Study Notes for Statistical Concepts and Market Returns

Statistical Concepts and Market Returns

Statistics is the science of analyzing data. When you are presented with the daily closing prices of a stock for the past one year, how do you make sense of this data? Using the tools and techniques offered by statistics, you can analyse the data in various ways. For example, you can find out the average price of the stock over the past one year. You can also calculate other statistics such as the dispersion of the stock prices around the mean. Statistics deals with all aspects of data including collecting data, organizing data, analysing it, interpreting it and presenting it in a useful forms.

All statistical methods can be classified as descriptive or inferential statistics.

<strong>Descriptive statistics</strong> refers to analysis of data in order to summarize the important characteristics of data in a meaningful way. However, descriptive statistics does not allow us to make any conclusions beyond the data. Two important types of descriptive statistics include the Measures of Central Tendency and Measures of Dispersion. For example, you may have the monthly savings data of 100 families and using that you can calculate descriptive statistics such as average savings and the dispersion of savings in this group of 100 families. However, descriptive statistics will describe the characteristics of only this group of 100 families. This group of data that contains all the data that you are interested in describing is called <strong>population</strong>. Another example of population is the returns of all stocks trading on NASDAQ. Note that the size of the population does not matter. As long as the data set, whether small or big, contains all the data that you are interested in, it represents your population.

<strong>Inferential statistics</strong> uses the sample data to reach some conclusion about the characteristics of the larger population. Using the same example of savings by families, we know that descriptive statistics cannot be used to make any conclusions about any families other that the 100 families in our data group. For example, what if you were interested in the savings pattern of an entire country, such as the U.S. It may not be feasible or practical to collect the monthly savings data of every family in the U.S. that constitutes your population. In that case, you will take a small <strong>sample</strong> of families from across the U.S. that will be used to represent the larger population of U.S. You will use this sample data to calculate its mean and standard deviation. We use inferential statistics techniques to make conclusions or inferences about the population that the sample represents. Two common methods of inferential statistics are Estimation of Parameters, and Hypothesis Testing.

Descriptive Vs. Inferential Statistics

Depending on the information we want the data to represent, we can choose one of the four measurement scales.

<strong>Nominal Scale</strong>
<ul>
 	<li>Used to classify data</li>
 	<li>Observations are put into categories based on some criteria.</li>
 	<li>The category labels can be numbers but they don’t have any numeric value.</li>
 	<li>Example 1: Classifying stocks as small-cap, mid-cap, and large-cap</li>
 	<li>Example 2: Classifying funds as equity funds, debt funds, and balanced funds.</li>
</ul>
<strong>Ordinal Scale</strong>
<ul>
 	<li>Used to classify and order (Ranking)</li>
 	<li>Observations are not just classified but also ordered</li>
 	<li>Example: Ranking top 10 stocks based on their P/E ratio</li>
 	<li>The numbers only represent the order. They do not say anything about how much better or worst a stock is at a given number compared to one at a lower number.</li>
</ul>
<strong>Interval Scale</strong>
<ul>
 	<li>Used to classify and order with an equal interval scale</li>
 	<li>The intervals between adjacent scale values are equal.</li>
 	<li>Scale has an arbitrary zero point and as a result you cannot calculate ratios.</li>
 	<li>Example: Temperature scales. A temperature of 40 degrees is higher than 35 degrees and is higher by 5 degrees.</li>
 	<li>The problem is that a temperature of 0 degrees does not imply absence of temperature. Because of this, a temperature of 20 degrees does not necessarily mean twice as hot as a temperature of 10 degrees.</li>
</ul>
<strong>Ratio Scale</strong>
<ul>
 	<li>All the above features along with an absolute zero.</li>
 	<li>Equal units of measurements and a rational zero point for the scale.</li>
 	<li>Example: Income of a group of people in dollars. If you have 0 dollars that means complete absence of money (what we are measuring). However, if A has $10 and B has $20, then B has twice as much money as A has.</li>
</ul>

Types of Measurement Scales

Earlier we took an example of measuring the average savings for a group of 100 families. This data of 100 families represents the population. The properties of a population such as mean and standard deviation are called parameters.

Instead if we wanted to measure the average savings of an entire country, we would take a sample and then use that sample whose characteristics will be used to represent the entire population. These properties of sample are not called parameters, but <strong>sample statistics</strong>.

The statistical data that we collect can be presented in the form of a frequency distribution. A frequency distribution refers to summarizing a large data set into a small number of intervals.

