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Chapter 3 Statistical Concepts 1 of 59 Simple Probability distribution Return calculation for a 1 period investment P CF PEND PBEG CF Pend CF r 1.0 PBEG PBEG PBEG Example: Today’s price is $40. We expect the price to go to $44 and receive $1.80 in dividends. What is the return? r 44 1.80 1.0 14.5% 40 2 of 59 Adjustments are necessary to annualize the return. P CF HPR end PBEG N = years of investment 2 years N = 2 1 N 1.0 6 months N = .5 What if it took 2 years to get the same result? 1/ 2 44 1.80 HPR 40 1.0 7.00% Beginning Value Ending Value Investment Cash Flows Investment Time (Yrs) $40.00 $44.00 $1.80 2.000 Return HPR (annualized return) 14.50% 7.00% 3 of 59 HPR 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. 4 of 59 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 E(r ) pi * ri 2) variance = potential deviation from the mean or average Sample 2 ri E(r ) * pi pi * ri2 E(r )2 1) Mean 2 r r i n 2) Variance r r 2 2 i n 1 5 of 59 Normal and Skewed (mean = 6% SD = 17%) Normal distribution have a skew of 0. Positive skew – standard deviation overestimates risk Negative – underestimates risk 6 of 59 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 7 of 59 Joint Probability Joint probability statistics Covariance Population COVab pi * ra E(ra ) * rb E(rb ) Sample COVab r ai ra * rbi rb n 1 Fig 3-3 pg 37 Pos. covariance • Note the data points are concentrated in Quadrants I & III Fig 3.4 pg 38 neg. Covariance • Data points in Quadrants II & I Fig 3.5 pg 38 zero covariance 8 of 59 Correlation Perfect positive Correlation Perfect positive correlation Fig 3.8 and 3.9 pg 41 Fig 3.10 pg 42 Imperfect correlations Perfect negative Perfect negative correlation ab COVab a * b (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 9 of 59 Correlation standardizes the COV 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 -1.0 0.0 +1.0 perfect negative zero perfect positive Can rewrite COV COVab ab * a * b 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 CD 2 ab 10 of 59 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 11 of 59 Stock and Market Portfolios Market portfolio contains every risky asset in the investment world Each asset is held in its proportion to all the others in the market (stocks, bonds, coins, real estate, etc.) 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 12 of 59 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. 13 of 59 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 COV j,m • Only the denominator j b coefficient 2 m Alpha This term represents the return on the stock if the market had a return of zero A j rj j * rm constan t 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 j,t rj,t A j j * rmt rj,t E(rj,t ) 14 of 59 How much error? Variance measures the deviation from the mean or expected return Residual variance measures the deviation from the characteristic line n2 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, 2 , j 2 j, t 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 15 of 59 Examples data pts 1 2 3 4 5 6 Mean Stnd Deviation Coef. of Var Return Asset A Returns Asset B Mkt returns Returns residuals A(mkt)residuals B(mkt) 5.00% -1.00% 4.00% 0.000011 0.000018 8.00% 3.00% 6.00% 0.000075 0.000012 9.00% 8.00% 8.00% 0.000000 0.000001 10.00% 12.00% 10.00% 0.000054 0.000078 12.00% 19.00% 12.00% 0.000028 0.000180 15.00% 22.00% 14.00% 0.000044 0.000018 9.833% 10.500% 9.000% 0.0002 0.0003 3.430% 8.961% 3.742% 0.3488 0.8534 0.4157 Correlation Matrix A with mkt B with Mkt Alpha 0.0173 (0.1097) Beta 0.9000 2.3857 Residual variance 0.0053% 0.0077% Portfolio Statistics weights 1 2 3 4 5 6 50.00% 2.50% 4.00% 4.50% 5.00% 6.00% 7.50% Correlation Matrix A B 1.000 0.973 0.982 A B C 20.00% -0.20% 0.60% 1.60% 2.40% 3.80% 4.40% C 1.000 0.996 1.000 30.00% sum returns sum squares 1.20% 3.50% 0.0039 1.80% 6.40% 0.0011 2.40% 8.50% 0.0001 3.00% 10.40% 0.0000 3.60% 13.40% 0.0014 4.20% 16.10% 0.0041 Mean 9.717% Std Dev 3.390% Coef of Var 2.87 Historical Data returns 16 of 59