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.