Category Archives: Quantitative Finance

Quantitative Finance is my professional field. I write columns for a well-known periodical in the field called The Wilmott Magazine. Here are those columns and more.

Columns, Quantitative Finance, The Wilmott Magazine

Risks and Rewards

Everything in life comes at a cost — with a price tag seldom denominated in dollars and cents, and almost always hidden.

In our profession as quants and traders, we know we cannot accumulate if we don’t speculate (as P. G. Wodehouse puts it). So we accept and even welcome some of these price tags. We take certain risks, which we hope are calculated and understood, so that we can bring unto our employers what is theirs. These are good risks.

Bad risks are those we cannot understand and quantify, or measure and hedge against. They are bad because, even if we rake in some profits, we are never sure that they are commensurate with the downside we are throwing ourselves open to.
Market risk is a good risk. We know how to measure and model it, hedge against and reap rewards from it. We have smart people with bulging foreheads solving stochastic differential equations for us and simplifying the risk-reward equation.

Operational risk is a bad one. We can put as many software locks and control processes as we want around it. But we cannot prevent the rogue elements amongst us from sharing their passwords over a beer in some French brasserie. Worse, we have no idea what the rewards are when we expose ourselves to certain levels of operational risk. Heck, we don’t even know what the levels are because we have no means of quantifying it.

Incomplete appreciation of the risks involved in many situations is an almost philosophical factor that comes around to haunt us. It is not that we underestimate the risks; it is more like we are not aware of certain ramifications. The inconvenient warming of our home planet, for instance, is a consequence that the Wright brothers and Henry Ford simply could not have been aware of.

No such thing as a free lunch — the seemingly unlimited and practically free supply of nuclear energy has a not-so-hidden cost: the necessity to dispose of or securely store dangerous waste for, say, twenty thousand years. How do you store something for that long? After all, twenty thousand years ago, we were only barely human!

But the list of such boons and associated banes is endless. Think of the prosperity that a flattened world (using Thomas Friedman’s lingo) brought to emerging economies like India and China, which came at the expense of the cultural values that took thousands of years of careful nurturing.

A personal ramification of our high-powered corporate life is the alarming level of stress that we put ourselves through. Stress comes from market movements. As the sub-prime market tanked and heads started to roll, some of us had to worry about our heads. Fat bonuses of the first quarter usher in tax worries; lean bonuses indicate uncertain corporate future. Rogue traders burn billions and expose everybody to scrutiny and associated stresses. Even the lack of stress brings in some worries that the corporate world is perhaps passing us by!

When I first switched to the finance industry in late 2005, I happened to flip through an issue of the Bloomberg Market magazine. On of the first things struck me was that most of the advertisements seemed to be of expensive cars or alcohol. Is alcoholism the cost we readily dish out so that we can afford a gleaming dream machine?

Is stress a price worth paying for our corporate success? Are the risks worth their rewards?

Married to the Job — Till Death Do Us Part?

Stress is as much a part of our corporate careers as death is a fact of life. Still, it is best to keep the two (career and death) separate. This is the message that was lost on some hardworking young souls here in Singapore who literally worked themselves to death. So do a lot of Japanese, if we are to believe the media.

The reason for death in sedentary jobs is the insidious condition called deep vein thrombosis. This condition develops because of extended hours spent sitting, when a blood clot forms in the lower limbs. The clot then travels to the vital organs in the upper body, where it wreaks havoc including death.

The trick in avoiding such an untimely demise, of course, is not to sit for long. But that is easier said than done, when job pressure mounts, and deadlines loom.

Here is where you have to get your priorities straight. What do you value more? Quality of life or corporate success? The implication in this choice is that you can’t have both, as illustrated in the joke in investment banking that goes like: “If you can’t come in on Saturday, don’t bother coming in on Sunday!”

You can, however, make a compromise. It is possible to let go a little bit of career aspirations and improve the quality of life tremendously. This balancing act is not so simple though; nothing in life is.

Undermining work-life balance are a few factors. One is the materialistic culture we live in. It is hard to fight that trend. Second is a misguided notion that you can “make it” first, then sit back and enjoy life. That point in time when you are free from worldly worries rarely materializes. Thirdly, you may have a career-oriented partner. Even when you are ready to take a balanced approach, your partner may not be, thereby diminishing the value of putting it in practice.

These are factors you have to constantly battle against. And you can win the battle, with logic, discipline and determination. However, there is a fourth, much more sinister, factor, which is the myth that a successful career is an all-or-nothing proposition, as implied in the preceding investment banking joke. It is a myth (perhaps knowingly propagated by the bosses) that hangs over our corporate heads like the sword of Damocles.

Because of this myth, people end up working late, trying to make an impression. But an impression is made, not by the quantity of work, but by its quality. Turn in quality, impactful work, and you will be rewarded, regardless of how long it takes to accomplish it. Long hours, in my view, make the possibility of quality work remote.

