This is the risk incurred due to failure by a party who owes money to make promised payments. In a risk management program, credit risk represents the risk’s portion managed by a firm. The firm must evaluate the counterparty’s riskiness and assess its effect on the value of derivatives positions.

# Credit Risks as Options

The Merton Model is the best model to value a risky debt by applying the pricing theory. Suppose a levered firm has one debt issue paying no dividends with no coupons and matures at time \(T\) with \({ V }_{ T }\) as the firm’s value and \({ S }_{ T }\) its equity value at date \(T\). The principal amount of debt \(F\) has to be paid on date \(T\). Equity holders will only receive amount \({ V }_{ T } – F\) if the face value of the debt \(F-{ V }_{ T }\) is exceeded by the firm value. Thus:

$$ { S }_{ T }=Max\left( { V }_{ T }-F,0 \right) $$

The payoff would be equivalent to \(F\) and for any value of the firm be the same had the debt been riskiness. However, the debt holders receive a value less than \(F\) by an amount equal to \(F-{ V }_{ T }\) because the debt was risky and the firm’s value fell lower than \(F\).

With exercise price \(F\), the debt can be considered as paying \(F\) less the put option’s payoff on the firm:

$$ { D }_{ T }=F-Max\left( F-{ V }_{ T },0 \right) $$

Where \({ D }_{ T }\) is the debts value at time \(T\).

# Merton’s Formula for the Value of Equity.

The firm’s equity value is \(S\left( V,F,t \right) \) at date \(t\), with \(F\) as the firm’s face value of zero-coupon debt maturing at \(T\), \(\sigma \) the firm’s value volatility, \(V\) the firm’s value, \({ P }_{ t }\left( T \right) \) the price paid by zero-coupon bond paying $1 at \(T\) and \(N\left( d \right) \) the cumulative distribution function evaluated at \(d\). Then:

$$ S\left( V,F,t \right) =VN\left( d \right) -{ P }_{ t }\left( T \right) FN\left( d-\sigma \sqrt { T-t } \right) $$

$$ d=\frac { ln\left( { V }/{ { P }_{ t }\left( T \right) F } \right) }{ \sigma \sqrt { T-t } } +\frac { 1 }{ 2 } \sigma \sqrt { T-t } $$

The pricing of a debt can be in 2 distinct ways. First, since the payoff of a risky debt is equal the payoff of a risk-free debt less the firm’s put option, this fact can be applied with the exercise price equal to the debts face value:

$$ D\left( V,F,T,t \right) ={ P }_{ t }\left( T \right) F-p\left( V,F,T,t \right) $$

Where \(p\left( V,F,T,t \right) \) is the put’s price with exercise price \(F\) on \(V\).

Secondly, the equity value can be subtracted from the equity value from the value of the firm:

$$ D\left( V,F,T,t \right) =V-S\left( V,F,T,t \right) $$

To ascertain the difference in yields across rating classes, investors can scrutinize credit spreads for different ratings:

$$ Credit\quad Spread=-\left( \frac { 1 }{ T-t } \right) ln\left( \frac { D }{ F } \right) -r $$

Where \(r\) is the risk-free rate.

## Finding Firm Value and Firm Value Volatility

Assume that a firm A has debt claim after a sell on credit of, say, $100million. If \({ V }_{ t+5 }\) is the firm’s value at debt maturity, then the debt pays \(F-Max\left( F-{ V }_{ t+5 },0 \right) \quad or\quad { V }_{ t+5 }-Max\left( { V }_{ t+5 }-F,0 \right) \).

A portfolio of securities presenting a claim to the whole firm cannot be traded directly. For a firm with non-traded securities, observing firm values directly, and trading the firm to hedge a claim whose value depends on the firm’s value is impossible.

To rectify this situation, the value of equity and the risk-free asset can be applied to construct a portfolio replicating the firm as a whole. A small change \(\Delta V\) of the firm value causes changes in equity by \(\delta \Delta V\).

Firm value \(V\), volatility, promised debt payment, the maturity of debt and the risk-free interest rate have to be known in order to calculate the \(\delta\) of equity; and with these values, the value of the debt can be computed.

# Subordinated Debt

In case of bankruptcy, it is only after full repayment of the senior debt that subordinated debt can be paid. Subordinated debt is not likely to be paid in full for a company in poor financial conditions and becomes more of an equity claim than a debt claim.

