The famous Monte Hall Problem poses the following question: “Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?”
While the intuitive answers seems to be “no,” as one might argue that the two remaining doors are equally likely to contain the car, the correct answer is actually “yes.” As vos Savant points out later on in the above link, the probability of winning if you switch is actually 2/3.
For instance, if we run the simulation 1,000 times, we see a fair amount of volatility in the results over the first 250 and even 500 trials. As we add more trials, however, the average of the results begin to converge to the true expected values: 2/3 chance of winning if we switch doors, and 1/3 chance if we don’t.
The results are even more concrete if we consider 10,000 trials:
If anyone ever tries to convince me that markets are rational, I will laugh politely and point to three telling events that have unfolded in the last month or so:
The increasingly absurd valuations of social media companies: Facebook ($104.7bn), Twitter ($24.2bn), Pinterest ($3.8bn), Snapchat ($3.0bn)
The three corresponding conclusions I draw from these stories are:
Markets are not only irrational, but also borderline stupid (for lack of better word) at times.
We have not learned much from the Financial Crisis of 2008. Finding novel ways of securitization without applying common sense can lead to potentially disastrous results.
We really have not learned much from the last technology/internet bubble. Markets should not be viewed as an archetype for the “wisdom of the crowd,” but rather as a broad manifestation of human psychology. In fact, human psychology is largely what drives cyclicality in the markets and the broader economy as a whole. This has been true throughout history, and will likely continue to hold true in the future.
Now, I could go into a whole rant on moral hazard and why I find the impending tech bubble so infuriating, but that’s best left for another day. Instead, I will turn my attention to point #3 and point out why current valuations of social media companies have gotten out of hand. In particular, I will focus on Twitter (TWTR), which has made much news in recent weeks for its IPO. Let’s start with a few facts:
November 7, 2013 – Twitter pops in the first day of trading, closing at $44.90 per share (a 73% increase from the IPO price).
One week later (and as of this blog post), shares have held steady ($43.98 per share) with a Market Cap of $24.0bn and Enterprise Value of $24.2bn.
Now, let’s pretend you know nothing about what Twitter does as a company. How would you justify such a dramatic increase in valuation over a mere two week span? Nothing material has occurred at the company to justify such a change. In fact, financial results were filed prior to the initial release of the IPO price range, and they were lukewarm at best. An argument could be made that the investment banks arranging the IPO purposely set a low price range in order to be conservative and generate pre-IPO buzz. Even if this is generally true for IPOs, a price jump of 73% in the first day of trading is highly uncommon and means that banks (and the company) have left a fair amount of money on the table. Typically, a 5-10% increase on the first day is the number that arrangers aim for, as it conveys a positive tone to the market while allowing the company to raise as much money as possible. Even if we can attribute all of the increase from the initial $17-$20 range to the final $26 price to marketing and syndication strategy, we still cannot explain the drastic 73% jump in valuation. So now what?
We know that the value of a company is equal to the discounted value of its expected future cash flows. An increase in valuation can be driven by either one or a combination of the following factors:
An increase in the amount of expected future cash flows
An acceleration of the timing of cash flows (time value of money)
An increase in the certainty of cash flows (i.e. a lower discount rate)
I would argue that in the last three weeks, none of these drivers has changed materially. Again, keep in mind that the company released financial results prior to setting its IPO price. Furthermore, considering that the Twitter’s current free cash flow position is negative, the company would have to make a quick and significant (re: miraculous) turnaround to justify its current $20bn+ valuation.
As some of you may recall, I kicked off the Fun with Excel series with a post on attraction, where I hoped to explore the mechanics of physical attraction from a statistical perspective. Due to the amount of feedback I have received (positive, skeptical, or otherwise), I have decided to write a follow-up post.
This couple is happy, but are you?
In the first part of The Laws of Attraction, I focused solely on physical attraction, and the impact of bias in our perception of attractiveness on seeking a compatible partner. In part two, I focus on the bigger picture: given a set of personal traits, what is the probability that you will find someone with those traits at the specific level that you desire?
