# How to compare mean-variance-skewness-kurtosis portfolios obtained by expected utility maximization?

Suppose I have some portfolios which are the result of maximizing the expected utility of different approximations of a utility function, how do you test these portfolio's out-of-sample and how do you compare these results?

More specifically, I am trying to compare the MV (mean-variance), MVS (mean-variance-skewness) and MVSK (mean-variance-skewness-kurtosis) portfolio's for a same set of assets, based on the CARA utility function. I maximized these three equations and for each I have a maximum expected utility and the solution in the form of a set of weights. So I now have three different portfolios, and these are based on the same sample set of 4 years (edit: 4 years of daily returns). Can I now test these portfolio's on a fifth year (one out-of-sample year, following the four year sample set) and compare some characteristics? If so which characteristics should I compare? Can you compare achieved utility, or is this not the right intention?

Edit: Some clarifications:

The fourth-order approximation of the expected value of the utility function is as follows:

$$E[U(W)] \approx U(\bar{W})+\frac{1}{2} U^{(2)}(\bar{W}) \sigma_{p}^{2}+\frac{1}{3 !} U^{(3)}(\bar{W}) s_{p}^{3}+\frac{1}{4 !} U^{(4)}(\bar{W}) \kappa_{p}^{4}$$

where $$U(W)$$ is the utility function, $$\bar{W}$$ is the expected wealth, and $$\sigma_{p}^{2}, s_{p}^{3}$$ and $$\kappa_{p}^{4}$$ are the portfolio variance, skewness and kurtosis.

The utility function I am using is the CARA utility function,

$$U(W)=-\exp (-\gamma W)$$,

where $$\gamma$$ represents the constant risk aversion coefficient.

This results in the following equation,

$$E[U(W)] \approx-\exp \left(-\gamma \mu_{p}\right)\left[1+\frac{\gamma^{2}}{2} \sigma_{p}+\frac{\gamma^{3}}{3 !} s_{p}^{3}+\frac{\gamma^{4}}{4 !} \kappa_{p}^{4}\right]$$,

which is our objective function to maximiz, in case of MVSK. If we consider MV, we drop the last two terms of the second operand of the last equation. For MVS, we only drop the last term.

These moments are represented by \begin{aligned} \mu_{p} &=\omega^{\prime} \mu \\ \sigma_{p}^{2} &=\omega^{\prime} \mathrm{M}_{2} \omega \\ s_{p}^{3} &=\omega^{\prime} \mathrm{M}_{3}(\omega \otimes \omega) \\ \kappa_{p}^{4} &=\omega^{\prime} \mathrm{M}_{4}(\omega \otimes \omega \otimes \omega) \end{aligned} where $$\mathrm{M}_{i}$$ represents the i'th matricized cumulant tensor, so for $$i = 2$$ this is the covariance matrix, for $$i = 3$$ the matricized coskewness tensor and for $$i = 4$$ the matricized cokurtosis tensor. With matricization of a tensor, the frontal slices of the tensor are placed sidewise, so for $$i = n$$ we get an $$n \times n^{(i-1)}$$ matrix.

Concerning the optimization, it can be solved by using multiple attempts of SLSQP. Although we are not guaranteed a global optimum, I have found that this works rather well, as most tries find the same optimum.

Now, whether or not we find a global optimum for the MVS and MVSK optimizations, I still do not get how to compare my resulting portfolios. As Pontus Hultkrantz mentioned, utility should b-e my measure of performance out of sample, but how do I measure realised utility?

I should also note that the moment tensors are being approximated. My research objective is to show that these approximations are adequate enough to be used in this application.

Thanks

• but portfolio skewness is non-convex and non-linear because it is a cubic, not a quadratic, objective function. you cannot solve it with quadratic programming, otherwise you are stuck at a local, not global, optimum for the higher moments. could you edit your question showing the formulas you used for the MVS and MVSK CARA models? without seeing what you are doing, an answer can't be properly provided regarding out-of-sample performance evaluation – develarist Dec 1 '20 at 13:14
• why are variance, skewness and kurtosis given the same sign coefficients ($+$)? are you aware that investors have positive preference for odd-numbered moments (mean and skewness), and negative preference for even-numbered ones (variance and kurtosis)? – develarist Dec 1 '20 at 13:37
• Also, in your formulation, are you making $\sigma_{p}^{2}, s_{p}^{3}$ and $\kappa_{p}^{4}$ all functions of the portfolio weights? What are their formulas. I do not see the portfolio weights anywhere in your model – develarist Dec 1 '20 at 13:40
• Thanks for the edits. Now i see where the portfolio weights are and sure enough portfolio skewness is a cubic function, kurtosis: quartic. Not quadratic. Your latest comment confirms moment preferences. Now its a question of whether they maximized or minimized the whole thing. What do you think? And did they make mention of which optimization algorithms they used for which model – develarist Dec 1 '20 at 15:02
• They maximize the object function as stated above, but in practice minimize (because optimizers minimize) the negation of the object function. In the literature Jondeau et al. just mention that the problem can be solved "using a standard optimisation package". – Jules Dec 1 '20 at 15:17

1. Empirical utility-based-optimization and moments

I would argue that comparing different degrees of a Taylor approximated utility optimization (a.k.a. a moment based model with two, three, four, ..., infinite moments) adds additional assumptions to your model when working with an asset universe whose statistics are not under your control, i.e. when using empirical asset returns.

