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Drake DRAKE UNIVERSITY UNIVERSITE D’AUVERGNE CreditMetrics

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DrakeDRAKE UNIVERSITY

UNIVERSITED’AUVERGNE

CreditMetrics

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Credit Risk

Generally Credit risk is related to the probability of default. However default does not necessarily imply that a zero recovery rate. Also changes in credit quality can cause changes in valueNeed a method to evaluate the changes in value related to changes in credit quality of the issuer in a portfolio context.

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Credit Metrics vs. Risk Metrics

RiskMetrics

Large amount of Data

Conditional Volatility

Depends on Normality Assumptions

Historical Data Produces Distributions of Returns

CreditMetrics

Limited Data

Unconditional Volatility

No dependence on Normality

Historical Data using Migration Analysis

Credit changes based on likelihood with outcomes related to value change

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Portfolio Considerations

Need to address portfolio impacts due to the possibility of concentration risk.Traditionally this is accomplished with intuitive exposure based credit limits.A more quantitative approach allows for investigation in terms of portfolio volatility.

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Reasons for Quantitative Approach

Complexity of Financial Products including more derivative instruments make managing exposures more difficult.Increased use of credit enhancements (3rd party guarantees, margin arrangement etc)Improved liquidty in secondary cash markets and increased use of credit derivativesNew products based upon migration.

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Portfolios of Credit Risk

Unlike equity returns, credit returns are not symmetric.The downside loss is larger (fat tails) and the mean is skewed to the rightCredit returns can be characterized as having a large likelihood of receiving a fairly small return based on NIM, combined with a small chance of a large loss.

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Credit Returns vs Equity Returns

0

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Credit Metrics

User Portfolio

Ratings & Equities Series

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Market Volatilities

Recov Rate in Default

Models (Correlatio

n)

Joint Credit rating

changes

Exposure Distributions

Portfolio Value at Risk

Due to Credit

Standard Deviation of value due to credit quality changes for 1 exposure

Exposures Value at Risk due to Credit

Correlations

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Step 1 VaR due to Credit

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Recov Rate in Default

Standard Deviation of value due to credit quality changes for 1 exposure

Value at Risk due to Credit

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Given the current credit rating, the asset may change in rating over the next year.The probability of a rating change can be calculated from historical experience

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Recov Rate in Default

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Example: Distribution of Returns

for a single bond

Consider a single BBB rated bond that matures in five years, pays a 6% yearly coupon payments and is a senior unsecured debt.If the rating changes over the course of the next year, there will be a corresponding change in the value of the bond.The value of the bond will depend upon the change in the rating and the level of interest rates.

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Rating Changes

The probability of a change in ratings can be observed historically based on the migration of bonds starting with a BBB ratings to other ratings classes.

Year end rating

Probability

AAA 0.02

AA 0.33

A 5.95

BBB 86.93

BB 5.30

B 1.17

CCC 0.12

Default 0.18

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Rating Changes

Similar rating change migrations can be found for any beginning rating.The next slide shows the rating migration based upon S&P data covering the 15 years prior to 1996.

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Rating at Year End (% )InitialRating AAA AA A BBB BB B CCC Defaul

t

AAA 90.81 8.33 0.68 0.06 0.12 0 0 0

AA 0.70 90.65 7.79 0.64 0.06 0.14 0.02 0

A 0.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

BBB 0.02 0.33 5.95 86.93 .30 1.17 0.12 0.18

BB 0.03 0.14 0.67 7.73 80.53 8.84 1.00 1.06

B 0 0.11 0.24 0.43 6.48 83.46 4.07 5.20

CCC 0.22 0 0.22 1.30 2.38 11.24 64.86 19.79

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If the bond defaults, it does not imply that there will be a zero recovery rate. The recovery rate by class of bond can be represents by the mean recovery rate and standard deviation by class of security.

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Recov Rate in Default

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Recovery Rates

Senority Class Mean (% ) Stnd Dev (% )

Senior Secured 53.80 26.86

Senior Unsecured 51.13 25.45

Sen Subordinated 38.52 23.81

Subordinated 32.74 20.18

Junior Sub 17.09 10.90

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Recovery Rate

In our example the bond is a senior unsecured debt. Assuming that the average recovery rate is received in the event of default, the bond would be worth $51.13 for every $100 of par value.

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If the bond experiences a change in credit rating it will experience a change in its value.This is also a component of credit risk. The change in value can be calculated based upon the new rating at the end of the year and the forward level of interest rates associated with the rating class.

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Recov Rate in Default

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PV of the bond

The basic bond pricing formula states that the value of the bond at a given point of time is equal to the PV of the cash flows.In this case we want to consider the value of the bond at a time in the future, therefore we need to discount the cash flows back to the time in the future that corresponds with our risk horizonAssume for the example that the horizon is one year.

