L04 Credit Risk Management

Introduction to Credit Risk

Settlement Risk

  • Credit risk is the risk of an economic loss from the failure of a counterparty to fulfill its contractual obligations.
    • its effect is measured by the cost of replacing cash flow if the other party defaults
    • it requires constructing the distribution of default probabilities（违约概率）, of loss given default (LGD)（违约损失率）, and of credit exposures（信用敞口）
    • traditionally, it is viewed as presettlement risk（结算前风险）, which arises during the life of the obligation
  • Presettlement risk is the risk of loss due to the counterparty's failure to perform on an obligation during the life of the transaction
    • it includes the default on a loan or bond or failure to make required payment on a derivative transaction
    • it exists over long periods
  • Settlement risk（结算风险/交割风险） is due to the exchange of cash flow and is of a much shorter-term nature
    • it arises as soon as an institution makes the required payment and exists until the offsetting payment is received
    • the risk is greatest when payments occur in different time zones, especially for foreign exchange transactions where notional are exchanged in different currencies
    • it can be caused by counterparty default, liquidity constraints, or operational problems
  • Status of a trade
    • Revocable（可撤销）: When the institution can still cancel the transaction without the consent of the counterparty
    • Irrevocable（不可撤销）: After the payment has been sent and before payment from the other party is due
    • Uncertain: After the payment from the other party is due but before it is actually received
    • Settled: After the counterparty payment has been received
    • Failed: After it has been established that the counterparty has not made the payment
  • Settlement risk occurs during the period of irrevocable and uncertain status (one to three days)
  • Tools for settlement risk management
    • real-time-gross settlement (RTGS) system（实时全额结算系统）
    • netting agreements（净额结算协议/轧差协议）
      • bilateral netting（双边净额结算/双边轧差）
      • multilateral netting system（多边净额结算系统/多边轧差系统）
      • continuous-linked settlements（连续关联结算/持续关联结算）
    • contract for differences (CFDs)（差价合约）

Overview of Credit Risk

Drivers of Credit Risk

  • Credit risk measurement systems attempt to quantify the risk of losses due to counterparty default
    • probability of default (PD)
    • credit exposure (CE)
    • loss given default (LGD) or recovery rate（回收率/违约回收率）
  • The borrower has the option to default, so the payment pattern is exactly equivalent to a short position in an option

Measurement of Credit Risk

  • Measurement of Credit Risk

    • Notional amounts, adding up simple exposures
    • Risk-weighted amounts, adding up exposures with a rough adjustment for risk
    • Notional amounts combined with credit rating, adding up exposures adjusted for default probabilities
    • Internal portfolio credit models, integrating all dimensions of credit risk

  • Credit Risk versus Market Risk

    Item Market Risk Credit Risk
    Sources of risk Market risk only Default risk, recovery risk, market risk
    Distributions Mainly symmetrical, perhaps fat tails Skewed to the left
    Time horizon Short-term (days) Long-term (years)
    Aggregation Business/trading unit Whole firm vs. counterparty
    Legal issues Not applicable Very important

Measuring Credit Risk

Credit Losses

Default mode: suppose all losses are due to the effect of defaults only.

The distribution of credit losses (CLs) from a portfolio of instruments issued by different obligators can be described as

If , and are all independent, the expected credit loss is

If assuming for all , we have

The variance can be derived using the following formula

So we have

When , the standard deviation is

Measuring Actuarial Default Risk

Credit Event

Definition of default by Standard & Pool's

The first occurrence of a payment default on any financial obligation, rated or unrated, other than a financial obligation subject to a bona fide commercial dispute; an exception occurs when an interest payment missed on the due date is made within the grace period.

