LO 6.6: Relate correlation risk to systemic and concentration risk.

A major concern for risk managers is the relationship between correlation risk and other types of risk such as market, credit, systemic, and concentration risk. Examples of major factors contributing to market risk are interest rate risk, currency risk, equity price risk, and commodity risk. As discussed earlier, risk managers typically measure market risk in terms of VaR. Because the covariance matrix of assets is an important input of VaR, correlation risk is extremely important. Another important risk management tool used to quantify market risk is expected shortfall (ES). Expected shortfall measures the impact of market risk for extreme events or tail risk. Given that correlation risk refers to the risk that the correlation between assets changes over time, the concern is how the covariance matrix used for calculating VaR or ES changes over time due to changes in market risk.

Risk managers are also concerned with measuring credit risk with respect to migration risk and default risk. Migration risk is the risk that the quality of a debtor decreases following the lowering of quality ratings. Lower debt quality ratings imply higher default probabilities. When a debt rating decreases, the present value of the underlying asset decreases, which creates a paper loss. As discussed previously, correlation risk between a reference asset and counterparty (CDS seller) is an important concern for investors. A higher correlation increases the probability of total loss of an investment.

Financial institutions such as mortgage companies and banks provide a variety of loans to individuals and entities. Default correlation is of critical importance to financial institutions in quantifying the degree that defaults occur at the same time. A lower default correlation is associated with greater diversification of credit risk. Empirical studies have examined historical default correlations across and within industries. Most default correlations across industries are positive with the exception of the energy sector. The energy sector has little or no correlation with other sectors and is, therefore, more resistant to recessions.

Historical data suggests that default correlations are higher within industries. This finding implies that systematic factors impacting the overall market and credit risk have much more influence in defaults than individual or company-specific factors. For example, if Chrysler defaults, then Ford and General Motors are more likely to default and have losses rather than benefit from increased market share. Thus, commercial banks limit exposures within a specific industry. The key point is that creditors benefit by diversifying exposure across industries to lower the default correlations of debtors.

Risk managers can also use a term structure of defaults to analyze credit risk. Rating agencies such as Moodys provide default probabilities based on bond ratings and time to maturity as illustrated in Figure 8.

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Figure 8: Default Term Structure for A- and CC-Rated Bonds

A-Rating

CC-Rating

25%

20%

15%

10%

5%

0%

g o u

$

Q

1

2

3

4

5

6

7

8

9

10

Years to Maturity

Notice in Figure 8 that the default term structure increases slightly with time to maturity for most investment grade bonds (solid line). This is expected because bonds are more likely to default as many market or company factors can change over a longer time period. Conversely, for non-investment grade bonds (dashed line), the probability of default is higher in the immediate time horizon. If the company survives the near-term distressed situation, the probability of default decreases over time.

Lehman Brothers filed for bankruptcy in September of 2008. This bankruptcy event was an important signal of the severity of the financial crisis and the level of systemic risk. Systemic risk refers to the potential risk of a collapse of the entire financial system. It is interesting to examine the extent of the stock market crash that began in October 2007. From October 2007 to March 2009, the Dow Jones Industrial Average fell over 50% and only 11 stocks increased in the entire Standard & Poors 500 Index (S&P 500). The decrease in value of 489 stocks in the S&P 500 during this time period reflected how a systemic financial crisis impacts the economy with decreasing disposable income for individuals, decreasing GDP, and increasing unemployment.

The sectors represented in the 11 increasing stocks were consumer stables (Family Dollar, Ross Stores, and Walmart), educational (Apollo Group and DeVry Inc.), pharmaceuticals (Edward Lifesciences and Gilead Pharmaceuticals), agricultural (CF Industries), entertainment (Netflix), energy (Southwestern Energy), and automotive (AutoZone). The consumer staples and pharmaceutical sector are often recession resistant as individuals continue to need basic necessities such as food, household supplies, and medications. The educational sector is also resilient as more unemployed workers go back to school for education and career changes.

Studies examined the relationship between the correlations of stocks in the U.S. stock market and the overall market during the 2007 crisis. From August of 2008 to March of 2009, there was a freefall in the U.S. equity market. During this same time period, correlations of stocks with each other increased dramatically from a pre-crisis average

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correlation level of 27% to over 50%. Thus, when diversification was needed most during the financial crisis, almost all stocks become more highly correlated and, therefore, less diversified. The severity of correlation risk is even greater during a systemic crisis when one considers the higher correlations of U.S. equities with bonds and international equities.