Such melancholy long hours are best left to workaholics; they keep working because they cannot help it. It is not so much a career aspiration, but a force of habit coupled with a fear of social life.

To strike a work-life balance in today’s dog eat dog world, you may have to sacrifice a few upper rungs of the proverbial corporate ladder. Raging against the corporate machine with no regard to the consequences ultimately boils down to one simple realization — that making a living amounts to nothing if your life is lost in the process.

Spousal Indifference — Do We Give a Damn?

After a long day at work, you want to rest your exhausted mind; may be you want to gloat a bit about your little victories, or whine a bit about your little setbacks of the day. The ideal victim for this mental catharsis is your spouse. But the spouse, in today’s double income families, is also suffering from a tired mind at the end of the day.

The conversation between two tired minds usually lacks an essential ingredient — the listener. And a conversation without a listener is not much of a conversation at all. It is merely two monologues that will end up generating one more setback to whine about — spousal indifference.

Indifference is no small matter to scoff at. It is the opposite of love, if we are to believe Elie Weisel. So we do have to guard against indifference if we want to have a shot at happiness, for a loveless life is seldom a happy one.

“Where got time?” ask we Singaporeans, too busy to form a complete sentence. Ah… time! At the heart of all our worldly worries. We only have 24 hours of it in a day before tomorrow comes charging in, obliterating all our noble intensions of the day. And another cycle begins, another inexorable revolution of the big wheel, and the rat race goes on.

The trouble with the rat race is that, at the end of it, even if you win, you are still a rat!

How do we break this vicious cycle? We can start by listening rather than talking. Listening is not as easy as it sounds. We usually listen with a whole bunch of mental filters turned on, constantly judging and processing everything we hear. We label the incoming statements as important, useful, trivial, pathetic, etc. And we store them away with appropriate weights in our tired brain, ignoring one crucial fact — that the speaker’s labels may be, and often are, completely different.

Due to this potential mislabeling, what may be the most important victory or heartache of the day for your spouse or partner may accidentally get dragged and dropped into your mind’s recycle bin. Avoid this unintentional cruelty; turn off your filters and listen with your heart. As Wesley Snipes advises Woody Herrelson in White Men Can’t Jump, listen to her (or him, as the case may be.)

It pays to practice such an unbiased and unconditional listening style. It harmonizes your priorities with those your spouse and pulls you away from the abyss of spousal apathy. But it takes years of practice to develop the proper listening technique, and continued patience and deliberate effort to apply it.

“Where got time?” we may ask. Well, let’s make time, or make the best of what little time we got. Otherwise, when days add up to months and years, we may look back and wonder: Where is the life that we lost in living?

Stress and a Sense of Proportion

How can we manage stress, given that it is unavoidable in our corporate existence? Common tactics against stress include exercise, yoga, meditation, breathing techniques, reprioritizing family etc. To add to this list, I have my own secret weapons to battle stress that I would like to share with you. These weapons may be too potent; so use them with care.

One of my secret tactics is to develop a sense of proportion, harmless as it may sound. Proportion can be in terms of numbers. Let’s start with the number of individuals, for instance. Every morning, when we come to work, we see thousands of faces floating by, almost all going to their respective jobs. Take a moment to look at them — each with their own personal thoughts and cares, worries and stresses.

To each of them, the only real stress is their own. Once we know that, why would we hold our own stress any more important than anybody else’s? The appreciation of the sheer number of personal stresses all around us, if we stop to think about it, will put our worries in perspective.

Proportion in terms of our size also is something to ponder over. We occupy a tiny fraction of a large building that is our workplace. (Statistically speaking, the reader of this column is not likely to occupy a large corner office!) The building occupies a tiny fraction of the space that is our beloved city. All cities are so tiny that a dot on the world map is usually an overstatement of their size.

Our world, the earth, is a mere speck of dust a few miles from a fireball, if we think of the sun as a fireball of any conceivable size. The sun and its solar system are so tiny that if you were to put the picture of our galaxy as the wallpaper on your PC, they would be sharing a pixel with a few thousand local stars! And our galaxy — don’t get me started on that! We have countless billions of them. Our existence (with all our worries and stresses) is almost incomprehensibly small.

The insignificance of our existence is not limited to space; it extends to time as well. Time is tricky when it comes to a sense of proportion. Let’s think of the universe as 45 years old. How long do you think our existence is in that scale? Eight seconds if we are very lucky!

We are created out of star dust, last for a mere cosmological instant, and then turn back into star dust. DNA machines during this time, we run unknown genetic algorithms, which we mistake for our aspirations and achievements, or stresses and frustrations. Relax! Don’t worry, be happy!

