The risk-aversion component of a portfolio utility function is expressed as the variance of the portfolio. Why the variance, instead of standard deviation, is used in here?
I'm asking this question because of the following calculation: suppose I only have a single stock and also a fixed risk-aversion parameter. Then I use mean-variance tradeoff to determine the optimal amount of the stock to hold. If the variance is used in the utility function, then I can get an optimal position because the variance is quadratic of the position. However, if standard deviation is used, then both the expected return and risk are linear in the position, hence we cannot get an optimal position in this setup. On the other hand, if indeed standard deviation is also a reasonable choice of expression of risk aversion, then it seems the "optimal position" obtained using the variance is purely an artifact of the functional form selected.