Market Beta

Market Beta ($\beta$)

• Consider:
  • Security $i$ with $(R_{i,},{\sigma }_i,{\mu }_i)$
  • Market $(R_M,{\sigma }_M,{\mu }_M)$
  • & Market Beta measures the degree to which the security price correlates with the market price
    • Market beta is a coefficient of regression, derived from past data
  • Market’s beta is $1$.
• Definition: ${\beta }_i:=\frac{Cov(R_i,R_M)}{{\sigma }_M^2}={\rho }_{i,M}\cdot \frac{{\sigma }_i}{{\sigma }_M}$
• ${\beta }_i=\frac{{\mu }_i-{\mu }_f}{{\mu }_M-{\mu }_f}$ or equivalently ${\mu }_i-{\mu }_f=\beta ({\mu }_M-{\mu }_f)$ ← ”CAPM Model”
  • Under assumptions $V$ is positive definite, $\vec{\mu },\vec{e}$ are linearly independent and $0\le {\mu }_f<max(\vec{\mu })$
  • $\beta <0$: security return is opposite of market
  • $\beta =0$: security is uncorrelated to market
  • $0<\beta <1$: security return moves same as market, but fluctuates less (defensive)
  • $1<\beta$: security return moves same as market, but fluctuates more (aggressive)

Security Market Line

Plotting ${\mu }_i-{\mu }_f=\beta ({\mu }_M-{\mu }_f)$ with return vs market beta: