1. Chapter 3
Statistical Concepts

2. Simple Probability distribution
Return calculation for a 1 period investment
Example: Today’s price is $40. We expect the price to go to $44 and receive $1.80 in dividends. What is the return?

3. Adjustments are necessary to annualize the return.
N = years of investment
2 years N = months N = .5
What if it took 2 years to get the same result?
HPR

4. The Normal Distribution
Absence Symmetry, standard deviation as a measure of risk is inadequate.
If normally distributed returns are combined into a portfolio, the portfolio will be normally distributed.

5. Descriptive Statistics
In order to describe what a distribution looks like we need to measure certain parameters
Population
1) Mean = expected return = weighted average
P = probability r = return
2) variance = potential deviation from the mean or average
Sample
1) Mean
2) Variance

6. Normal and Skewed (mean = 6% SD = 17%)
Normal distribution have a skew of 0.
Positive skew – standard deviation overestimates risk
Negative – underestimates risk

7. Normal and Fat Tails Distributions (mean = .1 SD =.2)
Kurtosis of a normal distribution is 0.
Kurtosis greater than 0 (fat tails), standard deviation will underestimate risk

8. Joint Probability Joint probability statistics Covariance Population
Sample
Fig 3-3 pg Pos. covariance
Note the data points are concentrated in Quadrants I & III
Fig 3.4 pg neg. Covariance
Data points in Quadrants II & I
Fig 3.5 pg zero covariance

9. Correlation Correlation Perfect positive correlation
Perfect negative
Correlation
Perfect positive correlation
Fig 3.8 and 3.9 pg 41
Perfect negative correlation
Fig 3.10 pg 42
Imperfect correlations
(Fig 3.11 and 3.12 pg 43)
There is some error in this technique (hence the imperfect)
(Use regression coefficient b = correlation) = slope of the line

10. Correlation standardizes the COV Correlation has a range of values
covariance can have an unlimited magnitude
It produces a line of best fit that minimizes the sum of the squared deviations
Correlation has a range of values
perfect negative zero perfect positive
Can rewrite COV
Coefficient of determination
Describes the amount of variation in one invest that can be associated with or explained by another
how much does one explain about the movement of another

11. How does the CD work?
Correlation between Asset A and B = 1.0; CD = 1.0 (100%)
If you can predict the returns of asset A; you can exactly predict the returns in Asset B
Correlation between Asset C an D = .70; CD = .49 (49%)
If you can predict the returns of asset C; you can predict 49% of the returns in Asset D

12. Stock and Market Portfolios
Market portfolio contains every risky asset in the investment world
(stocks, bonds, coins, real estate, etc.)
Each asset is held in its proportion to all the others in the market
Value weighted index
To create your own index you would buy .01% of the total market value of each and every risky asset. You would own the same amount of control in each company, but each asset would have a different value

13. Characteristic Line Fig 3.13 pg 45
plot the return of the asset at a certain time against the return of the market at the same time
can create a line that best describes the relationship (Line of best fit) using regression statistic
Shows the return you would expect to get in the asset given a particular return in the market index
The line is not a perfect fit, it does not show the exact return you will get. There are errors made in the estimation.

14. Variables for Characteristic Line
Beta factor
indicator of the degree to which the stock responds to changes in the return in the market
slope of the characteristic line
Note the similarity between the formulas for Beta and the correlation coefficient
Only the denominator
Alpha
This term represents the return on the stock if the market had a return of zero
Residual
since we will seldom if ever see perfect relationships, there will always be some “error” in the representation of the line of best fit
It is a line of “best” fit not “perfect” fit

15. How much error?
Variance measures the deviation from the mean or expected return
Residual variance measures the deviation from the characteristic line
As the residual variance approaches zero, the correlation between the returns of the stock and the returns of the market approaches either +1 or -1,
also then the coefficient of determination would approach 1.0
the closer to +/- 1 the better the explanation of the data, so you could use this as a prediction of the stocks returns, given an estimate of the markets returns
There would be less error in the prediction, if the residual variance was very low

16. Examples
Historical Data returns