Background: In the song One In A Million, Ne-Yo sings about a girl who he calls “one in a million.” Of course, not content with just enjoying the music, I wondered to myself what it actually meant to be “one in a million.” One way of measuring this is by breaking down attraction into a larger set of personality traits and trying quantify our desires, which is essentially what online dating services do with their “matching formulas.” For purposes of our exercise, let’s say you have a list of 10 distinct characteristics that you believe to be important and that you actively look for when searching for a partner. You might be more picky on some traits than others, but it isn’t too hard to quantify your objectives. Similar to my previous project, I quantify these objectives in terms of percentile, which, at least from a guy’s perspective, is pretty straightforward. For example, I might say, “I’m only interested in a girl who’s in the 80% percentile for Trait 1, 90th percentile for Trait 2, 50th percentile for Trait 3…” and so on and so forth. Now, the question is “what are the chances that such a girl exists?” A closely related question is “how many such girls are out there?”, followed by the not-so-fun reality-check of “what are the odds that I’ll actually find such a girl?”
The Model: While we won’t tackle the last question in this post, the first two are pretty straightforward to simulate from a mathematical standpoint. For each trait, the probability of finding someone who is at the X-percentile or higher of that trait is (100-X)%. For multiple traits, all we have to do is multiply these probabilities together, but the key assumption here is that all the traits are independent. Obviously, this isn’t true in real life, but we’ll revisit this point in a little while.
Assuming we start with a set of 10 traits, I will define a person having N “Perfect” Traits as someone who ranks at the 90th percentile or higher in N traits, and at the 50th percentile or higher in the remaining 10 minus N traits. Thus, assuming a world population of 7.12 billion, a male/female split of 50/50, and that you are heterosexual, the number of potential partners with 0 “Perfect” Traits walking on the planet is 3,476,563, or 1 in 1,024 (the mathematically inclined should immediately realize that 1,024 = 2^10). On the other hand of the spectrum, there are theoretically only 18 people with 9 “Perfect” Traits, or 1 in 200 million. Note that a person with 10 “Perfect” Traits technically doesn’t exist, as probability indicates a 1 in 10 billion chance. At this point, the astute reader will note one possible answer to Ne-Yo’s earlier problem: if you consider a smaller set of 6 traits rather than 10, a “one in a million” girl would simply be a girl who has 6 “Perfect” Traits (all 6 traits at the 90th percentile or higher) in that scenario.
The Results: I plotted the entire spectrum of N “Perfect” Traits in the scenario of 10 traits, to arrive at the following graph:
It should be no surprise that our graph strongly resembles a normal curve, as we are working with a binomial distribution.
I suppose the lesson here is that it doesn’t pay to be picky, but recall the very important (and incorrect) assumption we made earlier that all traits occur independently of one another. In the real world, however, this couldn’t be further from the truth. Creativity may be correlated with Curiosity, Honor may be correlated with Kindness, and Intelligence may be highly correlated with (or the cause of) all the other traits. Accounting for the dependencies between and among all 10 traits would require us to estimate both marginal and conditional probabilities, which would not only be difficult, but also complicate our model very quickly. Statistical mumbo jumbo aside, what this means is that the probabilities estimated by a simple binomial model are far too conservative (too low). This should be great news for all the picky daters out there.
An alternative way of tackling the dependent traits problem is to simply consider a smaller set of traits. For example, if we created a list of 10 traits, and then realized that two of them were very highly correlated with each other, then we could eliminate one of them and simply consider a 9 trait model, which in turn would be a more accurate simulation of what the actual probabilities might look like in real life. To that point, I also plotted out graphs for scenarios involving 7 traits and 5 traits:
Note that as we decrease the number of traits, the number of potential candidates increases exponentially. So if you only considered 5 main traits, and furthermore were only picky about 3 of them (3 “Perfect” Traits in the graph above), then you would only be looking at a probability of 1 in 400. Not bad.
Great Success!
Conclusion
At the end of the day, it is perhaps silly to attempt to model real life human dynamics with 50 lines in Excel. But that would also be missing the point of the exercise. Thinking about real world problems from a different perspective (whether it is psychologically, statistically, or otherwise) can shed new light on the issue, or simply affirm something we already knew or suspected. Even if it is only the latter, there is still value derived from being able to connect the dots between a variety of different frameworks.
