Einstein meets Markowitz: Relativity Theory of Risk-Return

When working with Gaussian processes, the observation that you can interpret covariance geometrically comes in very handy (see Visualizing Market Risk: A Physicist’s Perspective). Of course, when you’re given a new toy, you’ll want to take it apart. In this post, we’ll extend the analogy

to

The basic idea is to recall that Gaussian processes form a vector space. For instance, given Gaussian processes , the linear combination

is also a Gaussian process. Gaussian distributions are particularly nice because everything you can know about them is encoded in the two parameters (mean of the process) and (standard deviation of the process).

Each of the differentials may be expressed as

where is a standard Brownian motion with and .

We can interpret the as spanning a cotangent space of some risk manifold.

If you have such processes, the dimension of the space they span depends on the rank of the covariance matrix

The matrix is symmetric and positive semi-definite. However, if the rank of is , we can find uncorrelated differentials and construct a covariance matrix

This matrix is symmetric and positive definite. Therefore, on the space spanned by the , we can think of the covariance matrix as a metric tensor.

Now, we can re-express any Gaussian process via

Comparing this with the previous expression we see that

Furthermore,

which is just the familiar expression for the variance of the sum of uncorrelated processes.

The neat thing comes when you bring back into the picture.

Since we can interpret the covariance matrix as a metric tensor on a space, we can extend this to a Lorentzian metric on spacetime by specifying

and

With this extension of the metric, we have

Therefore at each point of our manifold, we have a risk-return cone in analogy to the light cone of special relativity with a velocity-like value given by

Note that is the radius of the risk-return cone at the particular security.

If you have two processes and , we can also look at their difference

Then the relative process

gives rise to a relative risk-return cone

Note that is the radius of the relative risk-return cone.

The relative risk-return cone has a velocity-like value given by

Now, this applies to finance by letting be the log of the price of some security and letting be the log of the price of some benchmark. With this financial interpretation, the process is the return of the security and is the excess return.

The “velocity” of the security

is known as the Sharpe ratio and the “relative velocity” of the relative security

is known as the information ratio. The radius is the tracking error.

To further the analogy, you could define an absolute velocity for which all other light cones are compared. This absolute light cone is closely related to an investor’s risk aversion.

Daily S&P 500 Value-at-Risk Estimates

A few people have commented about the methodology used to produce the charts in my last post. Keep in mind, I threw those together quickly for Felix based on charts already put together for a seminar at UCLA. If you want to see what I actually look at on a regular basis, I put the following chart together:

This is the 99%, 1-day VaR using a weighting scheme that places more weight on the most recent data.

Again, note the divergence between the two charts in recent months. Risk systems (like most third party vendors) based on normal distributions are likely indicating that risk continues to decrease. However, the stable distribution indicates the opposite, i.e. risk has begun increasing again.

80 Years of Daily S&P 500 Value-at-Risk Estimates

For Felix:

And the last 10 years…

Update: Felix has updated his post with a link to my charts above, but makes some comments that I thought I should address.

If we use a shorter horizon that better captures what is going on at this moment, we see that risk plateaued in June and has actually ticked up significantly in the past several weeks as measured via the “stable” distribution. On the other hand, volatility has actually decreased during the past several weeks. What this means, i.e. the discrepancy between “stable” and “normal” is that the tails have become fatter recently.

Also keep in mind that although risk appears to have decreased since the beginning of the year, it is still at extremely high levels. We would have to go back to 1934 to see comparable risk levels, so it is no time to become complacent.

Barclays quants error on leveraged ETFs

In a recent article, Cheng and Madhaven from Barclays Global Investors published a good article on leveraged ETFs

The Dynamics of Leveraged and Inverse Exchange-Traded Funds
April 8, 2009

Check it out.

The Error

They begin from a fairly standard starting point

However, they proceed to state that since

“holds for any period”, then it follows that

where is the ETF NAV and is the leverage factor.

Unfortunately, that is not correct. The problem is that

only holds when is 1 day. Otherwise, we could let be 1 year and this would say that the 1-year return of the ETF is times the 1-year return of the index, which we already know is not true.

