# Upside beta

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In [investing](/source/Investment), '''upside beta''' is the element of traditional [beta](/source/Beta_(finance)) that investors do not typically associate with the true meaning of [risk](/source/Financial_risk).<ref name="The Entrepreneur's Cost of Capital">{{cite web|title=The Entrepreneur's Cost of Capital: Incorporating Downside Risk in the Buildup Method|url=http://www.macrorisk.com/wp-content/uploads/2013/04/MRA-WP-2013-e.pdf|accessdate=26 June 2013|author=James Chong |author2=Yanbo Jin |author3=G. Michael Phillips |page=2|date=April 29, 2013}}</ref> It is defined to be the scaled amount by which an asset tends to move compared to a benchmark, calculated only on days when the benchmark's return is positive.

==Formula==
Upside beta measures this [upside risk](/source/upside_risk). Defining <math>r_i</math> and <math>r_m</math> as the excess returns to security <math>i</math> and market <math>m</math>, <math>u_m</math> as the average market excess return, and Cov and Var as the [covariance](/source/Covariance_operator) and [variance](/source/variance) operators, the CAPM can be modified to incorporate upside (or [downside](/source/Downside_beta)) beta as follows.<ref name="capm">{{cite journal|last=Bawa|first=V.|author2=Lindenberg, E.|title=Capital market equilibrium in a mean-lower partial moment framework|journal=Journal of Financial Economics|year=1977|volume=5|issue=2|pages=189–200|doi=10.1016/0304-405x(77)90017-4}}</ref>

:<math>\beta^+=\frac{\operatorname{Cov}(r_i,r_m \mid r_m>u_m)}{\operatorname{Var}(r_m \mid r_m>u_m)},</math>

with downside beta <math>\beta^-</math> defined with the inequality directions reversed. Therefore, <math>\beta^-</math> and <math>\beta^+</math> can be estimated with a regression of excess return of security <math>i</math> on excess return of the market, conditional on excess market return being below the mean (downside beta) and above the mean (upside beta)."<ref name="Capital market equilibrium">{{cite journal|last=Bawa|first=V.|author2=Lindenberg, E. |title=Capital market equilibrium in a mean-lower partial moment framework|journal=Journal of Financial Economics|year=1977|volume=5|issue=2|pages=189–200|doi=10.1016/0304-405x(77)90017-4}}</ref> Upside beta is calculated using asset returns only on those days when the benchmark returns are positive. Upside beta and downside beta are also differentiated in the [dual-beta](/source/dual-beta) model.

==See also==
*[Cost of capital](/source/Cost_of_capital)
*[Downside risk](/source/Downside_risk)
*[Macro risk](/source/Macro_risk)

==References==
{{Reflist}}

==External links==
*[http://www.grin.com/en/e-book/146638/a-study-of-sharpe-s-asymmetric-beta-model A Study of Sharpe's asymmetric beta model]
*[http://store.elsevier.com/Rethinking-Valuation-and-Pricing-Models/isbn-9780124158757/ Rethinking Valuation and Pricing Models, 1st Edition]

Category:Financial risk modeling

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Adapted from the Wikipedia article [Upside beta](https://en.wikipedia.org/wiki/Upside_beta) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Upside_beta?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.