Portfolio Credit Risk - RiskLab Toronto

Portfolio Credit Risk Luis Seco University of Toronto seco@math.utoronto.ca The Goodrich-Rabobank swap: 1983 5.5 million (11% fixed) Once a year (LIBOR – x) % Semiannual 5.5 million Once a year (LIBOR – y) % Semiannual 11% annual LIBOR + 0.5% ( Semi ) Belgian dentists U.S.Savings Banks Morgan Guarantee Trust Rabobank AAA rated B.F.Goodrich BBB- rated Swap Swap Review of basic concepts Cash flow valuation Credit premium The discounted value of cash flows, when there is probability of default, is given by q i denotes the probability that the counter-party is solvent at time t i The larger the default risk (q small), the smaller its value. The higher the credit risk (q small), the higher the payments, to preserve

the same present value The credit spread Since , we can write the loan is now valued as Default-prone interest rate increases. First model: two credit states What is the credit spread? Assume only 2 possible credit states: solvency and default Assume the probability of solvency in a fixed period (one year, for example), conditional on solvency at the beginning of the period, is given by a fixed amount: q According to this model, we have which gives rise to a constant credit spread: The general Markov model In other words, when the default process follows a Markov chain, the credit spread is constant, and equals Solvency Default Solvency q 1 - q Default 0 1 Goodrich-Morgan swap The fixed rate loan G-RB CreditMetrics analysis: setup  The leg to consider for Credit Risk is the one between JPMorgan and BF Goodrich  Cashflows of the leg (in million USD):  0.125 upfront  5.5 per yr, during 8 years  Assume:  constant spread h = 180 bpi  2 state transition probabilities matrix G-RB CreditMetrics: expected cashflows  Since Expected[cashflows] = ($cashflows) * Prob{non_default}  Then E[cashflows] = .125 + Sum( 5.5 * P{nondefault @ each year})  But at the same time E[cashflow] = G-RB CreditMetrics: probability of default  Under our assumptions: P {non-default} = exp(-h) = exp(-.018) = .98216  constant for each year  The 2 state matrix: BBB D BBB .9822 .0178 D 0 1 G-RB CreditMetrics: compute cashflows  Inputs  P{default of BBB corp.} = 1.8%; 1-exp(0.018)=0.9822  The gvmnt zero curve for August 1983 was r = (.08850,.09297,.09656,.0987855,.10550, .104355,.11770,.118676) for years (1,2,3,4,5,6,7,8) G-RB CreditMetrics: cashflows (cont)  E[cashflows]  Risk-less Cashflows G-RB CreditMetrics: Expected losses  Therefore E[loss] = 1 – ( E[cashflows] / Non-Risk Cashflow) = .065776 i.e. the proportional expected loss is around 6.58% of USD 24.67581 million  Or roughly E[loss] = 1.623 (USD million) Non-constant spreads A default/no-default model (such as CreditRisk+) leads to constant spreads, unless probabilities vary with time In order to fit non-constant spreads, and be able to fit the model to market observations, one needs to assume either: • Time-varying default probabilities • Multi-rating systems (such as credit- metrics) Markov Processes 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6 7 Transition Probabilities Constant in time t=0 t=1 t=2 Transition probabilities...

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