The following is a note I sent to my PhD advisor, Professor Weng Cho Chew, on September 13, 2011 after a discussion over dinner as he was headed back to UIUC from a 4-year stint as the Dean of Engineering at the University of Hong Kong.
Decomposing Finite Dimensional Inner Product Spaces
Given finite-dimensional inner product spaces , and a linear map , the adjoint map is the unique linear map satisfying the property
for all and .
In this section, we show that can be decomposed into two orthogonal subspaces
This is a fairly simple exercise as any finite-dimensional inner product space can be decomposed into a subspace and its orthogonal complement, i.e.
The only thing to show is that .
To do this, note whenever , then
for all . Thus, is also in , i.e. . Similarly, whenever , then
for all . Thus, is also in , i.e. . Since both and , it follows that .
Given finite-dimensional inner product spaces , , and linear maps , such that , we wish to show that the inner product space may be decomposed into three orthogonal subspaces
To show this, note that if , then
but this implies and . Conversely, if and , then is trivially in . In other words,
Finally, since , we also have . Consequently, when , then so
Applying the decomposition from the previous section twice, we conclude that
and since , it follows that
which may be expressed simply as
Putting this together we see the desired Hodge-Helmholtz decomposition
The preceding discussion is quite general and holds for any finite-dimensional inner product spaces , , and any linear maps , satisfying . In this section, we specialize to computational electromagnetics.
Consider a discretization of a surface consisting of vertices, directed edges, and oriented triangular faces. If we associate a degree of freedom to each vertex, the span of these degrees of freedom form an -dimensional vector space . Associating a degree of freedom to each directed edge forms an -dimensional vector space and associating a degree of freedom to each oriented face forms an -dimensional vector space . For concreteness, vectors in will be expanded via
where denotes the degree of freedom on the ith vertex, vectors in will be expanded via
where denotes the degree of freedom on the ith directed edge, and vectors in will be expanded via
where denotes the degree of freedom on the ith oriented face.
To turn , , and into inner product spaces, we need to define three respective inner products. This can be done by defining three sets of basis functions , , and . and take values defined at vertices and faces, respectively, and maps these to functions defined over each face. Similarly, takes values defined along each edge and maps these to vector fields defined over each face.
The basis functions linearly turn vectors in , , and into functions and vector fields defined over the surface via
The inner products may then be defined in terms of basis functions via
Letting , , and denote column matrix representations of vectors in , , and , the inner products may be expressed in terms of matrix-vector products via
The matrix-vector representation is helpful for explicitly expressing the adjoint of a linear map via
Similarly, the adjoint of a linear map may be represented in matrix form via
In computational electromagnetics, a fundamental linear map is the exterior derivative, which will be denoted for . Since , , and are finite dimensional, has a sparse matrix representation .
For the sake of interpretation, the matrix may be thought of as the gradient along the respective directed edge, may be thought of as the curl of the edge vector field around each oriented face, may be thought of as the transverse gradient across each directed edge, and may be thought of as the divergence of the edge vector field.
As a result, we have the inner product space of edge vector fields decomposes into
where . In other words, any edge vector may be expressed as
for some , , and . The above may be thought of as a discrete version of Hodge-Helmholtz decomposition for computational electromagnetics.
This note is an informal (and quickly drafted) document intended to help explain Hodge-Helmholtz decomposition in computational electromagnetics. No claim of any original content is intended and a proper literature search was not performed. For pointers to some related material with more complete references, see the following:
- PyDEC: Software and Algorithms for Discretization of Exterior Calculus
- Least Squares Ranking on Graphs
 If the degree of freedom associated to an oriented face is interpreted as the magnitude of vector normal to the face, may be thought of as the curl of this normal vector field along the directed edge.
This post is part of a series
The first equation is essentially the discrete calculus version of Ohm’s Law, where
is a discrete 1-form representing conductance,
is a discrete 0-form representing voltage, and
In components, this becomes
The second equation is a charge conservation law which simply says
is the sum of all currents into node and
is the sum of all currents out of node . This is more general than it may first appear. The reason is that directed graphs are naturally about spacetime, so the currents here are more like 4-dimensional currents of special relativity. The equation
is related to the corresponding Maxwell’s equation
where is the adjoint exterior derivative and is the 4-current 1-form
This also implies the discrete Ohm’s Law appearing above is 4-dimensional and actually a bit more general than the usual Ohm’s Law.
