Adding-up rules and Hockey Sticks
So I’m at the math program HCSSiM, teaching for three weeks in a “workshop,” which means I am responsible for teaching 12 teenagers the basic language and techniques of math- things like induction, proof by contradiction, the pigeon-hole principle, and how to correctly use phrases like “without loss of generality we can assume…” and “the following is a well-defined function…”, as well as familiarity with basic group theory, graph theory, number theory, cardinality, and fun things like Pascal’s triangle.
It’s really beautiful, classical math, and the students are eager and fantastically bright. They are my temporary brood, and I adore them and feed them chocolate at evening problem sets.
It’s also a fine opportunity to do some silly math doodling just for fun, the only rules being you can’t use a computer to look anything up until you’re done, and you can only use the stuff your kids at the program already learned. I’m going to describe what my mom and I, and then a junior (Amber Verser) and senior (Benji Fisher) staff member at the math program, figured out in the last couple of days. It’s super cool and turns out is at least 400 years old.
One of the most common examples of proof by induction is the formula for the sum of the counting numbers up to n:
1 + 2 + 3 + … + n = n(n+1)/2
And then, once you figure that out, you move on to the next case:
1^2 + 2^2 + 3^2 + … + n^2 = n(n+1)(2n+1)/6.
If you’re really into it, you can put the next case on the problem set:
1^3 + 2^3 + 3^3 + … + n^3 = (n(n+1)/2)^2.
Two obvious patterns are emerging when you add up successive dth powers up to n.
- It’s a polynomial of degree d+1, and
- The roots of the polynomial are symmetric about -1/2 (mom noticed this!).
How do you prove those two facts?
If you think it’s totally easy, stop reading now and give it a shot. There are about a million things you could try and none of them seem to work. I’ll wait.
…okay, let’s say you gave up, or already know, or don’t care. (Why are you reading still if you don’t care?!)
First let’s generalize the question to, if we add up values of some degree d polynomial for values i=0, 1, 2, …, n, then we want to prove the result is a degree d+1 polynomial in n. That this is equivalent to the first statement above is pretty easy to see by just re-arranging the terms of the double sum over i and over the terms of the polynomial in question. But it still seems like you need to know at least the answer to the question of what is a formula for 0^d + 1^d + 2^d + … + n^d, which is of course where we started.
But that’s where Pascal’s triangle comes in! We can generate Pascal’s triangle by the familiar “add up two consecutive numbers and put the answer below,” but we also can think of the element on the nth row and kth (tilted) column of Pascal’s triangle as the number of ways to choose k things from n things, which is referred to as “n choose k”, and where we start both the row and column counts at 0, not at 1. That definition satisfies the addition law because, if we have n things, we can label one as “special,” and then the choice of size k subsets of the n things divide into two categories: the size k subsets that contain the special guy and the ones that don’t. If they do, then we need only find k-1 other things in the remaining n-1 size set, and the number of ways to do that is given by the element on row n-1 and column k-1. If they don’t contain the special guy, we need to find k things in the remaining n-1 size set, and the number of ways to do that is given by the element on row n-1 and column k.
On the other hand, we also know a formula for the numbers in Pascal’s triangle: the guy on the nth row and kth column is given by a degree k polynomial in n, namely n!/k!(n-k)!. (This is because we can label all of the guys 1 through n, and just take the first k guys, and there are n! ways to label n things, but we don’t actually care about the order among the first k or among the last n-k.)
For example, in the second column, where we are looking at “n choose 2” for various n, we have the equation n(n-1)/2. This is a LOT like n^2 but has extra terms sticking on the end of lower order. When you’re looking at the third column, you’re working with the formula n(n-1)(n-2)/6, which is like the basic polynomial n^3 with extra stuff. In other words, the formula for “n choose k” is a degree k polynomial in n which we can think of as being a stand-in for n^k. Awesome.
The last ingredient is something called the “Hockey Stick Theorem,” which you gotta love just because of the name. It states that if we add up the values along a column, from the top of the rows down to the nth row, then the sum will be the number just below and to the right, and the entire picture will resemble a hockey stick.
The proof of the Hockey Stick Theorem is trivial- the answer is of course the sum of the two above it, and we have one in the sum already, but the other isn’t… but that other is the sum of the two above it, one of which is again already in the sum but the other isn’t… and you keep going until you get to the top edge of Pascal’s triangle, where the missing number is just 0.
Why does the Hockey Stick Theorem give us what we want? Going back to our generalized statement, we want to show the sum of values on a (any) degree d polynomial for i = 0, 1, 2, …, n is a degree d+1 polynomial. Well, use the dth column and make a hockey stick from the top to row n. Then the sum is on the (n+1)st row, in the (d+1)st column, which we know is a degree d+1 polynomial in n. Woohoo!
One way of looking at this is that we were actually asking the wrong question: instead of asking what the sum of the dth powers is we should have perhaps been asking what the sum of the dth column of Pascal’s triangle is; in other words, there is a better basis for the vector space of polynomials than x^d, namely “x choose d”. In fact, if there were an agreement in the world that actually the “x choose d” polynomials should be the standard basis, (by the way, these basis polynomials would be called “Pascalinomials”!) then the hockey stick theorem would be the last word on how do those things add up. As it stands, to figure out the actual formula for the sum of the dth powers for i=0, 1, 2, …, n, we need to write the first row of the change-of-basis matrix from one basis to the other.
As for the second question, we simply need to extend the definition of the sum F(n) of dth powers from 0 to n to the case where n is negative, by iteratively using the relation:
F(n) = F(n-1) + n^d, or
F(n-1) = F(n) – n^d.
Then we have F(0) = 0, F(-1) = 0, F(-2) = (-1)^(d+1), F(-3) = (-1)^(d+1) – (-2)^d = (-1)^(d+1)(1^d + 2^d) …, and it’s easy to prove that, for any n,
F(n) = (-1)^(d+1)F(-n-1).
This means that if we have a root at -1/2 + a, we also have a root at -1/2 – a = -(-1/2 +a) -1.
Does an academic job in math really suck?
My cousin recently sent me a link to this article about women in science. Actually it’s really about jobs in science, and how much they suck, and how women are too practical to want them. It’s definitely interesting- and pretty widely read, as well, although I’d never seen it. It makes a few excellent points, especially about the crappy amount of money and feedback one gets as an academic, two issues which were definitely part of my personal decision to leave my academic career.
I think his overall argument, though, is simultaneously too practical-minded and not practical-minded enough. And although his essay is about science, I’ll concentrate on how it relates to math.
It’s too practical in that it doesn’t really understand the attraction- the nearly carnal desire- people have to math. It essentially assumes that after some amount of time, maybe 20 years, people will lose interest in their subject, perhaps because they are getting poorly paid.
Is this really true? Maybe for some people this is true, but the nerds I know are nerds for life – they don’t wake up one day thinking math isn’t cool after all. And from what I know about people, they acclimate pretty thoroughly to their standard of living by the time they are 40.
It’s not practical enough, though, because it doesn’t get at one of the most important reasons women leave math, namely because they are married and maybe have kids and they simply can’t be that person who moves across the country for a visiting semester in Berkeley because their husband has a job already and it’s not in Berkeley.
[As a side note, if someone wants to actually encourage women in math, and they are loaded, I would encourage them to set up a fund that would pay costs for quality childcare and airplane tickets for kids when woman go to math conferences. You don’t even need to help organize the babysitting, just pay for it. It would help out a lot of young women and free them up to go to way more conferences, evening the playing field with young men.]
In fact there are plenty of women who are super nerdy and would love to go do math across the country, but when it comes to choosing between that lifestyle and having a family life, they will choose the family life more times than not. Really it’s the “nomadic monk” system itself that is crappy for women at that moment, even if they are theoretically happy to be a poor nerd for the rest of their lives.
I have another complaint (which will make it sound like I don’t like the essay but actually I do). It says that people in science don’t have the ability to switch careers, essentially because they don’t have the money. But that’s really not true, at least in math, and I’m a testament to the possibility of switching careers. One thing a nerd is really good at is learning new things quickly.
I also thought that there was something missing about the alternative jobs he mentions, in industry or otherwise, which is that, yes you do get paid better outside of academics, but on the other hand pretty much any nonacademic job requires you to have a boss, which can be really fine or really horrible, and restricts your vacation time to 3 or 4 weeks. By contrast the quality of life as an academic is, if not luxurious, at least much more under one’s control.
Glucose Prediction Model: absorption curves and dirty data
In this post I started visualizing some blood glucose data using python, and in this post my friend Daniel Krasner kindly rewrote my initial plots in R.
I am attempting to show how to follow the modeling techniques I discussed here in order to try to predict blood glucose levels. Although I listed a bunch of steps, I’m not going to be following them in exactly the order I wrote there, even though I tried to make them in more or less the order we should at least consider them.