Sure, you may get reprimanded if that report doesn’t go out tomorrow. Or, your trader may bite your head off if that pricing model is delayed again. Or, your colleague may send out that backstabbing email (and Bcc your boss) if you displease them. But, don’t you get it, in this mind-numbingly humongous universe, it doesn’t matter an iota. In the big scheme of things, your stress is not even static noise!

Arguments for maintaining a level of stress all hinge on an ill-conceived notion that stress aids productivity. It does not. The key to productivity is an attitude of joy at work. When you stop worrying about reprimands and backstabs and accolades, and start enjoying what you do, productivity just happens. I know it sounds a bit idealistic, but my most productive pieces of work happened that way. Enjoying what I do is an ideal I will shoot for any day.

Stress and Metaphysics

Realizing that our existence is a mere blink of an eye in time, and less than a speck of dust in space is a powerful way of cutting our stress to size. My favorite weapon, however, is even more potent. I ask myself a basic question — what are space and time to begin with?

These may sound like silly metaphysical musings that have no relevance to real life. But they have been the subject matter of many lifelong quests over the ages. If we, humanity as a whole, cannot stop pondering over such things, it is probably because they form the basis of our existence. Besides, our stress takes place in space and time.

Philosophical grand-standing aside, let’s get to the meat of the problem: What is space? Space seems to be closely associated with our sense of sight. It also forms the basis of our reality — everything happens in space and time. For this reason, “What are space and time?” is a question that cannot be reduced to simpler elements in our reality.

We can, however, approach the issue by posing a similar question “What is sound?” Sound is an experience associated with hearing, clearly. But what is it? The answer is hinted at in the age-old conundrum of a falling tree in a deserted forest. Does it make sound? A popular topic of conservation in cocktail parties, this question is also a serious contemplative inquiry for a Zen monk.

The knee-jerk response to the question is, yes, the tree does make sound. It’s just that there is nobody to hear it. But hear what exactly?

Sure, the falling tree creates air pressure waves. But, the waves are not sound. These waves create an electrical signal in the ear, if an ear is present. Electrical signals are electrical signals, not sound. These signals, when transported to the brain, induce neuronal firing, which is still not sound. It is a fallacy to think of sound as anything physical, anything real. Sound is an experience or a cognitive representation associated with the input signals (which are the pressure waves, we think. But are they?)

We can draw similar analogies between other sensations and the corresponding signals — taste and smell to chemical composition, for instance. What about sight? What is the “sensation” or the cognitive representation associated with sight? It is what we think of as space.

Of course, we think of space as real, as the basis of our reality. It takes more than this short column to shake our belief in it. That’s why I wrote my book — The Unreal Universe.

To me, the unreal nature of what we consider reality is more than a constant contemplation. It is a source of a Zen-like immunity against stress and other worldly worries.

Yes, stress is the cost exacted by the corporate chain of command. It is a cost most of us happily pay, for the rewards are abundantly clear. But we have to be aware of the risks associated with the rewards — both in accepting them and in declining them.

Columns, Quantitative Finance, The Wilmott Magazine

Quant Talent Management

The trouble with quants is that it is hard to keep them anchored to their moorings. Their talent is in high demand for a variety of reasons. The primary reason is the increasing sophistication of the banking clients, who demand increasingly more structured products with specific hedging and speculative motives. Servicing their demand calls for a small army of quants supporting the trading desks and systems.

Since structured products are a major profit engine on the trading floor of most banks, this demand represents a strong pull factor for quants from competing institutions. There is nothing much most financial institutions can do about this pull factor, except to pull them back in with offers they can’t refuse.

But we can try to eliminate the push factors that are hard to identify. These push factors are often hidden in the culture, ethics and the way things get done in institutions. They are, therefore, specific to the geographical location and the social settings where the banks operate.

Performance Appraisal — Who Needs It?

Performance appraisal is a tool for talent retention, if used wisely. But, if misused, it can become a push factor. Are there alternatives that will aid in retaining and promoting talent?

As it stands now, we go through this ordeal of performance appraisal at least once every year. Our career progression, bonus and salary depend on it. So we spend sleepless nights agonizing over it.

In addition to the appraisal, we also get our “key performance indicators” or KPIs for next year. These are the commandments we have to live by for the rest of the year. The whole experience of it is so unpleasant that we say to ourselves that life as an employee sucks.

The bosses fare hardly better though. They have to worry about their own appraisals by bigger bosses. On top of that, they have to craft the KPI commandments for us as well — a job pretty darned difficult to delegate. In all likelihood, they say to themselves that their life as a boss sucks!

Given that nobody is thrilled about the performance appraisal exercise, why do we do it? Who needs it?

The objective behind performance appraisal is noble. It strives to reward good performance and punish poor shows — the old carrot and stick management paradigm. This objective is easily met in a small organization without the need for a formal appraisal process. Small business owners know who to keep and who to sack. But in a big corporate body with thousands of employees, how do you design a fair and consistent compensation scheme?

