In order to understand finance and make prudent investment decisions it is imperative to know statistics, and have a basic understanding. Even in politics statistics is used when public opinion is asked on political issues, it is used in election campaigns and during elections. Statistics is used in medicine, in sports, and in the economy, and in the stock market.
In order to understand
finance and make prudent investment decisions it is imperative to know
statistics, and have a basic understanding. Even in politics statistics is used
when public opinion is asked on political issues, it is used in election
campaigns and during elections. Statistics is used in medicine, in sports, and
in the economy, and in the stock market.
In an organized economy,
the stock markets’ prices change daily, fluctuate constantly as a result of
various parameters. As a whole, the stock market fluctuates daily as a result
of macro or micro economic conditions, or even political decisions. These
parameters that fluctuate constantly, that take on different numbers are called
variables. So if on any giving day we observe stock market prices of the
general index, or any specific stock, we will observe that they take on
different prices.
How many times has a nation
taken a census on its population, on personal income changes, or even the
migration of population to different cities? If these census involve the total
number of observations for any given variable, then we are talking about the
population. If they involve a part of the observations, then we are talking
about a sample. For example, if we are to take for a month the price of a
particular stock trading on the stock market, we will observe the daily
fluctuations of the stock price closing. These are samples of the variable
stock price. The total population is the total data of the stock price closing
price from the day the stock enlisted in the stock exchange. So the sample is a
subtotal of the total population.
Statistics is the science
that is involved with statistical data, the collection of data and its assortment,
and presentation, and the conclusions as a result of the analysis. Going back
to the example of the closing price of a stock during any giving month, the
trend that the stock price closing tends to have, can be giving by the
arithmetic average. It is a statistical measure that shows the average price
trend of a giving variable during a specific time period.
The formula used is X= χ1 +χ2
+…+χn / V where X is
the arithmetic average (mean), is the price of the
variable, and V is the total number of observations.
When we determine the
arithmetic mean from a set of say a stock price we tend to assign the same
weight in each observation. However, we can assign different weights to each
observation. To illustrate and go ahead of ourselves, if we hold a portfolio of
stocks we tend to place money in different assets at different proportions. If
we had a portfolio of two stocks, obviously to minimize risk, we will not place
all of our money in one stock, but diversify the risk. So in our example of two
stocks, A and B, and given we have $100, we place say 60% in stock A, and 40%
in stock B.
In another example, say
we have data for the price of a stock in the last three years. The most recent
stock prices would have a higher weight than those in the distance past. So if
we wanted to find the weighted average, the weights assigned range from 0 to 1.
The formula used to calculate the weighted average is given by x (weight) =χ1 *w1 + x2 * w2+…..+χn * wn / w1+w2+…
where x (weight) is the weighted average, 
are the variables w1, w2, are the weights, and n is the number of
observations.
where x (weight) is the weighted average, are the variables w1, w2, are the weights, and n is the number of
observations.
To illustrate what is the
weighted average of stock prices of four banking stocks in the Athens Stock
Exchange, I took the closing prices as of August 12, 2015 along with the volume
of shares in millions.
Stock
|
Closing
price
|
Number
of shares(volume) millions, round off
|
Alpha
|
€0.135
|
25
|
National
|
0.640
|
11
|
Piraeus
|
0.161
|
19
|
Eurobank
|
0.068
|
29
|
x(weight) = 0.135 *25 +
0.640 *11 +0.161 *19 +0.068 *29 / 25+11+19+29 = 15/84= €0.20
Now let us suppose that
the above stocks are in a portfolio, and we have invested 15% in Alpha, 20% in
National, 25% in Piraeus, and 40% in Eurobank. Let us also suppose (real from
Athens Stock Exchange) that the change in stock prices are +3%, +0.79%, -3.0%,
+1.49%. What would be the change in the value of this portfolio?
x(weight) = 0.03*0.015
+0.01*0.20 +(-0.03)*0.25 +0.01*0.40 / 0.15+0.20+0.25+0.40 = -0.1%
The weighted average is a
very useful statistical tool in investments since it is consisted of a measure
of return of investments.