For example, it says first to clean the data. However, until you decide a bit about what your model will be attempting to do, you don’t even know what dirty data really means or how to clean it. On the other hand, you don’t want to wait too long to figure something out about cleaning data. It’s kind of a craft rather than a science. I’m hoping that by explaining the steps the craft will become apparent. I’ll talk more about cleaning the data below.
Next, I suggested you choose in-sample and out-of-sample data sets. In this case I will use all of my data for my in-sample data since I happen to know it’s from last year (actually last spring) so I can always ask my friend to send me more recent data when my model is ready for testing. In general it’s a good idea to use at most two thirds of your data as in-sample; otherwise your out-of-sample test is not sufficiently meaningful (assuming you don’t have that much data, which always seems to be the case).
Next, I want to choose my predictive variables. First, we should try to see how much mileage we can get out of predicting future blood glucose levels with past glucose levels. Keeping in mind that the previous post had us using log levels instead of actual glucose levels, since then the distribution of levels is more normal, we will actually be trying to predict log glucose levels (log levels) knowing past log glucose levels.
One good stare at the data will tell us there’s probably more than one past data point that will be needed, since we see that there is pretty consistent moves upwards and downwards. In other words, there is autocorrelation in the log levels, which is to be expected, but we will want to look at the derivative of the log levels in the near past to predict the future log levels. The derivative can be computed by taking the difference of the most recent log level and the previous one to that.
Once we have the best model we can with just knowing past log levels, we will want to add reasonable other signals. The most obvious candidates are the insulin intakes and the carb intakes. These are presented as integer values with certain timestamps. Focusing on the insulin for now, if we know when the insulin is taken and how much, we should be able to model how much insulin has been absorbed into the blood stream at any given time, if we know what the insulin absorption curve looks like.
This leads to the question of, what does the insulin (rate of) absorption curve look like? I’ve heard that it’s pretty much bell-shaped, with a maximum at 1.5 hours from the time of intake; so it looks more or less like a normal distribution’s probability density function. It remains to guess what the maximum height should be, but it very likely depends linearly on the amount of insulin that was taken. We also need to guess at the standard deviation, although we have a pretty good head start knowing the 1.5 hours clue.
Next, the carb intakes will be similar to the insulin intake but trickier, since there is more than one type of carb and different types get absorbed at different rates, but are all absorbed by the bloodstream in a vaguely similar way, which is to say like a bell curve. We will have to be pretty careful to add the carb intake model, since probably the overall model will depend dramatically on our choices.
I’m getting ahead of myself, which is actually kind of good, because we want to make sure our hopeful path is somewhat clear and not too congested with unknowns. But let’s get back to the first step of modeling, which is just using past log glucose levels to predict the next glucose level (we will later try to expand the horizon of the model to predict glucose levels an hour from now).
Looking back at the data, we see gaps and we see crazy values sometimes. Moreover, we see crazy values more often near the gaps. This is probably due to the monitor crapping out near the end of its life and also near the beginning. Actually the weird values at the beginning are easy to take care of- since we are going to work causally, we will know there had been a gap and the data just restarted, so we we will know to ignore the values for a while (we will determine how long shortly) until we can trust the numbers. But it’s much trickier to deal with crazy values near the end of the monitor’s life, since, working causally, we won’t be able to look into the future and see that the monitor will die soon. This is a pretty serious dirty data problem, and the regression we plan to run may be overly affected by the crazy crapping-out monitor problems if we don’t figure out how to weed them out.
There are two things that may help. First, the monitor also has a data feed which is trying to measure the health of the monitor itself. If this monitor monitor is good, it may be exactly what we need to decide, “uh-oh the monitor is dying, stop trusting the data.” The second possible saving grace is that my friend also measured his blood glucose levels manually and inputted those numbers into the machine, which means we have a way to check the two sets of numbers against each other. Unfortunately he didn’t do this every five minutes (well actually that’s a good thing for him), and in particular during the night there were long gaps of time when we don’t have any manual measurements.
A final thought on modeling. We’ve mentioned three sources of signals, namely past blood glucose levels, insulin absorption forecasts, and carbohydrate absorption forecasts. There are a couple of other variables that are known to effect the blood glucose levels. Namely, the time of day and the amount of exercise that the person is doing. We won’t have access to exercise, but we do have access to timestamps. So it’s possible we can incorporate that data into the model as well, once we have some idea of how the glucose is effected by the time of day.
Cookies
About three months ago I started working at an internet company which hosts advertising platforms. It’s a great place to work, with a bunch of fantastically optimistic, smart people who care about their quality of life. I’m on the tech team along with the team of developers which is led by this super smart, cool guy who looks like Keanu Reeves from the Matrix.
I’ve learned a few things about how the internet works and how information is collected about people who are surfing the web, and the bottom line is I clear my cookies now after every session of browsing. Now that I know the ways information travels the risks of retaining cookies seem to outweigh the benefits. First I’ll explain how the system works and then I’ll try to make a case for why it’s creepy, and finally, why you may not care at all.
Basically you should think of yourself, when you surf the web, as analogous to someone on the subway coming home from Macy’s with those enormous red and white shopping bags. You are a walking advertisement for your past, your consumer tastes, and your style, not to mention your willingness to purchase. Moreover, beyond that, you are also carrying around information about your political beliefs, religious beliefs, and temperament. The longer you browse between cookie cleanings, the more precise a picture you’ve painted of yourself for the sites you visit and for third parties (explained below) who get their hands on your information.
Just to give you a flavor of what I’m talking about, you probably are already aware that when you go to a site like, say, Amazon, the site assigns you a cookie to recognize you as a guest; when you return a week later it knows you and says, “Hi, Catherine!”. That’s on the low end of creepy since you have an account with Amazon and it’s convenient for the site to not ask you who you are every time you visit.
However, you may not be aware that Amazon can also see and parce the cookies that other sites, like Google (correction: a reader has pointed out to me that Google doesn’t let this happen, sorry. I was getting confused between the cookie and the “referring url”, which tells a site where the user has come from when they first get to the site. That does contain Google search terms), places on your web signature. In other words Amazon, or any other site that knows how to look, can figure out what other sites’ label of you says. Some cookies are encrypted but not all of them, and I think the general rule is to not encrypt- after all, the people who have the tools to read the cookies all benefit from that information being easy to read. From the perspective of Google, moreover, this information is helping improve your user experience. It should be added that Google and many other companies give you the option of opting out of receiving cookies, but to do so you have to figure out it’s happening and then how to opt out (which isn’t hard).
One last layer of cookie collection is this: there are other companies which lurk on websites (like Amazon, although I’m not an expert on exactly when and where this happens) which can also see your cookies and tag you with additional cookies, or even change your existing cookies (this is considered rude but not prevented). This is where, for me, the creep factor gets going. Those third parties certainly have less riding on their brand, since of course you don’t even see them, so they have less motivation to act honorably with the information they collect about you. For the most part, though, they are just looking to see what kind of advertisement you may be weak for and, once they figure it out, they show you exactly that model of showerhead that you searched for three weeks ago but decided was too expensive to buy. If you want to stop seeing that freaking showerhead popping up everywhere, clear thy cookies.
Here’s why I don’t like this; it’s not about the ubiquitous showerhead, which is just annoying. Think about rich people and how they experience their lives. I touched on this in a previous post about working at D.E. Shaw, but to summarize, rich people think they are always right, and that’s a pretty universal rule, which is to say anyone who becomes rich will probably succumb to that pretty quickly. Why, though? My guess is that everyone around them is aware of their money and is always trying to make them happy in the hope that they at some point could have some of that money. So they effectively live in a cocoon of rightness, which after a while seems perfectly logical and normal.
How that concept manifests itself in this conversation about cookies is that, in a small but meaningful way, that’s exactly what happens to the user when he or she is browsing the web with lots of cookies. Every time Joe encounters a site, the site and all third-party advertisers have the ability to see that Joe is a Republican gun-owner, and the ads shown to Joe will be absolutely in line with that part of the world. Similarly the cookies could expose Dan as a liberal vegetarian and he sees ads that never shake his foundations. It’s like we are funneled into a smaller and smaller world and we see less and less that could challenge our assumptions. This is an isolating thought, and it’s really happening.
At the same time, people sometimes want to be coddled, and I’m one of those people. Sometimes I enjoy it when my favorite yarn store advertises absolutely gorgeous silk-cashmere blends at me, or shows me to a rant against greedy bankers, and no I’d rather not replace them with Viagra ads. So it’s also a question of how much does this matter. For me it matters, but I also like New York City because it is dirty and gritty and all these people from all over the world live there and sweat on each other on the subway and it makes me feel like part of a larger community- I like to mix it up and have it mixed up.
I’d also like to mention another kind of reason you may want to clear your cookies: you get better deals. A general rule of internet advertising is that you don’t need to show good deals to loyalists. So if you don’t have cookies proving you have an account on Netflix, you may get an advertisement offering you three free months of membership. Or if you want to get more free articles on the New York Times website, clear your cookies and the site will have no idea who you are. There are many such examples like this.
