# Default Correlation

**When a bank provides credit, a chance that an obligor cannot meet its obligations and defaults always exists. By providing credit to multiple obligors, the chance of simultaneous default also arises. This so called default correlation risk has appeared to be a major risk for banks and has gained growing attention in the current economic situation.**

**Introduction**

During my internship at Rabobank, I provided exemplifying insight into the impact of default correlation on Rabobank’s portfolio. To be able to survive adverse economic situations, banks have to hold a certain amount of capital as a safety buffer. From the bank’s perspective, it is important to determine this level of capital with care; holding capital is expensive, being solvent is crucial. Moreover, the capital level has to meet international regulatory standards that are described in the Basel Accords (Basel I (1988), Basel II (2004), Basel III (2010)).

Being one of the input parameters for determining risk capital, default correlation has appeared to be an important issue to banks. Default correlation is the probability that, given one obligor defaults, another obligor will default at the same time. In case of high default correlation extreme losses become more likely to occur, resulting in an increased amount of risk capital. Low default correlation implies less extreme losses, because there is a higher dependency on random events. Since the impact of default correlation on the firm’s portfolio loss might be significant, the size of default correlation should be determined accurately. However, due to low numbers of (simultaneous) default events, it is difficult to determine the default correlation between obligors. They hardly ever default at the same time. In order to get improved insight into the default correlation risk, Rabobank is currently revising its correlation framework.

Rabobank is the market leader in the Dutch mortgage market and a large part of Rabobank’s portfolio consists of mortgages. Therefore, evaluation of correlation with respect to the mortgage portfolio is an important part of the correlation framework revision. Next to the fact that the order of magnitude of the correlation between provided mortgages is of large interest, its influence on the diversification level on an overall portfolio level needs to be investigated. Therefore, I have conducted research on the impact of the mortgage correlation on the overall portfolio’s diversification level.

Economic Capital

In order to calculate risk capital, Rabobank uses internal models for calculating Economic Capital (EC). In accordance with the model described in the Basel accords, Rabobank uses the Vasicek model [1] as a basis for computing the EC for the mortgage portfolio. Vasicek’s model states that an obligor is in default if its asset value falls below a defined default threshold. The asset value is determined by a single factor model including a systematic factor and an idiosyncratic part. Every asset value is driven by a certain factor and has an idiosyncratic part representing the obligor’s specific risk, like divorces, illness, or just bad luck. The obligor’s exposure to the systematic factor can also be interpreted as the degree of dependence on the market represented by the factor. This factor dependence is also known as the R2 of a factor model. Vasicek assumes that every obligor has the same amount of correlation with the systematic factor, implying that there is only one correlation factor. During the rest of this article, this single correlation factor will be called intra correlation.

Although Vasicek’s model is widely used, the model has gained deserved criticism from both the scientific and practical point of view. An advantage of Vasicek’s model is its allowance for analytical calculation and the fact that the results on an exposure basis are additive. Yet, its assumption about one single factor driving the entire portfolio appears to be too much of a simplification. Vasicek’s assumption about one `state of the economy’ contradicts with practical situations, since apart from the financial situation in the world, portfolios can have several different fundamental factors, like country or industry specific risk. For this reason, in the internal model, a multifactor model is introduced.

By using a (multi)factor model and a defined default threshold, a default state can be assigned to each obligor. Using the default states in combination with obligor specific risk parameters (Loss Given Default (LGD) and Exposure at Default (EaD)), the loss distribution and subsequently the EC of the entire portfolio can be derived. The EC is equal to the loss in case of a stressed scenario (e.g. the loss at a 99.99% quantile) minus the expected loss. The expected losses are covered by provisions et cetera, while losses exceeding the chosen quantile are extreme losses and are not covered. These extreme losses simply result in a default for the bank itself. Therefore, the quantile has often a high value (e.g. 99.99% for an AAA-status).

**Obligors Correlated**

In most of the cases, a bank’s portfolio does not only consist of mortgages, but also contains other sub-portfolios like corporate loans (large firms), loans to SMEs (small and medium enterprises), and consumer loans (credit cards, saving accounts, et cetera). Amongst these sub-portfolios, not only the amounts of exposure can differ enormously, but also the corresponding risk profiles can take different forms. For instance, the sub -portfolio consisting of consumer loans is characterized by a large number of exposures that are rather small and constant, while the corporate sub -portfolio consists of a small subset of obligors having very large exposures.