The solution, of course, is to pay a small fortune to consultants who design appraisal forms and define a uniform process — too uniform, perhaps. Such verbose forms and inflexible processes come with inherent problems. One problem is that the focus shifts from the original objective (carrot and stick) to fairness and consistency (one-size-fits-all). Mind you, most bosses know who to reward and who to admonish. But the HR department wants the bosses to follow a uniform process, thereby increasing everybody’s workload.

Another, more insidious problem with this consultancy driven approach is that it is necessarily geared towards mediocrity. When you design an appraisal process to cater to everybody, the best you can hope to achieve is to improve the average performance level by a bit. Following such a process, the CERN scientist who invented the World Wide Web would have fared badly, for he did not concentrate on his KPIs and wasted all his time thinking about file transfers!

CERN is a place that consistently produces Nobel laureates. How does it do it? Certainly not by following processes that are designed to make incremental improvements at the average level. The trick is to be a center for excellence which attracts geniuses.

Of course, it is not fair to compare an average bank with CERN. But we have to realize that the verbose forms, which focus on averages and promote mediocrity, are a poor tool for innovation management, especially when we are trying to retain and encourage excellence in quant talent.

A viable alternative to standardized and regimented appraisal processes is to align employee objectives with those of the institutions and leave performance and reward management to bosses. With some luck, this approach may retain fringe geniuses and promote innovation. At the very least, it will alleviate some employee anxiety and sleepless nights.

To Know or Not To Know

One peculiar push factor in the Asian context is the lack of respect for technical knowledge. Technical knowledge is not always a good thing in the modern Asian workplace. Unless you are careful, others will take advantage of your expertise and dump their responsibilities on you. You may not mind it as long as they respect your expertise. But, they often hog the credit for your work and present their ability to evade work as people management skills.

People management is better rewarded than technical expertise. This differentiation between experts and middle-level managers in terms of rewards is a local Asian phenomenon. Here, those who present the work seem to get the credit for it, regardless of who actually performs it. We live in a place and time where articulation is often mistaken for accomplishments.

In the West, technical knowledge is more readily recognized than smooth presentations. You don’t have to look beyond Bill Gates to appreciate the heights to which technical expertise can take you in the West. Of course, Gates is more than an expert; he is a leader of great vision as well.

Leaders are different from people managers. Leaders provide inspiration and direction. They are sorely needed in all organizations, big and small.

Unlike people mangers, quants and technical experts are smart cookies. They can easily see that if they want to be people managers, they can get started with a tie and a good haircut. If the pickings are rich, why wouldn’t they?

This Asian differentiation between quants and managers, therefore, makes for a strong push factor for some quants who find it worthwhile to hide their technical skills, get that haircut, grab that tie, and become a people manager. Of course, it comes down to your personal choice between fulfilment and satisfaction originating from technical authority on the one hand, and convenience and promotions arising from people skills on the other.

I wonder whether we have already made our choices, even in our personal lives. We find fathers who cannot get the hang of changing diapers household chores. Is it likely that men cannot figure out washing machines and microwaves although they can operate complicated machinery at work? We also find ladies who cannot balance their accounts and estimate their spending. Is it really a mathematical impairment, or a matter of convenience? At times, the lack of knowledge is as potent a weapon as its abundance.

How Much is Talent Worth?

Banks deal in money. Our profession in finance teaches us that we can put a dollar value to everything in life. Talent retention is no different. After taking care of as much of the push factors as we can, the next question is fairly simple: How much does it take to retain talent?

My city-state of Singapore suffers from a special disadvantage when it comes to talent management. We need foreign talent. It is nothing to feel bad about. It is a statistical fact of life. For every top Singaporean in any field — be it finance, science, medicine, sports or whatever — we will find about 500 professionals of equal calibre in China and India. Not because we are 500 times less talented, just that they have 500 times more people.

Coupled with overwhelming statistical supremacy, certain countries have special superiority in their chosen or accidental specializations. We expect to find more hardware experts in China, more software gurus in India, more badminton players in Indonesia, more entrepreneurial spirit and managerial expertise in the west.

We need such experts, so we hire them. But how much should we pay them? That’s where economics comes in — demand and supply. We offer attractive expatriate packages that the talents would bite.

I was on an expatriate package when I came to Singapore as a foreign talent. It was a fairly generous package, but cleverly worded so that if I became a “local” talent, I would lose out quite a bit. I did become local a few years later, and my compensation diminished as a consequence. My talent did not change, just the label from “foreign” to “local.”

This experience made me think a bit about the value of talent and the value of labels. The local quant talents, too, are beginning to take note of the asymmetric compensation structure associated with labels. This asymmetry and the consequent erosion of loyalty introduce another push factor for the local quant talents, as if one was needed.