I’ve been thinking about this off and on since then as time allows, but questions seem to be growing exponentially.
For one, the equation
is curious because it implies that is a derivative, i.e.
Further, although by pure coincidence, in my paper with Urs, we introduced the graph operator
and showed that for any directed graph and any discrete 0-form that
Is it possible that and are related?
I think they are. This brings thoughts of spin networks and Penrose, but I’ll try to refrain from speculating too much beyond mentioning it.
If they were related, this would mean that the discrete Ohm’s Law above simplifies even further to
In components, the above becomes
This expresses an effective conductance in terms of the total number of directed edges connecting the two nodes in either direction, i.e.
If the ‘s appearing in the conductance 1-form are themselves effective conductances resulting from multiple more fundamental directed edges, then we do in fact have
Applications from here can go in any number of directions, so stay tuned!
When I look out at the world, one of the major risks to the markets that I see is, ironically, risk management. I suspect that one of the primary employers of junior quants in the last 5 years has been in risk analytics (HHs, please correct me if I’m wrong). If there is any truth to that, it means there is literally an army of quants who have not lived through a business cycle building risk systems on markets that no one really understands, e.g. CDS/CDOs.
If things are at all like what I have seen, then we’ve got a bunch of fairly clueless risk managers out there with an army of fairly green quants developing sophisticated risk models that are probably pretty useless in a crisis. Nonetheless, there seems to be this completely ludicrous false sense of security.
Across the boards, vols seem to be historically low which would mean that most VaR engines are saying “smooth sailing”. What happens if vol increases? Everyone’s VaR model is going to start sending out little red flags. Assets are going to start getting reallocated. Since everyone has almost identical VaR models, the signals will be pretty much identical at all firms. I know it is not an original argument, but this could easily lead to a negative feedback. A small red flag due to increased VaR could signal everyone to make very similar reallocations. If everyone does it at the same time, the market will obviously be affected. In essence, the impact of risk management could actually increase systemic risk in the markets and amplify vol movements.
With that in mind, you can probably imagine how much I enjoyed this article by Avinash Persaud at VoxEU:
4 April 2008
I also enjoyed his 2000 article referenced in the above paper
Sending the herd off the cliff edge:
The disturbing interaction between herding and market-sensitive risk
Both are well worth reading.
Physicists learn at an early age (sometimes while still in diapers) about vectors. In fact, I supported myself financially through undergrad largely by explaining vectors to physical therapy students. Physics was the “weeder” course and PT students basically needed an “A” to get into the program. Tutoring them was quite lucrative, but that is another story. Here, I present a very neat way to visualize market risk in terms of arrows… err… I mean vectors.
What is a vector?
A vector can be thought of conceptually as an “arrow”.
What is the information contained in an arrow? There are two basic bits of information contained in an arrow
- Length or magnitude
In terms of direction, probably what is more useful than “absolute” direction, is the “relative” direction, which can be quantified as the angle between two arrows.
Multiplying a Vector by a Number
The next thing you need to know about vectors is that you can multiply a vector by a number to get another vector .
Multiplying a vector by a number merely “rescales” the arrow, i.e. changes its length, while keeping the direction unchanged.
Next, given two vectors and , there is a rule for adding the arrows to get a new vector . There are a couple different ways to visualize the addition of arrows. One involves drawing the vectors “tail to tail” and forming a parallelogram. The new vector is the arrow going across the diagonal as shown below
Alternatively, but equivalently, you can draw the vectors “head to tail” and the new vector is the arrow pointing from the tail of the first arrow to the head of the second arrow as shown below
It doesn’t matter which you choose. Just remember that the addition of arrows is a little funky and involves parallelograms.
If I haven’t lost anyone up to this point, I will be quite impressed if you stick around for what comes next.