As already mentioned, the correlations between obligors inside a portfolio (intra-correlation) influence the bank’s risk. However, not all correlation is covered by intra-correlation. Since a portfolio consists of several sub-portfolios, the correlation between obligors of different sub -portfolios should also be taken into consideration. To be more precise, this is exactly the correlation that results in diversification effects on a portfolio level and is determined by the sub-portfolio’s intra-correlation and the correlation between the sub-portfolio’s fundamental factors (interfactor correlation). Being interested in the impact of the intra-correlation of the mortgage portfolio as well as in the inter factor correlation, the analyses are based on both types of correlation.

Diversification

Using intra- and interfactor correlations for the asset valuation model results in a portfolio specific loss distribution as well as an EC level. Since diversification is the main topic in this research, the next step is the transfer from EC to EC diversification. Diversification effects arise if sub-portfolios are not fully correlated and complement each other in some way. In order to compute the level of diversification, first the required amount of EC for every sub -portfolio is determined on a stand-alone basis. This means that each of the sub-portfolios is not liable for the shortfall of the other one. By simply taking the sum of the stand-alone EC of all sub-portfolio’s, the summed stand-alone portfolio EC is obtained. This is the total amount of EC needed in case sub-portfolios are assumed to be all individual. Being interested in the EC in case sub-portfolios are integrated in one large portfolio, the EC is also determined on an aggregated portfolio level. A diversification effect arises when the EC level for the aggregated portfolio deviates from the summed stand-alone EC and is formulated by:

Diversification effect=1-(Aggregated portfolio EC)/(Summed stand-alone portfolio EC).

Since generally it is assumed that portfolio aggregation reduces EC and EC is always positive, the diversification effect is a number between zero and one and can be interpreted as the decrease in EC as a result of aggregating sub-portfolios in the total portfolio. In other words, the larger the diversification effect, the larger the impact of aggregation on the portfolio’s EC.

**Portfolio Definition**

By now, a theoretical framework is formulated. However, the `ingredients’, or input parameters, are still needed. These parameters are all based on a portfolio, which contains a set of obligors having their own risk profile. In order to meet Rabobank’s need for relevance and desire for privacy, based on its portfolio, a hypothetical (simplified) portfolio is defined. Since its portfolio exposure consists mainly of corporate loans and mortgages, the hypothetical portfolio contains only these two sub-portfolios, which differ in their characterizations. The mortgage portfolio typically contains a large number of exposures, which are rather small compared to the large exposures in the corporates portfolio. The corporate portfolio, however, includes a small selection of obligors. Furthermore, the sub-portfolios are assumed to be dependent on just two factors, one representing the Dutch market and the other one representing the world economy. Obviously, the mortgage portfolio is (almost) completely dependent on the Dutch market, while the obligors in the corporate portfolio are likely to have (large) international exposure. In this portfolio, interfactor correlation is regarded as the correlation between the Dutch market and the world market, which is assumed to be quite high.

**Results**

Inserting the defined hypothetical portfolio parameters into the theoretical framework, gives the diversification effects for varying mortgage intra-correlation and inter factor correlation. The first result is that the sub-portfolios stand-alone EC is highly sensitive to the correlation between the contracts that are inside the sub-portfolio. An explanation for this is the increased probability for extreme losses in case obligors become more correlated. The increased chance on large losses results in a thinner tail of the loss distribution, thus a higher quantile’s loss. This result implies that diversifying sub-portfolios carefully is worthwhile in order to decrease the amount of risk capital needed. However, the level of diversification intra-correlation does not have a significant impact and mostly stays within a range of only a few percents.

When investigating the impact of the interfactor correlation, it appears to have more effect on the diversification level. When the interfactor correlation grows to one, the diversification level declines to zero as a result of the convergence to a single factor model. Then, no benefit can be obtained by merging sub-portfolios as a result of dependency on the same market.

However, both types of correlation do not influence the level of diversification remarkably. The small shift in the level of diversification might be attributed to the composition of the hypothetical portfolio, which appeared to be skewed with respect to risk exposure. The risk exposure to the mortgage portfolio is large compared to the total portfolio, resulting in rather low sensitivity of the diversification level with respect to changing intra- correlation of the mortgage portfolio. Since the results seem to be highly portfolio-dependent, further portfolio sensitivity analysis is done. A change in portfolio composition appears to be accompanied by a change in the diversification level.

**Conclusions**

From the results obtained, it can be concluded that correlations should be determined with great care. Different correlation values can result in wide-ranging values for stand-alone EC for intra-portfolio correlation and varying diversification levels for different inter factor correlations. Furthermore, regarding the observation that the results are highly portfolio dependent, a review of the impact analysis is recommended in case of changing portfolio compositions.

*Text by: Emy van der Wielen*

**References**

(1) Vasicek, O. (1987), Probability of loss on loans portfolio (Technical Report). San Francisco (USA): KMV Corporation.