The solution to this problem is not a stricter enforcement of the confidentiality of salaries, but a more transparent compensation scheme free of anomalies that can be misconstrued as unfair practices. Otherwise, we may see an increasing number of Asian nationals using Singapore-based banks as a stepping stone to greener pastures. Worse, we may see (as indeed we do, these days) locals seeking level playing fields elsewhere.

We need to hire the much needed talent whatever it costs; but let’s not mistake labels for talent.

Handling Goodbyes

Losing talent is an inevitable part of managing it. What do you do when your key quant hands in the dreaded letter? It is your worst nightmare as a manager! Once the dust settles and the panic subsides, you should ask yourself, what next?

Because of all the pull and push factors discussed so far, quant staff retention is a challenge. New job offers are becoming increasingly more irresistible. At some stage, someone you work closely with — be it your staff, your boss or a fellow team member — is going to say goodbye. Handling resignations with tact and grace is no longer merely a desirable quality, but an essential corporate skill today.

We do have some general strategies to deal with resignations. The first step is to assess the motivation behind the career choice. Is it money? If so, a counter offer is usually successful. Counter offers (both making them and taking them) are considered ineffective and in poor taste. At least, executive search firms insist that they are. But then, they would say that, wouldn’t they?

If the motivation behind the resignation is the nature of the current or future job and its challenges, a lateral movement or reassignment (possibly combined with a counter offer) can be effective. If everything fails, then it is time to bid goodbye — amicably.

It is vitally important to maintain this amicability — a fact often lost on bosses and HR departments. Understandably so because, by the time the counter offer negotiations fail, there is enough bitterness on both sides to sour the relationship. Brush those wounded feelings aside and smile through your pain, for your paths may cross again. You may rehire the same person. Or, you may end up working with him/her on the other side. Salvage whatever little you can for the sake of positive networking.

The level of amicability depends on corporate culture. Some organizations are so cordial with deserting employees that they almost encourage desertion. Others treat the traitors as the army used to — with the help of a firing squad.

Both these extremes come with their associated perils. If you are too cordial, your employees may treat your organization as a stepping stone, concentrating on acquiring only transferable skills. On the other extreme, if you develop a reputation for severe exit barriers in an attempt to discourage potential traitors, you may also find it hard to recruit top talent.

The right approach lies somewhere in between, like most good things in life. It is a cultural choice that an organization has to make. But regardless of where the balance is found, resignation is here to stay, and people will change jobs. Change, as the much overused cliché puts it, is the only constant.

Summing Up…

In a global market that demands ever more customization and structuring, there is an unbearable amount of pull factor for good quants. Quant talent management (acquisition and retention) is almost as challenging as developing quant skills yourself.

While powerless against the pull factor, banks and financial institutions should look into eliminating hidden push factors. Develop respect and appreciation for hard-to-replace talents. Invent innovative performance measurement metrics. Introduce fair and transparent compensation schemes.

When it all fails and the talent you so long to retain leaves, handle it with tact and grace. At some point in the future, you may have to hire them. Or worse, you may want to get hired by them!

Columns, Quantitative Finance, The Wilmott Magazine

Benford and Your Taxes

Nothing is certain but death and taxes, they say. On the death front, we are making some inroads with all our medical marvels, at least in postponing it if not actually avoiding it. But when it comes to taxes, we have no defense other than a bit of creativity in our tax returns.

Let’s say Uncle Sam thinks you owe him $75k. In your honest opinion, the fair figure is about the $50k mark. So you comb through your tax deductible receipts. After countless hours of hard work, fyou bring the number down to, say, $65k. As a quant, you can estimate the probability of an IRS audit. And you can put a number (an expectation value in dollars) to the pain and suffering that can result from it.

Let’s suppose that you calculate the risk of a tax audit to be about 1% and decide that it is worth the risk to get creative in you deduction claims to the tune of $15k. You send in the tax return and sit tight, smug in the knowledge that the odds of your getting audited are fairly slim. You are in for a big surprise. You will get well and truly fooled by randomness, and IRS will almost certainly want to take a closer look at your tax return.

The calculated creativity in tax returns seldom pays off. Your calculations of expected pain and suffering are never consistent with the frequency with which IRS audits you. The probability of an audit is, in fact, much higher if you try to inflate your tax deductions. You can blame Benford for this skew in probability stacked against your favor.

Skepticism

Benford presented something very counter-intuitive in his article [1] in 1938. He asked the question: What is the distribution of the first digits in any numeric, real-life data? At first glance, the answer seems obvious. All digits should have the same probability. Why would there be a preference to any one digit in random data?

figure1
Figure 1. The frequency of occurrence of the first digits in the notional amounts of financial transactions. The purple curve is the predicted distribution. Note that the slight excesses at 1 and 5 above the purple curve are expected because people tend to choose nationals like 1/5/10/50/100 million. The excess at 8 is also expected because it is considered a lucky number in Asia.

