Question
- I found a simplified inequation to decide whether the new asset A should be added to my current portfolio B. If the following inequation is satisfied, the new asset A should be added to my portfolio. (source: Mackenzie Investment's research report Correlation vs. Beta: What is the difference)
$$\frac{E(R_{a})}{\sigma_{a}} > \frac{E(R_{b})}{\sigma_{b}} \times corr(R_{a}, R_{b})$$
- One colleague of mine suggested to me that the inequation shown above seems to be derived from the mathematical equation written below, when the condition $W_{a}> 0$ is satisfied.
$W_{a}$: how much percentage of my total wealth is invested in the asset A
$E(R_{a})$: the expected return of the asset A
$\sigma_{a}$: the standard deviation of the returns of the asset A
$r_{f}$: risk-free return such as the US government bonds
I assume that $R_{A}$ is the same thing as $R_{a}$, which means the return of the asset A.Is there anyone who can show me how the equation written at the bottom can be simplified to the inequation written at the top, when the condition $W_{a}> 0$ is satisfied?
I think it has another origin. Consider the linear model: $$r_{A}=\beta_{r_p,r_A}r_p+\varepsilon=\frac{\textrm{Cov}(r_A,r_p)}{\sigma_{r_p}^2}r_p+\varepsilon$$ where $E[\varepsilon]=0$ and the error is independent of $r_p$. The expected return of asset $A$ is then $E[r_A]=\beta_{r_p,r_a}E[r_p]$. The rule states that if the return/volatility ratio (sort of Sharpe ratio) of the actual expected return of asset $A$ is greater than the model's, then you should add the asset to the portfolio: $$\frac{E[r_p]\rho_{A,p}}{\sigma_{r_p}}=\frac{E[r_p]\textrm{Cov}(r_A,r_p)}{\sigma_{r_p}^2\sigma_{r_A}}=\frac{E[r_{p}]\beta_{r_p,r_A}}{\sigma_{r_A}}<\frac{E[r_A]}{\sigma_{r_A}}$$