Hi guys,
I am having trouble with the algebra of how this equation works. I know it was explained for previous exams but I can’t find a good explanation anywhere.
How does Bi,m= cov(I,m)/ var(m) = p(i,m)(o i/ o m)
Specifically I am having trouble understanding how to move from covariance to correlation coefficient and how the equations balance out.
Cov(i,m) = corr(i,m)(o i)(o m)… so if we substitute the correlation into the covariance equation how are these equal to each other. In other words where does the var(m) go and what is done to make them equivalent. I know there was some trick taught on previous levels but I can’t figure it out and I am getting hung up on this.
Please help!
By definition,
\rho\left(x,y\right) = \frac{Cov\left(x,y\right)}{\sigma_x\sigma_y}
Thus,
Cov\left(x,y\right) = \rho\left(x,y\right)\sigma_x\sigma_y
Therefore,
\beta_{i,m} = \frac{Cov\left(i,m\right)}{\sigma_m^2}
= \frac{\rho\left(i,m\right)\sigma_i\sigma_m}{\sigma_m^2}
= \rho\left(i,m\right)\left(\frac{\sigma_i}{\sigma_m}\right)
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Wow… Idk how I missed that thank you.
Could you also by chance help me figure out how we arrive at RPi= p(i,m)(o i)(RPm/o m) by rearranging CAPM and using the previous beta formula b(i,m)= p(i,m)(o i/o m)
Why does the RPm go above the standard deviation for the market? Just trying to figure out how formula is manipulated. I do understand it’s the sharpe ratio… but why
R_i = R_f + \beta_i(R_m - R_f)
R_i = R_f + \rho_{i,m} \frac{\sigma_i}{\sigma_m} \times (R_m - R_f)
R_i = R_f + \rho_{i,m} \times \sigma_i \times (\frac{R_m - R_f}{\sigma_m})
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This makes perfect sense. Thank you for that very much appreciated