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Example

The bond in the example makes payments of $6 at the end of each year for the next five years.At the end of the year the next coupon payment would be received, it would have a PV at the end of the year of $6. Each of the other cash flows then needs to be discounted back to the end of the first year. The value is then:

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Interest rates

The value of the bond will depend upon the new rating and the level of rates corresponding to the zero spot forward rate curve that corresponds to that rating category.The forward rate represent the interest rate that is implied to be received in the future given the current yield curve for the risk class.Assume that we know the forward yield curve based upon risk class(shown on next slide)

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One year forward zero rate curves

Category Year 1 Year 2 Year 3 Year 4AAA 3.60 4.17 4.73 5.12AA 3.65 4.22 4.78 5.17A 3.72 4.32 4.93 5.32

BBB 4.10 4.67 5.25 5.63BB 5.55 6.02 6.78 7.27B 6.05 7.02 8.03 8.52

CCC 15.5 15.02 14.03 13.52

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Value of the bond

Assume that the bond is upgraded to a rating of A over the next year. Its value is then:

Similarly the value for any of the ratings changes could be calculated. The value represents the mean value of the rating class given the characteristics of the bond

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One year forward values plus coupon

Year End Rating ValueAAA $109.37AA 109.19A 108.66

BBB 107.55BB 102.02B 98.10

CCC 83.64Default 51.13

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Expected value of Bond

The expected value of the bond is then simply the sum of the values multiplied by the probabilities

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Year End Rating Value Prob(% ) VxProb

AAA $109.37 .02 .02

AA 109.19 .33 .36

A 108.66 5.95 6.47

BBB 107.55 86.93 93.49

BB 102.02 5.30 5.41

B 98.10 1.17 1.15

CCC 83.64 .12 1.10

Default 51.13 .18 0.09

Sum = 107.09

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Standard Deviation

The standard deviation of the value can be found based on the probabilities of rating migration, the average (or expected) value and the values of the bond for each ratings class.

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09.107))13.51%(18.0)64.83%(12.0

)10.98%(17.1)02.102%(3.5)55.107%(93.86

)66.108%(95.5)19.109%(33.0)37.109%(02.0(

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iiitotal valueaverageValueprob

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Accounting for uncertainty

The values for each value represent an average outcome for each state. It is possible to account for the uncertainty in each sate by including a term based on the standard deviation of returns by state.For each of the up or down grade states this will be set to zero, for the default state we will use the calculated standard deviation from before.

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New calculation of Standard Dev

18.3

09.107)45.2513.51%(18.0)064.83%(12.0

)010.98%(17.1)002.102%(3.5)055.107%(93.86

)066.108%(95.5)019.109%(33.0)037.109%(02.0(

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Credit Metrics

User Portfolio

Ratings & Equities Series

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Market Volatilities

Recov Rate in Default

Models (Correlatio

n)

Joint Credit rating

changes

Exposure Distributions

Portfolio Value at Risk

Due to Credit

Standard Deviation of value due to credit quality changes for 1 exposure

Exposures Value at Risk due to Credit

Correlations

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Portfolios

The first example looked at the individual risk of an asset, however most institutions will be concerned with the risk associated with a portfolio of assets.Example 2 keeping the first bond we looked at add a 3 year 5% coupon bond that makes annual interest payments and is a senior subordinated debt that is currently rated A.

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Value of the new bond

The value of the bond can be calculated across various ratings changes by discounting the future cash flows just like we did in the previous case. A portfolio of one $100 par value of each bond would then have 64 possible outcomes based upon the ratings of both bonds.

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Portfolio Values

AAA AA A BBB BB B CCC D

106.59 106.49 106.3 105.64 103.15 101.39 88.71 51.13

AAA 109.37 215.96 215.86 215.67 215.01 212.52 210.76 198.08 160.50

AA 109.19 215.78 215.68 215.49 214.83 212.34 201.58 197.90 160.32

A 108.66 215.25 215.15 214.96 214.30 211.81 210.05 197.37 159.79

BBB 107.55 214.14 214.04 13.85 218.19 210.70 08.94 196.37 158.79

BB 02.02 208.61 208.51 208.33 07.66 205.17 203.41 190.73 153.15

B 98.10 204.69 204.59 204.40 203.74 201.25 199.49 186.81 149.23

CCC 83.64 190.23 190.13 189.94 189.28 186.79 185.03 172.35 134.77

D 51.13 157.72 157.62 157.43 156.77 154.28 152.52 39.84 102.26

.