Definition of credit event by International Swaps and Derivatives Association (ISDA)

  • Bankruptcy, which is defined as a situation involving either of the following:
    • The dissolution of the obligor (other than merger)（债务人的解散）
    • The insolvency, or inability to pay its debt（无力偿债或无法偿还债务）
    • The assignment of claims（债权转让/债权让与）
    • The institution of bankruptcy proceeding（破产程序的启动/破产程序的提起）
    • The appointment of receivership（指定接管人/委任接管人）
    • The attachment of substantially all assets by a third party（第三方查封实质上全部资产/第三方扣押基本全部资产）
  • Failure to pay, which means failure of the creditor to make due payment; this is usually triggered after an agreed-upon grace period and when the payment due is above a certain amount.
  • Obligation/cross default（义务违约/交叉违约）, which means the occurrence of a default (other than failure to make a payment) on any other similar obligation.
  • Obligation/cross acceleration（义务加速到期/交叉加速到期）, which means the occurrence of a default (other than failure to make a payment) on any other similar obligation, resulting in that obligation becoming due immediately.
  • Repudiation（拒绝履行/否认债务）/moratorium（延期偿付/暂停偿付）, which means that the counterparty is rejecting, or challenging, the validity of the obligation.
  • Restructuring（债务重组）, which means a waiver, deferral, or rescheduling of the obligation with the effect that the terms are less favorable than before.

Other events sometimes included are

  • Downgrading（信用评级下调/评级下降）, which means the credit rating is lower than previously, or is withdrawn.
  • Currency inconvertibility（货币不可兑换/汇率管制）, which means the imposition of exchange controls or other currency restrictions by a governmental or associated authority.
  • Governmental action（政府行为/政府措施）, which means either (1) declarations or actions by a government or regulatory authority that impair the validity of the obligation, or (2) the occurrence of war or other armed conflict that impairs the functioning of the government or banking activities.

Default Rates & Credit Ratings

  • 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 Investors Service
    - Standard and Pool's (S&P)
    - Fitch Ratings

  • Moody's definition of a credit rating

    Opinion of the future ability, legal obligation, and willingness of a bond issuer or other obligor to make timely payments on principal and interest due to investors.

  • Ratings represent objective (or actuarial) probabilities of default
    - published default frequencies can be used to convert ratings to default probabilities
Explanation Standard &
Poor's		 Moody's Investors
Service
Investment grade:
Highest grade AAA Aaa
High grade AA Aa
Upper medium grade A A
Medium grade BBB Baa
Speculative grade:
Lower medium grade BB Ba
Speculative B B
Poor standing CCC Caa
Highly speculative CC Ca
Lowest quality, no interest C C
In default D
Modifiers: A+, A, A−, and A1, A2, A3

  • Ratings

    • investment grade
    • speculative grade, or below investment grade

  • Classes & modifiers (also called notches)
  • Accounting ratios & credit ratings
    Rating Leverage:
    (Percent)    		 Cash Flow Coverage:
    (Multiplier)
    Total Debt/Capital EBITDA/Interest EBIT/Interest
    AAA 12 32.0 26.2
    AA 35 19.5 16.4
    A 37 13.5 11.2
    BBB 45 7.8 5.8
    BB 53 4.8 3.4
    B 73 2.3 1.4
    CCC 99 1.1 0.4

  • Multiple Discriminant Analysis (MDA)

    • MDA constructs a linear combination of accounting data that provides the best fit with the observed states of default and non-default for the sample firms
    • score is an example of MDA

  • score, variable used are:

    • working capital over other assets,
    • retained earnings over total assets,
    • EBIT over total assets,
    • market value of equity over total liabilities,
    • net sales over total assets.

Cumulative and Marginal Default Rates




  • Cumulative default rates measure the total frequency of default at any time between the starting date and year , while marginal default rates measure default during year
  • Notations
    • : the number of issuers rated at the end of year that default in year
    • : the number of issuers rated at the end of year that have not default by the beginning of year
  • Calculating rates
    • Marginal Default Rate during Year :
    • Survival Rate:
    • Marginal Default Rate from Start to Year :
    • Cumulative Default Rate:
    • Average Default Rate:

Transition Probabilities

Recovery Rates

The Bankruptcy Processes

Pecking order for a company's creditor:

Seniority Type of Creditor
Highest (paid first) 1. Secured creditors (up to the extent of secured collateral)
2. Priority creditors
   • Firms that lend money during bankruptcy period
   • Providers of goods and services during bankruptcy period (e.g., employees, lawyers, vendors)
   • Taxes
3. General creditors
   • Unsecured creditors before bankruptcy
Lowest (paid last)    • Shareholders

Estimates of Recovery Rates

The recovery rate depends on the following factors:

  • The status or seniority of the debtor. Claims with higher seniority have higher recovery rates.
  • The state of the economy. Recovery rates tend to be higher when the economy is in an expansion, and lower when in a recession.
  • The obligor's characteristics. Recovery rates tend to be higher when the borrower's assets are tangible and when the previous rating was high.
  • The type of default. Distressed exchanges, as opposed to bankruptcy proceedings, usually lead to higher recovery rates.

