Measurement of Credit Risk
Credit Risk versus Market Risk
Default mode: suppose all losses are due to the effect of defaults only.
The distribution of cresit losses (CLs) from a portfolio of
If
If assuming
The variance can be derived using the following formula
So we have
When
Definition of default by Standard & Pool's
Definition of credit event by International Swaps and Derivatives Association (ISDA)
Other events sometimes included are
A credit rating is an ''evaluation of creditworthiness'' issued by a credit rating agency (CRA).
The major U.S. bond rating agencies are
Moody's definition of a credit rating
Ratings represent objective (or actuarial) probabilities of default
Ratings
Classes & modifiers (also called notches)
Multiple Discriminant Analysis (MDA)
Cumulative default rates measure the total frequency of default at any time between the starting date and year
Notations
Pecking order for a company's creditor:
The recovery rate depends on the following factors:
The recovery rate for corporate debt.
The legal environment is also a main driver of recovery rates.
Trading prices of debt shortly after default can be used as an estimator of recovery rate, however, they are on average lower than the discounted recovery rates
An opportunity: buying the defaulted debt and working through the recovery process should create value
Suppose a bond has a single payment $100 in one period, the market-determined yield
We apply risk-neutral pricing:
We compound interest rates and default rates over each period.Let
If we use the cumulative default probability
A very rough approximation:
In the previous analysis we assume risk neutrality. As a result,
Assuming
The risk premium (
The transition from Treasuries to AAA credit most likely reflects other factors, such as liquidity and tax effects, rather than actuarial credit risk
We can use information in corporate bond yield to make inferences about credit risk
Movements in corporate bond prices tend to lead changes in credit ratings
Part of default risk can be attributed to common credit risk factors such as
General Economic conditions
Volatility
The effect of volatility through an option channel
The Merton (1974) model views equity as akin to a call option on the assets of the firm, with an exercise price given by the face value of debt
Consider a firm with total value
Firm value follows the geometric Brownian motion
The value of firm can be decompose in to the value of equity (
The equity value is
Stock Valuation
where
Firm Volatility
Bond Valuation
Risk-Neutral Dynamics of Default
Pricing Credit Risk
Credit Option Valuation
the KMV approach**: the company sells expected default frequencies (EDFs) for global firms
Advantages
Disadvantages
假设某3年期企业债券每年支付7%的券息,每半年付息一次,收益率为5%(以每半年复利计)。所有期限的无风险债券的收益率均为4%(以每半年复利计)。假设违约事件可能每半年发生一次(刚好在债券每次付息之前),回收率为45%。请在以下假设下估计违约概率:
在每个可能违约的日期,无条件违约概率均相同;
在每个可能违约的日期之前无违约的条件下,发生违约条件概率均相同。
请根据以下条件分析债券的违约概率和到期收益率:
无风险利率为每年4%,某信用债券的收益率为每年6%。假设若该债券违约,回收率为70%。请估计该债券一年内发生违约的概率为多少?;
某风险分析师尝试估计一个BB级债券的收益率。如果无风险利率为每年3.5%,BB级债券的违约概率为7%,违约损失率(Loss given default)为70%。请估计该债券的到期收益率。
Credit exposure:
Loans or Bonds
Garantees
Commitments
Swaps or Forwards
Long Options
Short Options
The expected credit exposure (ECE) is the expected value of the asset replacement value
The worse credit exposure (WCE) is the largest (worst) credit exposure at some level of confidence. It is implicitly defined as the value that is not exceeded at the given confidence level
To model the potential credit exposure, we need to
model the distribution of risk factors
evaluate the instrument given these risk factors
the process is identical to a market value at risk computation
the aggregation takes place at the counterparty level if contracts are netted
The average expected credit exposure (AECE) is the average of the expected credit exposure over time, from now to maturity
The average worst credit exposure (AWCE) is defined similarly:
Marking-to-Market (MTM)
involves settling the variation in the contract value on a regular basis
Daily MTM reduces the current credit exposure to zero, however there is still potential exposure because the value of the contract would change before the next settlement. Potential exposure arises from:
MTM introduces other types of risks
Margins
Margins represent the cash or securities that must be advanced in order to open a position
Margins are set in relation to price volatility and to the type of position, speculation or hedging
Collateral
OTC markets may allow posting securities as collateral instead of cash
Exposure Limits
Recouponing
Recouponing refers to a clause in the contract requiring the contract to be marked to market at some fixed dates. It involves
Netting Arrangements
It reduces the exposure to the net value for all the contracts covered by the netting agreement
Nettings can be classified into three types:
Other Modifiers
Credit triggers specify that if either counterparty's credit rating falls below a specified level, the other party has the right to have the swap cash settled
Time puts, or mutual termination options, permit either counterparty to terminate the transaction unconditionally on one or more dates in the contract.
Triggers and put, which are types of contingent requirements, can cause serious trouble
Credit derivatives provide an efficient mechanism to echange credit risk
Credit derivatives are over-the-counter contracts that allow credit risk to be exchanged across counterparties. They can be classified in terms of the following
In a credit default swap contract, a protection buyer (say A) pays a premium to the protection seller (say B), in exchange for payment if a credit evet occurs
A CDS is a option instead of a swap
Most CDS contracts are quoted in terms of an annual spread, with the payment made on quarterly basis
Default swaps are embedded in many financial products, for example:
The payment (
CDS contracts can be priced by considering the present value of the cash flows on each side of the contract.
The value
The default probabilities used to price the CDS contracts must be risk-neutraal probabilities, not real-world probabilities.
A CDS is unfunded
The first-of-basket-to-default swap gives the protection buyer the right to deliver one and only one defaulted security out of a basket of selected securities
With an
CDS indices are widely used to track the performance of this market
A total return swap (TRS) is a contract where one party, called the protection buyer, makes a series of payments linked to the total return on a reference asset. In exchange, the protection seller makes a series of payments tied to a reference rate, such as the yield on an equivalent Treasury issue (or LIBOR ) plus a spread.
In a credit spread forward contract, the buyer receives the difference between the credit spread at maturity and an agreed-upon spread, if positive. Conversely, a payment is made if the difference is negative. The payment is,
Or, equivalently
In a credit spread option contract, the buyer pays a premium in exchange for the right to put any increase in the spread to the option seller at a predefined maturity:
Credit-linked notes (CLNs) are structured securities that combine a credit derivative with a regular bond
The waterfall structure of CDO
In this example, 80% of the capital structure is apportioned to tranche A, which has the highest credit rating of Aaa, using Moody’s rating, or AAA. It pays LIBOR + 45bp, for example. Other tranches have lower priorities and ratings. These intermediate, mezzanine, tranches are typically rated A, Baa, Ba, or B (A, BBB, BB, B, using S&P's ratings). For instance, tranche C would absorb losses from 3% to 10%. These numbers are called, respectively, the attachment point and the detachment point.
Default mode (DM): considering only losses due to defaults instead of changges in market values
For a portfolio of
The net replacement value (NRV)
A typical distribution of credit profits & losses (P&L)
The distribution of P&L is highly skewed to the left
Major features
The effect of correlations
Correlations across default event
Correlations across default event and exposure
Assuming independency,
The present value of expected credit losses (PVECL):
It can be simplified by adopting the average default probability and average exposure over the life of the asset:
An even simpler approach, when ECE is constant, considers the final maturity
Credit VaR over a Target Horizon
Using Credit VaR to Manage the Portfolio
Model Type
Risk Definitions
Models of Default Probability
Models of Default Correlations