Quantitative Methods  Correlation and Regression
Financial variables are often analyzed for their correlation to other variables and/or market averages. The relative degree of comovement can serve as a powerful predictor of future behavior of that variable. A sample covariance and correlation coefficient are tools used to indicate relation, while a linear regression is a technique designed both to quantify a positive relationship between random variables, and prove that one variable is dependent on another variable. When you are analyzing a security, if returns are found to be significantly dependent on a market index or some other independent source, then both return and risk can be better explained and understood.
Scatter Plots
A scatter plot is designed to show a relationship between two variables by graphing a series of observations on a twodimensional graph  one variable on the Xaxis, the other on the Yaxis.
Figure 2.15: Scatter Plot
Sample Covariance
To quantify a linear relationship between two variables, we start by finding the covariance of a sample of paired observations. A sample covariance between two random variables X and Y is the average value of the crossproduct of all observed deviations from each respective sample mean. A crossproduct, for the ith observation in a sample, is found by this calculation: (ith observation of X  sample mean of X) * (ith observation of Y  sample mean of Y). The covariance is the sum of all crossproducts, divided by (n  1).
To illustrate, take a sample of five paired observations of annual returns for two mutual funds, which we will label X and Y:
Year  X return  Y return  CrossProduct: (X_{i}  X_{mean})*(Y_{i}  Y_{mean}) 
1st  +15.5  +9.6  (15.5  6.6)*(9.6  7.3) = 20.47 
2nd  +10.2  +4.5  (10.2  6.6)*(4.5  7.3) = 10.08 
3rd  5.2  +0.2  (5.2  6.6)*(0.2  7.3) = 83.78 
4th  6.3  1.1  (6.3  6.6)*(1.1  7.3) = 108.36 
5th  +12.7  +23.5  (12.7  6.6)*(23.5  7.3) = 196.02 
Sum  32.9  36.7  398.55 
Average  6.6  7.3  398.55/(n  1) = 99.64 = Cov (X,Y) 
Average X and Y returns were found by dividing the sum by n or 5, while the average of the crossproducts is computed by dividing the sum by n  1, or 4. The use of n  1 for covariance is done by statisticians to ensure an unbiased estimate.
Interpreting a covariance number is difficult for those who are not statistical experts. The 99.64 we computed for this example has a sign of "returns squared" since the numbers were percentage returns, and a return squared is not an intuitive concept. The fact that Cov(X,Y) of 99.64 was greater than 0 does indicate a positive or linear relationship between X and Y. Had the covariance been a negative number, it would imply an inverse relationship, while 0 means no relationship. Thus 99.64 indicates that the returns have positive comovement (when one moves higher so does the other), but doesn't offer any information on the extent of the comovement.
Sample Correlation Coefficient
By calculating a correlation coefficient, we essentially convert a raw covariance number into a standard format that can be more easily interpreted to determine the extent of the relationship between two variables. The formula for calculating a sample correlation coefficient (r) between two
random variables X and Y is the following:
Formula 2.39 r = (covariance between X, Y) / (sample standard deviation of X) * (sample std. dev. of Y). 
Example: Correlation Coefficient
Return to our example from the previous section, where covariance was found to be 99.64. To find the correlation coefficient, we must compute the sample variances, a process illustrated in the table below.
Year  X return  Y return  Squared X deviations  Squared Y deviations 
1st  +15.5  +9.6  (15.5  6.6)^{2} = 79.21  (9.6  7.3)^{2} = 5.29 
2nd  +10.2  +4.5  (10.2  6.6)^{2} = 12.96  (4.5  7.3)^{2} = 7.84 
3rd  5.2  +0.2  (5.2  6.6)^{2 }= 139.24  (0.2  7.3)^{2} = 50.41 
4th  6.3  1.1  (6.3  6.6)^{2} = 166.41  (1.1  7.3)^{2} = 70.56 
5th  +12.7  +23.5  (12.7  6.6)^{2 }= 146.41  (23.5  7.3)^{2} = 262.44 
Sum  32.9  36.7  544.23  369.54 
Average  6.6  7.3  136.06 = X variance  99.14 = Y variance 
Answer:
As with sample covariance, we use (n  1) as the denominator in calculating sample variance (sum of squared deviations as the numerator)  thus in the above example, each sum was divided by 4 to find the variance. Standard deviation is the positive square root of variance: in this example, sample standard deviation of X is (136.06)^{1/2}, or 11.66; sample standard deviation of Y is (99.14)^{1/2}, or 9.96.
Therefore, the correlation coefficient is (99.64)/11.66*9.96 = 0.858. A correlation coefficient is a value between 1 (perfect inverse relationship) and +1 (perfect linear relationship)  the closer it is to 1, the stronger the relationship. This example computed a number of 0.858, which would suggest a strong linear relationship.
Hypothesis Testing: Determining Whether a Positive or Inverse Relationship Exists Between Two Random Variables
A hypothesistesting procedure can be used to determine whether there is a positive relationship or an inverse relationship between two random variables. This test uses each step of the hypothesistesting procedure, outlined earlier in this study guide. For this particular test, the null hypothesis, or H_{0}, is that the correlation in the population is equal to 0. The alternative hypothesis, H_{a}, is that the correlation is different from 0. The ttest is the appropriate test statistic. Given a sample correlation coefficient r, and sample size n, the formula for the test statistic is this:
t = r*(n  2)^{1/2}/(1  r^{2})^{1/2}, with degrees of freedom = n  2 since we have 2 variables.
Testing whether a correlation coefficient is equal/not equal to 0 is a twotailed test. In our earlier example with a sample of 5, degrees of freedom = 5  2 = 3, and our rejection point from the tdistribution, at a significance level of 0.05, would be 3.182 (p = 0.025 for each tail).
Using our computed sample r of 0.858, t = r*(n  2)^{1/2}/(1  r^{2})^{1/2} = (0.858)*(3)^{1/2}/(1  (0.858)^{2})^{1/2} = (1.486)/(0.514) = 2.891. Comparing 2.891 to our rejection point of 3.182, we do not have enough evidence to reject the null hypothesis that the population correlation coefficient is 0. In this case, while it does appear that there is a strong linear relationship between our two variables (and thus we may well be risking a type II error), the results of the hypothesis test show the effects of a small sample size; that is, we had just three degrees of freedom, which required a high rejection level for the test statistic in order to reject the null hypothesis. Had there been one more observation on our sample (i.e. degrees of freedom = 4), then the rejection point would have been 2.776 and we would have rejected the null and accepted that there is likely to be a significant difference from 0 in the population r. In addition, level of significance plays a role in this hypothesis test. In this particular example, we would reject the null hypothesis at a 0.1 level of significance, where the rejection level would be any test statistic higher than 2.353.
Of course, a hypothesistest process is designed to give information about that example and the prerequired assumptions (done prior to calculating the test statistic). Thus it would stand that the null could not be rejected in this case. Quite frankly, the hypothesistesting exercise gives us a tool to establish significance to a sample correlation coefficient, taking into account the sample size. Thus, even though 0.858 feels close to 1, it's also not close enough to make conclusions about correlation of the underlying populations  with small sample size probably a factor in the test.