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Portfolio values

The combined portfolio value was easy to calculate for the two portfolio case, however as the number of assets it he portfolio grows the number of combinations becomes too large to deal with effectively.For example for a 5 asset portfolio there are 32,768 portfolio values

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Rating Migration

Year End Rating Probability (% )AAA .09AA 2.27A 91.05

BBB 5.52BB 0.74B 0.60

CCC 0.01Default 0.06

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Simple Joint Probability

If the rating migration of the two bonds is independent of each other then the joint probability would be the simple product of the two probabilities.For example the probability of the A staying rated A is 91.05%, the probability of the BBB staying rated BBB is 86.93%. If independent the joint probability would be

(91.05%)(86.93) = 79.15%

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Joint Probability

In reality there needs to consideration for the correlation between the two debts. Both originators of the debt respond to the same economic conditions and it is possible that when one of the bonds decreases in rating, the other may also.

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Joint Probability

A positive correlation would cause the joint probability to be higher than in the simple case. The correlation can be estimated from looking at changes in the value of both firms (or cash flows, etc.) Assuming that the correlation of the two firms is 0.3 the joint probabilities are given no the next page.

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Joint Probabilities

AAA AA A BBB BB B CCC D

.09 2.27 91.05 5.52 0.74 0.26 0.01 0.06

AAA .02 0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00

AA 0.33 0.00 0.04 0.29 0.00 0.00 0.00 0.00 0.00

A 5.95 0.02 0.39 5.44 0.08 0.01 0.00 0.00 0.00

BBB 86.93 0.07 1.81 79.69 4.55 0.57 0.19 0.01 0.04

BB 5.30 0.00 0.02 4.47 0.64 0.11 0.04 0.00 0.01

B 1.17 0.00 0.00 0.92 0.18 0.04 0.02 0.00 0.00

CCC 0.12 0.00 0.00 0.09 0.02 0.00 0.00 0.00 0.00

D 0.18 0.00 0.00 0.13 0.04 0.01 0.00 0.00 0.00.

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Credit Risk Measures

Given the portfolio values and probabilities it is easy to calculate an expected value of the portfolio and standard deviation.

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Problems

Since the distribution of returns is not normally distributed it is difficult to interpret the meaning of the standard deviation. In the previous example the maximum upside (215.96) is only .70 standard deviation from the average (213.63) while the maximum downside value (102.26) is 33.25 standard deviations below the average.

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Alternative approach

You can also simulate results then rank order the outcomes. Given the probabilities and distribution it is possible to undertake a monte carlo simulation of possible values.Remember that each value is an average for the dual set of ratings.Once 10,000 simulations are performed it is easy to find a given percentile.

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Alternative approach

For the distribution in the example, the 1st percentile would result in a return of 204.40 9there are 100 observations less than this out of the 10,000. This implies a VaR type number, there is a 1 % chance that the value will fall below 204.40.This implies a credit risk of 213.63–204.40= 9.23

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Interpretations

Using the standard deviation we know that one standard deviation is 3.35, but that does not tell the whole story since the distribution is not normally distributed.The simulation provides a more intuitive result, but the standard deviation is usually quicker to calculate.

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Credit Metrics

User Portfolio

Ratings & Equities Series

Credit SpreadsSeniority

Credit Rating

Rat Migration

Likelihoods

PV Bond Revaluatio

n

Market Volatilities

Recov Rate in Default

Models (Correlatio

n)

Joint Credit rating

changes

Exposure Distributions

Portfolio Value at Risk

Due to Credit

Standard Deviation of value due to credit quality changes for 1 exposure

Exposures Value at Risk due to Credit

Correlations

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Exposures

So far we have only presented the case of credit risks in bondsOther types of exposures can be addressed using similar techniques.

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Exposures

Non interest bearing receivables (trade credit)Bonds and LoansCommitments to lendFinancial letters of credit Market driven instrument (Swaps, forwards, etc)

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Exposures

The bond model required a two step process

Specifying the likelihood and joint likelihood of experiencing a credit quality change Calculating the new values given each possible rating change

The first step is the same for all the other exposure types.

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Receivables

Often receivables will be due in a shorter time frame than the risk horizon, if this is the case a change in rating is not an issue, only default (non payment or partial payment) is an issue.Therefore the key to look at recovery rates. There is a lack of information on receivable recovery rates, but a good approximation may be that for senior unsecured bond recovery rates.

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Receivables

Assume that you have a $1 Million receivable outstanding. In ach non default state, it pays the entire $1 Million.If the receivable defaults, assuming a 30% recovery rate, any default state would have a value of $300,000. The key is then to estimate the probability of default

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Bonds and Loans

Bonds have been covered in detail, a loan is very similar. Given the likelihood that a loan will be repaid (just like the likelihood for a bond) based upon a rating the value of the loan can be found. If default occurs a recovery rate based on the principal of the loan can be used to estimate the amount recovered.