For a case with both senior and subordinated debts maturing on similar dates, with \(F\) being the face value of the senior debt and \(U\) the subordinated debt, the exercise price is \(U + F\). Unless the Firm Value exceeds \(U+F\), shareholders will not receive any amounts. Thus, \(V\) is:

$$ V=D\left( V,F,T \right) +SD\left( V,U,T,t \right) +S\left( V,U+F,T,t \right) $$

Where \(D\) is the senior Debt, \(SD\) the subordinate debt and \(S\) equity.

The call option pricing formula below gives the value of the equity:

$$ S\left( V,U+F,T,t \right) =c\left( V,U+f,T,t \right) $$

The value of the senior debt:

$$ D\left( V,F,T,t \right) =V-c\left( V,F,T,t \right) $$

And the value of subordinated debt:

$$ SD\left( V,U,T,t \right) =V-c\left( V,F+U,T,t \right) -\left[ V-c\left( V,F,T,t \right) =c\left( V,F,T,t \right) -c\left( V,F+u,T,t \right) \right] $$

This is the difference between the value of an option on the value of the firm with exercise price \(F+u\) and an option on the value of the firm with exercise price \(F\).

# The Pricing of a Debt when interest Rates Change Randomly

Accounting the interaction between interest rate changes and firm value changes is crucial when a debt position is to be hedged. For a risky debt, the change in spot interest rate following the Vasicek model over period \(\Delta t\) is:

$$ \Delta { r }_{ t }=\lambda \left( k-{ r }_{ t } \right) \Delta t+{ \sigma }_{ r }{ \varepsilon }_{ t } $$

Where \({ r }_{ t }\) is the current spot interest rate, \({ \varepsilon }_{ t }\) is a random shock, and a positive \(\lambda\) implies reversion in interest rates to a long-run mean of \(k\).

The risky debt is:

$$ D\left( V,r,F,t,T \right) =V-VN\left( { h }_{ 1 } \right) +F{ P }_{ t }\left( T \right) N\left( { h }_{ 2 } \right) $$

$$ Q=\left( T-t \right) \left( { \sigma }^{ 2 }+\frac { { \sigma }_{ r }^{ 2 } }{ { k }^{ 2 } } +\frac { 2\rho \sigma { \sigma }_{ r } }{ k } \right) +\left( { e }^{ -k\left( T-t \right) }-1 \right) \left( \frac { 2{ \sigma }_{ r }^{ 2 } }{ { k }^{ 3 } } +\frac { 2\rho \sigma { \sigma }_{ r } }{ k } \right) -\frac { { \sigma }_{ r }^{ 2 } }{ 2{ k }^{ 3 } } \left( { e }^{ -k\left( T-t \right) }-1 \right) $$

$$ { h }_{ 1 }=\frac { ln\left( \frac { V }{ { P }_{ t }\left( T \right) F } \right) +0.5Q }{ \sqrt { Q } } ,{ h }_{ 2 }={ h }_{ 1 }-\sqrt { Q } $$

# VaR and Credit Risks

Assuming that the only risky asset a firm has is its risky debt, and its \(VaR\) measure is to be calculated, then the delta-\(VaR\) is calculated by transforming the risky debt into a risk-free bond portfolio, or the Monte-Carlo \(VaR\) is calculated by simulating equity returns and valuing the debt for the equity returns.

All the difficulties experienced in the inclusion of the credit risk calculation of \(VaR\) are in the implementation. Crucial obstacles are created by the sophistication of the firm structures, and debt is often issued by firms lacking traded equity.

# Beyond the Merton Model

It becomes even more difficult implementing the Merton model when a firm that has multiple debt issues maturing at different rates or a debt making coupon payments.

Another difficulty is the high predictability of default. Unless firm value is infinitesimally close to the point of occurrence of default, it cannot occur. Therefore, different models class has been developed taking as their departure point a default likelihood evolving with time in accordance with a process that is well defined.

Assuming that in the event of default, the recovery is a fixed fraction of the principal amount \(\theta \) which is not time dependent, then, if the debt is in default, its value is \(\theta F\) and the next period bond value is \({ D }_{ t+\Delta t }\). With \(q\) as the risk-neutral probability, then:

$$ { D }_{ t }={ P }_{ t }\left( t+\Delta t \right) \left[ q\theta F+\left( 1-q \right) \left( { D }_{ t+\Delta t }+u \right) \right] $$

In the last period, the debt value is equal to the principal amount plus the last coupon payment, \(F+U\) or to \(\theta F\). Therefore:

$$ { D }_{ T+\Delta t }={ P }_{ T-\Delta t }\left( T \right) \left[ q\theta F+\left( 1-q \right) \left( F+u \right) \right] $$

Bank subordinated debt and price swap rates work quite remarkably with this approach as suggested by empirical evidence.

