{"id":3405,"date":"2015-04-12T00:57:25","date_gmt":"2015-04-12T05:57:25","guid":{"rendered":"http:\/\/www.ssc.wisc.edu\/~jfrees\/?page_id=3405"},"modified":"2015-08-18T13:49:46","modified_gmt":"2015-08-18T18:49:46","slug":"unusual-points","status":"publish","type":"page","link":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/regression\/basic-linear-regression\/2-7-application-capital-asset-pricing-model\/unusual-points\/","title":{"rendered":"Unusual Points"},"content":{"rendered":"

To summarize the relationship between the market and Lincoln’s return, a regression model was fit. The fitted regression is \\begin{equation*} \\widehat{LINCOLN}=-0.00214+0.973 MARKET. \\end{equation*} The resulting estimated standard error, s<\/em> = 0.0696 is lower than the standard deviation of Lincoln’s returns, \\(s_y\\)=0.0859. Thus, the regression model explains some of the variability of Lincoln’s returns. Further, the \\(t\\)-statistic associated with the slope \\(b_1\\) turns out to be \\(t(b_1)\\)=5.64, which is significantly large. One disappointing aspect is that the statistic \\(R^2=35.4\\%\\) can be interpreted as saying that the market explains only a little over a third of the variability. Thus, even though the market is clearly an important determinant, as evidenced by the high t<\/em>-statistic, it provides only a partial explanation of the performance of the Lincoln’s returns. <\/p>\n

In the context of the market model, we may interpret the standard deviation of the market, \\(s_x\\), as non-diversifiable risk<\/em>. Thus, the risk of a security can be decomposed into two components, the diversifiable component and the market component, which is non-diversifiable. The idea here is that by combining several securities we can create a portfolio of securities that, in most instances, will reduce the riskiness of our holdings when compared with a single security. Again, the rationale for holding a security is that we are compensated through higher expected returns by holding a security with higher riskiness. To quantify the relative riskiness, it is not hard to show that \\begin{equation} s_y^2 = b_1^2 s_x^2 + s^2 \\frac{n-2}{n-1}. \\end{equation}
\nThe riskiness of a security is due to the riskiness due to the market plus the riskiness due to a diversifiable component. Note that the riskiness due to the market component, \\(s_x^2\\), is larger for securities with larger slopes. For this reason, investors think of securities with slopes \\(b_1\\) greater than one as “aggressive” and slopes less than one as “defensive.” <\/p>\n

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To summarize the relationship between the market and Lincoln’s return, a regression model was fit. The fitted regression is \\begin{equation*} \\widehat{LINCOLN}=-0.00214+0.973 MARKET. \\end{equation*} The resulting estimated standard error, s = 0.0696 is lower than the …<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":3395,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"jetpack_post_was_ever_published":false},"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/P8cLPd-SV","acf":[],"_links":{"self":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3405"}],"collection":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/comments?post=3405"}],"version-history":[{"count":4,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3405\/revisions"}],"predecessor-version":[{"id":4913,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3405\/revisions\/4913"}],"up":[{"embeddable":true,"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/pages\/3395"}],"wp:attachment":[{"href":"https:\/\/users.ssc.wisc.edu\/~ewfrees\/wp-json\/wp\/v2\/media?parent=3405"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}