Upside beta

☆ Save On Wikipedia ↗

In investing, upside beta is the element of traditional beta that investors do not typically associate with the true meaning of risk.[1] 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. Defining r i {\displaystyle r_{i}} {\displaystyle r_{i}} and r m {\displaystyle r_{m}} {\displaystyle r_{m}} as the excess returns to security i {\displaystyle i} {\displaystyle i} and market m {\displaystyle m} {\displaystyle m}, u m {\displaystyle u_{m}} {\displaystyle u_{m}} as the average market excess return, and Cov and Var as the covariance and variance operators, the CAPM can be modified to incorporate upside (or downside) beta as follows.[2]

β + = Cov ⁡ ( r i , r m ∣ r m > u m ) Var ⁡ ( r m ∣ r m > u m ) , {\displaystyle \beta ^{+}={\frac {\operatorname {Cov} (r_{i},r_{m}\mid r_{m}>u_{m})}{\operatorname {Var} (r_{m}\mid r_{m}>u_{m})}},} {\displaystyle \beta ^{+}={\frac {\operatorname {Cov} (r_{i},r_{m}\mid r_{m}>u_{m})}{\operatorname {Var} (r_{m}\mid r_{m}>u_{m})}},}

with downside beta β − {\displaystyle \beta ^{-}} {\displaystyle \beta ^{-}} defined with the inequality directions reversed. Therefore, β − {\displaystyle \beta ^{-}} {\displaystyle \beta ^{-}} and β + {\displaystyle \beta ^{+}} {\displaystyle \beta ^{+}} can be estimated with a regression of excess return of security i {\displaystyle i} {\displaystyle i} on excess return of the market, conditional on excess market return being below the mean (downside beta) and above the mean (upside beta)."[3] 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 model.

See also

References

  1. James Chong; Yanbo Jin; G. Michael Phillips (April 29, 2013). "The Entrepreneur's Cost of Capital: Incorporating Downside Risk in the Buildup Method" (PDF). p. 2. Retrieved 26 June 2013.
  2. Bawa, V.; Lindenberg, E. (1977). "Capital market equilibrium in a mean-lower partial moment framework". Journal of Financial Economics. 5 (2): 189–200. doi:10.1016/0304-405x(77)90017-4.
  3. Bawa, V.; Lindenberg, E. (1977). "Capital market equilibrium in a mean-lower partial moment framework". Journal of Financial Economics. 5 (2): 189–200. doi:10.1016/0304-405x(77)90017-4.