The recovery rate for corporate debt.

Priority Count Mean S.D. Min. 10th Median 90th Max.
All bank loans 310 61.6 23.4 5.0 25.0 67.0 90.0 98.0
Equipment trust 86 40.2 29.9 1.5 10.6 31.0 90.0 103.0
Senior secured 238 53.1 26.9 2.5 10.0 34.0 82.0 125.0
Senior unsecured 1,095 37.4 27.2 0.3 7.0 30.0 82.2 122.6
Senior subordinated 450 32.0 24.0 0.5 5.0 27.0 60.5 123.0
Subordinated 477 30.4 21.3 0.5 5.0 27.1 60.0 102.5
Junior subordinated 22 23.6 19.0 1.5 3.8 16.4 48.5 74.0
All bonds 2,368 36.8 26.3 0.3 7.5 30.0 80.0 125.0

Source: Adapted from Moody's, based on 1982-2002 defaulted bond prices.

The legal environment is also a main driver of recovery rates.

Instrument Europe North America
Bank loans 47.6 61.7
Bonds
   Senior secured 52.2 52.7
   Senior unsecured 25.6 37.5
   Senior subordinated 24.3 32.1
   Subordinated 13.9 31.3
   Junior subordinated NA 24.5
All bonds 28.4 35.3
Preferred stock 3.4 10.9
All instruments 27.6 35.9

Source: Adapted from Moody's, from 1982-2002 defaulted bond prices.

Instrument Trading Prices 15-45 Days Discounted Recovery
Bank loans 58.0 81.6
Senior secured bonds 48.6 67.0
Senior unsecured bonds 34.5 46.0
Senior subordinated bonds 28.4 32.4
Subordinated bonds 28.9 31.2

Source: Adapted from S&P, from 1988-2002 defaulted debt.

  • 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
    • clienteles for the two markets are different
    • risk premium in trading price
  • An opportunity: buying the defaulted debt and working through the recovery process
    • distress securities funds

Measuring Default Risk from Market Prices

Corporate Bond Prices

Spreads and Default Risk: Single Period

Suppose a bond has a single payment $100 in one period, the market-determined yield can be derived from its price

We apply risk-neutral pricing:


Spreads and Default Risk: Multiple Periods

We compound interest rates and default rates over each period. Let be the average annual default rate.

If we use the cumulative default probability

A very rough approximation:

算例：从债券利差提取违约概率

Given: A 5-year corporate bond yields 6.5%, while the 5-year risk-free rate is 5.0%. The recovery rate is 40%.

Step 1: Compute the annual credit spread

Step 2: Solve for the annual risk-neutral default probability

Step 3: Verify using multi-period formula

Interpretation: The annual risk-neutral default probability is approximately 2.5%. Note this is a risk-neutral measure — the physical (real-world) default probability would be lower, with the difference reflecting the risk premium.

Risk Premium

In the previous analysis we assume risk neutrality. As a result, is a risk neutral measure, which is not necessarily equal to the objective, physical probability of default.

Assuming and be the physical probability of default and the discount rate. We have the following

The risk premium () must be tied to some measure of bond riskiness as well as investor risk aversion. In addition, this premium may incorporate a liquidity premium and tax effects.