Personal Finance
How To Choose A Financial Advisor
Many advisors display similar skillsets that can make distinguishing between them difficult. The following guidelines can help you better understand their qualifications and services. 
Investing
Asset Manager Ethics: Investment Process and Actions
Managers, in developing their investment process, need to determine some “general rules” that make it meaningful. We offer six. 
Professionals
Career Advice: Financial Analyst Vs. Investment Banker
Read an indepth comparison about working as a Financial Analyst vs. working as an Investment Banker, two highly prestigious business careers. 
Professionals
Advisors: Which Certifications Are Essential?
The right advisor credentials can make all the difference, but wading through some 100 certifications can be a challenge. Here's some help. 
Investing Basics
Asset Manager Ethics: Valuation Is A Tricky Business
Asset managers must accurately represent all of a clients assets in the client portfolio. This can be tricky for unique and hardtovalue assets. 
Personal Finance
Top 10 Most Valuable Sports Teams in 2015
Cleats, pads and profits: we take a look at the top 10 most valuable sports teams in the world. 
Professionals
Chinese Slowdown Affects Iron Ore Market
The Chinese economy's ongoing slowdown is having a major impact on iron ore demand. 
Personal Finance
Invest in Costco? First Understand Its Balance Sheet
A strong balance sheet sets a company apart and boosts investor confidence. How healthy is Costco based on an analysis of its balance sheets from the last two years? 
Investing Basics
Brokers and RIAs: One and the Same?
Brokers and registered investment advisors have some key differences. Here's what you need to know. 
Professionals
DCF Vs. Comparables: Which One To Use
DCF and Comparables models are widely used in equity valuation. We explain the pros and cons of each method.

Personal Financial Advisor
Professionals who help individuals manage their finances by providing ... 
CFA Institute
Formerly known as the Association for Investment Management and ... 
Chartered Financial Analyst  CFA
A professional designation given by the CFA Institute (formerly ... 
Security Analyst
A financial professional who studies various industries and companies, ...

What are the differences between a Chartered Financial Analyst (CFA) and a Certified ...
The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >> 
How do I become a Chartered Financial Analyst (CFA)?
According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >> 
What types of positions might a Chartered Financial Analyst (CFA) hold?
The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >> 
Who benefits the most from prepaid expenses?
Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >> 
If I am looking to get an Investment Banking job. What education do employers prefer? ...
If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >> 
Can I still pass the CFA Level I if I do poorly in the ethics section?
You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>