The Dot Product
Both the length and relative direction of a vector, or arrow, can be determined by something that we lovingly refer to as the “dot product”. Later in academic life, I learned that the “dot product” was actually a twice covariant symmetric non-degenerate tensor, but I digress. For our purposes, we just need know that the dot product takes two arrows and spits out a number (in a bilinear fashion). In other words, if
A fundamental property of the dot product is that when you take the dot product of a vector with itself, the result is the square of the length of the vector, i.e.
Therefore, the dot product provides us with one of the most important characteristics of a vector: it’s length. To whet your appetite a little, I hope that the expression above makes you think of “variance” or more specifically “standard deviation/volatility”. That is no coincidence!
The dot product of two distinct vectors also gives some very useful information
Note that lies between +1 and -1. Kind of like correlation. Again, not a coincidence!
Projecting One Arrow Onto Another
Once you start doodling with a bunch of arrows, you may start to think about the relationship between two arrows and, like a shadow, how might one arrow project onto another as shown below
If you recall a bit of trigonometry, the length of is given by
which may be written in terms of the dot product as
How about the ratio of that projection? That is determined by simply dividing the above by |A|, i.e.
This ratio is particularly of interest in finance, as I will show below, and is related to “portfolio beta”.
Portfolio Return as an Arrow
Now the fun starts and I hope at least one person in the universe reads this far down :)
The return of a portfolio is a vector.
Well, hear me out. Consider a portfolio consisting on securities. The return of the portfolio can be expressed in terms of the returns of the securities as follows
where is the weight of the th security in the portfolio.
Hmm, the security returns are multiplied by numbers and then added together. That sounds a whole lot like a vector to me!
Ok. If returns can be thought of as arrows, what are the meanings of the length and relative directions of two returns?
As we saw above, to determine the length and relative direction, we needed a dot product. What would be a good dot product for the vector space of security returns?
Covariance as a Dot Product
Technically speaking, covariance does not satisfy all the axioms of a dot product, but fortunately physicists like to shoot from the hip and tend to be right under most practical circumstances and in exotic cases where we are not right, we can usually change things around a bit so that we are right. So technical arguments aside, I am simply going to tell you that covariance can be thought of as a dot product on the vector space of security returns.
Woohoo! That is great. Why? Because all the nice pictures above can be re-interpreted in the context of portfolios of securities.
Here is an arrow depicting the return of a portfolio
What information is conveyed by the length of this arrow? Recall that
but if the dot product is covariance, then we get the very interesting association
i.e. the length of the arrow is the volatility of the return.
Next, consider the return of a benchmark against which this portfolio is to be measured. The benchmark return can similarly be represented as an arrow and we have
What is the meaning of the angle between the “portfolio arrow” and the “benchmark arrow”? Recall that
However, we also just learned that
so that the above expression can be rewritten as
which you might recognize as the definition of correlation. The correlation between the portfolio return and the benchmark return is the cosine of the angle between the portfolio arrow and the benchmark arrow.
That is pretty neat, eh? We now have a nice way to visualize both the volatility and correlation of security returns. The length of a “return arrow” is its volatility and the angle between two returns relates to their correlation. I couldn’t make up a better story if I tried :)
But what can we do with this knowledge?
The excess return of a portfolio of its benchmark is simply
which we can trivially rearrange as
The right hand side of the equation of above represents the addition of two arrows, which we already described above. Therefore, we can represent the expression visually as
This little trick seems kind of neat, but fairly trivial, so what did it give us? Recall that the length of an arrow is its volatility. Recall also that tracking error is defined as the volatility of the excess return, therefore we have the very cool consequence
We have a very concise way to visualize tracking error that captures both the volatility of the portfolio and the volatility of the benchmark as well as the correlation between the two in one simple diagram.
“Now how much would you pay? But wait! There’s more!”
The same diagram above that gave you tracking error also gives you your portfolio beta. To see this, simply recall the definition of the projection of an arrow along another arrow and reinterpret that in terms of returns
where is the correlation between the portfolio and benchmark returns.
Next, if we consider the ratio of this projection to the length of the benchmark arrow, we have
which you might recognize as the definition of the portfolio beta! Therefore,
We’ve demonstrated, first, that returns can be visualized as arrows where the length of the arrow represents its volatility and the angle between two arrows represents the correlation of the two respective returns. Second, by comparing the portfolio and benchmark returns pictorially, we automatically get a very informative picture of both tracking error and portfolio beta (in one shot) that also contains information about the absolute market risk in terms of the volatilities (lengths) of the portfolio and benchmark arrows as well as the correlation between them.