Concentration risk is the financial loss that arises from the exposure to multiple counterparties for a specific group. Concentration risk is measured by the concentration ratio. A lower (higher) concentration ratio reflects that the creditor has more (less) diversified default risk. For example, the concentration ratio for a creditor with 100 loans of equal size to different entities is 0.01 (= 1 / 100). If a creditor has one loan to one entity, the concentration ratio for the creditor is 1.0 (= 1 / 1). Loans can be further analyzed by grouping them into different sectors. If loan defaults are more highly correlated within sectors, when one loan defaults within a specific sector, it is more likely that another loan within the same sector will also default. The following examples illustrate the relationship between concentration risk and correlation risk.

Example: Concentration ratio for bank X and one loan to company A

Suppose commercial bankX makes a $5 million loan to company A, which has a 5% default probability. What is the concentration ratio and expected loss (EL) for commercial bank X under the worst case scenario? Assume loss given default (LGD) is 100%.

Answer:

Commercial bank X has a concentration ratio of 1.0 because there is only one loan. The worst case scenario is that company A defaults resulting in a total loss of loan value. Given that there is a 5% probability that company A defaults, EL for commercial bank X is $250,000 (= 0.05 x 5,000,000).

Example: Concentration ratio for bank Y and two loans to companies A and B

Suppose commercial bank Y makes a $2,500,000 loan to company A and a $2,500,000 loan to company B. Assuming companies A and B each have a 5% default probability, what is the concentration ratio and expected loss (EL) for commercial bank Y under the worst case scenario? Assume default correlation between companies is 1.0 and loss given default (LGD) is 100%.

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Answer:

Commercial bank Y has a concentration ratio of 0.5 (calculated as 1 / 2). The expected loss for commercial bank Y depends on the default correlation of companies A and B. Note that changes in the concentration ratio are directly related to changes in the default correlations. A decrease in the concentration ratio results in a decrease in the default correlation. The default of companies A and B can be expressed as two binomial events with a value of 1 in default and 0 if not in default.

Figure 9 illustrates the joint probability that both companies A and B are in default, P(AB).

Figure 9: Joint Probability of Default for Companies A and B

P(AB)

The following equation computes the joint probability that both companies A and B are in default at the same time:

P(AB) = pAB JP D a (1 -P D a ) x PDb (1 – P D b) + PDA x PDB

where: Pa b ^/PD^(1 PD a ) = standard deviation of the binomial event A

= default correlation coefficient for A and B

The default probability of company A is 5%. Thus, the standard deviation for company A is:

V0.05(l – 0.05) = 0.2179

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Company B also has a default probability of 5% and, therefore, will also have a standard deviation of 0.2179. We can now calculate the expected loss under the worst case scenario where both companies A and B are in default. Assuming that the default correlation between A and B is 1.0, the joint probability of default is:

P(AB) = 1.0^0.05(0.95) x 0.05(0.95) + 0.05 x 0.05 = lW O.00226 + 0.0025 = 0.05

If the default correlation between companies A and B is 1.0, the expected loss for commercial bank Y is $250,000 (0.05 x $5,000,000). Notice that when the default correlation is 1.0, this is the same as making a $5 million loan to one company.

Now, lets assume that the default correlation between companies A and B is 0.5. What is the expected loss for commercial bank Y? The joint probability of default for A and B, assuming a default correlation of 0.5, is:

P(AB) = 0.5^000226 + 0.0025 = 0.02625

Thus, the expected loss for the worst case scenario for commercial bank Y is:

EL = 0.02625 x $5,000,000 = $131,250

If we assume the default correlation coefficient is 0, the joint probability of default is 0.0025 and the expected loss for commercial bank Y is only $12,500. Thus, a lower default correlation results in a lower expected loss under the worst case scenario.

Example: Concentration ratio for bank Z and three loans to companies A, B, and C

Now we can examine what happens to the joint probability of default (i.e., the worst case scenario) if the concentration ratio is reduced further. Suppose that commercial bank Z makes three $1,666,667 loans to companies A, B, and C. Also assume the default probability for each company is 5%. What is the concentration ratio for commercial bank Z, and how will the joint probability be impacted?

