Page 535 Eq ( 5 ) is for optimal weight w* of Portfolio A, Page 536 optimal weight w0 is for which portfolio not mentioned and explantion below Eq ( 7 )
“w * increases when Beta A increases because the greater the systematic risk, Beta A, of the active portfolio, A, the smaller is the benefit from diversifying it with the index, M, and the more beneficial it is to take advantage of the mispriced securities.”
If there is more systematic Risk in Portfolio A, How there will be more benefit to take advntge of mispriced securitied as mentioned above.
I didn’t study into that much of detail, but I tried to follow the textbook from page 534 - 536 and gave it a shot:
The context is that they have obtained w0 - the weight to be placed into the alpha portfolio assuming Beta A = 1. Alpha portfolio refers to point A on figure 2, page 534.
We could maximise Sharpe ratio by including the market portfolio into the Alpha portfolio holding. Let’s recall that as Beta tends to 1, the higher the correlation between Alpha portfolio and Market portfolio (i.e. point A and point M). When two are perfectly correlated, there is no benefit from diversifying from Alpha portfolio, right.
But Beta A (Beta of Alpha Portfolio) might not be 1. Thus, we need to adjust.
w* = w0 / [(1+(1-BetaA)w0] (eq 7)
And try some numbers into eq 7:
If BetaA = 1 (perfect correlation to market portfolio). W0 = w* i.e. no adjustment is required. No benefit from diversifying with the index.
If BetaA= negative (alpha portfolio move in opposite dirrection with market), w0>w*. i.e. we should lessen our weight in alpha portfolio and put some of that into the market portfolio
If BetaA > 1 (high tech, cyclical stocks) w*>w0 increase our weight in alpha portfolio.
I had a really long response generated, but it was a little too long.
In short, a high Beta (>1) implies the active portfolio already has high exposure to the market (systematic risk). But, we also know that portfolio A generates excess return over the market while generating unsystematic risk (specific to the securities in the portfolio that cant be diversified away). Systematic risk can be represented by Beta
So, the weight of A we should hold is a function of:
1.) W_0 = the excess return per unit of risk of the portfolio A divided by the excess market return per unit of risk of the market. If this is high, it means we should hold more of A by definition;
2.) and Beta
So, we look at 1.) and say - wow, this portfolio generates more return per unit of risk, I should put weight portfolio A higher.
But how much higher? Remember, with a high Beta, A has high systematic risk already by definition. It doesn’t do us any good to hold more of the market portfolio for diversification purposes, because A already holds that risk (as the market moves, A will move with it). But A also generates a higher return.
Equation 7 tells us that if A generates a higher excess return, and also has high systematic risk, we should hold more of it. Why? If we’re going to hold systematic risk, we may as well hold it in a portfolio that generates higher return. But, if those excess returns are generated at a cost of higher variance (ie, W_0 is below 1), we may want to hold less of A.