With a full-utility-function mean-variance optimization, you are not only identifying an (ex-post) mean-variance optimal portfolio under some strong assumptions regarding the iid-ness of your data, but also under the assumption that the asset universe is well represented by its (pairwise) first two moments. This is reconciled by restricting the utility function to be quadratic as well, though. As you then begin to add moments, you introduce additional implicit assumptions regarding the existence and stability of higher moments of the (unobservable) data generating process (DGP).

Let me put it yet differently: Performing statistical tests on some utility-based optimal portfolios could be re-interpreted as some (most probably quite weak) joint test of the assumptions for your DGP i.e. its mean, variance, skew, kurt, higher moments.

2. Comparing across utility functions Say that we have now successfully identified $$N$$ utility-based optimal portfolios under $$N$$ different utility functions, say quadratic, cubic, quartic, ... You can then analyse realised utility (or the certainty equivalents), but you cannot compare realised utilities across utility functions. What you can do is to compare the realised utilities of a given utility function, say quadratic, across the portfolios and see which performed best. The real deal is of course to plug your estimated portfolios into the full utility function and compare those, but keep my comments under 1. above in mind.

• As a comment to your second point: as I have maximized each of the three equations, the portfolio for that equation (MV, MVS or MVSK) will be the max in that equation compared to the other two equations. So for "What you can do is to compare the realised utilities of a given utility function, say quadratic, across the portfolios and see which performed best", the MV portfolio's utility will be higher than the MVS and MVSK portfolios (as this one is maximized for this equation). – Jules Dec 1 '20 at 12:39
• Likewise, if we decide to compare utilities of the three portfolios using the third-order Taylor series approximation of the utility function, the MVS portfolio's utility will be higher than the other two. So comparing that way does not make any sense to me. Am i misunderstanding your answer? – Jules Dec 1 '20 at 12:39
• Hi Jules, what I am saying is: Plug the portfolios into the utilities. This gives you a ranking (across portfolios) per utility function. Of course, you cannot compare utilities as these are only ordinal, not cardinal. In order to compare utilities you could transform each utility level into a certainty equivalent, though. But: Beware that with quadratic and cubic forms, you may come up with multiple certainty equivalents. – Kermittfrog Dec 1 '20 at 12:48
• Instead of per-utility function, the op needs to decide which performance evaluation criterion he wants to use to put the three models on a level playing ground. Each model will outperform the others for their own respective utility functions by design. A definition of out-performance has to be decided on that can be computed for each model's own portfolio weight vector. Computed doesn't mean optimized. – develarist Dec 1 '20 at 15:09
• simply plug the realized return (consumption) into the utility equation, no? If it was quadratic / cubic...then of course you need to compare realised return versus expected return / variance etc. – Kermittfrog Dec 1 '20 at 16:10

If utility is your measure of performance, then it will still be your measure of performance out of sample, since it is what you care about. You can see utility as a measure of the balance between profit and risk, where risk is some combination of variance, skew, kurtosis...

Your wealth is a random variable $$X$$ that can be described by its moments. First moment is related to the mean, second moment related to variance, third moment related to skew, fourth moment related to kurtosis.

If you do a Taylor expansion of your utility function, and taking expectation, you will see that it can be written in terms of its moments. So the choice of utility function dictates how you look at risk, is it only variance or also higher moments?

In Mean-Variance, you risk is the variance, since skew and kurtosis are all zero (normal distribution assumption). If the distribution is not normal, then a very negative skew might cause extreme losses for you, which you want to avoid, and hence include in your utility. So in Mean-Variance, you ignore any moment higher than 2nd, in Mean-Variance-Skew you ignore moments higher than 3rd, in Mean-Variance-Skew-Kurt you ignore any moments higher than 4th.

Edit: By definition, the portfolio that was found using a cost function that best approximate your utility function will on expectation be best.

So when you evaluate your portfolios OOS, you just plug in the final wealth $$\mathcal{U}(W_T) = -e^{\gamma W_T}$$ (no need to make it "Taylor-made"). So the problem is now that you only observe a single observation, hence a single utility value. So how can you estimate the expected utility? If you care about monthly utility, then you can obviously split the 1Y into 12 months and get 12 observations. Else, you can do some kind of resampling of the data, see for instance two common resampling procedures But you would also have the same problem with finite samples for your in sample data set, so I am not sure how you solved that?

• The first sentence is the answer to the op's question of what to do for out-of-sample performance evaluation: "If utility is your measure of performance, then it will still be your measure of performance out of sample, since it is what you care about." The utility function remains unchanged for both the IS or OOS evaluation. All that is different IS vs OOS is that IS data is input into the utility function during IS performance evaluation, whereas OOS data is input into the utility function during OOS performance evaluation. He is overthinking – develarist Dec 1 '20 at 17:35