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Answer:

Commercial bank Z has a concentration ratio of 0.333 (calculated as 1 / 3). Figure 10 illustrates the joint probability of all three loans defaulting at the same time, P(ABC) (i.e., the small area in the center of Figure 10 where all three default probabilities overlap). Note that as the concentration ratio decreases, the joint probability also decreases.

Figure 10: Joint Probability of Default for Companies A, B, and C

Professor Note: The assigned reading did not cover the calculation o f the joint probability for three binomial events occurring. The focus here is on understanding that as the concentration ratio decreases, the probability of the worst case scenario also decreases. Both a lower concentration ratio and lower correlation coefficient reduce the joint probability o f default.

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K e y C o n c e p t s

LO 6.1

Correlation risk measures the risk of financial loss resulting from adverse changes in correlations between financial or nonfinancial assets. For example, financial correlation risk can result from the negative correlation between interest rates and commodity prices. For almost all correlation option strategies, a lower correlation results in a higher option price.

LO 6.2

In May of 2005, several large hedge funds had losses on both sides of a hedged position short the collateralized debt obligation (CDO) equity tranche spread and long the CDO mezzanine tranche. The decrease in default correlations in the mezzanine tranche led to losses in the mezzanine tranche.

American International Group (AIG) and Lehman Brothers were highly leveraged in credit default swaps (CDSs) during the recent financial crisis. Their financial troubles revealed the impact of increasing default correlations with tremendous leverage.

LO 6.3

A correlation swap is used to trade a fixed correlation between two assets with the realized correlation. The payoff for the investor buying the correlation swap is:

notional amount x (prealized – pfixed)

where: realized P

realized

LO 6.4

Value at risk (VaR) for a portfolio measures the potential loss in value for a specific time period for a given confidence level:

VaRp = crpaVx

The VaR for a portfolio increases as the correlation between assets increase. The Basel Committee on Banking Supervision requires banks to hold capital for assets in the trading book of at least three times greater than 10-day VaR (i.e., VaR capital charge = 3 x 10-day VaR).

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LO 6.5

The covariance matrix of assets is an important input for value at risk (VaR) and expected shortfall (ES). These risk management tools are sensitive to changes in correlation.

A lower default correlation is associated with greater diversification of credit risk. Creditors benefit by diversifying exposure across industries to lower the default correlations of debtors. The default term structure increases with time to maturity for most investment grade bonds. The probability of default is higher in the immediate time horizon for non- investment grade bonds.

LO 6.6

Systemic risk refers to the potential risk of a collapse of the entire financial system. The severity of correlation risk is even greater during a systemic crisis considering the higher correlations of U.S. equities with bonds and international equities.

Changes in the concentration risk, which is measured by the concentration ratio, are directly related to changes in default correlations. A lower concentration ratio and lower correlation coefficient both reduce the joint probability of default.

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C o n c e p t C h e c k e r s

1.

2.

3.

4.

Suppose an individual buys a correlation swap with a fixed correlation of 0.2 and a notional value of $ 1 million for one year. The realized pairwise correlations of the daily log returns at maturity for three assets are p2 j = 0.7, p3 j = 0.2, and p3 2 = 0*3. What is the correlation swap buyers payoff at maturity? A. $100,000. B. $200,000. C. $300,000. D. $400,000.

Suppose a financial institution has a two-asset portfolio with $7 million in asset A and $5 million in asset B. The portfolio correlation is 0.4, and the daily standard deviation of returns for asset A and B are 2% and 1%, respectively. What is the 10- day value at risk (VaR) of this portfolio at a 99% confidence level (a = 2.33)? A. $ 1.226 million. B. $1,670 million. C. $2,810 million. D. $3,243 million.

In May of 2003, several large hedge funds had speculative positions in the collateralized debt obligations (CDOs) tranches. These hedge funds were forced into bankruptcy due to the lack of understanding of correlations across tranches. Which of the following statements best describes the positions held by hedge funds at this time and the role of changing correlations? Hedge funds held a: A. long equity tranche and short mezzanine tranche when the correlations in both long equity tranche and short mezzanine tranche when the correlations in both tranches decreased.