Lastly, I’d like to point out that you probably don’t need to worry about this. After all, many browsers will clear your cookies but also clear your usernames and passwords, and you may never be able to get some of those back. And maybe you don’t mind being coddled while online. Maybe it’s the one place where you get to feel understood. Why question that?
Fair Foods
This post will only be indirectly quantitative, and not a rant, so I guess that means I will have to either apologize or change my mission statement. Sorry. Oh and by the way I do have lots of ideas for quantitative blogs coming up, topics to include:
- clear your cookies! how internet companies track your every click
- update on the diabetes model
- is being a mathematician just a crappy job?
- shout-outs to other nerd bloggers who are sending me readers
So yesterday I loaded up the (rental) car to the brim, with my mom, my two older sons, a guitar (for me) and an air conditioning unit (for my mom), and drove out to Amherst for the math program I’m teaching in for three weeks.
Before I left I visited my friend Nancy at Fair Foods in Dorchester.
I drove to her house early, getting there at maybe 8:30am. She wasn’t home- she had me meet her at a church near Codman Square, where she was making a drop. When I got there I helped her unload a van full of maybe 40 or so boxes of vegetables and fruit, with a few 50-pound bags of carrots and potatoes. She got on the van and handed me the boxes and I carried them over to a sidewalk, while the woman, Marie, who was accepting the drop, carried some smaller boxes into the basement. Nancy introduced me to Marie as her daughter, and introduced Marie to me as the beautiful, wise Haitian woman who was a professional cook and would turn all of these vegetables into a delicious feast for her congregation. Nancy and Marie talked about the church, and the fact that it was shared between two different congregations, one Haitian immigrant and one African-American, and how the church was run.
After a while it didn’t seem like Marie was going to get the help she was expecting to carry the larger boxes into the basement, so Nancy and I moved all of the boxes down there, temporarily rigging a window to be a de facto dumb waiter to avoid three corners and some stairs. There were tomatoes, white potatoes, red potatoes, carrots, ugli fruit, limes, lettuce, string beans, wax beans, and others I can’t remember. Almost all of these were in great condition, but some needed sorting before going into the feast. Marie asked for corn for the 4th of July- since the food that is collected is surplus, a given request may be hard to fill, especially around a holiday, which Nancy explained. But then she said that if we got corn we would call Marie right away.
After we finished unloading the van I was soaked in sweat; it reminded me of how incredibly strong I’d gotten working one summer for Nancy, unloading trucks all day (as well as loading them at the Chelsea Produce Market every morning at 7) and driving around the city in the big yellow truck making drops to churches, senior centers, and youth centers, and holding dollar-a-bag sites in vacant parking lots and sidestreets. That was in 1992; and Nancy, who was born in 1950, has been doing the program ever since, with various peoples’ help.
Nancy mentioned that before I’d gotten there she had gone into the church and listened to the singing and the praying of the Haitian congregation, and that it had been seriously beautiful. Marie insisted on us coming inside. We sat in the pews as the woman leading the small prayer group of about 8 people, mostly women, was talking to one woman who was clearly in distress. Perhaps she was in mourning. They were speaking in Creole, which I don’t understand (although I know some French so every now and then I can pick up a word or two), but it was viscerally moving how kindly the leader was speaking to the sad woman seated in front of her. After she allowed that woman to finish, she looked up and welcomed us in English and asked us our names. Marie explained in Creole something about us, probably that we had just brought in the food for the July 4th meal, and we were instantly welcomed by the entire group. After that they told us they were wrapping up their prayer session and would stand and have a group prayer.
Everyone stood, except for the mourning woman who was holding her head in her hands. And at once everyone started praying, but the interesting thing was they were all saying different prayers, and it was fascinating to watch and listen to how they could be both praying together and praying individually. I could make out a few words from Marie’s prayer, which near the beginning was quiet and included lots of words like “please” and “hope”, but which, like everyone else’s, became louder and more fervent and contained more words like “thank you” and “hallelujah”. It ended by everyone holding their hands up to the front of them and giving thanks. Everyone ended at exactly the same time.
After the prayer group ended, there were lots of hugs and hand shaking. Many of the women wanted to talk to Nancy and she probably ended up hugging and being hugged by everyone there. There was a deep human connection inside that little church, which is pretty different from my normal assumptions about piousness and rules-based religions. Connection and empathy.
After we left the church we went to a playground and sat and had coffee together, and Nancy laid something down that was pretty thick. She talked about her disillusionment with her generation- the hippy generation- how they made all these promises but then didn’t follow through- the words she uses is didn’t apply themselves. She talked about having faith in her generation up to the “We Are the World” moment, and then waiting, and seeing nothing come out of it, and how bitter that had made her feel, how disappointed. She said it took her years to get over that, and now she feels like those years of her life, until recently in fact, are in some sense unaccounted for, both because she’s been sick and because she was somewhat paralyzed with anger.
She went on to say that she’s in a new phase now, she’s accepted the lazy fact of life that the people she was counting on, if anything, have made the world a worse place, not a better one, but that she’s decided to love them and love the world anyway, and to continue to make human connections with individuals, because it makes her have faith in a different way, a more diffuse but a stronger faith that won’t be disappointed.
It’s interesting to me that Nancy would ever describe her life as unaccounted for or her feelings as bitter. When I met her in 1989, she had been diagnosed with MS and lived in a huge old house with very little working anything (and what was working she’d installed herself- wired the electricity and installed plumbing). She had a great Dane and a broken-down donated truck, and when I came to her we spent the whole night cleaning out and reorganizing the truck. Whenever the truck’s insurance was due, or the phone was about to be cut off, we’d get a check for $50 and it would be a miracle, and I always felt like if I was ever going to believe in something it would be because of her.
I fell in love with her and with her approach to problem solving- namely, do the right thing, and go figure how to with bare knuckles and sweat. Over the years she’s been better or worse off with her health, but she’s never given up and, to be honest, I never sensed bitterness from her. Maybe these are relative notions, that bitterness from her is like frustration from someone else. Unaccountability from the woman who moves tons of food a week, that will otherwise be thrown away, into the homes of impoverished, mostly immigrant households, who know her and appreciate her act of kindness and take part in that act, would mean… what? to other people. Hard to say.
Did you have a happy childhood?
For whatever reason, I’ve been thinking about my childhood recently. Partly it’s the post I wrote about why I chose to call myself “mathbabe”, partly it’s an old essay of Jonathan Franzen’s that got me all riled up (in a good way). Plus I’m traveling to the math camp of my youth to teach, and stopping on the way in Harvard Square at my parents’ house; that’s enough to make you reconsider your memories in short order.
I have never understood what people mean when they talk about carefree, happy childhoods. I think I’ve always assumed this to be some kind of ironic joke, or maybe a plastered-over memory, a convenient approach to pain management. While it’s true that children have fewer responsibilities than grown-ups, it’s really not the responsibilities of adulthood that weigh me down (says the woman with three kids), or ever did. For me it was the constant awareness of my helplessness and impotence, my inability to decide my own fate, my feeling of having to wait forever for freedom, that got to me. I was also teased, but not relentlessly, and I did have friends, and moreover I wasn’t thought of (I don’t think) as a worrying child. From the outside people may have imagined me as a normal albeit nerdy kid. However, I always identified with the oppressed, and I had a keen sense of fairness which was constantly being challenged by reality. When we studied the “Manifest Destiny” in third grade, it killed me to think of the white man’s assumptions. When I saw a kid getting bullied at school, it tore me up that I didn’t know how to put an end to it and no teachers bothered. The list goes on, you get the idea. Also, I had an internal standard that was painfully high- I wanted to become a musician, a pianist, but never thought I’d be good enough, and I questioned my creativity, since what I really wanted to do was compose. When I decided to become a mathematician I started worrying about my thesis (I was 16). By the way, lest people get the wrong impression, my parents never put pressure on me to play music (in fact they openly discouraged me since it was expensive) and thought my worrying about my thesis was downright amusing. This was all internally generated. In short, I was a struggler, at best of times a striver, but never ever carefree and happy.
I have always been attracted to other people who struggle and strive; for the most part my closest friends are, like me, in constant flux with respect to their identities and their goals and even the interpretation of the most basic cultural assumptions like toenail polish and the role of the FDIC.
This brings me to the Franzen essay, where he talks about being isolated in childhood as a reader, and spending the rest of your life trying to find and form a community with other isolated readers. As an aside, Franzen makes a distinction in this essay between isolated readers and isolated math or technology nerds. He basically said that math nerds are isolated because they are autistic, incapable of social interaction, whereas readers are isolated because they feel more deeply and can’t relate to artificiality. I’m not sure whether to argue that math nerds aren’t all autistic or just count myself as both a reader and a math nerd and be proud of out-isolating Franzen, no easy task. Basically, I agree with Franzen. From my perspective upon meeting someone I am always looking for that inner torture, the hallmark of an examined life. It doesn’t make you happy, perhaps, but it makes you real, and moreover interesting.