Let’s take an example to understand how to construct a frequency distribution. Let’s say we have the following 20 observations with us.

1.5, 2.5, 3, 2.3, 4.3, 5.6, 4.2, 6.7, 5.9, 1.2, 5.4, 9.8, 8.5, 5.5, 2.9, 1.7, 8.8, 6.2, 9.5, 3.8

To construct a frequency distribution for this data, we will follow the following steps.

<strong>Step 1: Sort the data in ascending order</strong>

The sorted data is presented below:

1.2, 1.5, 1.7, 2.3, 2.5, 2.9, 3, 3.8, 4.2, 4.3, 5.4, 5.5, 5.6, 5.9, 6.2, 6.7, 8.5, 8.8, 9.5, 9.8

<strong>Step 2: Calculate the range of data</strong>

The range refers to the lower and upper limit for the data intervals. The minimum value is 1.2 and the maximum value is 9.8. This is the range of the data.

<strong>Step 3: Decide on the number of intervals in the frequency distribution</strong>

This must be done carefully so that the number of intervals is neither to high not too less. If we take very few intervals, our distribution will classify data very broadly. If we take too many intervals, then it won’t really be a summary of data.

In our example, we have 20 observations ranging from 1 to 10. If we use a value interval of 1, then we will have 9 intervals. For 20 values, 9 intervals are too many. So, we can decide to have 5 intervals with a width of 2.

<strong>Step 4: Determine the intervals.</strong>

Starting from 0, we can have 5 non-overlapping intervals as follows:

0 &lt;= r &lt; 2

2 &lt;= r &lt; 4

4 &lt;= r &lt; 6

6 &lt;= r &lt; 8

8 &lt;= r &lt; 10

Note that the above intervals are all-encompassing and non-overlapping. Each observation will fall under one (and only one) frequency interval.

<strong>Step 5: Tally and count the observations under each interval.</strong>

We will now assign each observation to one of these intervals and then count the total number of observations in each interval.
<table>
<tbody>
<tr>
<td width="308"><strong>Interval</strong></td>
<td width="308"><strong>Absolute Frequency</strong></td>
</tr>
<tr>
<td width="308"><strong>0 &lt;= r &lt; 2</strong></td>
<td width="308">3</td>
</tr>
<tr>
<td width="308"><strong>2 &lt;= r &lt; 4</strong></td>
<td width="308">5</td>
</tr>
<tr>
<td width="308"><strong>4 &lt;= r &lt; 6</strong></td>
<td width="308">6</td>
</tr>
<tr>
<td width="308"><strong>6 &lt;= r &lt; 8</strong></td>
<td width="308">2</td>
</tr>
<tr>
<td width="308"><strong>8 &lt;= r &lt; 10</strong></td>
<td width="308">4</td>
</tr>
</tbody>
</table>
Note that the interval 6-8 has the lowest frequency (number of observations) and interval 4-6 has the highest frequency. The frequency so calculated is referred to as <strong>Absolute Frequency</strong>.

Parameter, Sample Statistic, and Frequency Distribution

The data in a frequency distribution can also be presented using relative frequencies.

<img class="aligncenter wp-image-18952 " src="https://financetrain.com/wp-content/uploads/rf1.gif" alt="rf1" width="485" height="55">

Once we have relative frequencies, we can calculate cumulative relative frequencies where as we move from first frequency interval to the last, we keep adding the relative frequencies finally reaching 100%. Cumulative relative frequencies are useful in measuring what fraction of total observations are less than the upper limit of a frequency interval.

