MINYANVILLE ORIGINAL Ever since mathematics came to Wall Street, mainly in the 1980s, “quants” have been blamed for many things: dangerous innovations, overcomplexity, opacity, and using ivory-tower ideas to overrule common sense. Broadly speaking, there is truth to these accusations.

Most of the change in the financial system over the last 30 years has been driven by mathematical theory. Of course, we would have had plenty of financial disasters without mathematicians to help. They might have been less harmful than the disasters we had, or more harmful. But they definitely would have been different. I think the tremendous good wrought by the quantification of finance far outweighs any possible harm, but I do not deny that there was harm.

If I had to pick a mathematical error that was responsible for the global financial crisis, I’d name a different candidate. It was confusion among mathematicians and economists over the meaning of the word “capital” a quarter century ago that encouraged the creation of huge global institutions with business models guaranteed to blow up.

To understand how this happened, we have to go back to the early 1980s. The first “rocket scientists” arrived on Wall Street and began to trade using mathematical models. Unlike qualitative traders, we quants had a good idea of the probability distribution of our results. For example, suppose a trading system generated a +10% return on 51% of days, and a -10% return the other 49% of days. This is a toy example; real trading systems generate a range of potential returns and you have to worry about things like valuations, liquidity, and funding as well as accounting profit and loss. But the complexities do not change the basic mathematical point.

At first glance, this looks like a profitable trading system. On average it makes 0.2% per day which compounds to 68% per year. But despite having a positive expected return every day, this strategy is certain to blow up (in trader’s jargon, to “blow up” is to lose so much money you give back your accumulated profits and more, and have to stop trading).

The problem is something called “volatility drag.” Every time the strategy has a gain followed by a loss (or a loss followed by a gain) it loses 1%. A gain turns $1.00 into $1.10, a 10% loss from there means you end up with $0.99. A loss turns $1.00 into $0.90, a 10% gain from there means you end up with $0.99. The net loss from a gain and a loss is equal to the gain squared, 10% squared is 1% (10% is really 0.1, 0.1^{2} is 0.01 or 1%).

Suppose after 100 days you have the expected 51 gains and 49 losses. That means you have 49 pairs of gains and losses, each of which costs you 1%, reducing $1.00 to $0.61. You also have two unmatched 10% gains, which take your account up to $0.74. How can 100 trades, each with an expected profit of 0.2%, lead to the expectation of a 26% loss?

The answer is a bit subtle. After 100 days, you really do have an expected return of 1.002^100 – 1 = 0.22 or 22%. But all of your expected profit comes from 2.8% of the cases in which you have a return of 450% or better. This trading system is like a lottery ticket, with a small chance of a vast payout. The longer you run this system, the smaller the chance of a larger payout. After 10 years, for example, on average you will turn $1 into 1.002^2,600 = $180. But there is a 99.4% chance you will end up with less than $180, much less, only $0.92 on average. There is a 0.6% chance that you will strike it rich and turn your $1 into an average of $31,599.

It’s worse than that. The probability is negligible that you will survive ten years.

There is only one chance in 10,000 that you won’t have a drawdown of more than 90% at some point, and much smaller drawdowns than 90% will terminate your trading. Even the winning paths are very likely to have extreme drawdowns. Conditional on finishing the ten years above the $180 expectation, on average you will have a drawdown of 98% before you get there. Your unconditional average over all paths is to have a 99.9% drawdown.

Strategies with positive expected return less than their variance (or standard deviation squared) of return are guaranteed to blow up eventually. In practice, they produce tremendous profits for a period, but eventually they all fail. Their expected value is high, but volatility drag gets them all in the end. People who analyze them after-the-fact see boom-and-bust strategies, asset bubbles followed by crashes when the bubbles pop. But careful quantitative analysis reveals it is the tiny day-to-day losses that are toxic, not high volatility or greed or irresponsibility or any other macro-level characteristic that only appears in hindsight.

There is an easy way to solve the problem with this trading system. Give it five times the capital. Now it makes 2% on 51% of days and loses 2% on 49%. Its volatility drag is 2% squared, or 0.04% . That’s the same as the expected 0.04% daily profit, so the strategy will not blow up. The expected return is only 11% per year, instead of 68%, but it will survive to collect that 11%. It has a 95% chance of doing better than the ± 10% version over 10 years.

Running this strategy with any less capital means it will blow up. Running this strategy with any more capital is wasteful, it reduces expected return with no reduction in probability of blow up.

Much of the early success of quant traders was due to proper allocation of capital rather than finding exceptional edges.

It took a while, but qualitative traders started to notice. They understood the problem the quants had tackled, how aggressively to run a strategy in order to maximize profits, without blowing up. They were familiar with many stories of highly successful traders who mysteriously lost all their accumulated profits in one massive downswing (some traders managed to do this numerous times). But they also know of many more stories of traders with great day-to-day statistics who never managed to generate worthwhile profits.

Successful trading is like catching a wave in surfing. You have to hit the exponential risk level. Too little risk and the wave passes you by, too much risk and you wipe out. It was a welcome revelation to these qualitative traders that there was a mathematical solution to this problem. They no longer had to agonize over it.

Quantitative analysis of optimal risk levels for traders did not kill the financial system. But it set events in motion that led to the murder. That part of the story is in next week’s installment.

MINYANVILLE ORIGINALEver since mathematics came to Wall Street, mainly in the 1980s, “quants” have been blamed for many things: dangerous innovations, overcomplexity, opacity, and using ivory-tower ideas to overrule common sense. Broadly speaking, there is truth to these accusations.Most of the change in the financial system over the last 30 years has been driven by mathematical theory. Of course, we would have had plenty of financial disasters without mathematicians to help. They might have been less harmful than the disasters we had, or more harmful. But they definitely would have been different. I think the tremendous good wrought by the quantification of finance far outweighs any possible harm, but I do not deny that there was harm.