My father was a big fan of the Chicago Bulls back in the ’80s and ’90s, so I had the good fortune of watching some of the best playoff basketball (i.e. Michael Jordan) that the NBA (and the world) has ever witnessed. Perhaps that is the same reason why the last decade or so of NBA basketball has seemed to pale noticeably in terms of excitement. It is generally agreed upon among basketball fans that the game as it is played today is (a lot) less physical (and perhaps less exciting) than it once was.
Officiating has also seemingly become a bigger determinant of results, and like virtually all professional team sports, the blame often lands on the referees. “If it weren’t for that call, they would have won the game,” is a phrase we hear all too often, and one that I am guilty of committing as well. However, have changes in officiating really been that significant over the last few decades, and if so, how would we measure such a phenomenon? The answer, of course, lies in the numbers.
Luckily, statistics for the NBA are readily available, but for the purposes of my project, I decided to look at playoff statistics from the 1983-84 season to the latest 2012-13 season. However, even if the data is easily accessible, oftentimes the most time-consuming aspect of a project is collecting the data and organizing it in a way that makes it easy to analyze. This was no exception. Luckily, with a little vlookup and text parsing (the latter is needlessly complex in Excel) magic, I was able to largely automate the process of converting 30 years of raw playoff data into something I could process more easily.
My first goal was to see if there were any high level trends in the NBA playoffs through time, in particular the number of games played and the point differential in each game. Moreover, I wanted to analyze these metrics by playoff round (e.g. first round (1R), conference semifinals (2R), conference finals (3R), and finals (4R)). If we were to believe that officiating actually had a measurable impact on playoff results, we may expect to find the following:
An overall longer playoff campaign
Smaller average point differentials, to convey the appearance of “closer” games
I’ll make a few observations, but the data really speaks for itself here. In the first chart, we see that after adjusting for the NBA’s change in playoff format since the 2002-03 campaign, both the average numbers of games played in the playoffs and the average number of games played per round has not shown any noticeable shift through time. The average points differential chart shows the same story, and in fact both charts seem to suggest some cyclical trends through time. Lastly, the average free throw attempts and fouls chart actually displays a noticeable decrease through time on a per game adjusted basis. Perhaps this is a testament to just how physical the game was back in the 1980s and 90s, which MJ himself has suggested on many an occasion.
Conclusion
The data doesn’t seem to indicate any obvious playoff trends that may have been caused by officiating. However, more granular foul data (which may not be available) may help clarify the story. In particular, even if the average number of fouls per game has trended down over the last 30 years, have the types of fouls called changed in any significant way? Perhaps more calls are coming during particularly tight stretches of games, or conversely, during blow outs, to ensure that the losing team is “still in it.” Of course, all of this is pure speculation, and without hard evidence, it is difficult to move forward. As Sir Arthur Conan Doyle once said through his most famous character Sherlock Holmes, “It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” Until such facts are found, our theories will remain theories.
Sometimes, one of the biggest challenges of starting a new career is learning the jargon that is essential to your everyday work. While this is a legitimate concern for those entering a highly technical or specialized field such as medicine or engineering, it is less of an issue for those working in finance and in particular investment banking, where the majority of the “technical” terms used are either self-explanatory and/or used almost exclusively to give the impression of sounding smarter than one actually is. While this is generally acceptable in the context of work-related discussions, there are certain terms that bankers love that absolutely drive me up the wall. Below, I list five of my most hated expressions in finance/banking, and why I think they should not be allowed. To make things more interesting, I also try to come up with an even more obnoxious (and usually entire nonsensical) version of each one.
“Buddy (bud)/pal”: for whatever reason, everyone that a banker knows who is not immediate family is either a “buddy” or “pal.”
Why it’s stupid: It makes you sound like you’re in 3rd grade. Seriously. Not convinced? Just add the modifier “best” in front of those terms, and you realize that you can’t describe your best friend without sounding like you’re 10 years old. What ever happened to just “friend” or “acquaintance?” SMH.