We will extend our example to show the relative frequencies and cumulative relative frequencies.
<table>
<tbody>
<tr>
<td width="164"><strong>Interval</strong></td>
<td width="175"><strong>Absolute Frequency</strong></td>
<td width="139"><strong>Relative Frequencies</strong></td>
<td width="139"><strong>Cumulative Relative Frequencies</strong></td>
</tr>
<tr>
<td width="164"><strong>0 &lt;= r &lt; 2</strong></td>
<td width="175">3</td>
<td width="139">3/20 = 15%</td>
<td width="139">15%</td>
</tr>
<tr>
<td width="164"><strong>2 &lt;= r &lt; 4</strong></td>
<td width="175">5</td>
<td width="139">5/20 = 25%</td>
<td width="139">40%</td>
</tr>
<tr>
<td width="164"><strong>4 &lt;= r &lt; 6</strong></td>
<td width="175">6</td>
<td width="139">6/20 = 30%</td>
<td width="139">70%</td>
</tr>
<tr>
<td width="164"><strong>6 &lt;= r &lt; 8</strong></td>
<td width="175">2</td>
<td width="139">2/20 = 10%</td>
<td width="139">80%</td>
</tr>
<tr>
<td width="164"><strong>8 &lt;= r &lt; 10</strong></td>
<td width="175">4</td>
<td width="139">4/20 = 20%</td>
<td width="139">100%</td>
</tr>
<tr>
<td width="164"><strong>&nbsp;</strong></td>
<td width="175"><strong>20</strong></td>
<td width="139"><strong>100%</strong></td>
<td width="139"><strong>&nbsp;</strong></td>
</tr>
</tbody>
</table>
The cumulative relative frequency is equal to the some of the relative frequencies of all the previous intervals including the current interval. For example, the cumulative absolute frequency for the interval 4 &lt;= r &lt; 6 is 15% + 25% + 30% = 70%.

Relative Frequencies and Cumulative Relative Frequencies

<h5>Histogram</h5>
The data in a frequency distribution can be presented using a histogram. A histogram is a bar chart with different intervals on the X-axis and the absolute frequencies on the Y-axis. The histogram for our data is presented below:

<img class="aligncenter size-full wp-image-18955" src="https://financetrain.com/wp-content/uploads/ds1.gif" alt="Histogram" width="439" height="252">
<h5>Frequency Polygon</h5>
A frequency polygon is similar to a histogram, except that the x-axis plots the mid-point for each interval. Instead of bars, the neighbouring points are connected by lines.

The interval mid points for our frequency intervals are 1, 3, 5, 7, and 9. The frequency polygon will look as follows:

<img class="aligncenter size-full wp-image-18956" src="https://financetrain.com/wp-content/uploads/ds2.gif" alt="Frequency Polygon" width="434" height="262">

&nbsp;

Properties of a Data Set (Histogram / Frequency Polygon)

The measures of central tendency identify what is the center of a data set. These are the most widely used tools among all the statistical measures. The most commonly used measures of central tendency include: arithmetic mean, geometric mean, weighted average mean, median and mode.

We will use the following data set to calculate all these measures:

<strong>1.5, 2.5, 3, 2.3, 4.3, 5.6, 4.2, 6.7, 5.9, 1.2, 5.4, 9.8, 8.5, 5.5, 2.9, 1.7, 8.8, 6.2, 9.5, 3.8</strong>

Measures of Central Tendency

Arithmetic mean is the simple average of all observations and is calculated by adding all the observations and dividing it by the total number of observations.

We can calculate arithmetic mean for both the population and the sample.

<strong>Population Mean</strong>

This is the arithmetic mean of all observations in the population. The formula for population mean is given below:

<img class="aligncenter size-full wp-image-18964" src="https://financetrain.com/wp-content/uploads/ct1.gif" alt="Population Mean" width="536" height="60">

<strong>Sample Mean</strong>

This is the arithmetic mean of all observations in the sample of the population. The formula for sample mean is given below:

<img class="aligncenter size-full wp-image-18963" src="https://financetrain.com/wp-content/uploads/ct2.gif" alt="Sample Mean" width="515" height="56">

Notice the difference in notations between the two formulas:

Xi represents the observations in both formulas.

The number of observations in the population is represented by capital N, while the number of observations in the sample is represented by small <em>n</em>.

<strong>Example</strong>

<strong>Population Dataset:</strong> 1.5, 2.5, 3, 2.3, 4.3, 5.6, 4.2, 6.7, 5.9, 1.2, 5.4, 9.8, 8.5, 5.5, 2.9, 1.7, 8.8, 6.2, 9.5, 3.8

We can draw a sample from the above data set.

