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