stic line (SCL) for HP The next panel of Table 8.1 shows analysis of variance (ANOVA) for SCL. The sum of squares (SS) of the gression 1.3752) is the portion of the ¦ ante of the dependent variable (HP's return) that is explained by the independent varil (the S&P 500 return). it is equal to The MS column for the residual (. shows the variance of the onexplaisted portion of HP's return. that is, the portion of that is independent of the market index. The square mot of this value is the standard eo (SE) of the regression (.0767) reported in the first panel. If you divide the total SS of regression (.7162) by 59. you will obi 'n the estimate of the variance of the deperdi
variable (HP), ssion statistics ott•Packard .012 per month, equi t to a monthly standard deviation of 11%. Rog ion Multiple R R.square .7238 .5239 .5157 0767 60 df t 61%. nil" Adjusted 8-square Standard error Observations ANOVA SS I MS Regression Residual Total 1 58 59 .3752' .341 .71622;9 .3752 'U41\ Coefficients Standard Error eStat pelting Intercept 588500 0.0086 2.0348 .0099 .2547 0.8719 7.9888 .3868 .0000
annualized.' we obtain an annualized standard deviation of 38.17%. as reported earlier. Notice ihaLthe R-square (the ratio of explained to total variance) equals the explained (regression) SS divided by the total SS.9
The Estimate of Alpha Moving to the bottom panel. the intercept 1.0086 = .86% per month) is the estimate of HP's alpha for the sample period. Although this is an economically large value (10.32% on an annual basis), it is statistically insignificant. This can be seen from the three statistics mar to the estimated coefficient. The first is the standard error of the estimate (0.0099).9 This is a measure oldie imprecision of the estimate. If the standard error is large. the range likely estimation error is correspondingly large. The t-statistic reported in the bottom panel is the ratio of the regression parameter to its ndard error. This statistic equals the number of standard errors by which our estimate needs zero, and therefore can be used to assess the likelihood that the true but unob-mned value might actually equal zero rather than the estimate derived from the data). The intuition is that if the true value were zero, we would be unlikely to observe estimated values far away (i.e.. many standard errors) from zero. So large (-statistics imply low prob-abilities that the true value is zero. In the case of alpha. we are interested in the average value of HP's return net of the of market movements. Suppose we define the nonmarket component of HP's return its actual return minus the return attributable to market movements during any period. this HP's firm-specific return, which we abbreviate as Re,. = Rio = 13IV R511'•00 If Rf, were normally distributed with a mean of zero, the ratio of its estimate to its stan-error would have a t-distribution. From a table of the n.distribution (or using Excel's V function) we can find the probability that the true alpha is actually zero or even •er given the positive estimate of its value and the standard error of the estimate. This called the level of significance or. as in Table 8.1, the probability or p-mlue. The con-tional cutoff for statistical significance is a probability of less than 5%, which requires nstatistic of about 2.0. The regression output shows the t-statistic for HP's alpha to
twratalieine monthly den. asvrafc TINT and •Wilh't are multiplied by I2. Houever. because ...Hance is plied by 12. standard Soleil., is unikiplind by it/12 R-Squwe — 37'2 —,5239 rt,tet,a5 .7162
intently. floararc cowls I lion,us the fraolon at swan. that is nor explansed by matte, returns 1.e.. I mow ratio 01 linnoprallw risk irt total nal, FM HP. this is self*) 1 .3310 _ 5239
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can Main the standard Mm d thc alpha estimate la the standard error of the maiden's as folkaas.
(I lengSdeP5OHP - 0610 , 11n VantS4P70011 x (0 - t) .The r.danow 6 bawd on did anonunon that returns are normQlly &outland In &uncial. if ass standadire the neinate of d normal dwlibuied variable by compwins in difference km, a hyprdwored sable aid do alms by the olded emu of the emanate (Inexpert, thw tlitfince as a nemter ad seaubeslem,64. the r"uhing ',unable will have .-Ilonhotion. \kith u lunge nuttily, of oho:nation, northen
be .8719, indicating that the estimate is nol significantly different from zero cannot reject the hypothesis that the we value of alpha equals zero with an of confidence. The p-value for the alpha estimate (.3868) indicates that if the were zero, the probability of obtaining an estimate as high as .0086 (given the dard error of .0099) would be .3868, which is not so unlikely. We conclude that average of RI, is too low to reject the hypothesis that the we value of alpha is zero. But even if the alpha value were both economically and statistically significant sample. we still would not use that alpha as a forecast for a future period. Ov empirical evidence shows that 5-year alpha values do not persist over time. that is, to be virtually no correlation between estimate: from one sample period to the next words, while the alpha estimated from the regression tells us the average return on the when the market was flat during that estimation period it does nor forecast what perfomunce will be in future periods. This is why security analysis is so hard. The not readily foretell the future. We elaborate on this iswe in Chapter 11 on market
The Estimate of Beta The regression output in Table 8.1 shows the beta estimate for HP to be 2.0348. twice that of the S&P 500. Such high market sensitivity is not unusual for stocks. The standard error (SE) of the estimate is .2547.'' The value of beta and its SE produce a large t- statistic (7.9888). and a p-value tically zero. We can confidently reject the hypothesis that HP's true beta is zero. interesting r-statistic might test a null hypothesis that HP's beta is greater than the wide average beta of 1. This r-statistic would measure how many standard errors the estimated beta from a hypothesized Vailic of I. Here too. the difference is e enough to achieve statistical significance: Estimated value — Hypothesized value _ 2.03 — 1 — 400 Standard emir .2547 However. we should bear in mind that even here. precision is not what we might be. For example. if we wanted to construct a confidence interval that includes the unobserved value of beta with 95% probability, we would take the estimated v center of the interval and then add and subtract about two standard errors. This range between 1.43 and 2.53. which is quite wide.
Finn-Specific Risk The monthly standard deviation of HP's residual is 7.67%. or 26.6% annually. This large, on lop of HP's high-level systematic risk. The standard deviation of sys is 13 x cr(S&P 500) = 2.03 x 13.58 = 27.57%. Notice that HP's firm-specific large as its systematic risk. a common result for individual stocks.
Cavelation and Covariance Matrix Figure 8.4 graphs the excess returns of the pairs of securities from each of the three arz.40.efe.fafAW.reaffrese.erssraa-AzYk /*.en-vitavitk•/7:iztrea-e•iff•attri followed by the retail sector, and then the energy sector, which has the lowest v Panel 1 in Spreadsheet 8.1 shows the estimates of the risk parameters of the portfolio and the six analyzed securities. You can see from the high residual deviations (column El how important diversification is. These securities have usar0Aim-specific risk. Portfolios concentrated in these (or other) securities would have unnec-ly high volatility and inferior Sharpe ratios. Panel 2 shows the correlation matrix of the residuals from the regressions of excess *urns on the S&P 500. The shaded cells show correlations of same-sector stocks, which Ire as high as .7 for the two oil stocks (BP and Shell). This is in contrast to the assump-tion of the index model that all residuals are uncorrelated. Of course, these correlations me. to a great extent, high by design, because we selected pairs of firms from the same 'industry. Cross-industry correlations are typically far smaller, and the empirical estimates