<strong>Sample Dataset:</strong> 2.5, 5.6, 1.2, 9.8, 8.8

<img class="aligncenter size-full wp-image-18962" src="https://financetrain.com/wp-content/uploads/ct3.gif" alt="ct3" width="599" height="137">

<strong>Some observations:</strong>
<ul>
 	<li>As you can see, there is a lot of difference in population mean and sample mean. This can happen for small data sets or if the sample is not drawn correctly.</li>
 	<li>Arithmetic mean is very sensitive to extreme values as a very large or small value can significantly pull the mean on either side.</li>
 	<li>For arithmetic mean, the sum of deviations from the mean is always zero, i.e.,</li>
 	<li>Arithmetic mean is preferred over median and mode, as it uses all information about the observations such as size and magnitude.</li>
 	<li>There can be only one arithmetic mean for a data set.</li>
</ul>

Calculating Arithmetic Mean

One characteristic of an arithmetic mean is that all observations have equal weight (=1/N). However, this may not always be the case. In some cases, different observations may influence the mean differently. This has special relevance in portfolios where a portfolio is made up of different stocks each having a different weight.

Let’s assume that we have a portfolio comprising three stocks, A, B and C as follows:
<table>
<tbody>
<tr>
<td width="121"><strong>Stock</strong></td>
<td width="113"><strong>Returns</strong></td>
<td width="123"><strong>Weight</strong></td>
</tr>
<tr>
<td width="121"><strong>A</strong></td>
<td width="113">12%</td>
<td width="123">20%</td>
</tr>
<tr>
<td width="121"><strong>B</strong></td>
<td width="113">18%</td>
<td width="123">30%</td>
</tr>
<tr>
<td width="121"><strong>C</strong></td>
<td width="113">24%</td>
<td width="123">50%</td>
</tr>
</tbody>
</table>
We have the stock returns for each stock and the weight of each stock in the portfolio. For example, if the investor has a total of $1,000 invested in the portfolio, 20% or $200 is invested in Stock A, $300 is invested in stock B, and the remaining $500 is invested in Stock C.

The weighted average mean is calculated using the following formula:

<img class="aligncenter size-full wp-image-18989" src="https://financetrain.com/wp-content/uploads/wam1.gif" alt="wam1" width="540" height="75">

The weighted mean of our portfolio will be calculated as follows:

<img class="aligncenter size-full wp-image-18988" src="https://financetrain.com/wp-content/uploads/wam2.gif" alt="wam2" width="546" height="40">

Note that the weighted mean is closer to the returns from Stock C because Stock C has more influence (weight) on the portfolio.

Calculating Weighted Average Mean

One problem with arithmetic mean is that it assumes the returns on the investment made at the beginning of each period. So, for each period the beginning investment amount is assumed to be the same. It ignores the compounding effect of investment returns made in the previous years. Using arithmetic returns, our measure can be majorly flawed. Consider an investment of $100 at the beginning. Say in the first year the investment value rises to $200. The returns are 100%. In year 2, the investment falls back to $100, which will be a return of -50% in the year 2. If we take the average of two year returns, i.e., 100% in year 1 and -50% in year 2, it shows an average annual return of 25% on this investment, even though our investment value is back to $100 (from where we started). This problem can be solved by calculating geometric returns which incorporates the compounding effect.

Let’s taken an example to understand how geometric returns are calculated. Let’s say our portfolio generated the following returns in 5 years.

<img class="aligncenter size-full wp-image-18992" src="https://financetrain.com/wp-content/uploads/gm1.gif" alt="gm1" width="138" height="141">

<strong>Geometric returns = [(1+100%)*(1-50%)*(1+35%)*(1-20%)*(1+50%)]^(1/5) – 1 = 10%</strong>

The <a title="Calculating Arithmetic Mean" href="https://financetrain.com/calculating-arithmetic-mean/">arithmetic mean</a> on this portfolio would have been 23%.

As you can see, geometric return is lower than the arithmetic return, and is a better method for aggregating returns over multiple holding periods.

We can say that geometric mean is a more suitable method for aggregating returns over a period of time.

Calculating Geometric Mean

Harmonic mean is calculated by dividing the number of observations (n) by the sum of reciprocals of all observations.

<img class="aligncenter size-full wp-image-18995" src="https://financetrain.com/wp-content/uploads/hm1.gif" alt="hm1" width="526" height="74">

Harmonic mean has some applications in finance. One application is to calculate the average purchase cost of shares purchased over time.

Let’s say that an investor purchased a stock worth $100 for two months. The share price at the time of each purchase was 5 and 7. What will be the average purchase price? We can calculate this as follows.

The number of stocks purchased in the two months are $100/5 = 20 and $100/7 = 14.286. Total number of shares purchased is 34.286 for a total cost of $200. Average purchase price will be = $200/34.286 = 5.833. This is in fact the harmonic mean.

We can use the harmonic mean formula to calculate this.