But here’s the thing, I was blindsided this week by the discovery that my husband, of all people, had a happy childhood. He insists on this, even when I ask him if perhaps he’s misremembering his inner turmoil– he claims no. He moreover avers that, at the age of 12, he decided to become a mathematician and has never looked back, never once questioned that decision. Is this possible? That I’m married to a man who had a happy childhood? For all I know, it is true and moreover it may be exactly why I have a happy marriage. Maybe strugglers need to be married to non-strugglers to maintain some kind of balance. I don’t know, I’m still thinking about it. It does explain something that I’ve always been confused by, though- when my husband comes across an ethical or moral decision, he does so painlessly and makes a decision instantaneously. I now think this is because he just doesn’t think about things like that in between those moments, and so he’s got a clarity of consciousness which allows him to make snap decisions. When I come across such dilemmas, I am much more confused and ambivalent. I usually decide it’s a matter of opinion. I’m wondering if it’s this element of our differences that makes our marriage work.
Asymmetrical Information
From my experience, there are only a few basic kinds of trading models encountered on Wall Street. These are:
- chasing dumb money, which I’ve described already,
- asymmetrical information, which I want to talk about today,
- market-making,
- providing “insurance”,
- seasonality, which I’ve touched on, and
- taking advantage of macroeconomic misalignment (think Soros’s pound trade)
The concept of asymmetrical information is incredibly simple: I know more than you so I can make a more informed assessment of the value of some underlying contract. This could mean I know inside information about a company and trade before the announcement (illegal but common), or that I know the likelihood of bankruptcy for a company is higher than the market seems to think, or that the underlying mortgages of a packaged security are likely to default.
I could go on, and probably will in another post, but I’d like to make a very basic point, which is this: a lot of money is made every day via asymmetrical information, and in particular there’s a major motivation to obfuscate data in order to create asymmetry. One of the missions of this blog is to uncover and expose major, unreasonable examples of obfuscated information that I know about.
At this point it’s critical to differentiate between two things which typically get confused by non-nerds. Namely, the difference between a technical but thorough explanation and true information obfuscation. A technical explanation, if thorough, can be worked through eventually by someone with enough expertise, or someone who is motivated enough to get that expertise, whereas true information obfuscation just doesn’t provide enough details to really know anything.
The worst is when you are given pretty specific technical information, but which only explains half of the story. This leads to an imprecise false sense of security, which I suspect underlies most of the very large mistakes we’ve seen in finance in the last few years.
For example, let’s talk about the bank stress tests in the United States in 2009. They were conducted in two distinct phases. In the first, a bunch of economists were asked to write down two scenarios. The first was kind of a prediction of how 2009 and 2010 would play out, and the second was a more negative scenario. Okay so far, even though economists aren’t all that pessimistic as people (more on this on another post). The scenarios were averaged in some way and then publicly posted. The good news is, if you thought the scenarios were unrealistic, you’d at least know how to complain about them. The bad news is that they are pretty vague, only really specifying the GDP growth and the unemployment rate.
In the second phase, the banks were allowed to predict the impact of those two scenarios on their portfolios using their own internal models, which were not made public. Here’s the white paper if you don’t believe me. So, in the name of asymmetrical information, why is this a problem? Here are a few reasons:
- Banks had bad internal risk models
- Banks had clear motivation to mark their portfolios to their advantage
- The fact that their methods weren’t made public gives them ample cover to do whatever they wanted
There are two reasons I say that banks had bad internal risk models. The first reason is the one you know about already- they evidently bought a whole bunch of toxic securities leading up to 2008 and seemed to have no idea about the risks. But moreover, my personal experience working in the risk field is that banks used external risk modeling companies as a rubber stamp, essentially to placate those worrywarts who insisted on obsessing about risks. To be more precise without getting anyone into trouble, it was commonplace for banks to not notice when a model at a risk software company had very basic problems and would spit out nonsensical numbers. It was almost as if you couldn’t trust the banks to look at their risk numbers at all. This isn’t true of every bank at all times, but as a general rule when models had major problems it was hedge funds, not banks, who would bring attention to those problems. Moreover, the banks did not seem to have internal risk modeling across their desks. In other words, a trading desk which trades a certain kind of instrument may have some risk monitoring in place (mostly to bound the amount of trading of that type), but when it comes to understanding systemic risk across instrument types, the external risk companies were the source.
It is obvious that banks were motivated to mark their portfolios to their advantage. The ultimate result of bank stress tests were possible additional capital requirements, which they clearly wanted to avoid. This temptation meant it would benefit them to make every assumption of their risk model liberal to their cause.
Finally, they didn’t expose their methods- not even to explain in general terms how they dealt with, say, interest rate risks across instrument types. This meant that only the Fed people involved got to decide how honest the banks were. This is the opposite of what is needed in this situation. There is no reasonable need to keep these methodologies secret from the general public, since it is we who are on the hook if their methods are flawed, as we have seen.
Here’s where I admit that it’s actually really hard to come up with good methodologies to measure impact of vague GDP growth and unemployment estimates. But that admission is only going to add to my rant, because my overall point is that the instruments themselves have been created to make that hard. They are examples, especially tranched mortgage-backed securities but others as well, of intentional obfuscation for the sake of creating asymmetrical information. Instead of living in a world where banks who own things like this are allowed to measure them at their whim, and benefit from that obfuscation, we need to create a system where they are penalized for having illiquid or complex instruments.
And here’s where I admit that I’m not an expert on all of these instruments – some would say I don’t have the right to talk about how they should be assessed. Yet again, I choose to use that fact to add to my rant: if, after working for four years in finance as a quant at a hedge fund and then a researcher and account manager at a risk company, I can’t have an opinion about how to assess risk, then the system is too freaking complicated.
Women on a board of directors: let’s use Bayesian inference
I wanted to show how to perform a “women on the board of directors” analysis using Bayesian inference. What this means is that we need to form a “prior” on what we think the distribution of the answer could be, and then we update our prior with the data available. In this case we simplify the question we are trying to answer: given that we see a board with 3 women and 7 men (so 10 total), what is the fraction of women available for the board of directors in the general population? The reason we may want to answer this question is that then we can compare the answer to other available answers, derived other ways (say by looking at the makeup of upper level management) and see if there’s a bias.
In order to illustrate Bayesian techniques, I’ve simplified it further to be a discrete question. So I’ve pretended that there are only 11 answers you could possible have, namely that the fraction of available women (in the population of people qualified to be put on the board of directors) is 0%, 10%, 20%, …, 90%, or 100%.
Moreover, I’ve put the least judgmental prior on the situation, namely that there is an equal chance for any of these 11 possibilities. Thus the prior distribution is uniform:
We have absolutely no idea what the fraction of qualified women is.
The next step is to update our prior with the available data. In this case we have the data point that there a board with 3 women and 7 men. In this case we are sure that there are some women and some men available, so the updated probability of there being 0% women or 100% women should both be zero (and we will see that this is true). Moreover, we would expect to see that the most likely fraction will be 30%, and we will see that too. What Bayesian inference gives to us, though, is the relative probabilities of the other possibilities, based on the likelihood that one of them is true given the data. So for example if we are assuming for the moment that 70% of the qualified people are women, what is the likelihood that the board ends up being 3 women and 7 men? We can compute that as (0.70)^3*(0.30)^7. We multiply that by 1/11, the probability that 70% is the right answer (according to our prior) to get the “unscaled posterior distribution”, or the likelihoods of each possibility. Here’s a graph of these numbers when I do it for all 11 possibilities:
We learn the relative likelihoods of the outcome "3 out of 10" given the various ratios of women
In order to make this a probability distribution we need to make sure the total adds up to 1, so we scale to get the actual posterior distribution:
We scale these to add up to 1
What we observe is, for example, that it’s about twice as likely for 50% of women to be qualified as it is for 10% of women to be qualified, even though those answers are equally distant from the best guess of 30%. This kind of “confidence of error” is what Bayesian inference is good for. Also, keep in mind that if we had had a more informed prior the above graph would look different; for example we could use the above graph as a prior for the next time we come across a board of directors. In fact that’s exactly how this kind of inference is used: iteratively, as we travel forward through time collecting data. We typically want to start out with a prior that is pretty mild (like the uniform distribution above) so that we aren’t skewing the end results too much, and let the data speak for itself. In fact priors are typically of the form, “things should vary smoothly”; more on what that could possibly mean in a later post.
