Ch6 Credit Risks & Credit Derivatives (2)

3-4. Interest Rate Dynamics
Unaticipated changes in interest rates can affect debt value:
- an increase in interest rates reduces P.V. of promised coupon payments absent credit risk -> reduces the value of debt
- an increase in interest rates can affet firm value IAT empirical evidence


**Vasicek model
The change in the spot interest rate over a period of length deltat is:

where:
lambda = speed that interest rate reverts to the long-run mean, k
k = long run equilibrium value towards which the interest rate reverts
rt = current spot interest rate
sigmat = interest rate volatility
epsilont = random error term (random shock)

* mean reversion:both a stock's high and low prices are temporary and thend to have an average price over time.
* basice idea of Vasicek model: interest rates cannot rise/decrease indefinitely becaue at very high/low levels they would hamper economic activity & prompt a decrease/increase in interest rates => interes rates move in a limited range, showing a tendency to revert to a long run value.
* (lambda)(k - rt): expected instantaneous change in the interest rate at time t. => when rt <> generating a tendency for the interst rate to move upwards (toward equilibrium)

The value of risky debt is:

The value of the debt falls as the correlation between firm value and interest rate shocks increases. The impact of an increase in firm value on the value of the debt is more likely to be dampened by a simultaneous interest rate increase.
An increase in interest rate volatility and an increase in the speed of mean reversion reduce debt value.
At highly volatile interest rates, the value of the debt is less sensitive to changes in interest rate => hedge ratio depend on the parameters of the dynamics of interest rates

3-5. Application Difficulties
Empirical research:
- a naive model of redicting whether debt is riskless works better for investment grade bonds than the Merton model
- the Merton model works better than the naive moel for debt below investment grade
- the Merton model cannot predict credit spreads

4. Credit Risk Models
4-1. Merton Model
Challenge of measuring the risk of a debt portfolio:
- most debt instruments are not publicly traded
- the historical data is not reliable if securities are illiquid
- the distribution of bond returns is not normal
- debt is issued by creditors who do not have traded equity
- debt is not marked to market -> a loss is recognized only if default occrus

**assumptions:
- firm value is lognomally distributed with a constant volatility
- firm only has one liability, which is zero-coupon debt issue

PD (probability of default) & EL (expected loss):

**The following portfolio credit risk models resolve some of the difficulties of measuring a portfolio's probability of default and the amount of loss associated with default when using the Merton model.



4-2. Credit VaR
Credit VaR differs from market VaR in that it measures losses that are due specifically to default risk and credit deterioration risk.
**problems
- calculating changes in credit quality over a one-day period is difficult; credit VaR is usually calculated over a year
- changes in credit risk are highly skewed and do not follow a normal distribution; the loss distribution of changes in credit quality for investment grade bonds closely resembles a lognormal distribution



4-3. CreditRisk+
It measures the credit risk of a portfolio using a set of common risk factors for each obligor. It allows only two outcomes for each firm over the risk measurement period for a loss of a fixed size: default and no default. The probability of default for an obligor depends on its rating, the realization of K risk factors, and the sensitivity of the obligor to the risk factors. Conditional on the risk factor, defaults are uncorrelated across obligors. The risk factors can take only positive values and are scaled so that they have a mean of one. The model assumes that the risk factors follow a specific statistical distribution, gamma distribution. If the kth risk factor has a realization above one, this increases the probability of default of firm i in propotion to the obligor's exposure to that risk factor measured by wik. Once we have computed the probability of default for all the obligors, we can get the distribution of the total number of defaults in the protfolio.