Benford showed that the first digit in a “naturally occurring” number is much more likely to be 1 rather than any other digit. In fact, each digit has a specific probability of being in the first position. The digit 1 has the highest probability; the digit 2 is about 40% less likely to be in the first position and so on. The digit 9 has the lowest probability of all; it is about 6 times less likely to be in the first position.

When I first heard of this first digit phenomenon from a well-informed colleague, I thought it was weird. I would have naively expected to see roughly same frequency of occurrence for all digits from 1 to 9. So I collected large amount of financial data, about 65000 numbers (as many as Excel would permit), and looked at the first digit. I found Benford to be absolutely right, as shown in Figure 1.

The probability of the first digit is pretty far from uniform, as Figure 1 shows. The distribution is, in fact, logarithmic. The probability of any digit d is given by log(1 + 1 / d), which is the purple curve in Figure 1.

This skewed distribution is not an anomaly in the data that I happened to look at. It is the rule in any “naturally occurring” data. It is the Benford’s law. Benford collected a large number of naturally occurring data (including population, areas of rivers, physical constants, numbers from newspaper reports and so on) and showed that this empirical law is respected.

Simulation

As a quantitative developer, I tend to simulate things on a computer with the hope that I may be able to see patterns that will help me understand the problem. The first question to be settled in the simulation is to figure out what the probability distribution of a vague quantity like “naturally occurring numbers” would be. Once I have the distribution, I can generate numbers and look at the first digits to see their frequency of occurrence.

To a mathematician or a quant, there is nothing more natural that natural logarithm. So the first candidate distribution for naturally occurring numbers is something like RV exp(RV), where RV is a uniformly distributed random variable (between zero and ten). The rationale behind this choice is an assumption that the number of digits in naturally occurring numbers is uniformly distributed between zero and an upper limit.

Indeed, you can choose other, fancier distributions for naturally occurring numbers. I tried a couple of other candidate distributions using two uniformly distributed (between zero and ten) random variables RV1 and RV2: RV1 exp(RV2) and exp(RV1+RV2). All these distributions turn out to be good guesses for naturally occurring numbers, as illustrated in Figure 2.

figure2
Figure 2. The distribution of the first digits in the simulation of “naturally occurring” numbers, compared to the prediction.

The first digits of the numbers that I generated follow Benford’s law to an uncanny degree of accuracy. Why does this happen? One good thing about computer simulation is that you can dig deeper and look at intermediate results. For instance, in our first simulation with the distribution: RV exp(RV), we can ask the question: What are the values of RV for which we get a certain first digit? The answer is shown in Figure 3a. Note that the ranges in RV that give the first digit 1 are much larger than those that give 9. About six times larger, in fact, as expected. Notice how pattern repeats itself as the simulated natural numbers “roll over” from the first digit of 9 to 1 (as an odometer tripping).

figure3a
Figure 3a. The ranges in a uniformly distributed (between 0 and 10) random variable RV that result in different first digits in RV exp(RV). Note that the first digit of 1 occurs much more frequently than the rest, as expected.

A similar trend can be seen in our fancier simulation with two random variables. The regions in their joint distributions that give rise to various first digits in RV1 exp(RV2) are shown in Figure 3b. Notice the large swathes of deep blue (corresponding to the first digit of 1) and compare their area to the red swathes (for the first digit 9).

figure3b
Figure 3b. The regions in the joint distribution of two uniformly distributed (between 0 and 10) random variables RV1 and RV2 that result in different first digits in RV1 exp(RV2).

This exercise gives me the insight I was hoping to glean from the simulation. The reason for the preponderance of smaller digits in the first position is that the distribution of naturally occurring numbers is usually a tapering one; there is usually an upper limit to the numbers, and as you get closer to the upper limit, the probably density becomes smaller and smaller. As you pass the first digit of 9 and then roll over to 1, suddenly its range becomes much bigger.

While this explanation is satisfying, the surprising fact is that it doesn’t matter how the probability of natural distributions tapers off. It is almost like the central limit theorem. Of course, this little simulation is no rigorous proof. If you are looking for a rigorous proof, you can find it in Hill’s work [3].

Fraud Detection

Although our tax evasion troubles can be attributed to Benford, the first digit phenomenon was originally described in an article by Simon Newcomb [2] in the American Journal of Mathematics in 1881. It was rediscovered by Frank Benford in 1938, to whom all the glory (or the blame, depending on which side of the fence you find yourself) went. In fact, the real culprit behind our tax woes may have been Theodore Hill. He brought the obscure law to the limelight in a series of articles in the 1990s. He even presented a statistical proof [3] for the phenomenon.