Here’s the python code I wrote to make these graphs:
Here’s the R code that Daniel Krasner wrote for these graphs:
Cora Sadosky
I was looking through an old photo album (the kind where there are sticky pages and actual physical photos- it looks like an ancient technology now) and I came across one of my favorites of all time- a picture of me being embraced and supported by Cora Sadosky on one side and Barry Mazur on the other. This picture was taken in 1993 in Vancouver, where I received the Alice T. Schafer prize. It was a critical moment for me, and both of those people have influenced me profoundly. Barry became my thesis advisor; part of the reason I went into number theory was to become his student (the other part was this book).
Cora became my mathematical role model and spiritual mother. I already wrote earlier about how going to math camp when I was 14 changed my life and made me realize there is a whole community of math nerds out there and that I belonged to that nerd community. Well, Cora, whom I met when I was 21, was the person that made me realize there is a community of women mathematicians, and that I was also welcome to that world.
Actually it was something I didn’t even really want to know at the time. After all, I was happy to be a successful math undergraduate at UC Berkeley, frolicking in the graduate student lounge and partaking in tea every day at 3:00. Who cares that I was a woman? It seemed antiquated to me, almost crude, to mention my gender. When I got word that I’d won the prize, my reaction was essentially, “is there money?” (there was a bit).
And when I meet young women in math nowadays with that attitude, I am happy for them, really very happy for them. To live in that state of not caring what your gender is in mathematics is a kind of bliss, that lasts until the very moment it stops. My greatest wish for future generations of women in math is for that bliss to never stop.
And yet. I went to Vancouver and met Cora and learned about Alice Shafer and her struggles and successes as a trailblazer for women in math, and I felt really honored to be collecting an award in her name. And I felt honored to have met Cora, whose obvious passion for mathematics was absolutely awe-inspiring. She was the person who first explained to me that, as women mathematicians, we will keep growing, keep writing, and keep getting better at math as we grow older (unlike men who typically do their best work when they’re 29), and we absolutely have to maintain a purpose and a drive and fortitude for that highest call, the struggle of creation.
I kept up with Cora over the years. Every now and then she’d write to me and send me pushy little maternal notes reminding me to work hard and stay strong and productive. And I’d write to her with news of my life and my growing family and sometimes when I visited D.C. I’d meet her and we’d have lunch or dinner and talk about ideas and great books we’d read and how much we loved each other.
When I googled her this morning, I found out she’d died about 6 months ago. You can read about her difficult and inspiring mathematical career in this biography. It made me cry and made me think about how much the world needs role models like Cora.
Woohoo!
First of all, I changed the theme of the blog, because I am getting really excellent comments from people but I thought it was too difficult to read the comments and to leave comments with the old theme. This way you can just click on the word “Go to comments” or “Leave a comment” which is a bit more self-evident to design-ignorant people like me. Hope you like it.
Next, I had a bad day today, but I’m very happy to report that something has raised my spirits. Namely, Jake Porway from Data Without Borders and I have been corresponding, and I’ve offered to talk to prospective NGO’s about data, what they should be collecting depending on what kind of studies they want to be able to perform, and how to store and revise data. It looks like it’s really going to happen!
In fact his exact words were: I will definitely reach out to you when we’re talking to NPOs / NGOs.
Oh, and by the way, he also says I can blog about our conversations together as well as my future conversations with those NGO’s (as long as they’re cool with it), which will be super interesting.
Oh, yeah. Can I get a WOOHOO?!?
Better risk modeling: motivating transparency
In a previous post, I wrote about what I see as the cowardice and small-mindedness of the U.S. government and in particular the regulators for not demanding daily portfolios of all large investors. Of course this goes for the governments in Europe as well, and especially right now. The Economist had a good article this past Friday which attempted to quantify the results of a Greek default, but there were major holes, especially in the realm of “who owns the CDS contracts on Greek bonds, and how many are there?”. This fear of the unknown is a root cause of the current political wrangling which will probably end in a postponement of resolving the Greek situation; the question is whether the borrowed time will be used properly or squandered.
It’s ridiculous that nobody knows where the risk lies, but as a friend of mine pointed out to me last week at lunch, it probably won’t be enough to demand the portfolios daily, even if you had the perfect quantitative risk model available to you to plug them into. Why? Because if “transparency” is what the regulators demand, then “transparency” is what they would get – in the form of obfuscated lawyered-up holding lists.
In other words, let’s say a bank has a huge pile of mortgage-backed securities of dubious value on their books, but doesn’t want to accept losses on them. If they knew they’d have to start giving their portfolio to the SEC daily instead of quarterly, it would change the rules of the game. They’d have to hide these holdings by pure obfuscation rather than short-term month- or quarter-end legal finagling. So for example, they could invest in company A, which invests in company B, which happens to have a bunch of mortgage-backed securities of dubious value, but which is too small to fall under the “daily reporting” rules. This is just an example but is probably an accurate portrayal of the kind of thing that would happen with enough lead time and enough lawyers.
What we actually want is to set up a system whereby banks and hedge funds are motivated to be transparent. Read this as: will lose money if they aren’t transparent, because that’s the only motivation that they respond to.
In some sense, as my friend reminded me, we don’t need to worry about hedge funds as much as about banks. This is because hedge funds do their trades through brokerages, which force margin calls on trades that they deem risky. In other words, they pay for their risk through margins on a trade-by-trade, daily basis. If you are thinking, “wait, what about LCTM? Isn’t that a hedge fund that got away with murder and almost blew up the system and didn’t seem to have large margins in place?” then the answer is, “yeah but brokers don’t get fooled (as much) by hedge funds anymore”. In other words, brokers, who are major players in the financial game, are the policemen of hedge funds.
There are two major limits to the above argument. Firstly, hedge funds purposefully use multiple brokers simultaneously so that nobody knows their entire book, so to the extent that risk of portfolio isn’t additive (it isn’t), this policing method isn’t complete. Secondly, it is only a local kind of risk issue- it doesn’t clarify risk given a catastrophic event (like a Greek default), but rather a more work-a-day “normal circumstances” market risk.
Even so, what about the banks? Are there any brokers measuring the risk of their activities and investments? Since the banks are the brokers, we have to look elsewhere… I guess that would have to be at the government, and the regulators themselves, maybe the FDIC… in any case, people decidedly not players in the financial game, not motivated by pay-off, and therefore not prone to delving into the asperger-inspiring details of complicated structured products to search out lies or liberal estimates.
The goal then is to create a new kind of market which allows insiders to bet on the validity of banks’ portfolios. You may be saying, “hey isn’t that just the stock price of the bank itself?”, and to answer that I’d refer you to this article which does a good job explaining how little information and power is actually being exercised by stockholders.
I will follow up this post with another more technical one where I will attempt to describe the new market and how it could (possibly, hopefully) function to motivate transparency of banks. But in the meantime, feel free to make suggestions!
Women on S&P500 boards of directors
This is a co-post with FogOfWar.
Here’s an interesting article about how many board of directors for S&P500 companies consist entirely of men. Turns out it’s 47. Well, but we’d expect there to be some number of boards (out of 500) which consist entirely of men even if half of the overall set of board members are women. So the natural question arises, what is the most likely actual proportion of women given this number 47 out of 500?
In fact we know that many people are on multiple boards but for the sake of this discussion let’s assume that there’s a line of board seekers standing outside waiting to get in, and that we will randomly distribute them to boards as they walk inside, and we are wondering how many of them are women given that we end up with 47 all-men boards out of 500. Also, let’s assume there are 8 slots per board, which is of course a guess but we can see how robust that guess is by changing it at the end.
By the way, I can think of two arguments as to why the simplification that nobody is on multiple boards argument might skew the results. On the one hand, we all know it’s an old boys network so there are a bunch of connections that a few select men enjoy which puts them on a bunch of boards, which probably means the average number of boards that a man is on, who is on at least one board, is pretty large. On the other hand, it’s also well-known that, in order to seem like you’re diverse and modern, companies are trying to get at least a token woman on their board, and for some reason consider the task of finding a qualified woman really difficult. Thus I imagine it’s quite likely that once a woman has been invited to be on a board, and she’s magically dubbed “qualified,” then approximately 200 other boards will immediately invite that same woman to be on their board (“Oh my god, they’ve actually found a qualified woman!”). In other words I imagine that the average number of boards a given woman is on, assuming she’s on one, is probably even higher than for men, so our simplifying assumptions will in the end be overestimating the number of women on boards. But this is just a guess.
Now that I’ve written that argument down, I realize another reason our calculation below will be overestimating women is this concept of tokenism- once a board has one woman they may think their job is done, so to speak, in the diversity department. I’m wishing I could really get my hands on the sizes and composition of each board and see how many of them have exactly one woman (and compare that to what you’d expect with random placement). This could potentially prove (in the sense of providing statistically significant evidence for) a culture of tokenism. If anyone reading this knows how to get their hands on that data, please write!
Now to the calculation. Assuming, once more, that each board member is on exactly one board and that there are 8 people (randomly distributed) per board, what is the most likely percentage of overall women given that we are seeing 47 all-male boards out of 500? This boils down to a biased coin problem (with the two sides labeld “F” and “M” for female and male) where we are looking for the bias. For each board we flip the coin 8 times and see how many “F”s we get and how many “M”s we get and that gives us our board.