<img class="aligncenter size-full wp-image-18996" src="https://financetrain.com/wp-content/uploads/hm2.gif" alt="hm2" width="505" height="71">

The relationship between Harmonic Mean, Arithmetic Mean, and Geometric Mean is as given below:
<p style="text-align: center;"><strong>Harmonic Mean &lt; <a title="Calculating Geometric Mean" href="https://financetrain.com/calculating-geometric-mean/">Geometric Mean</a> &lt; <a title="Calculating Arithmetic Mean" href="https://financetrain.com/calculating-arithmetic-mean/">Arithmetic Mean</a></strong></p>

Calculating Harmonic Mean

<h3>Median</h3>
Median refers to the midpoint or the middle value of a data set after sorting the values in ascending or descending order.

Our data set sorted in ascending order is presented below:

1.2, 1.5, 1.7, 2.3, 2.5, 2.9, 3, 3.8, 4.2, 4.3, 5.4, 5.5, 5.6, 5.9, 6.2, 6.7, 8.5, 8.8, 9.5, 9.8

Note that our data set has 20 values, so we don’t have one middle point. There are two values namely, 4.3 and 5.4. The median in this case will be the arithmetic mean of these two values.

Median = (4.3+5.4)/2 = 4.85

This will be the case for all data sets containing an even number of observations.

When we have an odd number of observations (such as 5, 7, or 9), the median is simply the middle value. For example in case of 5 observations, the median will be the 3<sup>rd</sup> value.
<h3>Mode</h3>
Mode refers to the most frequently occurring value in a data set.

Let’s look at our data set:

1.2, 1.5, 1.7, 2.3, 2.5, 2.9, 3, 3.8, 4.2, 4.3, 5.4, 5.5, 5.6, 5.9, 6.2, 6.7, 8.5, 8.8, 9.5, 9.8

All the values in this data set are different. This means that this data set has no mode.

Let’s look at another data set:

1, 3, 4, 4, 5, 7

In this data set the value 4 occurs twice while the rest of the values occur only once. So, the value 4 has the highest frequency of occurring. The mode of this data set is 4. Such a distribution is called unimodal.

If two values had occurred in the data set with highest frequency, the distribution&nbsp; will be called bimodal. A data set having 3 values with highest frequency will be called trimodal, and so on. For example, the following data set has value 4 and 5 occurring with highest frequency.

1, 3, 4, 4, 5, 5, 7

Calculating Median and Mode of a Data Set

A quantile refers to a value at or below which a stated fraction of the data lies. Quantile is a general term, and we have different types of quantiles referring to different fractions.
<table>
<tbody>
<tr>
<td width="130"><strong>Quartiles</strong></td>
<td width="321">Divides the distribution into quarters.</td>
</tr>
<tr>
<td width="130"><strong>Quintiles</strong></td>
<td width="321">Divides the distribution into fifths.</td>
</tr>
<tr>
<td width="130"><strong>Deciles</strong></td>
<td width="321">Divides the distribution into tenths.</td>
</tr>
<tr>
<td width="130"><strong>Percentiles</strong></td>
<td width="321">Divides the distribution into hundredths.</td>
</tr>
</tbody>
</table>
Percentiles are the most commonly used quantiles and other quantiles are also expressed in terms of percentiles.

For example, the first quartile is the value at or below which a quarter or 25 percent of the values in the distribution lie. The second quartile will have 50 percent of values below it (50 percentile).

<strong>Example 1</strong>

Suppose we have a data set with 20 observations and we want to calculate third quartile or 75 percentile. Since the total number of observations (20) is divisible by 4, the calculation is simple. The 15<sup>th</sup> value is the third quartile because 3/4<sup>th</sup> of the values lie at or below the 15<sup>th</sup> value.

<strong>Example 2</strong>

The calculation of a quartile may however be difficult, if the total number of observations is not divisible by 4. For large data sets, this will not matter. However, for a small data set the quartile value will only be an approximation.

Let’s say our data set has 15 observations and we want to calculate the third quartile.

1.2, 1.5, 1.7, 2.3, 2.5, 2.9, 3, 3.8, 4.2, 4.3, 5.4, 5.5, 5.6, 5.9, 6.2

Since 15 is not divisible by 4, we cannot directly observe the third quartile. We will use a slightly different method to calculate this.

We know that third quartile is 75 percentile.

We can calculate the position of an observation at a given percentile using the following formula.