This should have also been obvious by plugging into their final expression

which violates the relation defining leveraged ETFs they started with. As a result of this error, their discussion of return dynamics in Section 4 must be re-examined

The Solution

The correct way to look at this is to let

and

If is 1 day, then

so that

If we assume is a geometric Brownian motion (as they do), then

where . With a slight abuse of notation, we can drop the indices and let

so that

This allows us to rewrite (using the definition of the binomial coefficient)

Noting that

and

we arrive at a disappointingly simple, yet important, expression

The expression above governing leveraged ETFs is the starting point for further analysis. We will come back to this in a subsequent post.

To be continued…

Bloggers Missing it on Leveraged ETFs

Back in December, I noticed several bloggers coming out against leverage ETFs. In response, I wrote

Leveraged ETF Math

to try to dispell some of the misunderstandings out there. Last week, Bespoke Investment Group, whom I generally admire, came out with an article:

Direxion 3x Financial ETFs Go Certifiably Crazy

The volume on Direxionshares’ 3x leveraged bull and bear financial ETFs shows that traders love the product.  However, the ETFs have returned some crazy numbers this year.  The 3x ETFs provide 3 times the daily change of the underlying index, and year to date, the financial index that FAS (long) and FAZ (short) track is down 14%.  However, the 3x long ETF (FAS) is down 68% year to date, but the 3x short ETF (FAZ) is down 65%!  And since the lows on March 9th, these things have returned some whopping numbers.  FAS is up 195%, while FAZ is down $102.78 (or 87%).  Rest assured that a lot of people have gotten burned with these leveraged ETFs, and even though they’re meant to track daily performance, their crazy longer-term returns won’t go unnoticed forever.

There is nothing crazy about the long-term returns of FAS and FAZ. The proper way to compare their performance is versus an index whose daily returns are exactly three times the unleveraged index. This is easy to do once you have the daily returns of the index. Here is the cumulative performance of FAS vs 3x the daily return of the Russell 1000 Financial Services index:

Here is the cumulative performance of FAZ versus -3x the daily return of the Russell 1000 Finance Services index:

I don’t think anyone can look at these charts and suggest Direxion is not tracking the indices well. Instead of spreading misinformation, perhaps it would be better if bloggers tried to explain these ETFs rather than set up strawman charts indicating how different cumulative returns of FAS and FAZ versus the cumulative return of the index. Of course with daily returns of 40%, the difference between cumulating 3x daily returns can deviate significantly from 3x the cumulative returns. This is perfectly normal and anyone investing in leverage ETFs should understand this. There is nothing “certifiably crazy” about it.

Direxion had the misfortune of introducing these ETFs during a financial crisis. Here is what the hypothetical cumulative performance of FAS would have looked like if it was around since 1995:

During bull markets, these ETFs suddenly do not seem so unattractive over long periods.

Disclosure: I own shares of BGU as a long-term investment. Let’s see where it is trading 3 years from now.

Disingenuous quant bashing

My last post:

Why do scientists go into finance?

started out as an introduction to this post, but grew a life of its own.

I am quite fed up with all the quant bashing we’ve seen in the media since the current crisis began. A recent article by Felix Salmon appeared in Wired Magazine:

Recipe for Disaster: The Formula That Killed Wall Street

I actually thought the article was quite good. I did take a small exception to the last paragraph though:

In the world of finance, too many quants see only the numbers before them and forget about the concrete reality the figures are supposed to represent. They think they can model just a few years’ worth of data and come up with probabilities for things that may happen only once every 10,000 years. Then people invest on the basis of those probabilities, without stopping to wonder whether the numbers make any sense at all.

I think this statement exaggerates the situation. I took it as a minor deviation from an otherwise good article. Unfortunately, he gleefully followed up his article with a reference to Paul Wilmott in:

When Quants Don’t Think

Wilmott is quite right that quants need to stop being so passive. But he also knows full well that they won’t be. It’s far too easy for them to go along with what everybody else is doing — and that’s exactly why the copula function turned out to be so disastrous.