In addition to causing our personal tax troubles, Benford’s law can play a crucial role in many other fraud and irregularity checks [4]. For instance, the first digit distribution in the accounting entries of a company may reveal bouts of creativity. Employee reimbursement claims, check amounts, salary figures, grocery prices — everything is subject to Benford’s law. It can even be used to detect market manipulations because the first digits of stock prices, for instance, are supposed to follow the Benford distribution. If they don’t, we have to be wary.

Moral

figure4
Figure 4. The joint distribution of the first and second digits in a simulation, showing correlation effects.

The moral of the story is simple: Don’t get creative in your tax returns. You will get caught. You might think that you can use this Benford distribution to generate a more realistic tax deduction pattern. But this job is harder than it sounds. Although I didn’t mention it, there is a correlation between the digits. The probability of the second digit being 2, for instance, depends on what the first digit is. Look at Figure 4, which shows the correlation structure in one of my simulations.

Besides, the IRS system is likely to be far more sophisticated. For instance, they could be using an advanced data mining or pattern recognition systems such as neural networks or support vector machines. Remember that IRS has labeled data (tax returns of those who unsuccessfully tried to cheat, and those of good citizens) and they can easily train classifier programs to catch budding tax evaders. If they are not using these sophisticated pattern recognition algorithms yet, trust me, they will, after seeing this article. When it comes to taxes, randomness will always fool you because it is stacked against you.

But seriously, Benford’s law is a tool that we have to be aware of. It may come to our aid in unexpected ways when we find ourselves doubting the authenticity of all kinds of numeric data. A check based on the law is easy to implement and hard to circumvent. It is simple and fairly universal. So, let’s not try to beat Benford; let’s join him instead.

References
[1] Benford, F. “The Law of Anomalous Numbers.” Proc. Amer. Phil. Soc. 78, 551-572, 1938.
[2] Newcomb, S. “Note on the Frequency of the Use of Digits in Natural Numbers.” Amer. J. Math. 4, 39-40, 1881.
[3] Hill, T. P. “A Statistical Derivation of the Significant-Digit Law.” Stat. Sci. 10, 354-363, 1996.
[4] Nigrini, M. “I’ve Got Your Number.” J. Accountancy 187, pp. 79-83, May 1999. http://www.aicpa.org/pubs/jofa/may1999/nigrini.htm.

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Columns, Quantitative Finance, The Wilmott Magazine

Quant Life in Singapore

Singapore is a tiny city-state. Despite its diminutive size, Singapore has considerable financial muscle. It has been rated the fourth most active foreign exchange trading hub, and a major wealth management center in Asia, with funds amounting to almost half a trillion dollars, according to the Monitory Authority of Singapore. This mighty financial clout has its origins in a particularly pro-business atmosphere, world class (well, better than world class, in fact) infrastructure, and the highly skilled, cosmopolitan workforce–all of which Singapore is rightfully proud of.

Among the highly skilled workforce are scattered a hundred or so typically timid and self-effacing souls with bulging foreheads and dreamy eyes behind thick glasses. They are the Singaporean quants, and this short article is their story.

Quants command enormous respect for their intellectual prowess and mathematical knowledge. With flattering epithets like “rocket scientists” or simply “the brain,” quants silently go about their jobs of validating pricing models, writing C++ programs and developing complicated spreadsheet solutions.

But knowledge is a tricky thing to have in Asia. If you are known for your expertise, it can backfire on you at times. Unless you are careful, others will take advantage of your expertise and dump their responsibilities on you. You may not mind it as long as they respect your expertise. But, they often hog the credit for your work and present their ability to evade work as people management skills. And people managers (who may not actually know much) do get better compensated. This paradox is a fact of quant life in Singapore. The admiration that quants enjoy does not always translate to riches here.

This disparity in compensation may be okay. Quants are not terribly interested in money for one logical reason–in order to make a lot of it, you have to work long hours. And if you work long hours, when do you get to spend the money? What does it profit a man to amass all the wealth in the world if he doesn’t have the time to spend it?

Besides, quants seem to play by a different set of rules. They are typically perfectionist by nature. At least, I am, when it comes to certain aspects of work. I remember once when I was writing my PhD thesis, I started the day at around nine in the morning and worked all the way past midnight with no break. No breakfast, lunch or dinner. I wasn’t doing ground-breaking research on that particular day, just trying to get a set of numbers (branching ratios, as they were called) and their associated errors consistent. Looking back at it now, I can see that one day of starvation was too steep a price to pay for the consistency.

Similar bouts of perfectionism might grip some of us from time to time, forcing us to invest inordinate amounts of work for incremental improvements, and propelling us to higher levels of glory. The frustrating thing from the quants’ perspective is when the glory gets hogged by a middle-level people manager. It does happen, time and again. The quants are then left with little more than their flattering epithets.