First, what would the expected number of all-male boards be if the coin is unbiased? Since expectation is additive and we are modeling the boards as independent, we just need to figure out the probability that one board is all-male and multiply by 500. But for an unbiased coin that boils down to (1/2)^8 = 0.39%, so after multiplying by 500 we get 1.95, in other words we’d expect 2 all-male boards. So the numbers are definitely telling us that we should not be expecting 50% women. What is the most likely number of women then? In this case we work backwards: we know the answer is 47, so divide that by 500 to get 0.094, and now find the probability p of the biased coin landing on F so that all-maleness has probability 0.094. This is another way of saying that (1-p)^8 = 0.094, or that 1-p is 0.744, the eighth root of 0.094. So our best guess is p = 25.6%. Here’s a table with other numbers depending on the assumed size of the boards:
If anyone reading this has a good sense of the distribution of the size of boards for the S&P500, please write or comment, so I can improve our estimates.
Step 0 Revisited: Doing it in R
A nerd friend of mine kindly rewrote my python scripts in R and produced similar looking graphs. I downloaded R from here and one thing that’s cool is that once it’s installed, if you open an R source code (ending with “.R”), an R console pops up automatically and you can just start working. Here’s the code:
gdata <- read.csv('large_data_glucose.csv', header=TRUE)
#We can open a spreadsheet type editor to check out and edit the data: edit(gdata) #Since we are interested in the glucose sensor data, column 31, but the name is a bit awkward to deal with, a good thing to do is to change it: colnames(gdata)[31] <- "GSensor" #Lets plot the glucose sensor data: plot(gdata$GSensor, col="darkblue") #Here's a histogram plot: hist(gdata$GSensor, breaks=100, col="darkblue")
#and now lets plot the logarithm of the data: hist(log(gdata$GSensor), breaks=100, col="darkblue")
And here are the plots:
One thing my friend mentions is that R automatically skips missing values (whereas we had to deal with them directly in python). He also mentions that other things can be done in this situation, and to learn more we should check out this site.
R seems to be really good at this kind of thing, that is to say doing the first thing you can think about with data. I am wondering how it compares to python when you have to really start cleaning and processing the data before plotting. We shall see!
Working with Larry Summers (part 2)
This is the second part of a description of my experiences working at D.E. Shaw, which was started here and continues here.
I want to describe the culture of working at D.E. Shaw during the credit crisis, so from June 2007 to June 2009, because I think it’s emblematic of something that most news articles and books written about hedge funds really miss out on when they fixate on the average I.Q. of the people working there, which is in the end a distraction and nothing more, or the bizarre or quirky personalities that exist there, which is only idiosyncratic and doesn’t explain anything deeply.
I promised myself I’d put focus on the following phrase, which struck me down when I first heard it used and still makes me shake my head, namely the concept of “dumb money.” The phrase was tossed around constantly and cleverly, and really, to understand what it means inside the context of the hedge fund culture, is to understand the culture. So I’ll try to explain it. First a bit of context.
Most of the quants at D.E. Shaw were immigrant men. In fact I was the only woman quant when I joined, and there were quite a few quants, maybe 50, and I was also one of the only Americans. What nearly all these men had in common was a kind of constant, nervous hunger, almost like a daily fear that they wouldn’t have enough to eat. At first I thought of them as having a serious chip on their shoulder, like they were the kind of guy that didn’t make the football team in high school and were still trying to get over that. And I still think there’s an element of something as simple as that, but it goes deeper. One of my colleagues from Eastern Europe said to me once, “Cathy, my grandparents were coal miners. I don’t want my kids to be coal miners. I don’t want my grandchildren to be coal miners. I don’t want anybody in my family to ever be a coal miner again.” So, what, you’re going to amass enough money so that no descendent of yours ever needs to get a job? Something like that.
But here’s the thing, that fear was real to him. It was that earnest, heartfelt anxiety that convinced me that I was really different from these guys. The difference was that, firstly, they were acting as if a famine was imminent, and they’d need to scrounge up food or starve to death, and secondly, that only their nuclear family was worth saving. This is where I really lost them. I mean, I get the idea of acts of desperation to survive, but I don’t get how you choose who to save and who to let die. However, it was this kind of us-against-them mentality that prevailed and informed the approach to making money.
Once you understand the mentality, it’s easier to understand the “dumb money” phrase. It simply means, we are smarter than those idiots, let’s use our intelligence to anticipate dumb peoples’ trades and take their money. It is our right as intelligent, imminently starving people to do this. Chasing dumb money can take various forms, but is generally aimed at anticipating lazy fund managers: if you know that they always wait until Friday afternoon to balance their books, or that they wait until the end of the month, or that they are required to buy certain kinds of things, you can anticipate their trades, make them yourself a bit before they do, thereby forcing them to pay more, and getting a nice little profit for yourself. In short this works in general, since statistically speaking the anticipated trade wasn’t driving up the intrinsic value of the underlying, but rather was being affected by trade impact for a short amount of time. If we can anticipate big trades by lots of dumb money, then the short-term market impact will be large enough and last long enough to buy in beforehand and sell at the top, while it still lasts, assuming there’s sufficient liquidity. The subtext of taking dumb money, going back to the football team issue, is: if we don’t somebody else will, and then we will feel like fools for not doing it ourselves.
To tell you the truth, I was completely naive when I went to work there. I had kind of accepted the job because I wanted to be a business woman, wanted a brisk pace after the agonizing slowness of academics, and I had really no moral judgment on the concept of a hedge fund; I thought it was morally neutral, at worst a scavenger on the financial system, like a market maker or someone who provides insurance for something. Well I’ve decided it’s more like a leech.
Getting to the part about actually working with Larry Summers. I did work on a couple of his ideas, although in order not to get sued I can’t be detailed about what his ideas were. And I had various meetings with him and a bunch of managing directors. One thing I remember about these meetings was the eery way the managing directors seemed intimidated by him, even though behind his back they kind of scoffed at the possibility that he could actually offer good modeling ideas. It was basically a publicity stunt, or at least rumored to be, to have him work there. It was after he had gotten pushed out of the Presidency at Harvard for talking out of his ass about women in math, and yes it was a bit surreal to be the only woman quant in the place, and to be working on his project considering that. Since I am pretty much never intimidated for some reason, I had no problem. He kept on grilling me about various things to try and I kept explaining what I’d done and how I’d already thought of that. It was fine, pretty combative and pushy, but actually kind of fun. I really have nothing to say about him treating me differently because I was a woman.
But when I think about that last project I was working on, I still get kind of sick to my stomach. It was essentially, and I need to be vague here, a way of collecting dumb money from pension funds. There’s no real way to make that moral, or even morally neutral. There’s no way to see that as scavenging on the marketplace. Nope, that’s just plain chasing after dumb money, and I needed to quit. I still don’t know if that model went into production.
Did someone say regulation?
FogOfWar has kindly offered the background below on the OTC market and an analogy with the bond market, inspired by this recent article describing the latest round of watering-down of derivatives regulations. The bottomline for me is that whenever you see people using the phrases “needlessly tying up capital that would otherwise be used to create jobs and grow the economy,” “would damage America,” or especially an emphasis on “U.S. firms,” it probably means they are trying to engender a local nationalistic fervor to camouflage a very basic greedy instinct. Here’s the background:
OTC derivatives, by definition, are not traded on an open exchange, but are entered into between two parties in a private transaction. We can use JPMorgan and United Airlines as a running example. United has some risk it has that it wants to hedge. Or maybe some banker has convinced United that they should be hedging a risk that they didn’t know they had until the banker showed up to tell them about it.
For some simple things, United could just go to an exchange (a stock market, but not limited to stocks). So, for example, United could buy a future on oil prices to lock in its cost of oil over the next year. The problem is that there actually aren’t that many different contracts traded on exchanges, and the risks don’t usually fit neatly into the contracts that are there to buy. There’s a whole chapter on how to get a best-possible hedge in this situation in Derivatives 101 (and probably a whole class after that and people who make a living doing it in real life). So you could ‘dirty hedge’ (do an imperfect hedge), or you could go to JPMorgan and ask for an exact hedge.
JPMorgan is happy to give you the hedge and either delta hedge out the risk and/or match against offsetting risk they have on their books (or even use the opportunity to take a speculative position they were thinking about anyway). The key point is that JPMorgan quotes you a price, but it isn’t a price on a transparent market–it’s just whatever price they think you’ll pay. If United is smart, they’ll farm out the hedging for a bunch of bids from different banks and try to get the best price, but they’ll never actually know if they got ripped off or not, because they don’t see how much it actually costs JPMorgan to cover that risk internally.
