Ch6 Credit Risks & Credit Derivatives (1)

1. Credit Risk
Two important roles in risk management programs:
- parts of the risks a firm tries to manage in a risk management program
e.g.) the riskiness of the debt claims it holds against thirt parties
- positions in derivatives for the express purpose of risk management
e.g.) the riskiness of counterparties in the position


2. Merton Model
2-1. Assumptions:
- only one debt issue with zero coupon
- no dividend
- perfect financial market
- no taxes, no bankruptcy costs, & no costs associated with enforcing contracts
- only debt holders & equity holders claims against the firm

2-2. Value of Equity
ST = Max (VT - F, 0)
where:
VT = the value of the firm at date T
F = the face value of the debt

2-3. Value of Debt
DT = F - Max(F - VT, 0)


2-4. Additional Assumptions to modify Black-Sholes-Merton Option-Pricing Model
- firm value characterized by a lognormal distribution with constant variance
- constant interest rate
- perfect financial market with continuous trading

2-5. Value of Equity at Time t

where:
V = value of the firm
F = face value of the firm's zero coupon debt maturing at T
Pt(T) = price at t of a zero-coupon bond that pays $1 at T
N(d) = cumulative distribution function evaluated at d

3. Credit Spreads, Time to Maturity, & Interest Rates
3-1. Credit Spreads: difference between the yield on a risky bond and the yield on a risk-free bond of same maturity

where:
D = current value of debt
F = face value of debt
T-t = remaining maturity

**as time increases, credit spreads (of both high-rated & low-rated) tend to widen. For a very risky debt, credit spreads narrow as maturity approaches. As interest rates increase, the expected value of the firm at muaturity increases, the risk of default decreases, & the credit spread decreases.

3-2. Firm Value & Volatility
Nontraded securities:
- we cannot observe firm value directly
- we cannot trade the firm to hedge a claim whose value depends on the value of the firm

With Merton's model the only random variable that affects the value of claims on the firm is the total value of the firm. A portfolio consisting of delta units of firm value plus a short position in the risk-free asset is equivalent to the value of firm. We can use equity and the risk-free asset to construct a portfolio that replicates the firm as a whole. -> the delta of equity must be estimated to do this.
To compute the delta of equity from Merton's formula, N(d), we need to know firm value, the volatility of firm value, the promised debt payment, the risk-free interest rate, and the maturity of the debt. -> If we have an estimate of delta & know the value of the firm's equity, then we can solve for firm value & the volatility of the firm value
e.g.) unkown delta, D(V, 100, t+5, t), the value of a share: $14.10, number of shares: 5M, interest rate: 10%
=> S(V, 100, t+5, t) = 14.10 x 5 = 70.5
S(V, 100, t+5, t) = c(V, 100, t+5, t)
if there are options traded on the firm's equity, then we can ge the volatility of equity using option pricing formula and deduce the volatility of the firm from the volatility of equity.
**Black-Sholes formula does not apply to the call option because it is a call option on equity => an option on an option(compound option). The distribution of equity values is not constant -> violation of the Black-Sholes-Merton model

Geske compound option model
- appropriate for compound options
- lognormal distribution with constant volatility
if we know the value of equity, then we can obtain the value of firm volatility using Geske's formula

e.g.) a firm value per share: $25, volatility: 50%
=> the value of call option using compound option model = $6.0349 & the value of equity = $15.50
if actual call option price = $6.72 & actual equity price = $14.10
=> the firm value is too high and the volatility of the firm is too low. To produce model values that are equal to the observed values, the firm value should be $21 per share and the firm volatility 68.36%.

3-3. Subordinate Debt
An increase in firm volatility makes it more likely that subordinated debt will be paid off and hence increases the value of subordinated debt. <-> Senior debt always falls in value when firm volatility increases.
The value of the firm:
V = D(V, F, T, t) + SD(V, U, T, t) + S(V, U + F, T, t)
where:
U = face value of the subordinated debt
D = senior debt
SD = subordinated debt
S = equity = c(V, U + F, T, t)

-> D(V, F, T, t) = V - c(V, F, T, t)
SD(V, U, T, t) = V - c(V, F + U, T, t) - [V - c(V, F, T, t)] = c(V, F, T, t) - c(V, F + U, T, t)

Subordinate debt can be valued in a portfolio as a long position in a call option on the firm with an exercise price equal to the face value of senior debt and a short positon on a call option on the firm with an exercise price equal to the total principal due on all debt.