As I alluded to in the last post, it really has nothing to do with going “along with what everybody else is doing”. It is more about doing what your boss tells you to do. It is wrong to point the finger at the quants when the real culprit was upper management. The quants had absolutely no authority and no incentive to reel in excessive risk taking. As I’ve said, the quants know quite well the strengths and the weaknesses of the models they employ. If you asked them if the models satisfactorily represented the risks in CDOs, they would quite plainly say “no”. Did anyone ask? Of course not. They were paid to build models and that is what they did. When the models expressed risk beyond what upper management thought it should be, the quants were forced to dumb down the models. Period. They were making too much money. In fact, as I’ve said in

CDOs and Risk Management,

CDOs were great because they represented a blind spot in existing risk management systems based on VaR. VaR essentially defines what a “tail event” would be given a particular financial model of a portfolio. Typically, this “tail event” is the loss at which you are not likely to exceed on average 1 out of 100 times. One out of 100 times, a CDO is not likely to lose ANYTHING. Hence, it’s risks do not show up on any risk system. This is what I refer to as “risk management arbitrage”. What these models do not tell you is the loss you can expect “if a tail event occurs”. As we now know, if a tail event occurs, a CDO can lose EVERYTHING.

Talking heads like Nassim Taleb are extremely unhelpful and are, in fact, counterproductive. He has been on the war path for years telling us that securities do not follow a normal distribution. HELLO!! We know that already. There is not one risk model I have ever seen (except some third party models that are not worth the disk space they use up) that is based on the assumptions that Nassim Taleb is attacking. This whole idea that quants assume normal distributions is completely bogus. Taleb has always been irritating to quants for his completely content-free rants and his straw man arguments. Now that we see him on Bloomberg almost every day, it is time someone puts him in his place.

It is true that the Black-Scholes model assumes returns are normally distributed, but that is pretty irrelevant. What the Black-Scholes model allows you to do is to convert a price into another number called “implied volatility”. The implied volatility shares a purpose similar to the yield of a bond. It helps you compare related securities. If two similar companies have outstanding bond issues with similar coupons and similar maturities, the yield of the bonds help determine which is the better value. Do people go on rants about how meaningless the “yield” is? No. The yield on a bond is based on a flimsier model than Black-Scholes and no one seems to care. To compute yield, you must assume the issuer is going to make every interest payment and that you can reinvest that interest payment at the same rate throughout the life of the bond. The yield is then the discount rate under this pretty bogus model that matches the theoretical price to the market price of the bond. When you really think about it, the yield of a bond is a pretty bogus concept. However, it does help make informed decisions about the relative valuation of bonds. The implied volatility serves precisely the same purpose. A fairly bogus model spits out a number that helps make informed decisions about the relative value of derivatives.

For Nassim Taleb to go on the media circuit complaining about black swans and the fallacy of normal distributions, it is as if a former bond trader suddenly decided to go on multi-decade crusade against the use of bond yields. Ok. We get it already.

It would be less disturbing if we didn’t see Paul Volcker going on the circuit parroting things that Taleb says. Toward the end of the video I posted on

A Call to all Finance Bloggers

we see Volcker referring to “black swans” and goes on a rant about financial engineers. His words are clearly straight out of Taleb’s mouth. They probably had dinner together at Davos or something. So now that Volcker has been infected by these bogus ideas, that is worrisome because Volcker obviously has a line to President Obama. The last thing we need is to get distracted by going after the quants.

Before leaving, let me turn some attention to Paul Wilmott. His recent turn against his own, i.e. his recent quant bashing, is disturbing. Paul Wilmott wrote some decent books on mathematical finance and derivatives pricing. But what he is probably more known for these days is his quant forum at Wilmott.com. Currently, there are over 66,000 registered users signed up at Wilmott.com. Long ago, I was a member myself and I even published my first finance-related research paper there. At the time, it was a decent place populated by knowledgeable quants. I got my first job on Wall Street due to the help of a friend I met on that forum. Unfortunately, in the last 5 years or so, the place has turned into a zoo. It is impossible to have a decent conversation there anymore as there is no form of moderation. He doesn’t care, just as long as more eyes turn to his web site. It is little more than a PR campaign.

Both Nassim Taleb and Paul Wilmott are opportunistically spouting nonsense about quants that really do a disservice to some really quite intelligent, experienced, and knowledgeable people who happen to be friends of mine.

Why do scientists go into finance?

Why do mathematicians and physicists go into finance?