Cross-Section of Yield Spreads

  • 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

Time Variation in Credit Spreads

Part of default risk can be attributed to common credit risk factors such as

  • General Economic conditions
    • growth
    • slow down
  • Volatility
    • investors require more risk premium in a more volatile environment
    • liquidity may dry up
  • The effect of volatility through an option channel
    • a callable bond = bond + (short) a call
    • the value of call increases in a more volatile environment

Equity Prices

The Merton Model

  • 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 that has one bond due in one period with face value

    • equity can be viewed as a call option on the firm value with strike price equal to the face value of debt

    • the current stock price embodies a forecast of default probability in the same way that an option embodies a forecast of being exercised
    • corporate debt can be viewed as risk-free debt minus a put option on the firm value

Pricing Equity and Debt

Firm value follows the geometric Brownian motion

The value of firm can be decomposed in to the value of equity () and the value of debt (). The corporate bond price is obtained as

The equity value is

Stock Valuation

where

Firm Volatility

Bond Valuation

Risk-Neutral Dynamics of Default

Pricing Credit Risk

Credit Option Valuation

Applying the Merton Model

  • the KMV approach: the company sells expected default frequencies (EDFs) for global firms
  • Advantages
    • it relies on the price of equities, which are more actively traded than bonds
    • correlations between equity prices can generate correlations between bonds
    • it generates movements in EDFs that seems to lead changes in credit ratings



  • Disadvantages
    • it can not be used to price sovereign credit risk
    • it relies on a static model of the firm's capital and risk structure
    • management could undertake new projects that increases not only the value if equity but also its volatility
    • the model fails to explain the magnitude of credit spreads we observe on credit-sensitive bonds

算例：Merton模型完整计算

Given: A firm has asset value , asset volatility . Debt face value , maturity year, risk-free rate .

Step 1: Compute and

Step 2: Equity value

Step 3: Bond value and credit spread

Risk-free bond value:

Credit spread:

Step 4: Risk-neutral default probability

Step 5: Expected loss

课堂练习

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。

课堂练习

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 Exposures

Credit Exposure by Instrument

  • Credit exposure:

    • current exposure is the value of the asset at the current time if positive
    • potential exposure represents the exposure on some future date, or set of dates

  • Loans or Bonds

    • loans or bonds are balance sheet assets whose current and potential exposure basically is notional, or amount lent or invested
    • the exposure is also notional for receivables and trade credits, as the potential loss is the amount due

  • Guarantees

    • guarantees are off-balance-sheet contracts whereby the bank has underwritten, or agrees to assume, the obligations of a third party.
    • the exposure is the notional amount
    • it is irrevocable

  • Commitments

    • Commitments are off-balance-sheet contracts whereby the bank commits to a future transaction that may result in creating a credit exposure at a future date
    • irrevocable commitments vs. revocable commitments
  • Swaps or Forwards
    • Swaps or forwards contracts are off-balance-sheet items that can be viewed as irrevocable commitments to purchase or sell some asset on prearranged terms
    • the current and potential exposure will vary from zero to a large amount depending on movements in the driving risk factors
    • similar arrangement are sale-repurchase (repos)

  • Long Options

    • Options are off-balance-sheet items that may create many credit exposure
    • the current and potential exposure will vary from zero to a large amount depending on movements in the driving risk factors
    • there is no possibility for options to have negative values

  • Short Options

    • the current and potential exposure for short options is zero because the bank writing the option can incur only a negative cash flow, assuming the option premium has been fully paid

Distribution of Credit Exposure

Expected & Worse Exposure

The expected credit exposure (ECE) is the expected value of the asset replacement value , if positive, on a target date:

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

Time Profile

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:

Exposure Modifiers（敞口调整因子/风险敞口修正系数）

Marking-to-Market (MTM)

  • involves settling the variation in the contract value on a regular basis

    • it is called two-way marking to market if the treatment is symmetrical across the two counterparties
    • if one party settles losses only, it is called one-way marking to market

  • 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:

    • the time interval between MTM periods
    • the time required for liquidating the contract when the counterparty defaults
  • MTM introduces other types of risks
    • operational risk, which is due to the need to keep track of contract values and to make or receive payments daily
    • liquidity risk, because the institution now needs to keep enough cash to absorb variations in contract value

Margins

  • Margins represent the cash or securities that must be advanced in order to open a position

    • the purpose of these funds is to provide a buffer against potential exposure
    • initial margin vs. maintenance margin

  • Margins are set in relation to price volatility and to the type of position, speculation or hedging

    • margins increase for more volatile contracts
    • margins are typically lower for hedgers