Market turmoil is still quite fascinating to me and I still believe the current environment will be one for the history books and I’m still trying to take as much as I can from this learning experience.
My professional work experience is in fixed income. For two years, I was very “plugged in” to the markets and was meeting regularly with some of the greatest thinkers out there, but now I’m more of a pure “quant” and most of my news comes from blogs, web news, etc. Unfortunately, I’m not yet spending as much time with the traders as I’d like (but that should change soon I hope). Most of the major news sources, e.g. Bloomberg, seem to concentrate more on equity markets than credit and fixed income. I pay more attention to the Dow now than ever before. That is why I am so perplexed by the stock market. I thought stocks were supposed to be easier than bonds, i.e all the smart guys are in fixed income, right?
So while the credit markets seem to be imploding, stocks are doing just dandy. Maybe people are taking cues from the market cheat sheet?
Anyway, I’ve blabbered quite a bit on this blog (and at my former employer) expressing my opinion of CDOs. I even managed to upset quite a few people while expressing my opinions as well. No regrets though. I’m happy to have this hugely public diary, both here and on NP, to later look back and see how I did in regards to thinking events through. Occasionally, I still like to poke my nose in over at NP and I see kr is still giving out the occasional nugget. Here is one of his latest:
A few thoughts:
– If you took all the writedowns at a single med-to-large bank rather than seeing them across the street, you could have reduced that entity’s equity to ZERO. For instance, MER has only something like USD54bn of mkt cap and USD39bn of book equity.
– If the view is that there will be another round of writedowns in the same amount as Q3 then you will have banks desperate to raise equity (i.e. it is not just the monolines). Who would buy that equity right now? Prince Alwaleed for example has floated his own holdings so I see him more as seller than buyer for example. I don’t see guys like JC Flowers or Cerberus well positioned for this job – in retrospect, even Barclays/RBS have not been with respect to ABN, as can be seen by the action in their share price and cost of jr capital.
– Another possibility would be the downgrade to BBB like the Japanese banks, with all the implications that brings with it. I.e. serious change in business model. That has contagion and macro effects. One example is that flow trading of financials has cost people a lot.
– I think investors will call foul on the FAS157 Level-3 assets, and it will hit guys like GS seriously as their L3 reporteds are a big multiple of their mkt cap.
– There was a funny comment in this month’s BBG mag about “nobody really knows how desks are hedging the CDO assets.” That is bull – the answer is that most people were NOT HEDGING AT ALL, BECAUSE THEY COULDN’T. Stuff was originated to sell, and the exit has vanished, or, it was originated to live forever on a trading book even though people tried to avoid saying that, and there is no decent MTM approach so instead banks are showing huge volatility, mostly to the downside.
– Implications of SIV / CDO / CP demise are pretty vast. There seem to be an increasing amount of trade receivables on the market, b/c there are no conduits to fund them… means corp cost of cap is going up in unexpected areas.
My hunch is that the fed cuts on the 11th b/c liquidity is dropping again, especially with year-end. It is out of control – specifically Ben’s control. It looks like political support for the various subprime fixes has stalled. What I think is that liquidity of all things financial (i.e. non-corporate) is going to get weaker and cause a full-on crisis for a market-traded institution. The talk about Citi cutting their div is one tremor, trading activity in Barclays is another, and the fact that even AFTER all the reported loss numbers, people still don’t feel comfortable, is yet another.
I think vols are still cheap, maybe looking to buy some.
All the while I was complaining about CDOs, I was coming at it from the angle of a “quant”, i.e. thinking about how to model CDOs and how those models are used in risk management, asset allocation, etc. Too bad I didn’t understand more about the legal/accounting aspects of CDOs. The term everyone has now heard of is SIV. I was blabbering about off balance sheet leverage and fair value accounting, but didn’t realize that the entire CDO market was (to a jaded eye) a big play on accounting in addition to the obvious play on ratings agencies. If I had known about SIVs, I might have been able to do more to help some who may have now lost a lot of money. Maybe not. That’s all in hindsight. But what am I missing now? Where is the next weakest link? How are corporations hiding off balance sheet debt? Has anyone looked at “Level 3″ assets in corporate, i.e. non-financial, balance sheets? Are they as scary as the big banks?