# Credit Risk Models

Determining the risk of a portfolio of debt claims and measuring the risk of a portfolio of other financial assets poses several differences. To start with, as credit instruments do not trade in liquid markets, historical data on individual credit instrument cannot be relied upon to measure risks.

Secondly, the distribution of returns differs. Also, debts issued by creditors without traded equity is a common occurrence in firms and the final difference is that in contrast to traded securities, debt is not marked to the market.

If \(m\) is the firm’s value expected rate of return, \(o\) them the probability of default is:

$$ Probability \quad of \quad default=N\left( \frac { ln\left( F \right) -ln\left( V \right) -\mu \left( T-t \right) +0.5{ \sigma }^{ 2 }\left( T-t \right) }{ \sigma \sqrt { T-t } } \right) $$

Where \(N\) is the cumulative normal distribution, \(F\) the face values of the debt, \(V\) the value of the firm, \(T\) the maturity date and \(\sigma \) the rate of change of \(V\)’s volatility.

# CreditRisk+

For each firm, default and no default are the only two outcomes allowed by CreditRisk+ over the period of risk measurement. The probability of default for a borrower is the conditional likelihood of default, while the unconditional likelihood of default is the likelihood obtained if the realization of the risk factors is unknown to us.

Suppose \({ p }_{ i }\left( X \right)\) is the \({ i }^{ th }\) obligor’s probability of default conditional on risk factors’ realization and X the vector of the risk factor realization and \({ \pi }_{ G\left( i \right) }\) the unconditional probability of default for obligor \(I\) given that it belongs to grade \(G\), then:

$$ { p }_{ i }\left( X \right) ={ \pi }_{ G\left( i \right) }\left( \sum _{ k=1 }^{ K }{ { X }_{ k }{ W }_{ ik } } \right) $$

Once \({ p }_{ i }\left( X \right)\) has been calculated, the portfolio’s total number of defaults distribution can be determined.

# CreditMetrics^{TM}

If we are to determine the risk of the value of the debt claim, with BBB rating, of a firm in a given time period, say 1 year, using the \(VaR\), then the \({ 5 }^{ th }\) quantile of the distribution has to be known.

For a BBB-rated debt claim, chances are 1.17% that the claim will be B-rated the following year. The value that the claim is expected to have for each rating in a year is calculated to obtain the debt’s value distribution.

If the assumption is that coupons are to be paid in exactly one year, then the forward zero curves can be applied to calculate the bond’s value for next year’s each possible rating. Should the bond default, a recovery rate is needed, which is the received amount as a fraction of the principal in the event of a default.

For each rating class the following year, the bond’s value can be calculated, and a likelihood that the bond will end up in each one of the rating classes is assigned. Calculating the joint distribution of the migrations of the bonds in the portfolio is the biggest challenge in using this approach.

# The KMV Model

For each borrower, default probabilities are derived by this model using expected default frequencies. The current equity value is used to determine the probability of default to ensure that any event the firm value is dependent and translates directly into a change in the likelihood of default.

Also, the change in the likelihood of default is continuous rather than only when the ratings change. In this approach, the change in the firm value can be substantial but the likelihood of default may be constant since the firm’s rating does not change.

# Credit Derivatives

These are financial instruments whose payoffs are contingent on credit risk realizations and are designed as hedging instruments for credit risks. Credit derivatives are not traded on exchanges apart from futures contracts.

A put that pays the loss on debt due to maturity default is the simplest credit derivative. Derivatives involving swap contracts are the most popular credit derivatives. The credit default swap is one such contact, and it can have physical delivery; in the event of a default, the firm would sign over the loan to the bank upon which a fixed payment will be received.

Another example is the credit default exchange swap, requiring each party to pay the default shortfall on a reference asset that is different. In the total return swap, the return on a risk-free investment is received against credit risk by the party seeking to purchase insurance and pays the return on investment with default risk. The futures contract is another credit derivative which is cash settled, and the number of bankruptcy filings in thousands is the index level during the quarter preceding the expiration of the contract.