In recent years, more specific allegations have become popular. To cite some examples at random, Felix Salmon made a splash claiming the Gaussian Copula model was "the formula that killed Wall Street." Ian Stewart argued instead that the Black-Scholes model was “the mathematical justification for the trading that plunged the world's banks into catastrophe." Pablo Triana wrote a book blaming value-at-risk—

The Number That Killed Us: A Story of Modern Banking, Flawed Mathematics, and a Big Financial Crisis. Scott Patterson wrotetwobooks—The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed Itpointing the finger at systematic trading algorithms andDark Pools: High-Speed Traders, A.I. Bandits, and the Threat to the Global Financial Systemblaming computerized execution systems.If I had to pick a mathematical error that was responsible for the global financial crisis, I’d name a different candidate. It was confusion among mathematicians and economists over the meaning of the word “capital” a quarter century ago that encouraged the creation of huge global institutions with business models guaranteed to blow up.

To understand how this happened, we have to go back to the early 1980s. The first “rocket scientists” arrived on Wall Street and began to trade using mathematical models. Unlike qualitative traders, we quants had a good idea of the probability distribution of our results. For example, suppose a trading system generated a +10% return on 51% of days, and a -10% return the other 49% of days. This is a toy example; real trading systems generate a range of potential returns and you have to worry about things like valuations, liquidity, and funding as well as accounting profit and loss. But the complexities do not change the basic mathematical point.

At first glance, this looks like a profitable trading system. On average it makes 0.2% per day which compounds to 68% per year. But despite having a positive expected return every day, this strategy is certain to blow up (in trader’s jargon, to “blow up” is to lose so much money you give back your accumulated profits and more, and have to stop trading).

The problem is something called “volatility drag.” Every time the strategy has a gain followed by a loss (or a loss followed by a gain) it loses 1%. A gain turns $1.00 into $1.10, a 10% loss from there means you end up with $0.99. A loss turns $1.00 into $0.90, a 10% gain from there means you end up with $0.99. The net loss from a gain and a loss is equal to the gain squared, 10% squared is 1% (10% is really 0.1, 0.1

^{2}is 0.01 or 1%).Suppose after 100 days you have the expected 51 gains and 49 losses. That means you have 49 pairs of gains and losses, each of which costs you 1%, reducing $1.00 to $0.61. You also have two unmatched 10% gains, which take your account up to $0.74. How can 100 trades, each with an expected profit of 0.2%, lead to the expectation of a 26% loss?

The answer is a bit subtle. After 100 days, you really do have an expected return of 1.002^100 – 1 = 0.22 or 22%. But all of your expected profit comes from 2.8% of the cases in which you have a return of 450% or better. This trading system is like a lottery ticket, with a small chance of a vast payout. The longer you run this system, the smaller the chance of a larger payout. After 10 years, for example, on average you will turn $1 into 1.002^2,600 = $180. But there is a 99.4% chance you will end up with less than $180, much less, only $0.92 on average. There is a 0.6% chance that you will strike it rich and turn your $1 into an average of $31,599.

It’s worse than that. The probability is negligible that you will survive ten years.

There is only one chance in 10,000 that you won’t have a drawdown of more than 90% at some point, and much smaller drawdowns than 90% will terminate your trading. Even the winning paths are very likely to have extreme drawdowns. Conditional on finishing the ten years above the $180 expectation, on average you will have a drawdown of 98% before you get there. Your unconditional average over all paths is to have a 99.9% drawdown.

Strategies with positive expected return less than their variance (or standard deviation squared) of return are guaranteed to blow up eventually. In practice, they produce tremendous profits for a period, but eventually they all fail. Their expected value is high, but volatility drag gets them all in the end. People who analyze them after-the-fact see boom-and-bust strategies, asset bubbles followed by crashes when the bubbles pop. But careful quantitative analysis reveals it is the tiny day-to-day losses that are toxic, not high volatility or greed or irresponsibility or any other macro-level characteristic that only appears in hindsight.

There is an easy way to solve the problem with this trading system. Give it five times the capital. Now it makes 2% on 51% of days and loses 2% on 49%. Its volatility drag is 2% squared, or 0.04% . That’s the same as the expected 0.04% daily profit, so the strategy will not blow up. The expected return is only 11% per year, instead of 68%, but it will survive to collect that 11%. It has a 95% chance of doing better than the ± 10% version over 10 years.

Running this strategy with any less capital means it will blow up. Running this strategy with any more capital is wasteful, it reduces expected return with no reduction in probability of blow up.

Much of the early success of quant traders was due to proper allocation of capital rather than finding exceptional edges.

It took a while, but qualitative traders started to notice. They understood the problem the quants had tackled, how aggressively to run a strategy in order to maximize profits, without blowing up. They were familiar with many stories of highly successful traders who mysteriously lost all their accumulated profits in one massive downswing (some traders managed to do this numerous times). But they also know of many more stories of traders with great day-to-day statistics who never managed to generate worthwhile profits.

Successful trading is like catching a wave in surfing. You have to hit the exponential risk level. Too little risk and the wave passes you by, too much risk and you wipe out. It was a welcome revelation to these qualitative traders that there was a mathematical solution to this problem. They no longer had to agonize over it.

Quantitative analysis of optimal risk levels for traders did not kill the financial system. But it set events in motion that led to the murder. That part of the story is in next week’s installment.