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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). enter image description here 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. enter image description here 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): enter image description here Here, the CPT portfolio in this example is 0.5559%, which is (given my constraints) close enough to the efficient frontier.

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    $\begingroup$ 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. $\endgroup$ Commented Nov 1, 2022 at 11:59
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    $\begingroup$ @Kermittfrog Thanks for your reply, yes, good points: I'll edit my question. $\endgroup$
    – T123
    Commented Nov 1, 2022 at 13:00
  • $\begingroup$ @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).. :-( $\endgroup$
    – T123
    Commented Nov 1, 2022 at 13:12
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    $\begingroup$ 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? $\endgroup$ Commented Nov 1, 2022 at 14:53
  • $\begingroup$ 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.. $\endgroup$
    – T123
    Commented Nov 1, 2022 at 14:55

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I just checked the portfolios that i get using various parameters and i think they are close enough for my purpose..enter image description here 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 :-) )

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