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SOME STATISTICAL CONCEPTS Chapter 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(r A ) =.10(5) +.20(10) +.40(20) +.20(40) +.10(70) = 25.5% E(r B ) =.10(10) +.20(20) +.40(40) +.20(50) +.10(60) = 37.0% Cov(r A,r B ) =.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 for Stock (j) and the Market (m) Return on the Stock Return on the Market Line passes through The means of both variables When the Market’s return is zero, the stock’s return is.355

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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:

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