3.1.Hull Ch 22
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Bond rating classification
Bonds above BBB rating are considered investment grade, while lower rated bonds are considered speculative.

3 ways to estimate bond default probabilities
 Historical: collect data from firms that start w/ same credit rating
 From bond price: bond price already reflects expected loss from default. Approximate: s = h(1  R). Exact: PV(loss) = PV(bond @ r_{f})  PV(bond @ y)
 Using equity price: unless the 2 previous techniques this one is not subject to infrequently updated ratings. Instead it infers the bond price at any time from firm's stock price, where p(default) = N(d_{2})

3 credit risk mitigation methods
 Netting: net any transaction for which money is due against amts that may be owed to that same counterparty
 Collateral requirementes: in form of cash or mktable securities
 Downgrade trigger: certain actions occur upon credit downgrade

2 approaches to model default bond correlation
 Structural models: correlate stochastic processes
 Reduced form models: assume hazard rates for different companies follow a stochastic process and are correlated w/ macroeconomic variables

Define copula
Multivariate distribution of 2 or more rdm variables which are both between 0 and 1. It can be used to describe the degree to which 2 or more probabilities are dependent on each other.

Gaussian Copula
 Suppose t_{A} and t_{B} are the times to default
 x = N^{1}[Q(t)], where Q(t) is the cum p(default)
 Then x is a normally dist rdm var
 The joint probability of A and B defaulting can be generated from a multivariate normal dist by calculating the prob of observing these transformed variables x_{A} and x_{B} from a joint standard normal dist w/ a correlation coeff ρ
 Onefactor model: x_{i} = a_{i} + √(1  a_{i}²)Z_{i}
 Q(TF) = N[N^{1}[Q(t)]  √(ρ)F] / √(1  ρ)

Credit VaR
 Attempts to determine a dollar amt that credit losses will not exceed w/ some high prob.
 Time horizon is usually a year or more.
 Gaussian copula: V(X,T) = N[N^{1}[Q(t)] + √(ρ)N^{1}(X)]/√(1ρ)
 Credit metrics model: use credit migration matrix to simulate credit rating changes