**Some Thoughts on Default Methods
**

*Summary: Asset defaults (ratio of events) are statistically different from dollar defaults (function of ratio of magnitudes).*

**Multiple Distributions:** I had originally thought I’d just discuss long tails, but found that some matters needed to be clarified before discussing long tails. Individual asset losses have fat and long tails; the result of default and loss severities that obey binomial, lognormal or gamma distributions.

**Default Methods:** There are only 2 broad methods of determining default probabilities in the mortgage industry. The first default method is *asset default Pa* defined as the number of assets defaulted divided by the total number of assets in the portfolio. This is a statistic of *proportion or ratio of events*.

The second is what I term *dollar defaults* *Pd* (a.k.a. structural models). A dollar default is said to have occurred when the ratio of the *default boundary value* to *original value* decreases below a specific value. I use term them *dollar default* because they are primarily driven by the *ratio of magnitudes* to estimate credit risk; *severity of loss* and *1 – severity of loss* are examples. These are statistics of *proportion or ratio of magnitude* and we can term these ratios *severity of loss type statistic*.

**Industry usage:** The 2 ways this is used in CMBS are:

**(1) CDR (Constant Default Rate):** CDR is the ratio of *outstanding balance at default* (*default boundary value*) divided by *original principal balance (original value).* In CMBS deal structuring CDRs are presented as a time series of ratios a.k.a. loss vectors; *severity of loss* in its most basic form. There is no need to model defaults as they are assumed to have occurred (very neat!) and severity of loss is predetermined by the CDR statistic.

**(2) DSCR loss models:** A default occurs when DSCR gets below 1.0. Property cash flows are reduced by 2% (or some suitable value) per annum until this default event occurs. The *ratio of magnitude*, the ratio of the *outstanding principal balance* (*default boundary value*) to the *original principal balance* (*original value*) is determined when DSCR drops below 1.0. To determine *severity of loss*, the default event is specified by a *rate* of deterioration of cash flows, which is itself a *ratio of magnitudes*.

___________________________________________________________________

**Empirical Data Confirms Biases**

*Summary: Empirical data confirms dollar default biases*

**Empirical Confirmation:** Empirical research (by others) confirm that dollar default methods (a.k.a. structural models) underestimate default probabilities. My own research on DSCR loss models concur with these results, that DSCR loss methods underestimate losses early in the life of a loan. My concern is not so much with these models’ expected values as with the shape of their tails.

**Illustration:**In a *non-rigorous* way we can illustrate why. Dollar defaults *Pd*, as a function of *proportion of magnitude*, have a different statistical behavior from asset defaults *Pa*, *a proportion of events*. We can see this by writing assets & dollar defaults, respectively, as some function of economic & industry factors *f(x)*:

Asset defaults as a function of economic and industry factors:

*Pa = f(x) = number of default events / total number of assets*

Dollar defaults as a function of economic, industry and asset size, s:

*Pd = g( f(x), s) = some function of ($ outstanding balance / $ original balance)*

* *

Using 2 portfolios to illustrate. Portfolio A consists of 2 assets of $100,000 each, and portfolio B consists of 3 assets of $100,000 each. Should one asset in each portfolio experience a loss (a good assumption if defaults are small) of $70,000, Portfolio A’s loss is 35% (70,000/200,000) & B’s is 23% (70,000/300,000).

**Different Statistics:** However, Portfolio A’s asset default rate is 50% (1/2) and Portfolio B’s is 33% (1/3) but their respective severities are 35% & 23%, and being less could result in an underestimation of asset default probabilities. But wait. Should the loss have been $20,000 then Portfolio A’s & B’s losses are 1% and 7% respectively, but the asset default would still be 50% & 33% respectively. That is, for each asset default there are multiple severity of losses, and therefore, dollar and asset defaults have different underlying statistical behaviors.

The 2 figures (click on figures to enlarge) above, Test 1 & Test 2, show very different statistical distribitions that are dependent on the underlying nature of the risk drivers. It is clear from the graphs that the probability distributions of these *severity of loss type statistics* used to generate defaults, do not exhibit Binomial behaviors; Test 2 is not Lognormal, and Test 2’s tail is much fatter and longer than Test 1’s.

**Alternative Explanation:** Researches currently believe that this consistent underestimation of default probabilities is due to missing factors such as liquidity and recovery. But including recovery will only reduce the severity of loss statistic and would further depress the dollar default estimations. My analysis, however, suggests an alternative explanation for the underestimation, that of different statistical properties.

