i'm facing a new and interesting task: We are calculating a time series of (hypothetical) behavioral portfolios, for which i need a few parameters to calculate the portfolio's weights in each asset. I'm using an observed portfolio as starting point, from which i need to extract the implied utility parameters (in the case at hand the CPT utility as seen in my screeenshot).
My idea is to find the parameters using a grid-search algorithm (as others such as Nelder-Mead don't reliably converge) and chose those parameter combinations for which the squared difference in weights (model-implied optimal portfolio weights-observed portfolio weights)^2 is minimized.
I tried to validate what i'm doing using Kahneman/Tversky(1992) parameters and calculated the optimal portfolio weights in my first step (i assume in the second step that these are my observed weights). In the second step i tried to calibrate my model-implied weights to these "observed" weights. I noted however that these implied CPT parameters are nowhere near the original parameters, which i used in the first stp, however, my model-implied weights fit more or less well to my "observed" weights. It apprears to me that i can reach the same optimal portfolio using various parameter combinations...
This is of course very unsatisfying so i wonder what else can be done? What other approach can i use to get to my implied parameters given i can only observe the real-world portfolio weights on a certain point in time?
Any ideas are appreciated :-) Thomas
EDIT: Optimization procedure: I'm maximizing CPT utility given the hist.vola for the observed portfolio. My CPT utility has the form: $$U(\Delta{W})=\sum_{i=-m}^{-1}v(\Delta{W_i})[w^{-}(\frac{i+m+1}{n+m})-w^{-}(\frac{i+m}{n+m})]+\sum_{i=1}^{n}v(\Delta{W_i})[w^{+}(\frac{n-i+1}{n+m})-w^{+}(\frac{n-i}{n+m})]$$ where the $n+m$ observed changes in wealth $\Delta{W}$ are sorted ascending with $-m$ being the largest loss up and $+n$ being the largest gain. Each change in wealth is observed with equal probability $p(\Delta{W_i})=(\frac{1}{n+m})$. $w^{+}$ and $w^{-}$ is the decision weighting function, which is subadditive and contains a parameter $\gamma$.The functional form i'm using for $v( )$ a power function of the form $(W-RP)^\alpha$ where $(W-RP)$ is $\Delta{W_i}>0$ for gains and for losses i used the form $\lambda(RP-W)^\alpha$. $\alpha$, $\gamma$ and $\lambda$ are my CPT parameters that i want to calibrate.
My portfolio optimizer maximizes this CPT utility given a) positive weights, b) sum of weights is equal or smaller than 100%, c) the portfolio variance given the hist. covariance matrix is equal to the historical vola of my observed portfolio.
I found that the effient frontier is close to my optimal portfolios (which is in line with https://academic.oup.com/rfs/article-abstract/17/4/1015/1570743?login=false):
Here, the CPT portfolio in this example is 0.5559%, which is (given my constraints) close enough to the efficient frontier.