The arithmetic average as
we discussed is a useful tool since it provides us with information as to the
central trend of a measurement. What if say we are taking a serious of
temperatures of a boiling substance. Sometimes we tend to find extreme (too
high or too low) measurements at both sides of the spectrum. Another words,
measurements that are too extreme, then the arithmetic average would be a
problem. To determine the tendency of the variable we use the Median. We arrange
the observations in increasing order, and if the number is odd, the median is
the value in the middle. If the number of observations are even, the median is
the average of the two values in the middle.
The median is another
measure of central tendency. A way to measure the median is that if in a sample
there appear values that are very high, or very low then the arithmetic mean
would be very high or low. From the arithmetic mean the median is used to
determine the central tendency of the variable. If the number of observations
is odd, the median is the value in the middle. If it even, the median is the
average 
where 
is the number of
observations. To illustrate the median,
say we take ten annual returns of the Dow Jones Industrials, and we come up
with a median 3.5%. That means that half the returns are higher and half are
lower than 3.5%.
where is the number of
observations. To illustrate the median,
say we take ten annual returns of the Dow Jones Industrials, and we come up
with a median 3.5%. That means that half the returns are higher and half are
lower than 3.5%.
Another measure is the
Mode, which be definition it is the value that occurs most frequently. If we
look at the Dow and take returns for any given years, there may be instances
where there is no mode, that is, no return appears more than once. But there
may be instances where returns may appear more than once. This is least likely
to appear in finance.
Now let us consider two
hypothetical assets with a mean return of 5%. Would assets A and B be equally
desirable if the observed values of returns for asset A is highly condensed
between 9 and 12%, while for asset B be dispersed between a low of -50% and a
high of 30%. The answer is No. Dispersion around the mean matters, and that is
what variance intends to explain. If we take the average of the squared
differences between each return and the mean, then the variance measures the
average of the squared deviations from the mean.
If we analyze the daily
percentage change in the price of a stock for a given period of time, we can
use the variance and the standard deviation to determine the volatility change,
the return and the risk associated with the stock. From a sample of
observations, the variance is where s squared (s2) is the variance of the
sample and χ
is the mean.
The variance of the
population is given by σ2
= (x1-μ) 2 + (x2 – μ) 2 + …. + (xn – μ) 2 / Ν. The standard deviation of the sample is
the squared root of s2, or s= x n –χ) 2/n-1 and the
standard deviation of the population is the squared root of σ2. That is σ =√(x1 – μ) 2 + (x2 – μ) 2 + …. + (xn – μ) 2/N. The standard deviation is the
squared root of the variance, V that is SD =V½. The higher the standard
deviation means the higher the dispersion around the mean.
Investments have to do
with returns and in order to make sound decisions we have to take into account
the risk factor. The more returns fluctuate over time, the greater the uncertainty
about say, stock prices and returns. This increased uncertainty is associated
with greater risk, and without going into further details, for an investor to
take the risk, one must be compensated for this with greater returns. In order
to capture this uncertainty is to compute the standard deviation (SD) of
returns.
The larger the SD is
the greater risk is associated with the asset. If we have a collection of data
say the returns of a stock market index for a month, a small SD indicates that
the returns fluctuate closely around the mean return, which means less
volatility. A large SD indicates that returns tend to depart more from the mean
return, or high volatility.
In finance it is
interesting to know the relationship between two variables. Imagine you had a
portfolio where you had only two stocks. Would it not be interesting to know
the relationship, if any, between the two stocks? Or suppose we want to know
the relationship between two indices, say the DOW Jones index and that of the
French CAC. Do they move together or in the opposite direction?
In statistics this is
accomplished by the Covariance. The relationship between two variables, i,j or
COVi,j, measures the linear relationship between them. There are two problems to
encounter here: the first is the units in which the variables are measured, and
the second, is that the variance has no bounds (upper or lower limits), and
thus you cannot conclude if the number you find is strong or weak in terms of
the relationship between the variables.
This is alleviated by
determining the correlation coefficient, CORR between variables i and j (CORRi,j).