I’ll say it again… this is not a subprime issue. Subprime contagion does not explain the current environment. Subprime was just the first to blow. We are experiencing the blowup of a global fixed income bubble. In fact, some would say we’re experiencing a general global asset bubble.
Who’s going to get hurt? Financial institutions for sure. Anyone who depends directly on the value of paper assets.
Who’s going to win? People whose wealth depends on physical assets.
I’ve already lost all hope in Bernanke. He is not going to let his monicker “Helicopter Ben” go by the wayside in a “time of need”. Bernanke is going to lower rates and weaken the USD until oil exporters are forced to break the peg to the USD and inflation skyrockets. I predict that all these gloom mongers about home prices dropping by 30% will turn out to be wrong in nominal terms even if they are correct in real terms. In other words, home owners are going to be saved by the dropping value of the USD. All those on Wall Street who were so gleeful every time rates dropped are suddenly going to feel the pain when the value of their paper securities go up in smoke.
Watch out for the “happy stage of inflation”, i.e. wage increases. It will be interesting to see what the world will look like when oil is priced in EUR and the USD is no longer the world currency. Fortunately, I still have faith that we’ll come out of the current mess stronger as a country, but there will certainly be pain felt at the higher end of the wealth spectrum.
I’m actually ironically optimistic about the outlook for suburban and rural economic development. A weaker dollar will make outsourcing less attractive. That will bring manufacturing jobs back home. I can imagine a boon in suburban and rural development. Just imagine if communities developed decent broadband via fiber-to-the-home/business. Suddenly, there will be attractive jobs and living standards in affordable places.
Maybe a weak dollar is what this country needs, i.e. a good kick in the pants. Pain is the best teacher, right?
[Edit: PS, the title of the post was inspired by a great article on Financial Armageddon, but I never got around to explaining why, but have a look and it might be obvious.]
Way way back a long time ago (in August), I clarified a semantic issue regarding one of my market forecasts (more like voodoo analysis) I made on July 24 here:
Here is a snippet:
But regarding that quote above from my last post, there is something I wanted to clarify. Particular the chain of events that I outlined, i.e.
subprime CDO imploding -> HY repricing -> CLO slowing -> LBO stopping -> private equity choking -> equities tanking -> M&A increasing -> ???
At the time I wrote that, we were only on step two, i.e. subprime had imploded and HY had repriced (a bit). I think their is still long way down to go for credit, but at the time I wrote that CLOs hadn’t been hit yet, LBOs were not on the radar screen (of most anyway), and the idea that private equity was in trouble seemed ludicrous. How about now?
Now, everything has occurred except “M&A increasing”. At the time I wrote that I was distinguishing LBOs from M&A, but have since learned that many people include LBOs under the M&A umbrella, so I thought I would clarify what I meant by that last item before the “???”.
LBO activity was driven by (at least) two factors: easy credit and value in acquiring companies. The first factor is now gone, which is why private equity is hurting. The second factor remains. Easy credit made it difficult for real-money people to take advantage of the value in acquiring companies because of competition from private equity. With easy credit gone and private equity struggling to get LBOs done, now people/corporations with real money on hand can take advantage of the situation.
By “M&A increasing” I meant that mergers and acquisitions that are not dependent on leverage, i.e. by people/corporations with “real money”, would increase. This, I thought, would prop up equities for a while longer. For example, on July 25, I said:
Since I expect LBO activiate to be significantly hampered, if not halted, and since there is value in stripping some of the companies, I expect more M&A activity to replace LBO activity.
I meant “non-leveraged” M&A. So watch for this. The first sign of this happening that I’ve seen comes from Bloomberg in regard to Warren Buffett. Certainly a “real money” guy.
Here’s an excerpt:
Berkshire Hathaway Inc. Chairman Warren Buffett is ready to spend $40 billion to $60 billion on an acquisition, and his opportunities are expanding as stocks fall and leveraged buyouts dry up.
