Ch6 Default Risk: Quantitative Methodologies (1)

1. Merton Model
*Structural or Firm-value-based model discribe the default process as the explicit outcome of the deterioration of the value of the firm. Given the capital structure of the firm, it is possible to price equity and debt using option pricing formula.

1-1. Firm value vs. securities value
Vt = St + Pt
where:
Vt = the value of the firm @t
St = the value of the firm's equity @t
Pt = the value of the pure discount bonds @t
** the value of the debt can serve as an indicator of the firm's default risk

Corporate securities are seen as contingent claims (options) on the value of the issuing firm.

1-2. Payoff to security holders
*Assumption: The debt consists of a single zero-coupon bond

Payment to Equity Holders = Max(VT - DT, 0) => long position in a call option
Payment to Debt Holders = DT - Max(DT - VT, 0) => short position in a put option & risk-free bond

VT follows a certain distribution over time -> the value of the stockholders' claim can be determined by the Black-Sholes-Merton option pricing model

1-3. Assumptions
- Bondholders cannot force bankruptcy prior to maturity
- The value of the firm is observable and follows the assumed time-series process
- No adjustment for liquidity is needed

**unrealistic assumptions
- only one issue of equity & debt with zero-coupon
- default can only occur at the maturity date
- the value of firm is observable and follows a lognormal diffusion process
dV = uVdt + SigmavVdZ (geometric Brownian motion)
- the risk-free interest rate is constant through time
- no need to adjust for liquidity

*lognormal: distribution of prices, diffusion: stochastic process


1-4. Strengths & Weaknesses
- Strenghths: simplicity
- Weaknesses: the number of assumptions required (unknow firm value without marked to market debt values & an accurate estimate of asset volatility)



2. KMV Approach (Moody's KMV Credit Monitor Model)
KMV Credit Monitor applies the structural approach to extracting probabilities of default at a given horizon from equity price. Given capital structure of the firms, it is possible to extract market-implied probabilities of default from their equity price.
The probability of default is called expected default frequency (EDF).

2-1. KMV model's assumptions:
- two debt issues: LT (long term) & ST (short term)
the maturity value(default threshold = par value of the firms liabilities) is a linear combination of the values

2-2. Distance to Default(DD)
*The number of standard deviations between current asset values and the debt repayment amount
DD = (Expected Asset Return - Default Threshold) / Standard Deviation of Expected Asset Return

2-3. Process
- KMV does not rely on the cumulative normal distribution => Calibration of EDF to match historical default frequencies recorded on its databases (EDFs & corresponding rating class).
- Buckets based on DD & mapping the DD to EDFs
- Drawbacks of EDFs: EDFs are at-the-point-in-time measures of credit risk focused on default probability at the 1-year horizon => ratings are through-the-cycle assesments of creditworthiness