One reason people may sympathize with is mere survival. Job prospects for mathematicians and physicists in academics is horribly bleak. Each PhD program churns out 10s if not 100s of PhDs each year. How many PhDs do these same institutions hire each year? Less than 10 for sure. Most likely none. Scientists are a different breed. Most pursue higher education simply because they love what they do with little thought about what will happen after school. It is often not until a few months before being cut loose that many graduate students think to themselves, “Oh %$#%! What am I going to do now?” A good picture to keep in mind is the classic absent-minded professor.

Although Wall Street rolled out the red carpet to scientists in the 70’s and 80’s, I would suspect that the idea was not high on most graduate student’s minds at the time. When quantitative risk management systems began being deployed on a large scale in the 90’s coincident with significant improvements in computational power, that marked a turning point. By the mid 90’s, Wall Street was becoming a clear beacon for mathematicians and physicists about to hit the job market. Leading up to Basel II setting capital requirements based on value-at-risk measurements in 2004, banks literally went on a hiring spree of PhDs. I know that when I first tested the waters on Wall Street in 2002, each advertised quant opening was receiving no less than 30 PhD resumes. Most of these had prior work experience in finance. Today, many physics and mathematics PhD programs offer minors in finance. Clearly, Wall Street is a destination for many graduating scientists these days.

Is survival the only reason scientists go to Wall Street? Clearly not. The real and only important reason physicists and mathematicians go into finance is that they can potentially make lots of money doing very interesting and rewarding work. Who wouldn’t want to work in finance? I know I absolutely fell in love with finance at first sight. The first time I stepped foot on a trading floor, I knew I had found my calling in life. It was a truly transformative experience.

There are about as many different kinds of quants as their are scientists. I have been fortunate enough to have seen the quant world from many perspectives. I started finance life as a “risk quant” working at a large bank with a group of 12 other PhDs building risk models that spanned trading desks across the globe. These guys have been getting a bad rap lately and I’ll have more to say about that another time. Make no mistakes though. The risk quants know quite well the strengths and limitations of their models and given more authority, they could have and would have kept the credit bubble from getting out of hand. Unfortunately, the reality is that risk quants have been relegated to secondary roles whose purpose is often to massage numbers to tell the risk managers what they want to hear. For example, at one point, a friend told me that their risk manager did not like the numbers produced for a particular trading desk. This trader had significant influence. So the risk manager came back and told them to recompute the correlation matrix until it output what the trader wanted to see. Did the quant have a choice? Not if he wanted to keep his job. At another point, another friend was told that they needed to modify the risk numbers coming out of the models because they were too high which forced the bank to retain too much capital. He was warned that people higher up were becoming unhappy and that the entire group could be eliminated if they didn’t do something about it. Since the job market was so competitive and since the pay was quite good, there really was no incentive to rock the boat. This has absolutely nothing to do with poor models or “black swans”. It has everything to do with greed. Period.

There are some really good aspects of being a risk quant. Usually, it is a good entry point to other things since you get a general introduction to a large variety of securities. The typical entry requirements are often lower as well. The downside is that you are effectively a NARC with absolutely no authority. You may think your job is to reign in excessive risk takers, but the reality is that you are most likely a puppet for upper management.

As a byproduct of proliferation of risk management systems, clients and investors are becoming increasing demanding in terms of risk reporting. This has trickled down from investment banks on the “sell side” to money managers on the “buyside”. Traditional asset managers who previously had no interest in quants or their models are now being forced to hire quants simply due to client demands. This can be a very good place for scientists to end up. You will often come across as a super star rocket scientist regardless of what you actually contribute. The downside is that many traditional investors may view you as a necessary evil and don’t really want you there. It is a challenge in such an environment to demonstrate the value of the work you do. Yes, I am speaking from experience There are definitely good things to be learned from investors who are firmly “anti-quant” though. I value the experience obtained from attempting to understand the way traditional investors think and invest. It has had a definite positive impact on the way I look at things. My advice to any quants moving into traditional asset management is to try to find a way to “quantify” your contributions. Make it clear that you are doing things that few others could do. My biggest mistake was assuming that my hard work and the contributions I was making would be obvious and rightfully recognized. Make sure you have champions and make sure these champions speak up for you. Working on the buyside can be quite rewarding both scientifically and financially. I know it is where I belong.

