Credit Value at Risk Chapter 18 Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Rating Transitions One year rating transition probabilities are published by rating agencies. If we assume that the rating transition in one period is independent of that in other periods we can calculate the rating transition for any period (see Appendix J and software) The “ratings momentum” phenomenon means that the independence assumption is not perfectly correct Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

One-Year Rating Transition Matrix (% probability, Moody’s 1970-2010) Table 18.1 page 401 Initial Rating Aaa Aa A Baa Ba B Caa Ca-C Default 90.42 8.92 0.62 0.01 0.03 0.00 1.02 90.12 8.38 0.38 0.05 0.02 0.06 2.82 90.88 5.52 0.51 0.11 0.19 4.79 89.41 4.35 0.82 0.18 0.41 6.22 83.43 7.97 0.59 0.09 1.22 0.04 0.14 5.32 82.19 6.45 0.74 4.73 0.16 0.53 9.41 68.43 4.67 16.76 0.39 2.85 10.66 43.54 42.56 100.00 Rating at year end Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Five-Year Rating Transition Matrix (calculated from one-year transitions) Table 18.2 page 401 Initial Rating Aaa Aa A Baa Ba B Caa Ca-C Default 61.12 29.99 7.70 0.89 0.21 0.05 0.01 0.00 0.03 3.45 61.89 28.70 4.71 0.73 0.25 0.07 0.19 0.44 9.72 65.78 18.88 3.24 1.06 0.24 0.04 0.60 0.22 1.69 16.38 60.98 12.93 4.64 0.97 0.13 2.06 3.40 18.20 44.69 20.07 3.70 0.52 8.92 0.20 0.83 3.27 13.28 43.05 11.49 1.64 26.21 0.08 0.23 0.93 3.52 16.80 18.67 2.93 56.84 0.02 0.06 0.31 1.39 5.89 6.78 2.40 83.15 100.00 Rating at end Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

One-Month Rating Transition Matrix (calculated from one-year transitions) Table 18.3 page 401 Initial Rating Aaa Aa A Baa Ba B Caa Ca-C Default 99.16 0.82 0.02 0.00 0.09 99.12 0.77 0.01 0.26 99.18 0.51 0.04 0.44 99.05 0.41 0.06 0.59 98.46 0.79 0.03 0.53 98.32 0.70 0.07 0.36 1.01 96.79 0.67 1.48 0.28 1.53 93.23 4.92 100.00 Rating at month end Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Credit VaR (page 321) Can be defined analogously to Market Risk VaR A one year credit VaR with a 99.9% confidence is the loss level that we are 99.9% confident will not be exceeded over one year Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Vasicek’s Model (Equation 18.1, page 402) For a large portfolio of loans, each of which has a probability of Q(T) of defaulting by time T the default rate that will not be exceeded at the X% confidence level is Where r is the Gaussian copula correlation Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

VaR Model (Equation 18.2, page 402) Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Credit Risk Plus (Section 18.3, page 403) This calculates a loss probability distribution using a Monte Carlo simulation where the steps are: Sample overall default rate Sample probability of default for each counterparty category Sample number of losses for each counterparty category Sample size of loss for each default Calculate total loss from defaults This is repeated many times to calculate a probability distribution for the total loss Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

CreditMetrics (Section 18.4, page 405) Calculates credit VaR by considering possible rating transitions A Gaussian copula model is used to define the correlation between the ratings transitions of different companies Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

The Copula Model : xA and xB are sampled from correlated standard normals Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Credit Risk in the Trading Book: The Specific Risk Charge To calculate the specific risk charge, banks must model credit spreads to calculate a 10-day 99% VaR Alternatives: Historical simulation 10-day transition matrix Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Credit Risk in the Trading Book: Incremental Risk Charge Banks must calculate a one year 99.9% VaR This is to ensure that capital is similar to the capital that would be charged if the instrument were in the banking book They are allowed to make a constant level of risk assumption (minimum liquidity horizon is three months) Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012

Constant Level of Risk Assumption Suppose a bank has a BBB bond and uses a liquidity horizon of 3 months At the end of each month period the bond, if it has deteriorated is assumed to be sold and replaced with a new BBB bond The one-year loss is then replaced by four three-month losses Risk Management and Financial Institutions 3e, Chapter 18, Copyright © John C. Hull 2012