<img class="aligncenter size-full wp-image-19002" src="https://financetrain.com/wp-content/uploads/qt.gif" alt="qt" width="528" height="51">

Where y is the percentile we want to find and n is the number of observations in a dataset. In our data set, n = 15, and y = 75.

<strong>Ly = (15+1)*(75/100) = 12</strong>

The 12<sup>th</sup> observation is 5.5. That means 75 percent of observations lie below 5.5.

In this example Ly was a whole number. If it was not, we would have used linear interpolation to find the quartile.

Quartiles, Quintiles, Deciles, and Percentiles

In investment management, one of the most important things for an investor is the trade-off between the returns and risk from an investment. The return or reward is measured using the measures of central tendency while the risk is measured using the [measures of dispersion](https://financetrain.com/measures-of-dispersion). The dispersion here refers to how the observations vary around the mean.

We will now look at the different types of measures of dispersion.
<h3>Range</h3>
Range is calculated as the difference between the highest value and the lowest value.

<strong>Range = Highest Value – Lowest Value</strong>

Refer to our data set:

1.2, 1.5, 1.7, 2.3, 2.5, 2.9, 3, 3.8, 4.2, 4.3, 5.4, 5.5, 5.6, 5.9, 6.2, 6.7, 8.5, 8.8, 9.5, 9.8

The lowest value is 1.2 and the highest value is 9.8

The range will be given as:

<strong>Range = 9.8 – 1.2 = 8.6</strong>
<h3>Mean Absolute Deviation</h3>
Mean absolute deviation is calculated as the average of the absolute values of difference between each observation and the arithmetic mean.

<img class="aligncenter size-full wp-image-19047" src="https://financetrain.com/wp-content/uploads/mad1.gif" alt="mad1" width="549" height="59">

Note that we take the absolute value of differences. So, any negative signs are ignored.

Let’s take the following data set of five values:

1.2, 1.5, 1.7, 2.3, 2.5

<img class="aligncenter size-full wp-image-19046" src="https://financetrain.com/wp-content/uploads/mad2.gif" alt="mad2" width="580" height="169">

Both Range and Mean Absolute Deviation are not very commonly used measures are dispersion. We will now look at the more robust measures, namely, variance and standard deviation.

Range and Mean Absolute Deviation

Risk is the possibility that actual returns might differ, or vary, from expected returns. In fact, actual returns will most likely differ from expected returns. It is important for decision-makers to estimate the magnitude and likelihood of the difference between actual and estimated returns. After all, there is a big difference if your predictions result in an error of only $100 versus an error of $1 million.

By using the concepts of variance and standard deviation, investors can judge not only how wrong their estimates might be, but also estimate the likelihood, or probability, of favorable or unfavorable outcomes. With the tools of expected return and standard deviation, financial decision-makers are better able to evaluate alternative investments based on risk-return tradeoffs, and their own risk preferences.

In general, the risk of an asset or a portfolio is measured in the form of the standard deviation of the returns, where standard deviation is the square root of variance. Let’s look at how standard deviation and variance is calculated.

Let’s say our observation data set is the returns data of a stock. Using this data, we calculate the mean/average returns. The variance of the asset returns will then be the average of the square of the difference between the returns and the mean.

<img class="aligncenter size-full wp-image-19051" src="https://financetrain.com/wp-content/uploads/sd1.gif" alt="sd1" width="589" height="121">

The standard deviation will simply be the square root of the variance.

The following is a simple example that illustrates the calculation:

<img class="aligncenter size-full wp-image-19050" src="https://financetrain.com/wp-content/uploads/sd2.gif" alt="sd2" width="426" height="383">

Note that in the above example we start with the stock price. Once we have the stock price data, the first step is to calculate the returns. What kind of returns we have depends on the periodicity of the data. For example, if we have daily prices, then we calculate the daily returns, which is calculated as (P(t1) – P(t0) /P(t0).

<strong>Notations</strong>

The population variance and standard deviation are written as s<sup>2 </sup>and s while the sample variance and standard deviation are written as s<sup>2 </sup>and s. For sample variance, we use n-1 in the denominator instead of N. This is done because as per the theory, using the entire sample set n, will underestimate the population variance.

Variance and Standard Deviation

Chebyshev’s Inequality is used to describe the percentage of values in a distribution within an interval centered at the mean.