Another type of quant is the “front-office quant” whose job it is to build derivatives models to assist traders directly. From my experience, this is where most quants would like to end up. It is often fast-paced and quite demanding. You have to be willing to be brutalized and cannot be sensitive to fowl language A part of me would love to work on a fast-paced desk. I almost look at these guys as the rock stars of quants. These guys can enjoy quite ridiculous compensation since they participate more directly in the profit sharing. Plus, the closer to the money you are, the better. This role can also lead to opportunities to become a trader. I think secretly (or not so secretly) most quants dream of becoming traders.

When I grow up, I hope to become a quantitative portfolio manager. I envision this as somewhat of a hybrid between the traditional asset manager and the traditional quant. People need some place to put their retirement investments. Traditional asset managers have let many retirees down in a bad way. They often charge high fees for unremarkable performance. Many asset managers saw the current crisis coming and positioned themselves appropriately. Others had their heads in the sand for far too long and ended up destroying a lot of hard-earned wealth.

I love finance. I do not feel like I’ve given anything up by leaving physics. The modeling is quite enjoyable and regardless of what some talking heads in the media would have you believe, can be quite valuable to investors. Any decent credit model was screaming that fixed-income securities were grossly overpriced leading up to the crash. I know that I literally begged my research directors to let me work with the high yield analysts when I saw the risk premium go negative in 2006. Every other quant I talked to knew it too. As long as the music plays, you need to keep dancing, right?

What do I think about markets now? I hope to say more in a separate post, but I started this blog on July 10, 2007 with a post entitled:

“The End is Near”

At the time, I claimed to be an optimist and I am. I was scared because very few others were scared. Now, everyone is scared as they should be, but I see that as the first step to recovery. You have to recognize how serious the situation is before it can get better. Spreads in fixed income have priced in some very gruesome scenarios. I think many of these gruesome scenarios will come to pass. Corporate defaults will obviously increase and this will put a strain on the CDS market. I was more scared about this before, but recent efforts to move CDS to clearinghouses has dramatically reduced my fears. There will be more blood before things hit a bottom, but investors are slowly beginning to see beyond it. We’re not out of the woods by any means and risks remain extremely elevated, but I am optimistic that in 2-3 years, the equity markets will be much higher than they are today regardless of how low they go in the interim.

A Call to all Finance Bloggers

It is clear we are in a financial crisis. If you have any doubt about how serious this crisis is, have a look at Volcker’s statement:

What should we do about it?

I am a mere four years into my finance career, but have been a scientist at top research institutions for a lot longer than that. One of the strengths that I can offer is knowing who to listen to. At this point in history, the best minds and the best analysis is coming from financial blogs. Unfortunately, there remains a stigma about blogs and serious policy makers tend to dismiss this form of media. That is a mistake that is slowly correcting itself, but not fast enough considering the needs of our times.

I propose that all financial bloggers join forces and work together collaboratively in the truest sense of “Web 2.0″ to come up with a plan that can help stabilize the global financial markets. Let’s take President Obama at his word when he said he would listen to all good ideas. Let’s give him a good idea that is endorsed by prominent financial bloggers. All would be welcome, but in particular, I would like to see a plan endorsed by

• Nouriel Roubini (RGE Monitor)
• Barry Ritholtz (The Big Picture)
• Brad Setser (Folow the Money)
• Yves Smith (naked capitalism)
• Calculated Risk
• Michael Panzner (Financial Armageddon)

Let’s be productive and find a solution.

Richter Scales Rock

Here they are again. This is AWESOME!

Leveraged ETF Math

There seems to be some confusion out there regarding leveraged ETFs and their ability to track the underlying index.

The key thing to keep in mind is that ETFs attempt to track the DAILY returns. This leads to some nonintuitive, but perfectly natural, behavior when looking at cumulative returns. For example, consider the return of an index over a two-day period:

Now consider the return of a triple-leveraged ETF over the same two-day period:

In other words, err… symbols

After just two days, you can see that the ETF return will naturally deviate from the index return by a factor of even if the ETF is perfectly tracking the index.

The same logic extends to ultra-short ETFs:

Or

You can now see that the deviation is not symmetric since the short ETF deviates by a factor of 12 as opposed to 6 for the long ETF. As a result, if you were to plot the cumulative returns for ultra-long and ultra-short ETFs versus their index, things may begin to look screwy over time.

THIS IS NOTHING MAGICAL. It doesn’t mean the ETF is not doing its job. It is just a perfectly natural consequence ETF math.