Introduction
The bank may use its internal model to calculate specific risk if it can demonstrate the model sufficiently captures specific risk. The capital rules also stipulate that the model should explain the historical price variation in the portfolio and capture potential concentrations, including magnitude and changes in composition. Finally, the model should be sufficiently robust to capture greater volatility due to adverse market conditions. If the bank's internal model cannot meet these requirements, the bank must use the standardized approach to measuring specific risk under the capital rules.
For the fixed income portfolio, total market risk is divided into general market risk and specific risk. General market risk is caused by the systematic credit spread movement (systematic risk). Individual credit spread movement (idiosyncratic risk) and credit migration of obligors (transition/default risk) are two main sources of specific risk. In practice, the risk managers have two questions related to the latter source of specific risk:
Can the bank assign zero correlation between the credit migration of distinct obligors to calculate the transition/default risk in short time horizon?
To calculate the transition/default risk, risk managers should define the distribution of credit migration events. The rating agencies have provided historical averages and volatilities (the so called transition matrix and its standard deviation), but not the correlation of credit migration events. If the correlation is significantly larger than zero, the transition/default risk is greater because default and downgrade events are concentrated in a small number of outcomes.
Can the bank assign zero correlation between the credit migration events and the systematic market risk factors in short time horizons when aggregating systematic risk and transition/default risk?
By definition, idiosyncratic risk is independent of the other two risks. But is transition/default risk independent of systematic risk? The most important systematic market risk factors for the fixed income portfolio are the yield curve and average credit spreads across ratings, which are in turn correlated to the stock market index. If the factors significantly affect the credit migration events, large systematic risk induces large transition/default risk. Then, aggregating the three risks under the assumption of independence underestimates the total market risk. If the answers to both above questions are YES, how can risk managers prove it? Following the capital rules, the bank has the burden of proof for its internal model of the reduced specific risk.
To relieve headaches of the risk managers, we try to answer two questions in this paper. Specifically, we test whether the credit migration correlation of obligors is significantly larger than zero and whether the correlation between the credit migration events and the systematic market risk factors is significantly larger than zero. It is worth noting that our tests focus on monthly or bi-weekly horizons and contemporaneous correlations, which The bank may use its internal model to calculate specific risk if it can demonstrate the model sufficiently captures specific risk. The capital rules also stipulate that the model should explain the historical price variation in the portfolio and capture potential concentrations, including magnitude and changes in composition. Finally, the model should be sufficiently robust to capture greater volatility due to adverse market conditions. If the bank's internal model cannot meet these requirements, the bank must use the standardized approach to measuring specific risk under the capital rules.
For the fixed income portfolio, total market risk is divided into general market risk and specific risk. General market risk is caused by the systematic credit spread movement (systematic risk). Individual credit spread movement (idiosyncratic risk) and credit migration of obligors (transition/default risk) are two main sources of specific risk. In practice, the risk managers have two questions related to the latter source of specific risk:
Can the bank assign zero correlation between the credit migration of distinct obligors to calculate the transition/default risk in short time horizon?
To calculate the transition/default risk, risk managers should define the distribution of credit migration events. The rating agencies have provided historical averages and volatilities (the so called transition matrix and its standard deviation), but not the correlation of credit migration events. If the correlation is significantly larger than zero, the transition/default risk is greater because default and downgrade events are concentrated in a small number of outcomes.
Can the bank assign zero correlation between the credit migration events and the systematic market risk factors in short time horizons when aggregating systematic risk and transition/default risk?
By definition, idiosyncratic risk is independent of the other two risks. But is transition/default risk independent of systematic risk? The most important systematic market risk factors for the fixed income portfolio are the yield curve and average credit spreads across ratings, which are in turn correlated to the stock market index. If the factors significantly affect the credit migration events, large systematic risk induces large transition/default risk. Then, aggregating the three risks under the assumption of independence underestimates the total market risk.
If the answers to both above questions are YES, how can risk managers prove it? Following the capital rules, the bank has the burden of proof for its internal model of the reduced specific risk.
To relieve headaches of the risk managers, we try to answer two questions in this paper. Specifically, we test whether the credit migration correlation of obligors is significantly larger than zero and whether the correlation between the credit migration events and the systematic market risk factors is significantly larger than zero. It is worth noting that our tests focus on monthly or bi-weekly horizons and contemporaneous correlations, which are relevant to the specific risk for the calculation of the total market risk. Therefore, this paper is not designed to completely cover the integration of market and credit risk. There are a very small number of previous studies about the default correlation. Lucas (1995) derived formulas to take the default correlation into account when calculating joint default probability. Li (2000) introduced a copula function approach to measure the default correlation as a function of time horizon. Erturk (2000) measured the default correlation using the relationship between default correlation and default volatility.
Since credit migration events in the short time horizon are too rare to calculate the correlation directly, we utilize both the factor model and asset value model. The factor model regresses the default and/or downgrade rate and the profit and loss (P&L) of an artificial fixed income portfolio on financial and macroeconomic variables. The systematic movement explained by the factors is interpreted as the source of the correlation between obligors rating migrations. Thus, the relative size of the portfolio's systematic and non-systematic movements shows the size of the credit migration correlation.
The asset value model assumes that the credit migration correlation reflects the asset value correlation based on the framework of Merton's option theoretic valuation of a firm. Then, the implied asset value correlation is calibrated to match the volatility of the historical credit migration rate.
The estimated results of both approaches show that the credit migration correlation between obligors decreases as the time horizon becomes shorter and the correlation is near zero at monthly and bi-weekly horizons. Furthermore, the correlation between credit migration events and systematic market risk factors is not significantly larger than zero at monthly or bi-weekly horizons. These results indicate that the use of positive credit migration correlations to calculate specific risk for trading portfolios results in an overestimation of required capital.
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