# A better calibration method available?

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.

• Hi there, could you please add some mathematical details to your portfolio optimization routine? There could be various parts at play, e.g. a) The optimization is using mean/variance, only and it is not considering efficient portfolios, only, or b) the goal function is not sufficiently convex, so that there are multiple 'optima', at least from a numerical point of view. Nov 1, 2022 at 11:59
• @Kermittfrog Thanks for your reply, yes, good points: I'll edit my question.
– T123
Nov 1, 2022 at 13:00
• @Kermittfrog I just checked it (see my screenshot after the Edit: CPT maximizing portfolios match with large sections of the efficient frontier, so i guess its probably point b).. :-(
– T123
Nov 1, 2022 at 13:12
• Given your update: Could you try an asset universe of / investement into only two instruments, only, and see whether input portfolio and output portfolio match? Nov 1, 2022 at 14:53
• Thanks for this idea: i tried this as well, it shows the same effect. I'm working on a way to analyze your point b) by plotting the surface for 2 of the 3 parameters and check whether there are saddlepoints in CPT utility and other interesting pattern..perhaps it will shed some light on that as $\lambda$ introduces some convexity in the optimization problem as far as i can say as $\lambda$ is negative..
– T123
Nov 1, 2022 at 14:55

I just checked the portfolios that i get using various parameters and i think they are close enough for my purpose.. However, i still appreciate any hints what can be done better or how i can improve my estimates. I also appreciate all helpful suggestions and ideas i received so far (thanks to @kermittfrog :-) )