Credit-Default-Curve Default Statistics

This paper explains how to build continuous or discrete credit curves from Hazard Rates, Expected Default Probabilities, Survival Rates, Marginal Conditional Probabilities or Cumulative Default Probabilities.

Both Marginal Conditional Probabilities and Hazard Rates can be used as discrete or continuous Building blocks respectively. This approach is akin to the well known forward rate building block found in yield curve construction. 

Marginal Conditional Probabilities also called (Forward Default) and Hazard Rates can be used interchangeably since they produce exactly the same results !
   
This document is part of the credit-curve add-in module.  

All the formulas explained below are part of the Risksvr(tm) Engine. Examples of this technology can also be found in the Excel (R) spreadsheet included in the credit-curve.xll add-in package.

Syntax:

t is the time of observation.

q_t= q_(t,t+1) is the Marginal Conditional default probability which is defined as the probability of default over the period t,t+1 considering it has already survived up to period t. 

P_(t,t+1)= Conditional default probabilities (or marginal no-default probability!), is the probability of survival over the period t considering it has survived up to period n and is simply 1- q_(t,t+1).

h_t  =>  h_{(t,t+1) =}Hazard Rate is the default intensity measure. The Hazard Rate is defined as the interval of time starting at time t and ending at time t+1, as a matter of convenience this is usually simplified with the notation h_t. for one year intervals, which is the norm in industry. 
Hazard rates form a series of non decreasing rates from time 0 up to time n-1.

H_t = H_(0,t) Is the Cumulative Hazard Rate which is simply defined as the cumulated interval of hazards from time 0 up to time t.

It is important to understand both measures are, at least in theory, instantaneous measures that are expressed as continuous functions. However in practice they are assumed constant over intervals of time.

P_(t,t+1)=1- q_(t,t+1) ---> exp(-ht)

For standard one year periods, Marginal default probabilities and Hazard Rates are easily converted accordingly:

q_{t(t)=1-exp(-h(t))}

h_{(t)=-Ln(1-q(t))}

f_t=is the probability of default or expected default probability from time 0 to time t.

The opposite of f_{t is the Survival probability (or no-default probability)   i.e. 100%-} f_t.

S_t =survival probability up to time t, since there are 100% chances that an asset which hasn’t defaulted now is not in default s₀=1.

The same holds true for f₀, since there are 0% chances that an asset that is alive now has defaulted, f₀=0.

In practice, the Credit Curve is built from a sequence of discrete marginal conditional default probabilities or hazard rates:

If Hazard rates are used, each Hazard rate will be cumulated over each interval. The Cumulated Hazard rate is then used directly to convert to Survivals or EDF.

In the Discrete Credit-Curve functions, Marginal Conditional Default Probabilities are used as the curve's building block. Every credit statistic is converted to a bootstrapped forward-like marginal conditional default / no-default probability. 

Each Marginal Probability is brought down to a one day probability and then carried over the interval sought.
 
Each interval is then compounded in order to produce Expected Default Probabilities from which Survivals or Hazards can be computed.


Calculation of Expected Default Probabilities From Marginal Conditional Probabilities:

Calculation of Expected Default Probabilities From Hazard Rates:       

Credit Curve Interpolator Algorithm:

The Credit-Curve Interpolator functions differs from the Continuous Credit Curve function based on Hazards. Instead of multiplying hazard pro-rata temporis, each statistic is converted into a one day marginal probability.

The one day marginal probabilities is then carried over the exact interval requested by the user. 
If the simulation horizon extends beyond the last provided interval, the marginal default probabilities are extended by extrapolation.

Expected Default Frequencies are converted into marginal default probabilities over the time steps provided accordingly.

   q_{(t-1, t)} = (Edf(t)-Edf(t-1)/(1.0-Edf(t-1))

 

The Marginal Conditional Default Probability is then converted into a one day Marginal default probability:

Marginal probabilities are then carried over each time interval pro-rata temporis and then aggregated into expected default probabilities.