Collateral

  • OTC markets may allow posting securities as collateral instead of cash
    • the amount of collateral will exceed the funds owed by an amount known as haircut（折扣率/减值系数）, which reflects both default risk and market risk.
    • collateral is typically managed within the International Swap and Derivatives Association (ISDA) credit support annex (CSA)（信用支持附件/担保品管理协议）

Exposure Limits（敞口限额）

  • Credit exposure can also be managed by setting position limits on the exposure to a counterparty
  • To enforce limits, information on transactions must be centralized in middle-office systems
  • These limits can also be set at the instrument level

Recouponing (债券重组/重新附息)

  • Recouponing refers to a clause in the contract requiring the contract to be marked to market at some fixed dates. It involves
    • exchanging cash to bring the MTM value to zero
    • resetting the coupon or the exchange rate to prevailing market value

Netting Arrangements（净额结算安排/轧差安排）

  • It reduces the exposure to the net value for all the contracts covered by the netting agreement

  • Nettings can be classified into three types:

    • payment netting（支付净额结算/付款轧差） involves the daily offsetting of several claims in the same currency
    • novation netting（更新轧差/新债轧差） involves the cancellation of several contracts between the two parties, resulting in a replacement contract with new, net payments
    • close-out netting（提前终止净额结算/违约轧差） involves the cancellation of all transactions under the master agreement in the event of bankruptcy or any other specified default event

Other Modifiers

  • third-party guarantees
  • purchasing credit derivatives

Credit Risk Modifiers（信用风险调节因子/信用风险修正因子）

  • Credit triggers（信用触发条件/信用触发机制） 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
    • these are not exposure modifiers, but rather attempt to reduce the probability of default on that contract
    • these triggers are useful when the credit rating of a firm deteriorates slowly, because few firms jump directly from investment grade into bankruptcy

  • Time puts（定时卖出期权/时间卖出条款）, or mutual termination options（双方终止期权/互相终止选择权）, permit either counterparty to terminate the transaction unconditionally on one or more dates in the contract.

    • the feature decreases both the default risk and exposure
    • it allows one counterparty to terminate the contract if the exposure is large and the other party's rating starts to slip

  • Triggers and put, which are types of contingent requirements（或有要求/条件性要求）, can cause serious trouble

Credit Derivatives and Structured Products

Introduction

  • Credit derivatives provide an efficient mechanism to exchange credit risk

    • credit risk can not be perfectly diversified for banks
    • banks can keep the loans on their books and to buy protection with credit derivatives

  • 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

    • The underlying credit, which can be a single entity (single name) or a group of entities (multiname)
    • The exercise conditions, which can be a credit event (such as default or rating downgrade, or an increase in credit spreads)
    • The payoff function, which can be a fixed amount or a variable amount with a linear or nonlinear payoff

Credit Default Swaps

  • 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 event occurs
    • the premium payment can be a lump sum or periodic
    • the contingent payment is triggered by a credit event (CE) on the underlying credit, say a bond issued by company Y
  • A CDS is a option instead of a swap
    • the main difference from a regular option: the cost of a option is paid in installments instead of up front
    • if the cost is paid up front, the contract is called a default put option
    • the annual payment is referred as the CDS spread




An example of CDS

The protection buyer, call it A, enters a one-year credit default swap on a notional of $100 million worth of 10-year bonds issued by XYZ. The swap entails an annual payment of 50bps. The bond is called the reference credit asset.

At the beginning of the year, A pays $500,000 to the protection seller. Say that at the end of the year, company XYZ defaults on this bond, which now trades at 40 cents on the dollar. The counterparty then has to pay $60 million to A. If A holds this bond in its portfolio, the credit default swap provides protection against credit loss due to default.

  • 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:

    A long position in a defaultable bond is economically equivalent to a long position in a default-free bond plus a short position in a CDS on the same underlying credit.