Shares of health insurers, steelmakers and department stores are as much as 22 percent cheaper than in May, when Buffett said he would “figure out a way” to come up with $60 billion for the right deal. WellPoint Inc., Nucor Corp., Kohl’s Corp. and dozens more companies are now closer to meeting his investment criteria.
It seems we are approaching the faintest regions of that earlier “prediction”. Real money seems to be stepping up and it is not necessarily from the obvious candidates. Watch out for sovereign wealth funds to keep things propped up for a while longer.
A bit dated due to my reduced blogging pace:
and a bit more recent via a guest appearance on Brad Setser’s blog
Michael Pettis seems to think sovereign wealth funds can prop up the market for years to come. I tend to agree with him (but not sure if the time span is years… or months). By the way, as a sidenote, when I asked someone at work about the possible positive influence of sovereign wealth funds on structured finance, I was shot down pretty publicly. It was brutal. Friendly new colleagues seemed to get a kick out of it and still rib me about it to this day. Pick on the new guy! :) Oh well. I’ve decided to keep my mouth shut about markets and economies during high-profile meetings. After all, I’m not an economist or even a market analyst. I’m a quant. As a quant, I am quite competent and should just stick with what I know when it comes to my actual career. On the “quant” front, however, I’ve recently developed some massively cool technique for rapidly producing analytics, e.g. OAS, OA duration, etc for a large pool of securities. In the “privacy” of my own blog, I can say whatever I want though ;)
Even before things in credit markets went completely haywire, there were technical concerns about the building pipeline of new high yield issues that were feeding into CLOs, which were in turn spurring the LBO boom. My own quantitative analysis had shown that high yield was particularly vulnerable going back to September 2006 (which my models were unfortunately only completed in March-April). That is why I was so certain that July 16 would be a big day. That was the day Bear Stearns was to report their hedge fund losses.
I think we all know what happened after that, but I still think not everyone is aware of the huge buildup of high yield issues that have resulted from delays as brokers “hope” things get better. This has created a bifurcation in the primary market (which is essentially non-existent) and the secondary market. Just as holding of CDOs will eventually be forced to face the music, the high yield market will need to face the music and eventually issue some new bonds. That will (in my opinion) bring in a pretty clear “mark to market” in high yield and there very well could be another general repricing.
Apparently, next week is the first real test as First Data issues $24B of debt ($16B in loans and $8B in bonds)
All Eyes Are On First Data
September 10, 2007
If I were a high yield investor (which I’m not, so don’t take my armchair analyst advice too seriously), I would not be eager to get back in just yet. I think things will still get worse before they get better, but like they say “only monkeys pick bottoms”, so I would certainly be shopping for bargains, while keeping in mind that the definition of a “bargain” has changed. For example, the assumption of continued historically low default rates should be discarded imho.
I hope I’m giving enough disclaimers so that no one can possibly be fooled into thinking I actually know what I’m talking about, but I continue trying to absorb as much as I can as events unfold.
I know very little about financial statements and most of my comments on the subject of corporate leverage have been guided by simplistically thinking about basic human nature. For example, in this post
I stated (emphasis added in bold)
I’ve heard economists blabber about how strong corporate balance sheets are as they have decreased leverage since 2000-2001, but these same economists have absolutely no clue about CDOs and other avenues for off balance sheet implied leverage. In my opinion, the only thing holding default rates down was the availability of easy credit, not some increased sense of corporate responsibility. I think we will find that most corporations are more highly leveraged than balance sheets would suggest.
And in this comment, I said:
What you say about FAS 140 and particularly securitized mortgage products makes perfect sense. I don’t disagree, but my thinking is that there is something else less obvious related to FAS 140 (or at least off-balance sheet exposures) that will crop up once default rates pickup again. We’ll see. Like I said, I don’t have anything more to support the idea than basic human greed on the part of corporate executives during a period of easy credit.
Maybe I was looking in the wrong place, but my instinct may not have been too far off. Here is a very interesting article via Portfolio.com
Maybe instead of looking for sneaky off-balance sheet stuff, maybe it is right there in front of our eyes. “Fair value accounting” seems quite amenable to “asset price inflation”. If asset prices are inflated, this would make corporate leverage seem muted. A situation that could be quickly reversed. Hmm…