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Loan Commitments

Composed of two parts, the drawn portion and undrawn portion.Interest is paid on the amount already drawn, and a fee is paid on the undrawn portion

The fee compensates the institution for maintaining liquidity to cover the loan if exercised.

The fee is based upon the rating of the creditor

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Year end ratingFee: Un drawn Portion (Basis

Points)

AAA 3

AA 4

A 6

BBB 9

BB 18

B 40

C 120

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Ratings and Drawdown

The borrower has the ability to exercise the loan commitment at any time.This implies that there can be sudden changes in the loan portfolio.It is likely that if the borrowers credit quality deteriorates, there will be a corresponding increase in the amount they borrow.

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Ratings and Drawdown

To account fro this some Loan commitments have covenants that allow for a floating interest rate based upon a credit spread tied to the rating of the borrower and the level of interest rates in the economy. The worst case scenario is that the entire loan commitment is drawn down and the borrower defaults.

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Example

Assume that you have a three year $100 million to lend at a fixed 6% interest rate (on the drawn portion) to a borrower currently rated A.There is currently $20 M drawn down and $80M unused which is charged a fee of 6 Bp.The revaluation based on credit rating will depend upon

Changes in draw down as credit quality changes Change in value of both portions

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Usage of Loan Commitment

Credit Rating Usage Usage of unused in Defualt

AAA 0.1% 69%

AA 1.6% 73%

A 4.6% 71%

BBB 20.0% 65%

BB 46.8% 52%

B 63.7% 48%

CCC 75% 44%

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An example

Assume that the credit rating decreases to BBB from the original value of A.Asarnow and Market (1996) found that the draw increased from 4.6 to 20%, or the undrawn portion changed from 95.4% to 80%This is a 100% - (80%/95.4%)=16.1% reductionGiven that we start at 80% undrawn, we would have a 16.1%(80%) =12.9% increase in borrowing.

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Year End Rating

Current Drawdown

Change in Drawdown

Est od New Drawdown

AAA 20 -19.6 .4

AA 20 -13 7

A 20 0 20

BBB 20 12.9 32.9

BB 20 35.4 55.4

B 20 49.6 69.6

CCC 20 59 79

Default 20 56.8 76.8

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Change in value

The change in value is a combination of the lost fee income (even though a higher fee is charged, it is on a lesser amount of unused portion of the Loan commitment) and lost income due to the credit spread widening on the fixed rate portion of the bond.

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Observations

Some things are observed in the calculationsThe expected percentage drawn down is the most important factorFees have a relatively small impact on the revaluationsIt is possible to have negative revaluations greater than the current draw downCovenants that reset the spread on up and down grades would help in offsetting the risk

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Market Driven Instruments

Usually off balance sheet items such as swaps.Market risk and credit risk are intertwined due to the option nature of many of these instruments.To fully capture the credit risk would require an integrated model of both market and credit risk. Instead Credit metrics attempts to capture the credit component of the risk

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Basic Intro

The swap is valued similar to the bond and loan assuming that the counter party is in an out of the money position. This implies that there is a risk of default (they owe in terms of net present value) Given the expected cash flow stream, the value of the swap can be calculated as if it was a bond including the probability of default and credit rating changes.

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Example from the tech document

Given a 20 asset portfolio across ratings classes, the mean return and standard deviation of the portfolio 9based on the individual rating migrations and recovery rates) was calculated.Based on the results they generated 20,000 scenarios of possible outcomes in one’s years time

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Charts 11.1 to 11.3

The most common occurrence (approximate value of $67 million assumes that there were no rating changes across all securities.Default events produce more significant value changes than the other scenarios.Mean portfolio value = 67,284,888Standard deviation = 1,136,077

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55 60 65 70

0

2000

4000

6000

8000

10000

12000

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Chart 11.4 to 11.8

As sample size increase confidence bands are tighter.

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Chart 11.10

Combines risk measure (marginal standard deviation with credit exposure).Provides information about risk reduction possabilities.

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Applications of results

Chart 12.1 graphs combinations of marginal standard deviation and absolute exposure, it is easy to see which assets have the greatest impact on risk.You can set limits on exposure and risk and adjust when exceeded.

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Choosing a Time Horizon

The time horizon in the example has been one year. Other horizons may be used but it is likely that the horizon should be shorter than a quarter. This is due to the frequency that rating changes occur and can be quantified.If other horizons are used, the results from before should be adjusted for the new time horizon.