Settlement

  • The payment () on default reflects the losses to the holders of the reference asset when the credit event occur. It takes a number of forms:
    • cash settlement, or a payment equal to the strike minus the prevailing market value of the underlying bonds
    • physical settlement of the defaulted obligation in exchange for a fixed payment
    • a lump sum, or a fixed amount based on some pre-agreed recovery rate. (if the CE occurs, the recovery rate is set at 40%, leading to a payment of 60% of the notional)
  • The payoff of a CDS is

    • binary credit default swap（二元信用违约掉期/数字式信用违约掉期）:
    • with physical settlement, the contract defines a list of bond, which can trade at different prices but must be exchanged for their face value, that can be delivered (delivery option)
    • cash settlement can be conducted through an auction, which defines the recovery rate

算例：CDS定价

Given: A 5-year CDS on a reference entity. Annual CDS spread bp. Notional = $10M. Risk-free rate , recovery rate .

Step 1: Risk-neutral default probabilities (assumed constant annual marginal rate )

Assume per year.

Year Survival Marginal default Discount PV Payoff PV Spread
1 0.970 0.030 0.9709 0.0175 0.0194
2 0.941 0.029 0.9426 0.0164 0.0189
3 0.913 0.028 0.9151 0.0154 0.0183
4 0.885 0.027 0.8885 0.0144 0.0178
5 0.859 0.027 0.8626 0.0137 0.0173
Total 0.0774 0.0917

Step 2: Fair spread calculation

Since the market spread (200bp) > fair spread (84.4bp), the CDS is expensive for the protection buyer but profitable for the protection seller.

Step 3: Value of existing CDS (at spread bp)

Pricing

CDS contracts can be priced by considering the present value of the cash flows on each side of

Year
t		 Probability (%) Discount
Factor
PVt		 Payoff Payments Spread Payments
Cumul.
Ct		 Annual
dt		 Marg.
kt		 Survival
St		 Expected
kt(1-f)		 PV Expected
sSt-1		 PV
12.642.6402.6400.97360.94341.5841.494s1.000s0.943
25.482.9172.8400.94520.89001.7041.517s0.974s0.867
38.573.2693.0900.91430.83961.8541.557s0.945s0.794
411.893.6313.3200.88110.79211.9921.578s0.914s0.724
515.434.0183.5400.84570.74732.1241.587s0.881s0.658
Total15.4304.21247.733s3.986

The value and the fair spread of the CDS contract should satisfy the following:

The default probabilities used to price the CDS contracts must be risk-neutral probabilities, not real-world probabilities.

The CDS swap spread should approximately equal the yield on a corporate bond issued by the same obligor minus the risk-free yield.

Counterparty Risk

  • A CDS does not eliminate credit risk entirely.
    - the protection buyer decreases exposure to the reference credit Y but assume new credit exposure to the CDS seller
    - correlation between the default risk of the underlying credit and of the counterparty is important
Correlation Counterparty Credit Rating
AAAAAABBB
0.0194194194194
0.2191190189186
0.4187185181175
0.6182178171159
0.8177171157134
  • A CDS is unfunded
    - unfunded（无资金担保）: each party is responsible for making payments without recourse to other assets
    - unfunded: the protection seller makes a payment that could be used to settle any potential credit event

Other Contracts

CDS Variants

  • The first-of-basket-to-default swap（追索权首次违约篮子互换/可追索首违篮子掉期） gives the protection buyer the right to deliver one and only one defaulted security out of a basket of selected securities
    - the contract will be more expensive than a single credit swap, all else kept equal
    - the price of protection also depends on the correlation between credit events

  • With an th-to-default swap, payment is triggered after defaults in the underlying portfolio, but not before

  • CDS indices are widely used to track the performance of this market
    - the iTraxx indices cover the most liquid names in European and Asian credit markets
    - the North American and emerging markets are covered by the CDX indices

Total Return Swaps

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.

