Question

Explain the concept and purpose of z-scores and calculate them for the performance of the following set of portfolio managers.

Portfolio	Portfolio Performance
Manager	(Annualised Return
Z	2%
W	11%
T	-4%
S	5%

Answer 1

1] Definition

Z - score is the standardized value that specifies the exact location of an X value within a distribution by describing its distance from the mean in terms of standard deviation units(s or sigma ,according to whether the data set is a sample or is the entire population.)

     

the formulas in the definition allow us to compute the z-score when x is known.

A positive Z score indicates that the observed value is Z standard deviations above the mean. Negative Z score indicates that the value is below the mean

2] Purpose of z-scores

Z-scores are used in statistics to measure an observation's deviation from the group's mean value.

It tells us wheather a score is typical for a specific dataset or not.

It also allows comparison of scores on different kinds of variables (or different samples )by standardizing the distribution. A standard normal distribution (SND) is a normally shaped distribution with a mean of 0 and a standard deviation (SD) of 1

3]

calculate the z-score for the performance of the following set of portfolio managers.

Portfolio	Portfolio Performance
Manager	(Annualised Return
Z	2%
W	11%
T	-4%
S	5%

Answer:-

For this data = 0.035 and standard deviation for sample ,which is denoted by s =0.06245. The first observation x=0.02 in the data set has z-score,

which means that x=0.02 is 0.24019 standard deviation below the sample mean. Similarly, for the second observation x= 0.11 the z-score is 1.20096 standard deviattion is above the sample mean.

Similarly , repeat the process we will get z score as follows--

Portfolio	Portfolio Performance	z- score
Manager	(Annualised Return)
Z	0.02	-0.24019
W	0.11	1.200961
T	-0.04	-1.20096
S	0.05	0.240192