I’m not painting all people managers with the same unkindly stroke; not all of them have been seduced by the dark side of the force. But I know some of them who actively hone their ignorance as a weapon. They plead ignorance to pass their work on to other unsuspecting worker bees, including quants.

The best thing a quant can hope for is a fair compensation for his hard work. Money may not be important in and of itself, but what it says about you and your station in the corporate pecking order may be of interest. Empty epithets are cheap, but it when it comes to showing real appreciation, hard cash is what matters, especially in our line of work.

Besides, corporate appreciation breeds confidence and a sense of self-worth. I feel that confidence is lacking among Singaporean quants. Some of them are really among the cleverest people I have met. And I have traveled far and wide and met some very clever people indeed. (Once I was in a CERN elevator with two Nobel laureates, as I will never tire of mentioning.)

This lack of confidence, and not lack of expertise or intelligence, is the root cause behind the dearth of quality work coming out of Singapore. We seem to keep ourselves happy with fairly mundane and routine tasks of implementing models developed by superior intelligences and validating the results.

Why not take a chance and dare to be wrong? I do it all the time. For instance, I think that there is something wrong with a Basel II recipe and I am going to write an article about it. I have published a physics article in a well-respected physics journal implying, among other things, that Einstein himself may have been slightly off the mark! See for yourself at http://TheUnrealUniverse.com.

Asian quants are the ones closest to the Asian market. For structures and products specifically tailored to this market, how come we don’t develop our own pricing models? Why do we wait for the Mertons and Hulls of the world?

In our defense, may be some of the confident ones that do develop pricing models may move out of Asia. The CDO guru David Li is a case in point. But, on the whole, the intellectual contribution to modern quantitative finance looks disproportionately lopsided in favor of the West. This may change in the near future, when the brain banks in India and China open up and smell blood in this niche field of ours.

Another quality that is missing among us Singaporean parishioners is an appreciation of the big picture. Clichés like the “Big Picture” and the “Value Chain” have been overused by the afore-mentioned middle-level people managers on techies (a category of dubious distinction into which we quants also fall, to our constant chagrin) to devastating effect. Such phrases have rained terror on techies and quants and relegated them to demoralizing assignments with challenges far below their intellectual potential.

May be it is a sign of my underestimating the power of the dark side, but I feel that the big picture is something we have to pay attention to. Quants in Singapore seem to do what they are asked to do. They do it well, but they do it without questioning. We should be more aware of the implications of our work. If we recommend Monte Carlo as the pricing model for a certain option, will the risk oversight manager be in a pickle because his VaR report takes too long to run? If we suggest capping methods to renormalize divergent sensitivities of certain products due to discontinuities in their payoff functions, how will we affect the regulatory capital charges? Will our financial institute stay compliant? Quants may not be expected to know all these interconnected issues. But an awareness of such connections may add value (gasp, another managerial phrase!) to our office in the organization.

For all these reasons, we in Singapore end up importing talent. This practice opens up another can of polemic worms. Are they compensated a bit too fairly? Do we get blinded by their impressive labels, while losing sight of their real level of talent? How does the generous compensation scheme for the foreign talents affect the local talents?

But these issues may be transitory. The Indians and Chinese are waking up, not just in terms of their economies, but also by unleashing their tremendous talent pool in an increasingly globalizing labor market. They (or should I say we?) will force a rethinking of what we mean when we say talent. The trickle of talent we see now is only the tip of the iceberg. Here is an illustration of what is in store, from a BBC report citing the Royal Society of Chemistry.

China Test
National test set by Chinese education authorities for pre-entry students As shown in the figure, in square prism ABCD-A_1B_1C_1D_1,AB=AD=2, DC=2\sqrt(3), A1=\sqrt(3), AD\perp DC, AC\perp BD, and foot of perpendicular is E,

  1. Prove: BD\perp A_1C
  2. Determine the angle between the two planes A_1BD and BC_1D
  3. Determine the angle formed by lines AD and BC_1 which are in different planes.
UK Test
Diagnostic test set by an English university for first year students In diagram (not drawn to scale), angle ABC is a right angle, AB = 3m BC = 4m

  1. What is the length AC?
  2. What is the area of triangle ABC (above)?
  3. What is the tan of the angle ABC (above) as a fraction?

The end result of such demanding pre-selection criteria is beginning to show in the quality of the research papers coming out of the selected ones, both in China and India. This talent show is not limited to fundamental research; applied fields, including our niche of quantitative finance, are also getting a fair dose of this oriental medicine.

Singapore will only benefit from this regional infusion of talent. Our young nation has an equally young (professionally, that is) quant team. We will have to improve our skills and knowledge. And we will need to be more vocal and assertive before the world notices us and acknowledges us. We will get there. After all, we are from Singapore–an Asian tiger used to beating the odds.

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