It states that for a distribution, the percentage of observations that lie within k <a title="Variance and Standard Deviation" href="https://financetrain.com/variance-standard-deviation/">standard deviations</a> is atleast 1 – 1/k<sup>2</sup>

This is illustrated below:

<img class="aligncenter  wp-image-19054" src="https://financetrain.com/wp-content/uploads/ci1.png" alt="ci1" width="348" height="141">

<strong>Example</strong>

The following table shows the minimum number of observations that lie within a certain number of standard deviations of the mean.
<table>
<tbody>
<tr>
<td width="206"><strong>Standard Deviations</strong></td>
<td width="227"><strong>% of observations</strong></td>
</tr>
<tr>
<td width="206"><strong>1.5</strong></td>
<td width="227">56%</td>
</tr>
<tr>
<td width="206"><strong>2</strong></td>
<td width="227">75%</td>
</tr>
<tr>
<td width="206"><strong>3</strong></td>
<td width="227">89%</td>
</tr>
<tr>
<td width="206"><strong>4</strong></td>
<td width="227">94%</td>
</tr>
</tbody>
</table>
An important feature of Chebyshev’s Inequality is that it works with any kind of distribution.

Chebyshev’s Inequality

We earlier learned about calculating the <a title="Variance and Standard Deviation" href="https://financetrain.com/variance-standard-deviation/">variance and standard deviation</a> for a set of data. Standard deviation as a measure of dispersion is much easier to interpret as it uses the same unit of measurement as the data itself. However, standard deviation is not a good measure if we are comparing the relative degree of variation of two sets of data or distributions. For this purpose we have another measure called the coefficient of variation. The coefficient of variation measures the degree of variation in a distribution relative to the mean of the distribution.

Suppose we have two data sets A and B. The coefficient of variation for these data sets is calculated as follows:

<img class="aligncenter size-full wp-image-19058" src="https://financetrain.com/wp-content/uploads/cv1.gif" alt="cv1" width="544" height="114">

The coefficient of variations of two distributions can be compared to find out which one has less dispersion per unit of risk.

Let’s say standard deviation of A is 2% with a mean of 6% and standard deviation of B is 1.5% with a mean of 4%.

<img class="aligncenter size-full wp-image-19057" src="https://financetrain.com/wp-content/uploads/cv2.gif" alt="cv2" width="539" height="124">

We can see that distribution A has less dispersion per unit of return than distribution B.

Coefficient of Variation

While deciding about what investments to make, one should weigh the rewards versus the risks of the investment opportunity. The Sharpe ratio is one popular measure of return on risk. It is named after Nobel Laureate professor William F. Sharpe.

The Sharpe ratio measures the reward (or excess return) of an asset per unit of risk.

The Sharpe ratio is also commonly expressed as:

<img class="aligncenter size-full wp-image-19061" src="https://financetrain.com/wp-content/uploads/sr1.gif" alt="sr1" width="555" height="58">

Both the return and the <a title="Variance and Standard Deviation" href="https://financetrain.com/variance-standard-deviation/">standard deviation</a> are annualized. To annualize returns, you multiply linearly by time. For example, a monthly return of 1% converts to an annualized return of 12%. Standard deviation of return is a measure of risk, or uncertainty, of returns. To annualize standard deviation, multiply by the square root of time.

For example, a monthly standard deviation of return of 1 % converts to annualized standard deviation of 1 % x SQRT(12) = 3.46%.

A higher Sharpe ratio indicates better portfolio performance. Sharpe ratios can be increased either by increasing returns or by decreasing risk.

As we know, a portfolio can achieve higher returns by taking on additional risks. Using the Sharpe Ratio one can determine the source of higher returns: better performance or from additional risks.

Historically, Sharpe ratios over long periods of time for most major asset classes have ranged from 0.3 to 2.

Sharpe ratio has two limitations.
<ol>
 	<li>When the Sharpe ratio is positive, if we increase the risk, the ratio decreases. When Sharpe ratio is negative, however, increasing the risk brings the Sharpe ratio closer to zero, i.e., a higher Sharpe ratio. However, it doesn’t mean better risk-adjusted performance.</li>
 	<li>It considers only the standard deviation as a measure of risk. If the portfolio returns are asymmetric such as strategies involving options, standard deviation is not a good risk measure. For such a portfolio using standard deviation will overestimate Sharpe ratio.</li>
</ol>

Sharpe Ratio

<p>We say that a distribution is symmetric, if it is equally balanced on both sides of the mean. This means that the frequency of observations on both sides is equal.</p>