But even more annoying… “comrade/crony”: you can either sound like a Communist or a pirate, take your pick.
“Out of pocket”: for 99.99% of the world’s population, the term “out of pocket” refers to a type of expense that is paid by an individual rather than by a business or insurance company, e.g., “My dental plan only covered $20 of the $100 bill, the rest I paid out of pocket.” But for those in finance (and perhaps the corporate world in general), “out of pocket” has recently evolved to mean “out of the office” or “unavailable.”
Why it’s stupid: It makes no sense. Period. Unlike the general accounting meaning of the term, where “out of pocket” refers to literally paying expenses out of one’s own pocket, none of the words in this phrase have anything to do with being unavailable. Yes, I’m aware of the football term of a quarterback being “out of THE pocket,” (where the pocket is a defined area) but note the critical addition of the word “the.” If someone actually has a semi-logical explanation of how “out of pocket” = “out of the office,” I’d love to hear it. But until then, the phrase is officially on the top 10 of the dumbest things I’ve ever heard in my life.
But even more annoying… “under the shed”: no, it means absolutely nothing. Now you know how I feel.
“Circle back“: it’s often used in lieu of “revisit” or “come back to,” as in, “I’d love to keep talking about this, but I have another call right now. Let’s circle back in an hour.”
Why it’s stupid: It’s unnecessary and misleading. Why are you moving in a circle in the first place? Can’t you come back to an item in a straight forward manner (like what people actually do in real life)? “Circle back” seems to imply that a lot of work is being done along the way (where no work is required), so I guess it’s no surprise that it is a banking favorite.
But even more annoying… “take the roundabout to”: continuing with our circular theme and combining it with my least favorite traffic intersection leads us to this gem. Again, it makes no sense…its only purpose is to (1) annoy and (2) point out how stupid the first phrase is.
“Touch base”: to make contact with/get in touch with someone.
Why it’s stupid: Okay, so this phrase gets props for having an etymology that actually makes sense. In case you haven’t figured it out yet, the term comes from baseball, where a runner is required to touch the base in order to make the run legal. Now, aside from the fact that I dislike baseball (perhaps a story for another day), the reason why this phrase is stupid is mainly because of how it is used in the modern day. You see, the only appropriate time to use an (annoying) phrase like this is either at the start of a conversation or at the end of one. However, if you use it at the beginning (“Hi Bob, I just wanted to call to touch base on…”), it becomes completely redundant. Yes, the fact that you’re calling me makes it clear that you’re indeed contacting me regarding something. On the other hand, using the phrase at the end of the conversation (“Ok Bob, let’s touch base on this later.”) is almost always too vague. Yes, I know we need to talk again, but when, and how? Touching base gives me absolutely no idea how the follow-up conversation is going to happen.
But even more annoying… “hit helmets”: see, not all sports analogies work.
“Color”: no, no, no, we’re not talking about “hue” (you know, the one that involves your eyes and light bouncing off stuff), or someone’s character (“he showed his true colors”), or even the hypothetical property of quarks. In fact, Merriam-Webster has 15 distinct definitions of the word “color,” NONE of which fit with how the term is actually used in finance. So what does it actually mean? “Details, or information.” Confused? Well maybe an example will help. If a research analyst asks the CEO of a company, “Can you provide some more color on last quarter’s capex numbers?” he is basically asking if the CEO can provide more information on the company’s capital expenditures.
Why it’s stupid: I don’t know who first started using this term, but if I ever find out, I will promptly go back in time and punch him in the face. How you could take such a basic word from the English language and turn it into something completely nonsensical is beyond me, but I swear if I had a dollar for every time I heard someone throw that word around in an attempt to sound smart, I would be able to retire and live the rest of my life in comfort.
But even more annoying… “gradience”: how do you beat out the most annoying word of all time? Come up with something that (literally) means the same thing but sounds 100 times more pretentious. The best part? Gradience isn’t even a real word, since gradient is already a noun. But hey, doesn’t it sound so much cooler and intellectual?
Can you think of more words that coworkers use that drive you crazy (cough, cough, *synergistically*)? Post them below!
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