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 @ rf) - 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(-d2)
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
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.
- Suppose tA and tB 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 xA and xB from a joint standard normal dist w/ a correlation coeff ρ
- One-factor model: xi = ai + √(1 - ai²)Zi
- Q(T|F) = N[N-1[Q(t)] - √(ρ)F] / √(1 - ρ)
- 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)]/√(1-ρ)
- Credit metrics model: use credit migration matrix to simulate credit rating changes
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