SOME STATISTICAL CONCEPTS Chapter 3

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SOME STATISTICAL CONCEPTSChapter 3

• Distributions of Data
• Probability Distribution
• Expected Rate of Return
• Variance of Returns
• Standard Deviation
• Covariance
• Correlation Coefficient
• Coefficient of Determination
• Historical Distributions
• Various Statistics
• Relationship Between a Stock and the Market
  Portfolio
• The Characteristic Line
• Residual Variance

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DISTRIBUTIONS OF DATA

• When evaluating security and portfolio returns,
  the analyst may be confronted with
• 1. possible returns in some future time period
  (probability distributions of possible future
  returns), or
• 2. past returns over some historical time period
  (sample distribution of past returns).
• The same statistics may be used to describe both
  types of distributions (probability and sample).
  For each type of distribution, however, the
  procedures for calculating the various statistics
  vary somewhat.
• In the following examples, statistics are
  discussed first with respect to probability
  distributions, and then with respect to sample
  distributions of historical returns.

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PROBABILITY DISTRIBUTION(Evaluating Possible
Future Returns)
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PROBABILITY DISTRIBUTION(Continued)
Probability
Possible Return ()
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• Expected Rate of Return (Best Guess)
• E(r) .05(-20) .10(-10) .20(5)
  .30(30)
• .20(55) .10(70) .05(80)
• 30
• Variance of Returns (Potential for deviation of
  the return from its expected value)

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• ??????2(r) .05(-20 -30)2 .10(-10 -30)2
  .20(5 -30)2
• .30(30 -30)2 .20(55 -30)2
  .10(70 -30)2
• .05(80 -30)2
• 820
• Standard Deviation
• Covariance (A measure of the interrelationship
  between securities)
• A positive number indicates positive correlation.
  A negative number indicates negative correlation.
  A value of zero indicates zero correlation.

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• Covariance - An Example

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• Covariance - An Example (Continued)
• E(rA) .10(5) .20(10) .40(20) .20(40)
  .10(70)
• 25.5
• E(rB) .10(10) .20(20) .40(40) .20(50)
  .10(60)
• 37.0
• Cov(rA,rB) .10(5 - 25.5)(10 - 37)
• .20(10 - 25.5)(20 - 37)
• .40(20 - 25.5)(40 - 37)
• .20(40 - 25.5)(50 - 37)
• .10(70 - 25.5)(60 - 37)
• 241.50 (Positive Covariance)

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Graphic Illustration of Positive Covariance
Return on Stock A
Return on Stock B
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• Correlation Coefficient Ranges between 1.0
  (perfect positive correlation) and -1.0 (perfect
  negative correlation).

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• Coefficient of Determination
• Percentage of the variability in returns on
  one investment that can be associated with the
  returns on another investment

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HISTORICAL DISTRIBUTIONS(Evaluating Past Returns)
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Graph of Past Returns
Return on Stock A
Return on Stock B
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• Mean Return
• Variance and Standard Deviation

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• Covariance
• Correlation Coefficient
• Coefficient of Determination

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Relationship Between a Stock and the Market
Portfolio
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• Mean Returns
• Variance and Standard Deviation

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• Covariance
• Correlation Coefficient

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The Characteristic Line
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The Characteristic Line forStock (j) and the
Market (m)
Return on the Stock
Line passes through The means of both variables
When the Markets return is zero, the stocks
return is .355
Return on the Market
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• Residuals
• Deviations from the characteristic line
• 1. -7 - .355 .665(-10) - .705
• 2. 6 - .355 .665( 5) 2.32
• 3. 15 - .355 .665(25) - 1.98
• 4. 9 - .355 .665(15) - 1.33
• 5. 22 - .355 .665(30) 1.695

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• Residual Variance
• Propensity to deviate from the line