Tranche Declension.

800px-CDO_-_FCIC_and_IMF_Diagram

With the CDO (collateralized debt obligation) market picking up, it is important to build a stronger understanding of pricing and risk management models. The role of the Gaussian copula model, has well-known deficiencies and has been criticized, but it continues to be fundamental as a starter. Here, we draw attention to the applicability of Gaussian inequalities in analyzing tranche loss sensitivity to correlation parameters for the Gaussian copula model.

We work with an RN-valued Gaussian random variable X = (X1, … , XN), where each Xj is normalized to mean 0 and variance 1, and study the equity tranche loss

L[0,a] = ∑m=1Nlm1[xm≤cm] – {∑m=1Nlm1[xm≤cm] – a}

where l1 ,…, lN > 0, a > 0, and c1,…, cN ∈ R are parameters. We thus establish an identity between the sensitivity of E[L[0,a]] to the correlation rjk = E[XjXk] and the parameters cj and ck, from where subsequently we come to the inequality

∂E[L[0,a]]/∂rjk ≤ 0

Applying this inequality to a CDO containing N names whose default behavior is governed by the Gaussian variables Xj shows that an increase in name-to-name correlation decreases expected loss in an equity tranche. This is a generalization of the well-known result for Gaussian copulas with uniform correlation.

Consider a CDO consisting of N names, with τj denoting the (random) default time of the jth name. Let

Xj = φj-1(Fjj))

where Fj is the distribution function of τj (relative to the market pricing measure), assumed to be continuous and strictly increasing, and φj is the standard Gaussian distribution function. Then for any x ∈ R we have

P[Xj ≤ x] = P[τj ≤ Fj-1j(x))] = Fj(Fj-1j(x))) = φj(x)

which means that Xj has standard Gaussian distribution. The Gaussian copula model posits that the joint distribution of the Xj is Gaussian; thus,

X = (X1, …., Xn)

is an RN-valued Gaussian variable whose marginals are all standard Gaussian. The correlation

τj = E[XjXk]

reflects the default correlation between the names j and k. Now let

pj = E[τj ≤ T] = P[Xj ≤ cj]

be the probability that the jth name defaults within a time horizon T, which is held constant, and

cj = φj−1(Fj(T))

is the default threshold of the jth name.

In schematics, when we explore the essential phenomenon, the default of name j, which happens if the default time τis within the time horizon T, results in a loss of amount lj > 0 in the CDO portfolio. Thus, the total loss during the time period [0, T] is

L = ∑m=1Nlm1[xm≤cm]

This is where we are essentially working with a one-period CDO, and ignoring discounting from the random time of actual default. A tranche is simply a range of loss for the portfolio; it is specified by a closed interval [a, b] with 0 ≤ a ≤ b. If the loss x is less than a, then this tranche is unaffected, whereas if x ≥ b then the entire tranche value b − a is eaten up by loss; in between, if a ≤ x ≤ b, the loss to the tranche is x − a. Thus, the tranche loss function t[a, b] is given by

t[a, b](x) = 0 if x < a; = x – a, if x ∈ [a, b]; = b – a; if x > b

or compactly,

t[a, b](x) = (x – a)+ – (x – b)+

From this, it is clear that t[a, b](x) is continuous in (a, b, x), and we see that it is a non-decreasing function of x. Thus, the loss in an equity tranche [0, a] is given by

t[0,a](L) = L − (L − a)+

with a > 0.

In Praise of Libertarianism. Drunken Risibility

The-True-Political-Spectrum

Devotion to free markets is a sin??? Nah!!!. Like quantitative induction and philosophical deduction, economics has always had a political purpose, and the purpose has usually been libertarian. Economists are freedom nuts, which is to say that they look with suspicion on lawyerly plans to solve problems with new state compulsions and longer jail sentences. Economics at its philosophical birth, among physiocrats in Paris and moral philosophers in Edinburgh, was in favor of free markets and was suspicious of overblown states. Mostly it still is. Let things be, laissez faire, has been the economists’ cry against intervention. Let the trades begin.

True, not all economists are free traders. The non-free traders, often European and disproportionately French, point out that you can make other assumptions about how trade works, A’, and get other conclusions, C’, not so favorable to laissez faire. The free-trade theorem, which sounds so grand, is actually pretty easy to overturn. Suppose a big part of the economy – say the household – is, as the economists put it, “distorted” (e.g., suppose people in households do things for love: you can see that the economists have a somewhat peculiar idea of “distortion”). Then it follows rigorously (that is to say, mathematically) that free trade in other sectors (e.g., manufacturing) will not be the best thing. In fact it can make the average person worse off than restricted, protected, tariffed trade would.

And of course normal people – meaning non-economists – are not persuaded that free trade is always and everywhere a good thing. For example most people think free trade is a bad thing for the product or service they make. But, the reality is to think the need to blockade entry into the profession of being an economist: it is, in all agreement, scandalous that so many unqualified quacks are bilking consumers with adulterated economics.

And very many normal people of leftish views, even after communism, even after numerous disastrous experiments in central planning, think socialism deserves a chance. They think it obvious that socialism is after all fairer than unfettered capitalism. They think it obvious that regulation is after all necessary to restrain monopoly. They don’t realize that free markets have partially broken down inequality (for example, between men and women; “partially”) and partially undermined monopolies (for example, local monopolies in retailing) and have increased the income of the poor over two centuries by a factor of 18. The felony lies in, the lefties think, in exactly its free-market bias.

But, my dearly beloved friends on the left, think, think again. There really is a serious case to be made against government intervention and in favor of markets. Maybe not knockdown; maybe imperfect here or there; let’s chat about it; hmm, a serious case that serious people ought to take seriously. The case in favor of markets is on the contrary populist and egalitarian and person-respecting and bad-institution-breaking libertarianism. Don’t go to government to solve problems, said Adam Smith. As he didn’t say, to do so is to put the fox in charge of the hen house. The golden rule is, those who have the gold rule: so don’t expect a government run by men to help women, or a government run by Enron executives to help Enron employees.

Libertarianism is typical of economics, especially English-speaking economics, and most especially American economics. Most Americans if they can get clear of certain European errors, are radical libertarians under the skin. Give me liberty. Sweet land of liberty. Live free or die. But alas, no time, no time. Libraries of books have been written examining the numerous and weighty arguments for the market and against socialism. Really, that the average literary person believes the first few pages of The Communist Manifesto suffice for knowledge of economics and economic history, in which he professes great interest, is a bit of a scandal. As Cromwell said wearily to the General Assembly of the Church of Scotland, 3 August, 1650, “I beseech you, in the bowels of Christ, think it possible you may be mistaken.” Oh, permit one short libertarian riff.

Nor is government obstruction peculiar to the present-day Third World. In one decade in the eighteenth century, according to the Swedish economist and historian Eli Heckscher in his book, Mercantilism, the French government sent tens of thousands of souls to the galleys and executed 16,000 (that’s about 4.4 people a day over the ten years: you see the beauty of statistical thinking) for the hideous crime of… are you ready to hear the appalling evil these enemies of the State committed, fully justifying hanging them all, every damned one of their treasonable skins? … importing printed calico cloth. States do not change much from age to age. In view of How Muches and Oh, My Gods like these – the baleful oomph of governmental intrusions world-wide crushing harmless (indeed, beneficial) exchange, from marijuana to printed calico – perhaps laissez faire does not seem so obviously sinful, does it now? Consider, my dear leftist friends. Read and reflect. I beseech you, think it possible that, like statistics and mathematics, the libertarianism of economics is a virtue.