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IFT Notes for Level I CFA® Program

R07 Statistical Concepts and Market Return

Part 1


 

1. Introduction

Statistics is a powerful tool for analyzing data and drawing conclusions. As investors, we are concerned about the returns on our investments and the distribution of those returns. We need this information to evaluate our investments.

For example, we often look for central tendency. Let’s say that the stock market has returned on average 14% over the last 10 years. Obviously, we are concerned with this average number but we are also concerned with the dispersion which tells us how spread out the returns have been. One of the simplest measures of dispersion is range. Let’s say that over the last 10 years the stock market has ranged between -20% and +35%. This tells us about the riskiness of our investments.

In this reading, we will study statistical methods that allow us to summarize return distributions.

Specifically, we will explore four properties of return distributions:

  • Where the returns are centered (central tendency).
  • How far returns are dispersed from their center (dispersion).
  • Whether the distribution of returns is symmetrically shaped or lopsided (skewness).
  • Whether extreme outcomes are likely (kurtosis).

2. Some Fundamental Concepts

2.1 The Nature of Statistics

The term statistics can have two broad meanings, one referring to data and the other to a method used to analyze data. Statistical methods include:

  • Descriptive statistics: It refers to how large data sets can be summarized effectively to describe their important characteristics.
  • Inferential statistics: It refers to making forecasts, estimates or judgments about a large data set based on a small representative set.

The focus of this reading is on descriptive statistics. We will cover inferential statistics in a later reading.

2.2 Populations and Samples

A ‘population’ is defined as all members of a specified group. A ‘parameter’ describes the characteristics of a population.

A ‘sample’ is a subset drawn from a population. A ‘sample statistic’ describes the characteristic of a sample.

All stocks listed on the exchange of a country is an example of population. If 30 stocks are selected from among the listed stocks, then this is a sample.

2.3 Measurement Scales

The different types of measurement scales are:

  • Nominal scales: These scales categorize data but do not rank them. Hence, they are often considered the weakest level of measurement. An example would be the classification of mutual funds according to the investment strategy followed – growth fund, value fund, income fund, emerging equity fund etc.
  • Ordinal scales: These scales sort data into categories that are ordered with respect to some characteristic. An example is Standard & Poor’s star ratings for mutual funds. One star represents the group of mutual funds with the worst performance. Similarly, groups with two, three, four, and five stars represent groups with increasingly better performance.
  • Interval scales: These scales not only rank data, but also ensure that the differences between scale values are equal. The Celsius and Fahrenheit scales are examples of such scales. The difference in temperature between 10oC and 11oC is the same amount as the difference between 40oC and 41o However, these scales do not have a true zero. For example, 0oC does not represent the absence of temperature. It is simply the freezing point of water; it is not a true zero.
  • Ratio scales: These scales have all the characteristics of interval scales as well as a true zero point as the origin. This is the strongest level of measurement. The rate of return on an investment is measured on a ratio scale. A return of 0% means the absence of any return.

3. Summarizing Data Using Frequency Distributions

A frequency distribution is a tabular display of data summarized into a relatively small number of intervals. In order to construct a frequency distribution, we can follow the following procedure:

  1. Define the intervals.
  2. Tally the observations i.e. assign observations to intervals.
  3. Count the observations in each interval.

Example

Say you are evaluating 100 stocks with prices ranging from 46 to 65 that are divided into the following four intervals of stock price each having a width of 5:

46-50, 51-55, 56-60 and 61-65. Assume the number of stocks whose prices fall in each of these intervals are 25, 35, 29, and 11 respectively.

Calculate the cumulative frequency, relative frequency, and the cumulative relative frequency for the stock prices given the set of intervals above.

Solution:

Stock Price Absolute

Frequency

Cumulative Frequency Relative Frequency Cumulative Relative Frequency
46-50 25 25 0.25 0.25
51-55 35 60 0.35 0.60
56-60 29 89 0.29 0.89
61-65 11 100 0.11 1.00

Absolute frequency: The actual number of observations in a given interval is called the absolute frequency, or simply the frequency, which is given here.

Cumulative frequency: For an interval, it is calculated as the sum of the absolute frequencies of all intervals lower than and including that interval.

Relative frequency: It is the absolute frequency of each interval divided by the total number of observations.

Cumulative relative frequency: For an interval, it is calculated as the sum of the relative frequencies of all intervals lower than and including that interval.

4. The Graphic Presentation of Data

4.1 The Histogram

It is a bar chart of data that has been grouped together into a frequency distribution. The height of each bar is equal to the absolute frequency of each interval.

The advantage of the visual display is that we can quickly see where most of the observations lie.

 

Histogram_1

4.2 The Frequency Polygon and the Cumulative Frequency Distribution

A frequency polygon plots the midpoints of each interval on the X-axis and the absolute frequency of that interval on the Y-axis. Each point is then connected with a straight line.

 

F.Polygon_1

Another graphical tool is the cumulative frequency distribution. Such a graph can plot either the cumulative frequency or cumulative relative frequency against the upper interval limit. The cumulative frequency distribution allows us to see how many or what percent of the observations lie below a certain value. The figure below is an example of a cumulative frequency distribution.

 

F.Polygon_1

 


Quantitative Methods Statistical Concepts and Market Returns Part 1