**Undesirable Statistic:** The dollar defaults statistical properties may even be undesirable. Using the form *sum of (probability of default x outstanding balance at default) *to estimate expected portfolio loss, we see that dollar defaults introduce asset size ** twice** and asset defaults only

*. Therefore, dollar default methodologies may not be desirable for determining default probability.*

**once****Beta Distribution:** An additional caution for those of you who model default & severity of loss. In my opinion, using the beta distribution is an assurance that your results are incorrect. Why? In my 30+ years working with large data sets, the beta distribution is the single most unstable distribution I have come across. This distribution will change shape when you are not looking! It is so unstable that small changes in its parameters can lead to significant changes in its shape.

___________________________________________________________________

**Reducing Impact of Loss Tails
**

*Summary: Portfolios alter the shape of the tail for the better.*

Therefore, we drop the use of dollar default methods. Most of us use portfolio diversification to reduce risk as measured by standard deviation of returns. But portfolios have little known properties, they can reduce the effect & change the shape of long tails.

**Severity Reduction:** A portfolio consists of many assets, and each asset will have default probabilities and loss severities associated with it. All other factors being equal, the impact of a portfolio’s tail loss can be reduce by increasing the number of assets in the portfolio. Using the 2 portfolios above to illustrate this; the severity of loss of Portfolio A is 35% but that of Portfolio B is 23%. The severity of loss to a portfolio is reduced by the size of the portfolio (given all other factors being equal).

**Shape Change:** Taking this a step further CMBS loss severities tend to be Gamma distributions while portfolio losses ought to approach Normal distributions (*but not quite*). Therefore for the same mean & standard deviation, the Gamma’s tail can be 25 to 35 times longer than the Normal’s tail. Why not quite?. In lay man’s terms, the Central Limit Theorem justifies the approximation of large-sample statistics with the normal distribution, and therefore large portfolio statistics should look Normal. However, default probabilities tend to be small, in the 1 to 2% range. Therefore, there aren’t enough observations to substantially shrink the loss tail, and therefore, appear lognormal or at least skewed to the right.

**Multiple Properties:** Likewise, having multiple properties (and beware of cross collaterized loans, they are usually synonymous with multi-property loans) under a single mortgage can lead to catastrophic failure if there are a few properties. The loan defaults if a single property’s loss of income causes the loan’s DSCR to drop below 1.0. In this case a multiple property mortgage magnifies the effect of a single default. To reduce this impact one needs to either reduce the number of these assets (loans) in the deal or increase the number of properties in the mortgage. However, the latter is not a good solution as it defeats the purpose of deal structuring.

**Spatial Correlations:** Another problem with multiple property mortgages is that these properties tend to be in the same MSA (*Metropolitan Statistical Area*), and therefore are at risk to *spatial correlations* (see for example Prof. Tom Thibodeau, CU Boulder) that properties in close proximity tend to rise and fall together.

**Multiple Liens:** More obviously, the reverse is also true, multiple mortgages on a single property causes all loans to be in default should the property’s income fall. In this case the mortgages should be assigned to different portfolios, thereby reducing the severity of loss to a specific portfolio.

**Wrong Signals:** Note that RBS has tried an approach to reduce underwriter’s risk by not closing the loans as they are pooled. Interesting. While it does not reduce investors’ risk, don’t you think this sends the wrong market signals?

___________________________________________________________________

**Some Lessons
**

*Summary: Some lessons from a loss perspective.*

1. Avoid single-mortgage-multiple-property (& cross collateralized) assets (loans).

2. Avoid CMBS deals with multiple cross collateralized assets as portfolio diversification may not be what it appears to be.

3. Multiple-mortgages-single-property assets reduce risk for the same total principal.

4. My experience with CMBS data suggests that CMBS deals should be in the 150+ asset range. The RBS $309.7 million, 81 property deal is small, and it should be interesting to see how a small deal at the bottom of the market fares in the future.

5. Check your methodology.

___________________________________________________________________

Disclosure: I’m a capitalist too, and my musings & opinions on this blog are for informational/educational purposes and part of my efforts to learn from the mistakes of other people. Hope you do, too. These musings are not to be taken as financial advise, and are based on data that is assumed to be correct. Therefore, my opinions are subject to change without notice. This blog is not intended to either negate or advocate any persons, entity, product, services or political position. Nor is this blog post to be construed as investment advice.

Contact: Ben Solomon, Managing Principal, QuantumRisk

___________________________________________________________________