For a portfolio with n underlying market variables

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- For a portfolio with n underlying market variables, with each instrument in the portfolio being dependent on only one of the market variables, Æ 2 2 1 1 1 ( ) 2 n n i i i i i i i i P S x S x δ γ = = Δ = Δ + Δ , where Si is the value of the ith market variable. - When some of the individual instruments are dependent on more than one market variable, this equation takes the more general form: 1 1 1 1 2 n n n i i i i j ij i j i i j P S x S S x x δ γ = = = Δ = Δ + Δ Δ ∑∑ , where gamma ij is a ‘cross gamma’. - Cornish – Fisher expansion : Æ estimate percentiles of a probability distribution from its moments that can take account of the skewness of the probability distribution. Using the first three moments of dP, the Cornish-Fisher expansion estimates the q-percentile of the distribution of dP as P q P μ ω σ + , where 2 1 ( 1) 6 q q q z z P ω ξ = + and Z is q-percentile of the standard normal distribution and P ξ is the skewness of dp. 7. The model-building approach is frequently used for investment portfolios. It is less popular for the trading portfolios of financial institutions because it does not work well when deltas are low and portfolios are nonlinear. Chapter 11 Credit Risk: Estimating Default Probabilities 1. Credit-risk arises from the possibility that borrowers, bond issuers, and counter-parties in derivatives transactions may default . In theory, a credit rating is an attribute of a bond issue, not a company. However, in most cases all bonds issued by a company have the same rating. A rating is therefore often referred to as an attribute of a company. - Ratings changes relatively infrequently . One of rating agencies’ objectives is ratings stability. They want to avoid ratings reversals where a firm is downgraded and then upgraded a few weeks later. Ratings therefore change only when there is reason to believe that a long-term change in the firm’s creditworthiness has taken place. The reason for this is that bond traders are major users of ratings. Often they are subject to rules governing what the credit ratings of the bonds they hold - 17 -
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Study Notes: Risk Management and Financial Institutions By Zhipeng Yan must be. If these ratings changed frequently they might have to do a large amount of trading just to satisfy the rules. - Rating agencies try to ‘rate through the cycle’ . Suppose that an economic downturn increases the probability of a firm defaulting in the next 6 months, but makes very little difference to the firm’s cumulative probability of defaulting over the next three to five years. A rating agency would not change the firm’s rating. 2. Internal credit ratings: most banks have procedures for rating the creditworthiness of their corporate and retail clients. The internal ratings based (IRB) approach in Basel II allows bank to use their internal ratings in determining the probability of default, PD. Under the advanced IRB approach, they are also allowed to estimate the loss given default, LGD, the exposure at default, EAD, and the maturity, M.
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