Assuming that time period used to calculate the beta and correlation between an index and an asset is the same, is it possible to observe low beta while having high correlation?
If yes, how would you explain it?
Yes, it's possible if one instrument has higher volatility that the other. Consider a low volatility series that is slightly lagged. For example, take a large sine wave and duplicate with tiny lag.
In practice it may be hard to find a well-correlated asset - perhaps alternatively consider a very high freq sine of a lower freq sine function (oscillating oscillations) versus the lower freq sine.
From these formula: Corr(1,2) = Cov(1,2)/(Var(1).Var(2))^1/2 and Beta = Cov(1,2)/Var(1)
Therefore Beta = Corr(1,2) * 1/Var(1) * [Var(1) * Var(2)]^0.5 = Corr(1,2) * [Var(2) / Var(1)]^0.5
A good reference may be: Do two stocks with the same beta have a correlation of 1?