" When the beta of the active portfolio is low (less than 1.0), there is more potential gain from diversification, hence a smaller position in the active portfolio is called for.
If the beta of the active portfolio iwere significantly greater than 1.0, a larger position would be called for.
Thus, there is a direct relationship between the beta of the active portfolio and the optimal weight in the optimal overall risky portfolio"
Can someone please explain why this is the case? I would think it’s better to size UP the active portfolio’s weight if beta is low?
I’m not 100% sure about my answer, but I will give it a shot:
The Treynor Black model is used to figure out the proportions that should be invested in a passive (diversified portfolio) and an active portfolio (consisting of only a few mispriced stocks). You want to invest more in the active portfolio when the potential returns, for the additional risk are higher. When the active portfolio has a beta of less than 1, its potential returns are lower, meaning that its advantage compared to the passive portfolio is lower. If the beta is greater than 1 than the potential gains from the active portfolio are much larger which would mean allocating a larger portion to the active portfolio.
I believe the problem assumes that in either case, beta below 1 or above 1, the risk for the active portfolio doesn’t change.
Beta is a determinant for expected returns an asset.
remember how when you are determining asset portfolios you look at expected returns and standard deviations… well once you have an active portfolio you generally need to figure out its expected return, so you refer to:
E® = rfr + B(Market risk premium)
higher the beta the higher the expected return, so most probably the high asset allocation on an active portfolio.
of course you need to consider other the variable in creating the efficient asset allocation frontier (between active and passive portfolio), which is standard deviations.