There are some areas where the hedges are common enough and enough people offer them OTC that the profit margins are pretty low (simple interest rate swaps are a good example). However, there is also a lot of money to be made from ripping off dumb customers like United when they wander outside of these areas into other areas where they get crap pricing. This is how derivatives trading desks make their bonus.
And that’s why JPMorgan cares about this legislation. They want to keep ripping off the United Airlines of the world, and if the government makes United go to an actual exchange with open prices, there’ll be competition and the profit margin will shrink. Adding a margin requirement is a bit more wonky, but at the end JPMorgan doesn’t like it because it might drive United to an exchange and away from an OTC derivatives trade with JPMorgan.
It may go without saying, but Jamie Dimon, JPMorgan, GS, BofA, etc. do not give a shit whatsoever about United, Shell, Alcoa, or any other corporate. They just want the profits from their OTC derivatives trading desk to keep rolling in–profits that come off the backs of their customers–and they’ll say whatever garbage they think Congress and the Agencies will swallow to keep the trades rolling.
Felix Salmon wrote all this up a ways back when Barney Frank was caving to the investment banks and putting the end-user exception into Dodd-Frank to begin with. That was around the time my opinion of Barney Frank went from “rock star” to “big fat pussy”. The history (Salmon honed in on this) tells the story in the world of bond trading–what follows is a very general overview from memory:
Once upon a time, if a corporate wanted to buy bonds, they went to their investment bank. They didn’t see exchange-listed prices, and maybe they got a few quotes to try to get good pricing, but at the end of the day, much like the OTC derivatives market described above, they either had to take a price offered by a bank or not.
Then the government came in and said bonds should be traded on open exchanges (with bid and ask prices available for participants to see). The banks said it would destroy the market, they said the corporates would suffer, they said the markets would move overseas, they probably said it would “hurt America” to do this. All of exactly the same horse shit Jamie Dimon and the banks are saying now about moving derivatives to exchanges.
Well, bond trading got moved to exchanges and exactly none of the things the banks warned about actually happened. Instead, the thing all of the banks were secretly fearing did happen: customers got good execution at lower prices and bank profit margins in the bond business slowly collapsed over time to a fraction of what they were back in the OTC-bond days. Go figure.
Inspirational speech for women in math
I wanted to tell you about my experience a few months back sitting on a “non-academic career” panel at a women’s math conference at IPAM on the UCLA campus*.
The panel consisted of four women who have Ph.Ds in math and had left academics for other things. I was the only person from finance represented- the other women worked for government sponsored research agencies (NSA, Aerospace Corporation, and Los Alamos National Labs). In fact considering the fact that I was unemployed you could say I was a bad role model but I decided to attend anyway because I thought I could contribute- and after all, being unemployed is part of life, especially outside of academics. The audience consisted of about 50 women (and one man who never announced himself but I think worked for the NSF) who were either finishing up their Ph.Ds or had recently gotten them and were in post-docs – the perfect moment for a little inspirational speech. The panelists were given about 10 minutes each to talk about their experiences and then the audience had a chance to ask questions. Here’s (more or less) what I said to them.
Hi, I’m your unemployed role model. I thought of not coming here today since, after all, I’m unemployed, and what kind of role model does that make me? Actually it makes me a good one, and here’s why. That job I left wasn’t good enough for me. I didn’t get fired, I quit (although plenty of great people I know have been laid off so that’s no proof of anything). The truth is, I deserve a job that I really like, where I’m challenged to grow and to learn and to do my best and I’m rewarded for doing so. After all, I have a super power, which is mathematics. So the reason I’m saying this is that you do too. All of you have a superpower, which is mathematics. You all deserve to work at good jobs that you actually enjoy- and if the jobs you have turn out to be bad, or if the become bad for some reason, then quit! Get another one! Get a better one! I actually got a job offer on the plane over here yesterday (true!). I know I’m going to get a good job, even in this economy, because I can do something other people actually regard as magical. Mathematical training and thinking is something that everybody needs and not everybody can achieve, so remember that. Never feel stuck. This is not to say that the specific training you have right now is sellable on the open market, but since you’re a mathematician the one thing you can count on being good at is learning new stuff. So if you decide to change fields, get ready to roll up your sleeves and work your butt off to learn the necessary stuff, but be sure that you can do it and that it will be really important to the people you work for. And if it isn’t, or if you don’t think your work is being appreciated, go get a better job. Thanks!
By the way, I thought I should give you a wee bit of context for the above speech. It’s my opinion that one of the main reasons you don’t see as many women as men in great positions in the sciences is that, just around this point in their careers, women lack the confidence to sell themselves strongly in the job market. In fact the confidence problem exists earlier- I don’t know how many times I’ve seen (at MIT and then at Barnard/Columbia) a woman in my class crying over a grade of 94 (out of 100) in office hours while at the same time I talk to a man who got a 64 who says, “no problem I’ll ace the final”. However, once the grades stop and the field becomes more about confidence in yourself than in other peoples’ grades, women often flounder.
Just as a measurement of how much women need to hear this kind of stuff, I must have been approached by nearly all the women in the audience on a person-by-person basis, as well as emailed by quite a few thanking me for my inspirational words. Many of them expressed relief and astonishment that their accomplishments are actually worthy of a good job. These women need to hear this stuff- not to become arrogant but just to know that they really have something to offer. Also many of them are married or engaged and many of them have small kids, so they also need to prioritize themselves, which is again something they have trouble with.
And one more thing: I love encouraging young men in math too.
* This has been posted at my friend Chelsea’s blog already.
Step 0: Installing python and visualizing data
A friend of mine has type I diabetes, and lots of data (glucose levels every five minutes) from his monitor. We’ve talked on and off about how to model future (as in one hour hence) glucose levels, using information on the current level, insulin intake, and carb intake. He was kind enough to allow me to work on this project on this blog. It’s an exciting and potentially really useful project, and it will be great to use as an example for each step of the modeling process.
To be clear: I don’t know if I will be able to successfully model glucose levels (or even better be able to make suggestions for how much insulin or carbs to take in order to keep glucose levels within reasonable levels), but it’s exciting to try and it’s totally worth a try. I’m counting on you to give me suggestions if I’m being dumb and missing something!
I decided to use python to do my modeling, and I went to this awesomely useful page and followed the instructions to install python and matplotlib on my oldish mac book. It worked perfectly (thanks, nerd who wrote that page!).
The data file, which contains 3 months of data, is a csv (comma separated values) file, with the first line describing the name of the values in the lines below it:
Index,Date,Time,Timestamp,New Device Time,BG Reading (mg/dL),Linked BG Meter ID,Temp Basal Amount (U/h),Temp Basal Type,Temp Basal Duration (hh:mm:ss),Bolus Type,Bolus Volume Selected (U),Bolus Volume Delive\ red (U),Programmed Bolus Duration (hh:mm:ss),Prime Type,Prime Volume Delivered (U),Suspend,Rewind,BWZ Estimate (U),BWZ Target High BG (mg/dL),BWZ Target Low BG (mg/dL),BWZ Carb Ratio (grams),BWZ Insulin Sens\ itivity (mg/dL),BWZ Carb Input (grams),BWZ BG Input (mg/dL),BWZ Correction Estimate (U),BWZ Food Estimate (U),BWZ Active Insulin (U),Alarm,Sensor Calibration BG (mg/dL),Sensor Glucose (mg/dL),ISIG Value,Dail\ y Insulin Total (U),Raw-Type,Raw-Values,Raw-ID,Raw-Upload ID,Raw-Seq Num,Raw-Device Type 1,12/15/10,00:00:00,12/15/10 00:00:00,,,,,,,,,,,,,,,,,,,,,,,,,,,,,28.4,ResultDailyTotal,"AMOUNT=28.4, CONCENTRATION=null",5472682886,50184670,236,Paradigm 522 2,12/15/10,00:04:00,12/15/10 00:04:00,,,,,,,,,,,,,,,,,,,,,,,,,,,120,16.54,,GlucoseSensorData,"AMOUNT=120, ISIG=16.54, VCNTR=null, BACKFILL_INDICATOR=null",5472689886,50184670,4240,Paradigm 522 3,12/15/10,00:09:00,12/15/10 00:09:00,,,,,,,,,,,,,,,,,,,,,,,,,,,116,16.21,,GlucoseSensorData,"AMOUNT=116, ISIG=16.21, VCNTR=null, BACKFILL_INDICATOR=null",5472689885,50184670,4239,Paradigm 522
I made a new directory below my home directory for this file and for the python scripts to live, and I started up python from the command line inside that directory. Then I opened emacs (could have been TextEdit or any other editor you like) to write simple script to see my data.