The formula is given by CORRi,j = COVi,j / SDi* SDj. Correlation coefficient
measures the strength of the two variables, and takes values of 1 or -1. When
the correlation is positive, it means that the two variables move together in
the same direction. If it is -1, it means that the two variables move in opposite
direction. A correlation of 0 means there is no linear relationship between the
variables.
In an organized economy,
the stock markets’ prices change daily, fluctuate constantly as a result of
various parameters. As a whole, the stock market fluctuates daily as a result
of macro or micro economic conditions, or even political decisions. These
parameters that fluctuate constantly, that take on different numbers are called
variables. So if on any giving day we observe stock market prices of the
general index, or any specific stock, we will observe that they take on
different prices.
How many times has a nation
taken a census on its population, on personal income changes, or even the
migration of population to different cities? If these census involve the total
number of observations for any given variable, then we are talking about the
population. If they involve a part of the observations, then we are talking
about a sample. For example, if we are to take for a month the price of a
particular stock trading on the stock market, we will observe the daily
fluctuations of the stock price closing. These are samples of the variable
stock price. The total population is the total data of the stock price closing
price from the day the stock enlisted in the stock exchange. So the sample is a
subtotal of the total population.
Statistics is the science
that is involved with statistical data, the collection of data and its assortment,
and presentation, and the conclusions as a result of the analysis. Going back
to the example of the closing price of a stock during any giving month, the
trend that the stock price closing tends to have, can be giving by the
arithmetic average. It is a statistical measure that shows the average price
trend of a giving variable during a specific time period.
The formula used is X= χ1 +χ2
+…+χn / V where X is
the arithmetic average (mean), is the price of the
variable, and V is the total number of observations.
When we determine the
arithmetic mean from a set of say a stock price we tend to assign the same
weight in each observation. However, we can assign different weights to each
observation. To illustrate and go ahead of ourselves, if we hold a portfolio of
stocks we tend to place money in different assets at different proportions. If
we had a portfolio of two stocks, obviously to minimize risk, we will not place
all of our money in one stock, but diversify the risk. So in our example of two
stocks, A and B, and given we have $100, we place say 60% in stock A, and 40%
in stock B.
In another example, say
we have data for the price of a stock in the last three years. The most recent
stock prices would have a higher weight than those in the distance past. So if
we wanted to find the weighted average, the weights assigned range from 0 to 1.
The formula used to calculate the weighted average is given by x (weight) =χ1 *w1 + x2 * w2+…..+χn * wn / w1+w2+… where x (weight) is the weighted average, χ1, χ2, ...χn are the variables, w1, w2, are the weights, and n is the number of
observations.
where x (weight) is the weighted average, χ1, χ2, ...χn are the variables, w1, w2, are the weights, and n is the number of
observations.
To illustrate what is the
weighted average of stock prices of four banking stocks in the Athens Stock
Exchange, I took the closing prices as of August 12, 2015 along with the volume
of shares in millions.
Stock
|
Closing
price
|
Number
of shares(volume) millions, round off
|
Alpha
|
€0.135
|
25
|
National
|
0.640
|
11
|
Piraeus
|
0.161
|
19
|
Eurobank
|
0.068
|
29
|
x(weight) = 0.135 *25 +
0.640 *11 +0.161 *19 +0.068 *29 / 25+11+19+29 = 15/84= €0.20
Now let us suppose that
the above stocks are in a portfolio, and we have invested 15% in Alpha, 20% in
National, 25% in Piraeus, and 40% in Eurobank. Let us also suppose (real from
Athens Stock Exchange) that the change in stock prices are +3%, +0.79%, -3.0%,
+1.49%. What would be the change in the value of this portfolio?
x(weight) = 0.03*0.015
+0.01*0.20 +(-0.03)*0.25 +0.01*0.40 / 0.15+0.20+0.25+0.40 = -0.1%
The weighted average is a
very useful statistical tool in investments since it is consisted of a measure
of return of investments.
The arithmetic average as
we discussed is a useful tool since it provides us with information as to the
central trend of a measurement. What if say we are taking a serious of
temperatures of a boiling substance. Sometimes we tend to find extreme (too
high or too low) measurements at both sides of the spectrum. Another words,
measurements that are too extreme, then the arithmetic average would be a
problem. To determine the tendency of the variable we use the Median. We arrange
the observations in increasing order, and if the number is odd, the median is
the value in the middle. If the number of observations are even, the median is
the average of the two values in the middle.
