# Interpretation of a uniform asset return distribution

Typically asset return distributions are bell-shaped with most mass occurring in and around the center, 0% returns, and less so in the tails, with the left tail representing the probability of large losses, and the right tail representing the probability of large gains. Despite the tails being small compared to the center mass, alot of problems arise due to return distributions being non-normal.

Now consider observing an asset whose return distribution is uniformly (equally) distributed in such a way that tail returns (large gains and large losses) are just as likely as centered returns (small around 0%).

Do such assets exist? If not, do they at least exist in economic models as some sort of theoretical ideal/non-ideal extreme? What can be said about the properties of that asset to an investor? Does its uniformity make it more uncertain than non-normal, but bell-shaped, assets? Would investors be more averse to this asset compared to non-normal, but bell-shaped, investments?

• that's a really interesting question. I don't know the answer but the answer to "why not" is not obvious either because, assuming that investors base their decisions on expectations, then an asset with a uniform distribution that could still have the same expectations in terms of mean and variance as a normally distributed RV. Aug 29 '20 at 14:06
• Roulette maybe? Aug 29 '20 at 16:41

Such assets do not exist due to market efficiency: people would trade such assets until the price was near the expected value which would tend to yield more returns near 0 and fewer returns that were larger in magnitude. Thus such a distribution is in no way an ideal. The effect of market efficiency also renders your other questions moot.

Even if that did not render the other questions moot, it is impossible to answer your other questions without much more information, e.g. support of the uniform distribution or the variance and other moments of the bell-shaped return distribution.

• isn't the support of a uniform distribution $[0,1]$, and that mean and variance can likewise be estimated for a uniform distribution as how it's done for a bell-shaped distribution and then compared? Regardless if there are uniform assets, couldn't we think of it as being much more uncertain than non-normal (fat-tailed) assets that everyone complains about? therefore establishing a much worst case product than fat-tailed assets for theoretical purposes Aug 29 '20 at 13:12
• No, they are not necessarily much more uncertain. A standard uniform has support on [0,1], but the uniform distribution can have support on any finite interval. Aug 29 '20 at 15:49
• an investment that always gives a 5% return with 100% probability would have a spike for a distribution. I would say this distribution has 0 uncertainty. the uniform distribution, on the other extreme, is more uncertain than the fully certain example just described because its outcomes are equally probable. that puts the bell-shaped returns somewhere between these two cases, making the uniform returns more uncertain. this is all assuming a snapshot in time rather than expectations that drive all distributions to normality though. and i should have said at the beginning standard uniform Sep 1 '20 at 3:21
• I did not say there is no uncertainty. If anything, we expect the risky assets to return at least as much as that 5% return since that would be your risk-free rate. The uniform is not an extreme. If I have an asset that has a return uniformly distributed on [0%, 10%], that is far less risky than holding an investment where it might go limit down four days in a row and I've lost 45% by the time I can sell. As for standard uniform, even [0,1] (meaning 0% to 100%) could be less risky than power contracts where you could lose 100% (or worse, cf negawatts) or gain 200$\times$. Sep 1 '20 at 4:37
• Sorry, I’m being thick, and just don’t get. Your asset has a starting value of 100, with a daily random walk of between -1 and +1. Where is the arb there? Silly me just doesn’t get... Sep 1 '20 at 21:30

Supplemental to copious previous discussion here: all based on this being an interesting, ie thought-provoking, question.

The crux of the problem with any asset having a uniform return distribution (as opposed to the standard assumption of normality) is that such an asset with such a distribution in one time horizon would have a very different return distribution, seen from a different time horizon.

Put simply. Over longer periods, uniform returns would then converge towards normality. But over shorter periods, returns would have to converge towards binary. And at the time horizon at which they were uniform, they would have to be bounded. So for the uniform to exist, one has to be believe in a time-dependant multiplicity of return distributions!

There need not (but might) be some arbitrage to prevent this from (theoretically) happening. However, good luck proposing the notion that your asset in question behaves "this way" daily, "that way" weekly; and "another way" monthly! This begs obvious questions about the transition dynamics between your three different asset pricing regimes. And if a clever trader can't work out how to arb that; then I'm a donkey ;-)

By claiming the uniform, you are telling me there's a point in time where the probability of price<L=0, of the price>H=0, but any point between L and H is uniformly likely. In this world, puts striked at L and calls striked at H should be free. Calls striked between these should be fractionally priced where the strike lies in the range between L and H. I can't necessarily arb you if you are 100% correct in your estimator of the return distribution... but I can get some very cheap lunches off you if you are less than 100% correct in your estimator.

Plus, I and other investors WILL leverage up, if your bounds required for your uniform hold firm. Given that we would all be simultaneous buyers/sellers at those bounds in unison, there needs to be some deus-ex-machina trader-of-last-resort willing to step in and guarantee those price levels, for the system to hold. That's obviously intuitively problematic, because it suggests that agent providing free options to market participants. If the distribution was uniform, I have NOTHING to lose buying insurance at the boundaries of your distribution; with potentially a lot to gain if the distribution is not the true one!

Over shorter time periods, one could have similar fun trading the absence of continuous behaviour. Over longer ones, trading normality versus uniformity. Put simply, the assumption of uniformity is quickly too constrained to work.

• was a previous answer where copious comments were exchanged deleted Sep 1 '20 at 3:27
• Yes, @mark leeds and I used more words on clarifications/follow-ups than on any actual answers! Which have been mostly mutually deleted. I hope to his satisfaction; but I’m sure he’ll let us know if not. Sep 1 '20 at 20:58