A really easy way of importing this kind of file into python is to use a DictReader. DictReader is looking for a file formatted exactly as this file is, and it’s easy to use. I wrote this simple script to take a look at the values in the “Sensor Glucose” field (note there are sometimes gaps and I had to decide what to do in that case):
And this is the picture that popped out:
Taking a quick look at the Glucose levels
I don’t know how easy it is to see this but there are lots of gaps (when there’s a gap I plotted a dot at -1, and the line at -1 looks pretty thick). Moreover, it’s clear this data is being kept in a pretty tight range (probably good news for my friend). Another thing you might notice is that the data looks more likely to be in the lower half of the range than in the upper half. To get at this we will draw a histogram of the data, but this time we will *not* fill in gaps with a bunch of fake “-1″s since that would throw off the histogram. Here are the lines I added in the code:
And this is the histogram that resulted:
This is a pretty skewed, pretty long right-tailed distribution. Since we know the data is always positive (it’s measuring the presence of something in the blood stream), and since the distribution is skewed, this makes me consider using the log values instead of the actual values. This is because, as a rule of thumb, it’s better to use variables that are more or less normally distributed. To picture this I replace one line in my code:
skip_gaps_datalist.append(log(float(row["Sensor Glucose (mg/dL)"])))
And this is the new histogram:
This is definitely more normal.
Next time we will talk more about cleaning this data and what other data we will use for the model.
Data Without Borders
How freaking cool is this?! I signed up today and wrote to the founder, Jake Porway. He seems fantastic. I’m very excited about his project and how we (meaning you and me, kind reader) can help use our data scientist hats to help NGOs think about what data to collect and how to analyze it once they have it. Please consider signing up!
Guest post: Tax Repatriation Day
I’m delighted to have my first guest blogger!
“FogOfWar” (named after the documentary) is someone I’ve known for some time who comes from a mathy background, with a detour through accounting, tax & law winding up in banking (not as a quant). FOW & I have jammed finance policy many times and we tend to agree on a lot of things–I hope it will bring a “what really happens on the ground” perspective to thoughts about modeling as well as some useful insight into some of the technical rules (like accounting) that can matter a lot. Here’s his post:
The NYT ran an article on tax repatriation yesterday. Often, as someone in the industry, these articles can be infuriating for their lack of accuracy, misdirection or imprecision. In this case, however, my hat is off to the NYT for some damn fine traditional journalism. They’ve taken a fairly complicated issue (one I happen to know more than a little about), understood the core points in play and laid them out in an interesting, informative and readable article. Yes, it really is as bad as they make it out to be.
The “repatriation holiday” makes my vague-and-unofficial list of “10 worst tax ideas out there”. Unfortunately, every bad idea ultimately finds its way to Congress & this one is back for seconds. The NYT article lays out the case well, but here’s are two additional reasons on why this idea seems to have lasting appeal, which come in the form of catchy phrases:
“The money is trapped overseas”
We all know what “money”, “trapped” and “overseas” mean, and we can form an immediate idea of how this would be a bad thing, and how freeing that trapped money and bringing it back to the US would be good for the economy. Thus we get the inference: “The money is trapped overseas, and if we could bring it back it would create jobs.” Unfortunately, the second half of the second sentence is completely false. A more accurate sentence would be “The money is trapped overseas, and if we could bring it back corporations would pay slightly larger dividends this year, but not create any jobs or invest in any US plants that weren’t already in their strategic planning.” Doesn’t have quite the same ring to it…
“Structural subordination”
Not nearly as catchy as the first phrase, and uses two words for which most people don’t have a quick definition (at least not when paired together). Relevant, however, and a quick wonkish example with illustrate the thrust:
Let’s take a hypothetical US company, called (just to pick a name at random) “Lehman Holdings”. Lehman Holdings has assets claimed at $900 on its books and debt of $800. Lehman Holdings also owns 100% of another company, who we’ll call “Lehman UK”, which has assets claimed at $100 on its books and no debt. So, at first blush, one might think that Lehman has a 20% equity buffer: $1,000 of assets and $800 of debt (or a 4:1 debt ratio). This is nice easy math, which happens to be wrong in practice. The hidden assumption is that the people who loaned Lehman Holdings $800 can get access to all $1,000 of assets. They certainly can access the $900 of assets (or whatever they’re worth by the time bankruptcy hits), but the UK subsidiary is subject to UK bankruptcy rules, not US bankruptcy rules. Thus, when US creditors try to pull the $100 of assets out of the UK, they may find it’s more difficult than they anticipated (international bankruptcy gets sticky fast). Perhaps they could sell the stock of the subsidiary, but in real life that would involve untangling a whole host of interconnected contractual arrangements between Lehman Holdings and Lehman UK, which could take years. Not to mention the fact that to pull the $100 back, they’d have to pay ($35) in US taxes, so really there may be only $65 net to work with (other facts could zero out the tax bill). Probably in the end they can get the $100 of assets ($65 post taxes), but it can mean a significant time delay, and when you’re dealing with an imminent default, delay in action can translate to financial loss.
So, for all of these reasons, having $100 in a subsidiary isn’t worth quite the same thing as having $100 in the parent company. The fancy name for this is “structural subordination”, a term used by the credit rating agencies. So, if you’re a tech or pharma company with many billions of USD in your tax-shelter Irish/Dutch/Singapore subsidiaries, this can become a problem for your credit rating (which can impact your cost of borrowing). It’s probably not the primary reason for the lobbying efforts on tax repatriation, but it is definitely a factor, as the ($35) in tax is what’s preventing Holdings in the above example from pulling the $100 out of UK.
-FOW
Quantitative risk management
After the credit crisis hit we all realized that there’s a lot more risk out there than can be described by trailing volatility measures. Once I decided to leave the hedge fund world, I was thinking about working for the “other side,” namely to help quantify risk and/or work on the side of the regulators. I applied to the SEC, the New York Fed, and Riskmetrics, a software company which had a good reputation. I never heard from the Fed, and the SEC didn’t seem to have something for me, but I landed a job at Riskmetrics.
I figured it this way: if you work on a risk in a good way, if you make a better risk model, then you can at least argue you are improving the world. If you are instead making a bad risk model, and you know it, then you’re making the world a worse, riskier place. For example if you are working for a rating agency and get paid to ignore signs of riskiness, then that would be the not improving the world kind.
I really enjoyed my job, and after some months I was put in charge of “risk methodology,” which meant I got to think about how to quantify risk and why. I worked on our credit default model, which was super interesting, and I got to talk to the head trader of one of the biggest CDS trading desks regularly to understand the details of the market. In fact many of the biggest hedge funds and banks and pension funds send their portfolios daily to companies such as Riskmetrics to get overnight assessments of the riskiness of their portfolios. Bottomline is that my job kind of rocked, but it didn’t last forever; we were acquired soon after that by a company which didn’t offer me the same kind of position and I left pretty soon.
Here’s an article that very clearly articulates some of the problems in the field of quantitative risk. In my opinion it doesn’t go far enough with respect to their last point, or maybe it misses something, where they talk about “forecasting extreme risks.” This refers to the kind of thing that happens in a crisis, when all sorts of people are pulling out of the market at the same time and there are cascading, catastrophic losses.
What gets to me about this is that everyone talks about moments like these as if they can’t be modeled, but of course they can be, to a limited extent. Namely, although we don’t know what the next huge crisis will be, there are a few obvious candidates (like the Greek, Portuguese, Irish, or U.S. defaulting on their debt) which we should be keeping an eye on to the best of our quantitative abilities. Many of the “panic” situations (like the mortgage-backed securities debacle) were pretty obvious risks weeks or months in advance of their occurring, but people just didn’t know how to anticipate the consequences. That’s fine for a given individual trader but shouldn’t be true for the government.
I think the first step should be to compile a longish list of possible disaster scenarios (include the ones we’ve already seen happen) and decide what the probability of each scenario is- these probabilities can be updated each week by a crew of economists or what have you. Secondly and separately, set up a quantitative model which tries to capture the resulting cascade of consequences that each scenario would create; this would be complicated and involve things like guessing the losses at which hedge funds start liquidating their books, but should be aided by amassing huge amounts of information of the underlying portfolios of the largest institutions.
In my opinion the regulators have made a huge mistake in the past three years by _not_ insisting on getting the entire portfolio from every major hedge fund and bank every night (which from above we know is possible for them since they already send them to Riskmetrics-like software companies, although I’ve read articles where they claim this would be way too onerous a task) and, with that deep information, model the effect of a crisis scenario from our above list; how would it affect the bond market? The CDS market? The model which already exists at quantitative hedge funds now, which measures the impact and decay on trades, is a great start. Moreover, this model is not impossible to train (i.e. the actual coefficients inside the model’s formulas aren’t that hard to estimate), in fact it wouldn’t be that big a deal if we had as much data as I’m talking about. To me it’s unbelievable that we aren’t getting this portfolio information every day (or even intraday) and creating a “systemic impact model,” because it would clearly make us better prepared for future events (although not of course perfectly prepared) and no hedge fund or bank could argue that we shouldn’t be worried – it should be one of the costs of doing business on Wall Street.
mathbabe 
