Beta is a measure of Systematic Risk for Highly diversified portfolios as we have to measure Risk w.r.t the Market or Index Portfolio, therefore, Beta is the measure of covariance of the security wrt the Market divided by the variance of the Market Portfolio
Let's start from the Coefficient of Correlation (rho)
Therefore: rho(x,y) = Cov(x,y)/std(x)*std(y); rho (x,m) = Cov(x,m)/std(x)*std(m)
Beta = rho(x,m) *std(x)/std(m); therefore, substituting from the above, [Cov(x,m)/(std(x)*std(m))] * std(x)/std(m)
So Beta = Cov(x,m)/var(m)
The Second formula that you have mentioned above seems to be that of Coefficient of Correlation between i and p, not Beta
Thanks @QuantMan2318 ! @Kavita.bhangdia Unlike correlation, beta is not symmetric. By beta, we mean "beta of x with respect to m" such that, as QuantMan says, β(x,m) = cov(x,m)/var(m); whereas beta of m with respect to x would be β(m, x) = cov(x,m)/var(x). Of course, our beta is typically "with respect to the market index" or "with respect to the benchark" such that their variances are in the denominator. Further, the important reduction is given by β(x,m) = cov(x,m)/var(m) = σ(x)*σ(m)*ρ(x,m)/σ(m)^2 = σ(x)*ρ(x,m)/σ(m) = ρ(x,m)*σ(x)/σ(m); i.e., beta = correlation * cross-volatility. Thanks!