Individuals process small and large numbers differently.

Small numbers are processed on a linear scale, while large numbers are processed on a logarithmic scale. In this paper, we show that financial analysts process small prices and large prices differently. When they are optimistic (pessimistic), analysts issue more optimistic (pessimistic) target prices for small price stocks than for large price stocks. Our results are robust when controlling for the usual risk factors such as size, book-to-market, momentum, profitability and investments. They are also robust when we control for firm and analyst characteristics, or for other biases such as the 52-week high bias, the preference for lottery-type stocks and positive skewness, and the analyst tendency to round numbers. Finally, we show that analysts become more optimistic after stock splits. Overall, our results suggest that a deeply-rooted behavioral bias in number processing drives analysts’ return expectations.

That is from Tristan Roger, Patrick Roger, and Alain Schatt, via the excellent Kevin Lewis.


Sounds reasonable, which is more than can be said for financial analysts.

I would go so far as to say that it is an obvious result for anyone who has observed people much. But it's still worth studying, of course.

June 9, 1964 (Tuesday)[edit]
A federal court jury in Kansas City, Kansas, found army deserter George John Gessner guilty of passing United States secrets to the Soviet Union and sentenced him to life imprisonment. Private Gessner had fled from Fort Bliss in Texas on December 6, 1960 and, two days later, went to the Soviet embassy in Mexico City.[40] He would admit that a month later, in January 1961, he had sold operational and design details of the Mark 7 nuclear bomb , the 280 mm atomic cannon and the 8-inch atomic mortar, in return for a payment of $200.

Perry Mason reruns are fun. The rich have estates of $5000-$10,000 justifying complex murder plots to inherit.

Should it follow that higher priced stocks outperform lower priced stocks over the long-term given analysts tendency to underestimate higher priced stocks?

Not to worry, hedge funds are replacing analysts with computer engineers and quants. The irony is that the quants predict price changes (up or down) based on past behavior of investors. What will the quants do if all stock pickers are bots? That's a problem.

One prediction is highly stable markets. Of course, that means less opportunity for gains, and less opportunity (if that's the correct term) for increasing levels of inequality (unstable asset prices being a big contributor to inequality).

I'm not going to buy the paper.

Do they identify the border between "small" and "large" numbers?

I have a fool for a financial analyst. I do my own.

Did the paper differentiate fundamental from technical analysis?

Two of my prejudices: one, it seems (I weekly only scan Barron's) as if 90+% of analysts' recommendations are positive or neutral. Two, lower price stocks have broader appeal to larger numbers of investors ergo more liquidity. I think that is why God created the stock split.

All that being said, I wish I owned more AMZN.

Still I rose (132), but I knew (105). 27/208=.32

WJC (208) LHD (166). Pay attention. WJC (LL=24), (II=18), (FF=12), (EE=10).

LHD=187 but TJM=224, besides 208/180=1.155556

Joseph Start (October 14, 1842 – March 27, 1927), nicknamed "Old Reliable", was one of the biggest stars of baseball's earliest era, and certainly the top first basemen of his time. He started his career before the American Civil War and continued to play professionally until 1886, when he was 43. Joseph Start died at the age of 84 in Providence.

I was having dinner with a centimillionaire, and while discussing costs of big projects or something, I lightheartedly said "a billion dollars isn't so much anymore." He truned suddenly serious and said "yes, it is."

Funny moment. Perhaps numbers stop being logarithmic when you are closer to them.

For me a hundred million and a billion are quite similar.

There are techniques to re-ground big numbers though, for instance consider them as seconds:

"I found that 1,000 seconds ago was equal to almost 17 minutes. It would take almost 12 days for a million seconds to elapse and 31.7 years for a billion seconds. Therefore, a trillion seconds would amount to no less than 31,709.8 years. A trillion seconds ago, there was no written history." - NYT

While I am not very close to being either a centi-millionare or a billionaire I have to deal with n every day, where n can be thousands or many millions, perhaps billions. It is orders of magnitude we are concerned with here.

Money isn't actually large or small- you could map money into a space between 0 and 1. Money is only relatively large.

Discrete items have actual magnitude.

"Centi" means one hundredth, so a centi-millionaire would I think be someone who has $10,000. Maybe you mean hectamillionaire?

Perhaps you have a way of staying grounded, but I am agreeing with the main idea above, that most people treat large numbers more .. vaguely.

This is a problem as trillion dollar debts or budgets are discussed.

Or billion dollar fighter jets.

Over what market?
The only thing small and large have in common is a similarly regulated set of price listings. Other than being listed at the same place, all the stocks are independent markets, subhect to nuance.
large prices are different in that they dominate the indices. They use greek letters, the large have large gamma, I think, the small one have none.

