This model incorporate other distributional assumption (many asset values follow distributions that are not strictly normal) to convert the distance to default into a PD measure.
- KMV: databases containing historical experience
- Moody's: neural network to analyze historical experience & current financial data
2-3. CreditGrades
This model assumes a stochastic, lognormally distributed default barrier. It allows for possible discrete jumps in the default point, thereby increasing the estimated PD as compared to the understated estimates obtained by Merton using the nomal distribution.
=> CreditGrades is able to estimate PD using a closed form function of six observable market parmeters
- Total debt obligations per share
- Average debt recovery rates
- Standard deviation of the constructured distribution
- Initial stock price
- Current stock price
- Implied stock volatility
**Geometric Brownian Motion
In Merton's structural model, default occurs only after a gradual decline in asset values to the default point => This implies that the PD steadily approaches zero as the time to maturity decliens. CrediGrades resolves this problem by matching equity volatilities to observed spreads on credit default swaps and assuming a stochastic default barrier.
2-4. Reduced Form or Intensity-Based Models
Reduced form models do not specify the economic process leading to default. Default is modeled as a point process. Default occur randomly with a probability determined by the hazard function. Intensity-based models decompose observed credit spreads on defaultable debt to ascertain both the PD and the LGD.
* Observed Credit Spead Equation:
CS = PD x LGD
where:
CS: difference between the yield on risky debt and the risk-free rate
* Probability of default using M.V. of bond expression:
Market Price of Bond = (Face Value of Bond) x (1 - PD) / (1 + Risk-free Rate)
e.g.) a B rated $100 face value, zero coupon debt security, security price observed: $87.96, Risk-free rate: 8%
=> 87.96 = 100 x (1 -PD) / (1 + 0.08) => PD = 5%
Alternatively, if we are given the market price of the bond and the face value of the bond, PD can be discovered by computing the risk adjusted return,y
Market Price of Bond = Face Value of Bond / (1 + y)
=> 87.96 = 100 / (1 + y) => y = 13.69%
If we assume that LGD = 100%
1 + Risk-free rate = (1 - PD) x (1 + y)
=> 1 + 0.08 = (1 - 0.05) x (1 + 0.1369)
Hence, probability of default is calculated using the following expression:
PD = 1 - (1 + risk-free rate) / (1 + y)
=> PD = 1 - 1.08 / 1.1369 = 5%
**Time-varying Credit Spread
using forward rate: (1 + 0y2)^2 = (1 + 0y1)(1 + 1y1)
e.g.) two-year spot rate on the B rated date: 16%
=> (1 + 0.16)^2 = (1 + 0.1369)(1 + 1y1) => one-year forward rate on the one-year B rated corporate bond 1y1 = 18.36%
one-year forward rate on the one-year Treasury bond:
(1 + 0.10)^2 = (1 + 0.08)(1 + 1r1) => 1r1 = 12.04%
==> 1 + 12.04% = (1 - PD)(1 + 18.36%) -> PD = 5.34% ->conditional probality: P(default year 2 no default in year 1)
**Cumulative PD = 1 - (1 - PD1)(1 - PD2) = 1 - (1 - 0.05)(1 - 0.0534) = 10.07%
***Expected Loss(EL) can be calculated using EL = PD x LGD
(1 + risk-free rate) = (1 - EL)(1 + y) = (1 - PD x LGD)(1 + y)
3. Proprietary VaR Models
3-1. CreditMetrics
- historical default rates -> transition matrix
- a mark to market model: credit migration matrix inc. upgrades & downgrades
- credit migration scenario with their likelihood of occurrence
3-2. Algorithmics Mark to Future (MTF)
A mark to market credit VaR model that incorporates elements of market risk, credit risk, & liquidity risk <--> CreditMetrics' credit VaR: static view of market risk
- estimating credit risk using Merton model (unconditional default probabilities)
- CWI (creditworthiness index) & geometric Brownian motion process
- generating conditional cumulative default probability distributions for each scenario
- estimating historical sensitivity of the asset class to the risk driver using multi-factor model
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