I try to understand the different ways to compensate for risk.
In the CAPM, when we plot the excess return against the risk, we find that portfolios of interest lie on the efficient frontier (i.e. the Markowitz-bullet). Among these, the tangency (i.e. market) portfolio has the highest Sharpe ratio, and is therefore to be preferred over the other ones.
The Sharpe ratio measures the compensation required for additional risk. It makes sense to me that if two portfolios have the same risk, but one has a higher excess return, it has a higher Sharpe ratio, and we should favor it over the other one. But, in my understanding, it also implies that portfolios with the same Sharpe ratio should be treated somewhat the same (e.g., if portfolio A is twice as risky as portfolio B, then we simply require twice the return, and we are compensated for the additional risk).
How is this linear relationship between risk and return justified? I can see that this could be explained by saying that identical Sharpe ratio is equivalent to identical Value at Risk (assuming normal distributions). Is this justification correct?
Another notion I encountered is the one of Value Added (VA), which, given a risk-aversion parameter $\lambda$, is $VA = r-\lambda \sigma^2$. Here, $r$ is the return, and $\sigma$ is the variance. In some contexts, it is stated that $VA$ tells us how much compensation we require for additional risk, and managers should seek to maximize VA (see Grinold & Kahn, Active Portfolio Management). The situation is similar to before: If two portfolios have the same VA, a manager with fixed risk preference would not prefer one portfolio over the other - even though one is riskier, it gives an adequate amount of higher return. (I understand that the concept of VA is quite heuristic, but I don't intend to question it here.) But obviously, VA is a nonlinear relation between risk and return, and in general, two portfolios with the same Sharpe ratio will not have the same VA.
In the CAPM, why don't we try to maximize VA instead of the Sharpe ratio? (and therefore obtain a different tangency/market portfolio).