Mispricing of Correlation Risk --- I Think it is Really that Simple

I'm not an option modeller or a mortage expert. My knowledge of financial economics is mostly focussed around equity markets. So my understanding of how MBS and CDO's work comes from what I've read in the papers essentially.

Let's take an equity guy's look at a MBS and see how it's alchemy is pulled off. We take a portfolio of risky securities (in this case the risk is that the mortgages go into default, for equity it would be the "common" risk we're familiar with --- systematic risk and idiosyncratic or residual risk). If we hold the portfolio as a simple group of assets we get the "portfolio effect." That is standard deviation per dollar of asset of the portfolio is less than the sum of the standard deviations of per dollar of the assets invidivually because the idosyncratic value fluctuations are not correlated and so sometimes they cancel each other out. This is the statistician's friend, the Law of Large Numbers, working some real genuine magic for us. Of course, if we have stocks that are dominated by market risk, i.e. stocks whose returns are highly correlated with eachother, then we don't get much diversification value; on the other hand for uncorrelated stocks or anticorrelated stocks we get a big effect.

One can state the "value" of the diversification effect as: St.Dev.(Portfolio) - Sum St.Dev.(Assets).

Now let's take our portfolio and put it in a trust. However, we write the trust documents so that the trust's assets are transferred back to several portfolio's at the end of a fixed period, not just the single original portfolio. We set up these portfolios by ranking all the constituent assets by their total return at the end of the period and then we give the top quintile to the A portfolio; the second quintile to the B portfolio; the third to the C portfolio; etc.

Clearly for "normal stocks" the present value of the "A" portfolio is much higher than that of the "E" portfolio and we've now worked the MBS magic on an equity portfolio. We should be able to sell the "A" portfolio rights for much more money than the "E" portfolio rights. This value comes from the fact that we have written a trust document that allows us to adjust the portfolio constituents after the relative returns are known.

This is essentially what's done with a MBS. A portfolio of mortgages is put in a trust and trust documents written so that the "AAA" tranche gets the payments from the mortgages that default last and the "equity" tranche gets the payments from the mortgages that defaults first.

Going back to our equity trust, look at the spread in value between the "A" portfolio and the "E" portfolio in the circumstance that the equity returns are all completely correlated. Clearly, in that circumstance all the stocks have exactly the same return, whatever that turns out to be, and there is no difference in any of the constituent returns and so:

value(A) - value(E) = 0 when common correlation = 100%.

The other end of the spectrum is when the stocks are uncorrelated (there are mathematical restrictions on the number of stocks that can share a common anti-correlation, for example three stocks cannot all be perfectly anti-correlated with each other, so we won't consider that case). In this case the dispersion between the constituent asset returns is maximized and so:

value(A) - value(E) = maximum when common correlation = 0%.

So, although I'm not going to work out the precise form here (that is going to be a function of how we actually specify our returns model), it's seems clear that the "tranching value" will turn out to be a decreasing function of the degree of common correlation.

I claim that the same will also be true for a MBS. On the assumption that default is purely idiosyncratic, there is value to the reordering of cash flows done within a MBS or CDO trust. On the assumption that default purely synchronous, there is no value whatsoever to such a device. In general the premium for a MBS "AAA" tranche above the "equity" tranche is a decreasing function of the degree of synchronous default.

It was previously claimed (by market participants) that default of a mortgagee was a highly idiosyncratic event, contingent on personal circumstances like the loss of a job or a death in a family etc. So the idea that everybody could default at the same time was not worth considering and so the price of this correlation, expressed as a discount to the "normal" premium of the "AAA" tranche over the "equity" tranche was zero. But a gigantic, worldwide, property bubble, gave us a situation in which there would be a lot of systematic defaults and this correlation discount has been repriced dramatically. Hence the falling apart of the MBS/CDO markets.

I believe it is really that simple.

An Aside - Ad-hoc Trades and the Take Profits Algorithm

Everybody makes ad-hoc trades, even the most rigourously algorithmic traders sometimes just pick up the mouse and click themselves into a position. If we're right, or if we're lucky, this starts as a good idea and generates a gain. However, although many people feel they have good insight about when to buy (or sell) --- following an earnings suprise, for example --- it's my observation that the decision to close a position out and take profits, or limit a loss, is a lot harder to make successfully. And this applies to myself as much as others.

So what happens is that the initial information fades and the trade turns against you, but you sit on the position waiting for it to come back. Everybody does this when they "punt" stocks; it's something about how the human brain processes decisions.

I going to describe here something that is at complete variance to the statistical-analytical trading methods I use for normal businesses. But I find that it helps. I use automated electronic trading to manage my systems, and I apply the following method to get me out of ad hoc trades with a system I call the take profits algorithm. It's a simple idea that doesn't really need the computer to be operated (although that does make it emotionally easier to deal with); you could use stop orders to do some of this.

Essentially. we're going to resign ourselves to not taking all of the profits nominally available to us. We're going to leave some profits "on the table" as discretionary traders would put it. I would claim that the profits we leave on the table, the opportunity cost of our trading, are the risk premium which we are paying out in return for our aversion to losing our gains.

I implement an algorithm in which, when a trade is profitable, we take a profit. Specifically if we have a gross gain of G on a position, we cut the position to a fraction of the initial position. I chose the fraction 1/(1+G)^2, but that is essentially an arbitary amount. The important point is that we take some profits and the more profit that exists the more of it we take. This means that if a stock goes straight up we cannot ever capture all of the gains it makes, because we will book profits on the way up. The chart "Effect of Take Profit Algorithm" shows how much you are theoretically giving up.

Sometimes, of course, we don't get it right and the stock we bought sinks instead of rising; or we bought when it was 10% up on the earnings suprise but it settles to 5% up so we actually get a 5% loss. Whenever the trade is losing I apply a different algorithm. This one is based on holding time because I'm all for giving the trade a little time to turn around (especially since academic research indicates that the initial pop on an earnings suprise generally underestimates the final net response to the news). So I cut the position to a fraction exp(-d/5) of the initial positions, where d is the number of days since the trade entry. Again the exact formalism is arbitary, but the idea is that the longer it's been since the inital trade the more you should take off. Essentially what we're saying is that if it's been a long time since the initial trade idea we have to accept the fact that we're probably wrong.

The final step is that when we have adjusted our position, we "reset the clock" and treat the new position as a new trade. (I also make all my decisions on a beta adjusted basis relative to the S&P 500 benchmark, because ad hoc trades are about residual returns not systematic risk.)

I apply this algorithm automatically to all the ad-hoc trades I do. I have a completely automated environment so the algorithm just places orders for me, I don't have to work through the rules for each trade every day.

For example, at the end of July this year I decided to take a punt on Lehman Brothers Inc. (I'm a client, and have been for years, and I also know people who work there.) I thought that the housing/credit crisis had got through the worst and that things were going to look up from now on. I bought 2,500 shares of LEH for my personal account at $15.90.

Today LEH is trading at $7.98. Although the financial stocks initially turned up, it seemed that market had underestimated the extent of the problems at Lehman and the loss from entry-to-date would have been around 50%. However, the TPA got me out at a profit. I'm pretty certain I would have been caught up in the euphoria of my gains and not closed out until it was too late. My actual trading activity is illustrated in the table "Trading in Lehman Brothers."