<p>However, if the frequency of occurrence of observations is more in a particular direction then the distribution is asymmetric. This asymmetry of the distribution on either side of the mean is called skewness.</p>



<p>A symmetric distribution has no tail on either side. Such a distribution is not skewed in any direction. This is the same as a normal distribution i.e. a distribution which has zero skewness.</p>



<p>Suppose we are looking at a distribution of returns with a mean return of 0. If there is a large frequency of occurrence of negative returns compared to positive returns then the distribution displays a fat left tail or <strong>negative skewness</strong>.</p>



<p>In case the frequency of positive returns exceeds that of negative returns then the distribution displays a fat right tail or <strong>positive skewness</strong>.</p>



<div class="wp-block-image"><figure class="aligncenter size-large"><img src="https://financetrain.com/wp-content/uploads/Skewness_PosNegPict.jpg" alt=""/><figcaption>Skewed Distribution</figcaption></figure></div>



<p><strong>Skewness</strong> is a measure of degree of asymmetry of a distribution. It measures the degree to which a distribution leans towards the left or the right side.</p>



<p>While skewness is a measure of asymmetry, <strong>kurtosis</strong> is a measure of the ‘peakedness’ of the distribution.</p>



<p>Both skewness and kurtosis are measured relative to a normal distribution. Just like a distribution can be negatively or positively skewed, it can be Leptokurtic or Platykurtic depending on whether the peakedness is more than or less than the normal distribution.</p>



<p><strong>Leptokurtic:</strong> The distribution is more peaked than a normal distribution.</p>



<p><strong>Platykurtic:</strong> The distribution is less peaked than a normal distribution.</p>



<div class="wp-block-image"><figure class="aligncenter size-large"><img src="https://financetrain.com/wp-content/uploads/KurtosisPict.jpg" alt=""/></figure></div>



<p>For a normal distribution, the computed kurtosis is 3. So, its excess kurtosis is 0. So, a Leptokurtic distribution will have excess kurtosis greater than 0 and a Platykurtic distribution will have excess kurtosis less than 0.</p>



<h3>Calculating Sample Skewness and Kurtosis</h3>



<p>The formulae for calculating skewness and kurtosis are given below:</p>



<figure class="wp-block-image size-large"><img src="https://financetrain.com/wp-content/uploads/sample-skewness-kurtosis-1024x269.png" alt="" class="wp-image-27970"/></figure>



<h3><strong>Some Observations</strong></h3>



<ul><li>Sample skewness can be positive or negative. Positive skewness means that the distribution is right skewed.</li><li>A skewness of 0.5 or more indicates significant skewness.</li><li>For a normal distribution kurtosis is 3. We calculate excess kurtosis as <em>Sample Kurtosus – 3</em>. A positive excess kurtosis indicates that the distribution is Leptokurtic and a negative excess kurtosis indicates that the distribution is Platykurtic.</li><li>An excess kurtosis of 1 or more is considered high.</li></ul>

Skewness and Kurtosis

<p>The skewness of a distribution can have an impact on the location of mean, median and mode.</p>



<p>The relative locations of these measures on symmetric, negatively skewed, and positively skewed distributions are shown below. We have assumed a unimodal distribution, i.e., it has only one mode.</p>



<h3>Symmetric Distribution</h3>



<p><strong>Mode = Median = Mean</strong></p>



<div class="wp-block-image"><figure class="aligncenter size-large"><img src="https://financetrain.com/wp-content/uploads/Picture1.png" alt="" class="wp-image-27974"/></figure></div>



<h3>Negatively Skewed Distribution</h3>



<p><strong>Mode > Median > Mean</strong></p>



<div class="wp-block-image"><figure class="aligncenter size-large"><img src="https://financetrain.com/wp-content/uploads/Picture2.png" alt="" class="wp-image-27975"/></figure></div>



<h3>Positively Skewed Distribution</h3>



<p><strong>Mode &lt; Median &lt; Mean</strong></p>



<figure class="wp-block-image size-large"><img src="https://financetrain.com/wp-content/uploads/Picture3.png" alt="" class="wp-image-27976"/></figure>



<p>Remember that skewness pulls the mean in the direction of the skew. Median is always between mean and mode.</p>

Relative Locations of Mean, Median and Mode

This quiz is a part of the CFA Level 1 reading 'Statistical Concepts and Market Returns'. 

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Statistical Concepts and Market Returns

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