Timeline for Fixes of quadratic utility when probability of decreasing utility is large
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Feb 4, 2023 at 8:03 | comment | added | Richard Hardy | My last comment misses an important (and retrospectively obvious) point that we should work with adjusted prices, not nominal ones, such that splits or reverse splits are taken proper account of. But I think that is a relatively minor point in this discussion. | |
| Feb 20, 2020 at 0:36 | history | edited | Dave Harris | CC BY-SA 4.0 |
comments
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| Feb 19, 2020 at 17:09 | comment | added | Richard Hardy | Oh, and the ratio of prices cannot blow up because price values very close to zero are unlikely: exchanges impose requirements on the minimum price. If the limit is hit, the company is supposed to do a reverse-split or something. So we are never dividing by small values --> the ratio cannot blow up. | |
| Feb 19, 2020 at 17:00 | comment | added | Richard Hardy | But if you are deriving your theory from deeper fundamentals, my question could/should be redirected towards them. If only I understood your theory better, I would post a question here on QF SE so that you could explain yourself concisely in an answer (if you were interested). As the utmost expert of your own theory, you could even do it yourself, debating in a Q&A style to provide your readers a lesson. Of course, this is extra work, so you might not be willing to invest your time in that. But perhaps you have some existing material you could use, so it would not be all that costly? | |
| Feb 19, 2020 at 16:58 | comment | added | Richard Hardy | Moreover, there is a long-run lower bound of prices being nonnegative. Now, we can approximate the price distribution with a normal one in some settings, but in other settings such an approximation is clearly harmful. Taking ratios belongs among the latter cases, because the poor approximation of the tails of levels really blows up the ratios. Also, simulations of Cauchy-distributed data do not look quite like the empirical rewards data, or do they? E.g. if you calculate variance over increasing sample size, you get a saw-like pattern for Cauchy. I doubt you would get this from empirical data. | |
| Feb 19, 2020 at 16:56 | comment | added | Richard Hardy | I got interested in your theory and would like to question it. More specifically, I would like to question the premise that leads to the conclusion that rewards are (truncated-) Cauchy distributed and so do not even have a finite mean. My gut feeling says this is not true, so I am trying to see why. Here is my intuition: prices are bounded both from below and from above, so their ratios are bounded, which means all moments are finite. Why are prices bounded? Because exchanges have daily trading limits, i.e. limits of how much a price can change. | |
| Feb 8, 2020 at 0:17 | history | edited | Dave Harris | CC BY-SA 4.0 |
more complete answer
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| Feb 7, 2020 at 20:53 | comment | added | Richard Hardy | OK, in the video you find infinite moments (naturally enough as rewards are ratios and returns are rewards minus one), and I suspect that must limit the choice of utility functions as expectation of utility will not exist for some classes such as quadratic. | |
| Feb 7, 2020 at 20:24 | comment | added | Richard Hardy | Still thinking things through. One thing that stands out is your linking of estimation (expectations, quadratic estimation loss) with portfolio optimization (expected utility or expected evaluation loss). Is that a historical thing? Because I do not see a fundamental connection between estimating/modeling return distributions and optimizing portfolios. In my understanding, estimation does not belong in this discussion. (Aside from cases where the estimand does not exist, as in infinite variance and such.) So that part is a bit confusing to me. Watching your video now, perhaps it will help. | |
| Feb 6, 2020 at 8:59 | comment | added | Richard Hardy | thank you very much! I will read carefully and come back to you. | |
| Feb 6, 2020 at 7:13 | comment | added | Dave Harris | @RichardHardy I did a rewrite for you. | |
| Feb 6, 2020 at 7:08 | history | edited | Dave Harris | CC BY-SA 4.0 |
Answered edits, solved errors, more complete answer
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| Jan 23, 2020 at 7:07 | comment | added | Richard Hardy | Finally (or not), (6) you seem to be knowledgeable of the context. Could you suggest any references? Nontechnical ones would be especially appreciated. (I never had a class on topics like this and have only accidentally picked up some pieces of information over time. A unified treatment could be quite helpful to patch the gaps / connect the islands.) And once again, thanks a lot for your input! | |
| Jan 23, 2020 at 7:07 | comment | added | Richard Hardy | Also less importantly, (5) Logically, $\lambda$ would not be a constant. As a gamble becomes large, the importance of precision and accuracy should increase. This is a normative statement that questions how well quadratic utility represents actual preferences, and it may be a valid piece of criticism. But given that quadratic utility is used quite a lot in finance, my original question is how to make it work. | |
| Jan 23, 2020 at 7:07 | comment | added | Richard Hardy | This is probably less important, but (4) Quadratic utility will generate the most efficient estimator. First, I wonder in which sense would quadratic utility generate something; but in any case, it will reward an estimator that minimizes expected squared error. Second, if we were to consider a statement "an estimator that is optimal under quadratic utility is an efficient estimator", it sounds pretty much tautological to me. | |
| Jan 23, 2020 at 6:34 | comment | added | Richard Hardy | (2) I am not sure why I put $\mu_x$ in the function; perhaps I was reverse-engineering the utility function from the mean-variance optimization problem. It could (should?) just be $u(x)=x-\frac{\lambda'}{2}x^2$ or $u(x)=x-\frac{\lambda''}{2}(x-c)^2$ for some $c$ that reflects an agent's preferences. Ideally, $c$ would be greater than $\max(x)$, but if the support of $x$ extends to $+\infty$, such a $c$ does not exist, which is likely the root of the problem. (3) How come $\lambda$ is supposed to be a scale parameter of some distribution? Where does this come from? | |
| Jan 23, 2020 at 6:34 | comment | added | Richard Hardy | Thank you so much for you answer! Very interesting! (1) What is valued with quadratic utility is accuracy and precision, not increasing wealth: where does this come from? Does it not go against the basics of utility theory? Also, Imagine your concern was to have \$3 at the end of the gamble, then you are worse off if you have \$3.50 or \$2.50. Now this really goes against the basics, since utility should be increasing in wealth. | |
| Jan 23, 2020 at 1:07 | history | answered | Dave Harris | CC BY-SA 4.0 |