The median is another
measure of central tendency. A way to measure the median is that if in a sample
there appear values that are very high, or very low then the arithmetic mean
would be very high or low. From the arithmetic mean the median is used to
determine the central tendency of the variable. If the number of observations
is odd, the median is the value in the middle. If it even, the median is the
average where is the number of
observations. To illustrate the median,
say we take ten annual returns of the Dow Jones Industrials, and we come up
with a median 3.5%. That means that half the returns are higher and half are
lower than 3.5%.
where is the number of
observations. To illustrate the median,
say we take ten annual returns of the Dow Jones Industrials, and we come up
with a median 3.5%. That means that half the returns are higher and half are
lower than 3.5%.
Another measure is the
Mode, which be definition it is the value that occurs most frequently. If we
look at the Dow and take returns for any given years, there may be instances
where there is no mode, that is, no return appears more than once. But there
may be instances where returns may appear more than once. This is least likely
to appear in finance.
Now let us consider two
hypothetical assets with a mean return of 5%. Would assets A and B be equally
desirable if the observed values of returns for asset A is highly condensed
between 9 and 12%, while for asset B be dispersed between a low of -50% and a
high of 30%. The answer is No. Dispersion around the mean matters, and that is
what variance intends to explain. If we take the average of the squared
differences between each return and the mean, then the variance measures the
average of the squared deviations from the mean.
If we analyze the daily
percentage change in the price of a stock for a given period of time, we can
use the variance and the standard deviation to determine the volatility change,
the return and the risk associated with the stock .From a sample of
observations, the variance is where s squared (s2) is the variance of the
sample and χ
is the mean.
The variance of the
population is given by σ2
= (x1-μ) 2 + (x2 – μ) 2 + …. + (xn – μ) 2 / Ν. The standard deviation of the sample is
the squared root of s2, or s= squared root (x1-x)2 + (x2 - x) 2 +....+(xn - x)2 / n-1 , and the
standard deviation of the population is the squared root of σ2. That is σ =√(x1 – μ) 2 + (x2 – μ) 2 + …. + (xn – μ) 2/N. The standard deviation is the
squared root of the variance, V that is SD =V½. The higher the standard
deviation means the higher the dispersion around the mean.
Investments have to do
with returns and in order to make sound decisions we have to take into account
the risk factor. The more returns fluctuate over time, the greater the uncertainty
about say, stock prices and returns. This increased uncertainty is associated
with greater risk, and without going into further details, for an investor to
take the risk, one must be compensated for this with greater returns. In order
to capture this uncertainty is to compute the standard deviation (SD) of
returns.
The larger the SD is
the greater risk is associated with the asset. If we have a collection of data
say the returns of a stock market index for a month, a small SD indicates that
the returns fluctuate closely around the mean return, which means less
volatility. A large SD indicates that returns tend to depart more from the mean
return, or high volatility.
In finance it is
interesting to know the relationship between two variables. Imagine you had a
portfolio where you had only two stocks. Would it not be interesting to know
the relationship, if any, between the two stocks? Or suppose we want to know
the relationship between two indices, say the DOW Jones index and that of the
French CAC. Do they move together or in the opposite direction?
In statistics this is
accomplished by the Covariance. The relationship between two variables, i,j or
COVi,j, measures the linear relationship between them. There are two problems to
encounter here: the first is the units in which the variables are measured, and
the second, is that the variance has no bounds (upper or lower limits), and
thus you cannot conclude if the number you find is strong or weak in terms of
the relationship between the variables.
This is alleviated by
determining the correlation coefficient, CORR between variables i and j (CORRi,j).
The formula is given by CORRi,j = COVi,j / SDi* SDj. Correlation coefficient
measures the strength of the two variables, and takes values of 1 or -1. When
the correlation is positive, it means that the two variables move together in
the same direction. If it is -1, it means that the two variables move in opposite
direction. A correlation of 0 means there is no linear relationship between the
variables.