This seems to align broadly with the Weber-Fechner law (–Fechner_law) which relates to the physical senses. Small changes in physical stimuli (small sounds, faint objects) may be perceived in a linear fashion - research divided on this - but large changes (loud sounds, large collection of dots) are perceived logarithmically. Not surprising that how we perceive the abstract world should mirror how we perceive the physical world.

Our brains have a hard time overcoming the visual cues even though we understand the abstraction. A 9 digit number is 9/10th the visual size of a 10 digit number. As a result we interpret the two numbers as fairly similar in size when really they are incredibly different. By rounding to the millions we get a better visual represantion of relative size - still way off - but better.

Smaller companies will often report financial statement in actual dollars - ie lots of digits or rounded to the thousands, so lots of 3 and 4 digit numbers. Mid to large companies will have everything rounded in millions and lots of 2 and 3 digit numbers in their financial statements. My totally unsubstantiated guess is that the different visual cues lead to some of the problems that the resarchers are observing.

if you watch baseball for fifty years you realize that not much that happens hasn't already happened in almost exactly the same way, even a one in a million occurrence happens pretty often, because there are lots of ways for a pitch to wind up a one in a million pitch. After fifty years you get a real good feeling for what one in a million means.

the trick is you need to know something really well, most wall streeters are youthful rubes compared to some guy who has watched literally millions of fastballs and curveballs and so on tracking down that same sort of rectangular but invisible corridor of blank air-space between the pitchers release point at some given point above the pitchers mound and the "strike zone" directly above "home plate": picture it: "I am just a humble country slugger" Snoopy said when he was portrayed as a baseball player, a good one, but he would have understood, our dear comical canine pal

there are old comedians even tonight watching some of those lame "late night" comedy shows with "men ironically wearing suits and ties" on a "late night Hollywood sound stage" who watched Jack Parr back in the day and who, while watching the current crop of "late night hosts", bless their little millionaire hearts, amuse themselves by wondering about how much one millionth and, effectively combined, even one billionth variations on the multiplied comic timings of the current "late night hosts" would make said hosts funnier or less funny.
OK maybe I am wrong maybe they are thinking about differentials in timing at just the one thousandth level but maybe I am right after 60 years they know if I am right or not

my favorite explanation of how someone can understand big and little numbers accurately and in tune comes from an old review I once read of one of those passionate tousled-hair opera conductors from back in the day - "Maestro", said the lovely lady after the performance, "that was perfect" and the Maestro replied "thank you" and after a moment's hesitation said "but did not the French horn sound just a little flat in the tenor's first aria, just slightly flatly reprising the second and third of the last three notes of his first phrase" and the lady replied "well I thought so too at the time but at the intermission I realized that the tenor sang a little sharp - I saw that effect when I was much younger when two out of all the fireflies, on a summer night when me and my friends were strolling in the lanes, settled at the same time on a white daisy - the fireflies, not me or my friends - and shined their little yellow lights just slightly brighter, one thought at the moment only God and the angels might know why - and then we noticed that the daisy they had settled on had caught the reflection of the moonlight to a slightly brighter degree than all the others of the hundreds of the daisies in the field" and the conductor said "how wonderful that you remember those daisies in that moonlight" and the lady said "I remember all of them".

thousands, millions, billions, they are just numbers, but we are creatures of God

I remember, too. See what I mean when I say cor ad cor loquitur? Thanks for reading.

Can anyone explain to me why Trump voters on average have lower IQs than Hillary voters? I'm upset since I voted for Trump. If I switch votes next election, will it undo any damage or do you think its permanent?

Permanent. Hahahaaaaaahaha!

Brains use the new AI techniques. They find the minimum transaction cost between a semi-random sequence of stimulations and the response, a pattern matching, like a bank. Instead of interest charges the brain charges neurotransmitter charges across nerve synapses.
Just like alpha zero.

Pension fund managers and largebokers do the same, they bet the index. A Calpers manager is a quant in the sense the manager knows large trades alter the market structure. Large wealth funds and pensions funds have observable impact.
We have two types of betters. The small caps are actually disconnected from the large caps. Large caps form a value added chain, part of the Wall Street/ Treasury/Fed interaction. Small caps are trying to invest in some disconnected small valuable things, they just happen to use the same government regulated listing systems.

OK, then to finish up, why are large cap log?
Because you want to know where they fit in the finite value chain, log gives you trade size. A quick example. If my trade window is limited to eight bit trades, and some prices are in the range of 100, then I know large trades take up a bunch of local bandwidth, and happen rarely. The key assumption, value added chains can be treated as trade flow through a constricted channel. Back to the TV guy and the mess on the shop floor. large cap have to be treated as part of a shop floor.

Is this why economist now talk about the federal deficit as a percent of GDP rather than in dollars?

Is the methodology of the researchers valid? The paper is behind a paywall so I don't know.

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