B. short equity tranche and long mezzanine tranche when the correlations in both

tranches increased.

C. short senior tranche and long mezzanine tranche when the correlation in the

mezzanine tranche increased.

D. long mezzanine tranche and short equity tranche when the correlation in the

mezzanine tranche decreased. 4

Suppose a creditor makes a $4 million loan to company X and a $4 million loan to company Y. Based on historical information of companies in this industry, companies X and Y each have a 7% default probability and a default correlation coefficient of 0.6. The expected loss for this creditor under the worst case scenario assuming loss given default is 100% is closest to: A. $280,130. B. $351,680. C. $439,600. D. $560,430.

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The relationship of correlation risk to credit risk is an important area of concern for risk managers. Which of the following statements regarding default probabilities and default correlations is incorrect? A. Creditors benefit by diversifying exposure across industries to lower the default

correlations of debtors.

B. The default term structure increases with time to maturity for most investment

grade bonds.

C. The probability of default is higher in the long-term time horizon for non-

investment grade bonds.

D. Changes in the concentration ratio are directly related to changes in default

correlations.

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C o n c e p t C h e c k e r An s w e r s

1. B First, calculate the realized correlation as follows:

Prealized =

~ X (0-7 + 0.2 + 0.3) = 0.4

The payoff for the correlation buyer is then calculated as:

$1,000,000 x (0.4 – 0.2) = $200,000

2. A The first step in solving for the 10-day VaR requires calculating the covariance matrix.

= 0.022 = 0.0004 0 2 = 0 – 0 1 2 = 0 . 0 0 0 1 covn = c o v 22 = covi2 = Pi2XCTi Xct2 = 0.4×0.02×0.01 = 0.00008

Thus, the covariance matrix, C, can be represented as: ‘ 0.0004 0.00008 V

0.00008’ 0.0001 /

Next, the standard deviation of the portfolio, crp, is determined as follows:

Step 1: Compute (3h x C:

0.0004 0.00008

0.00008 0.0001

5]
[(7×0.0004)+ (5×0.00008) [0.0032 0.00106]
(7×0.00008)+ (5×0.0001)]
Step 2: Compute ((3h x C) x (3 :

0.00106] 7 5

[0.0032 = (0.0032 x 7) + (0.00106×5) = 0.0277

Step 3: Compute a p: ctp = ^/[3h x C x (3V = V0.0277 = 0.1664 or 16.64%

The 10-day portfolio VaR (in millions) at the 99% confidence level is then computed as: VaRP = crPaVx = 0.1664x 2.33x VlO = $1,226 million 3

3. D A number of large hedge funds were short on the CDO equity tranche and long on the CDO mezzanine tranche. Following the change in bond ratings for Ford and General Motors, the equity tranche spread increased dramatically. This caused losses on the short equity tranche position. At the same time, the correlation decreased for CDOs in the mezzanine tranche, which led to losses in the mezzanine tranche.

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4. B The worst case scenario is the joint probability that both loans default at the same time. The

joint probability of default is computed as:

P(A B) = 0 .6 V 0 .0 7 (0 .9 3 )x 0.07(0.93) + 0.07 x 0.07 = 0.6V 0.00424 + 0.0049 = 0.04396

Thus, the expected loss for the worst case scenario for the creditor is:

E L = 0.04396 x $8,000,000 = $351,680

5. C The probability of default is higher in the immediate time horizon for non-investment grade bonds. The probability of default decreases over time if the company survives the near-term distressed situation.

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The following is a review of the Market Risk Measurement and Management principles designed to address the learning objectives set forth by GARP. This topic is also covered in:

E m p i r i c a l Pr o p e r t i e s o f C o r r e l a t io n : H o w D o C o r r e l a t io n s B e h a v e i n t h e R e a l Wo r l d ?

Topic 7

E x a m F o c u s

This topic examines how equity correlations and correlation volatility change during different economic states. It also discusses how to use a standard regression model to estimate the mean reversion rate and autocorrelation. For the exam, be able to calculate the mean reversion rate and be prepared to discuss and contrast the nature of correlations and correlation volatility for equity, bond, and default correlations. Also, be prepared to discuss the best fit distribution for these three types of correlation distributions.

C o r r e l a t i o n s D u r i n g D i f f e r e n t E c o n o m i c S t a t e s