# Credit Risks of Derivatives

Evaluation of credit risks of options can be simple as compared to those of swaps. A vulnerable option is an option with default risk. Only when the promised payment gets smaller than \(V\) can the holder receive the payment for vulnerable options. The payoff’s call is:

$$ Max\left[ Min\left( V,S-K \right) ,0 \right] $$

The credit derivative that pays the difference between a call lacking default risk and the vulnerable call is the appropriate credit derivative.

$$ Max\left( S-K,0 \right) -Max\left[ Min\left( V,S-K \right) ,0 \right] $$

Alternatively, if default occurs, the recovery rate can be used. Assuming a probability \(p\) of default for a holder receiving a portion \(z\) of the option’s value and the value of option without default risk is \(c\), then:

$$ The\quad options\quad value\quad today=\left( 1-p \right) c+pzc $$

Consider a market maker entering a swap with risky credit receiving fixed amount \(F\) at maturity of the swap and pays \(S\). If the value of the risky credit net of all the debt senior to the swap is \(V\), the market receives \(S-F\) only if the amount is less than \(V\) in the absence of default risk and pays \(F-S\). The pay off of the swap to the market maker is:

$$ -Max\left( F-S,0 \right) +Max\left[ Min\left( S,V-F \right) ,0 \right] $$

The payment \(F\) made by the risk-free counterparty is such that the swap has no value at inception and the present value of the swap payoff to the market is calculated to determine \(F\).

# Practice Questions

1) The face value of the firm’s only zero-coupon debt maturing in one year is $108 million, and the price of a put with exercise price \(F\) is $7.1 million. Today, the value of the zero-coupon bond paying $1in a year’s time is $0.89. What is the firm’s debt value?

- $96.12 million
- $100.90 million
- $101.68 million
- $89.02 million

The correct answer is **D**.

The value of the debt is given by theformula:

$$ D\left( V,F,T,t \right) ={ P }_{ t }\left( T \right) F-p\left( V,F,T,t \right) $$

From the problem we have that:

\({ P }_{ t }\left( T \right)\)=$0.89,

\(p\left( V,F,T,t \right)\)=$7.1 million,and

\(F\)=$108 million

Therefore:

$$ D\left( V,F,T,t \right) =$0.89\times $108 million\times $7.1 million $$

$$ \Rightarrow D\left( V,F,T,t \right) =$89.02 million $$

2) Assuming that the probability of default for a firm is 0.15 and the recovery rate is 13percent., what is the value of the vulnerable call without default risk?

- 86.67%
- 1.95%
- 15.38%
- 86.95%

The correct answer is **D**.

Recall that the current value of the option is given by the formula:

$$ \left( 1-p \right) c+pzc $$

where

\(p\) = 0.11 and

\(z\)= 0.3, and

\(c\) is the option’s value without default risk

Therefore:

The vulnerable call=(1-0.15)c+0.15×0.13×c=0.86c

$$ The\quad vulnerable\quad call=(1-0.15)c+0.15\times 0.13\times c=0.86c $$

$$ \Rightarrow The\quad vulnerable\quad call=86\%without\quad default\quad risk $$

3) Suppose that the equity of Alonso Transports is valued at $31 million and the cumulative distribution function \(N\left( d \right) \) evaluated at \(d\) is 1.8. The face value of the firm’s only zero coupon bond maturing in \(T\) years is $103 million.Calculate the value of the firm if its volatility is 28% and price of a zero-coupon bond paying $1 in a year’s time is $0.89 while \(T – t\) is 3 years.

- $200.63M
- $149.59M
- $156.37M
- $199.05M

The correct answer is **B**.

Recall that when the formula of the value of equity is inverted, the value of the firm is given by the formula:

$$ V=\left( \frac { 1 }{ N\left( d \right) } \right) S\left( V,F,T,t \right) +{ p }_{ t }\left( T \right) F\left( \frac { N\left( d \right) -\sigma \sqrt { T-t } }{ N\left( d \right) } \right) $$

From the problem we have that

\(N\left( d \right) \)=1.985,

\(S\left( V,F,T,t \right)\)=$31 million,

\(\sigma \)=0.28,

\({ p }_{ t }\left( T \right)\)=$0.79,

\(F\)=$103 million

Therefore:

$$ V=\left( \frac { 1 }{ 1.8 } \right) \times $31million+0.89\times $103million\left( \frac { 1.8-0.28\times \sqrt { 3 } }{ 1.8 } \right) $$

$$ \Rightarrow V=$84.19 \quad million $$