  • TRS provides protection against credit risk in a mark-to-market (MTM) framework
  • For the protection buyer, the TRS removes all the economic risk of the underlying asset without selling it
  • Unlike a CDS, the TRS involves both credit risk and market risk, the latter reflecting pure interest rate risk



案例：Archegos Capital爆仓事件（2021）

背景： 2021年3月，对冲基金Archegos Capital Management通过总收益互换（TRS）持有大量集中头寸后爆仓，导致多家投行遭受巨额损失。

交易结构：

  • Archegos通过多家投行使用TRS进行杠杆投资，名义头寸超过$300亿
  • TRS使Archegos无需实际购买股票即可获得经济敞口，隐藏了其真实持仓规模
  • 各投行看不到Archegos在其他机构的头寸

风险暴露：

投行 估计损失
Credit Suisse $55亿
Nomura $30亿
Morgan Stanley $9.1亿
UBS $7.7亿

关键问题：

  • 对手方信用风险： TRS对手方风险集中在Archegos一家机构
  • 保证金管理失败： Archegos频繁违反保证金追缴要求，投行未能及时平仓
  • 隐蔽持仓： 各投行未意识到其他机构同时持有大量类似敞口
  • 集中度风险： Archegos对少数几只股票的持仓极为集中

教训：

  • 对手方尽职调查（KYC）的重要性——需要穿透了解客户的真实风险敞口
  • TRS等OTC衍生品的透明度问题——场外交易使得风险暴露难以被全面追踪
  • 保证金制度需要更加严格的执行——延迟追缴保证金会导致损失急剧扩大
  • 集中度限额必须适用于客户层面而非仅仅交易层面

Credit Spread Forwards and Option

In a credit spread forward contract（信用利差远期合约/信用价差远期协议）, 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:

Securitization and Structured Products

Principles of Securitization

  • Procedures of securitization (off-balance-sheet)

    • create a special purpose vehicle (SPV), or special-purpose entity (SPE)
    • the originator pools a group of assets and sells them to the SPV
    • SPV issues tradable claims, or securities, that are backed by the financial assets

  • Advantage of this structure

    • it shields the asset-backed security (ABS) investor from the credit risk of the originator
    • pooling offers ready-made diversification across many assets
  • All sorts of assets (collateral) can be included in ABSs
    • mortgage loans, student loans, credit card receivables, account receivables, debt obligations, and etc.
    • RMBSs: backed by residential mortgage loan
    • CMBSs: backed by commercial mortgage loan
    • the cash flow of RMBSs and CMBSs is subject to interest rate risk, prepayment risk, and default risk

  • Structure of securitization

    • pass-through: the SPV issues single class of security
    • tranching: the SPV issues different classes of securities

  • On-balance-sheet securitizations (covered bonds or Pfandbriefe in Germany)

    • banks originates the loans and issues securities secured by these loans, which are kept on its books
    • investors have recourse against the bank in the case of defaults on the mortgage
    • banks provides a guarantee against credit risk

Issues with Securitization

  • The moral hazard problem

  • The adverse selection problem

  • credit rating fails for securitization that with complex structures

  • Securitization pushes housing prices away from their fundamental values

  • When the securitization markets froze, many banks and loan originators were stuck with loans that were warehoused, or held in a pipeline that was supposed to be temporary

Tranching

Concept

  • Cash flows are redistributed to fit investors' need

    • however, cash flows and risks are fully preserved
    • they are only redistributed across tranches
    • weighted duration & convexity of the portfolio of tranches must add up to the original duration and convexity

  • This structure applies to collateralized mortgage obligations (CMOs), collateralized bond obligations (CBOs), collateralized loan obligations (CLOs), collateralized debt obligations (CDOs)

Credit-Linked Notes (CLNs)

  • Credit-linked notes (CLNs)（信用关联票据/信用挂钩债券） are structured securities that combine a credit derivative with a regular bond
    - the buyer of protection transfers credit risk to an investor via an intermediary bond-issuing entity
    - the entity can be the buyer itself or a special purpose vehicle (SPV)
    - this structure may carry a higher yield if the CDS spread is greater than the bond yield spread
    - this structure may also be attractive to investors who are precluded from investing directly in derivatives

Collateralized Debt Obligations（担保债务凭证/抵押债务债券）

The waterfall structure of CDO
  • 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（夹层）, tranches（分层） 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（附着点/起赔点） and the detachment point（分离点/脱离点）.

案例：2008年次贷危机

链条传导： 次级抵押贷款 → MBS → CDO → CDO²

关键环节：

环节 问题
贷款发放 "发起-分销"模式下，贷款机构放松审查标准（NINJA贷款：无收入、无工作、无资产）
证券化 SPV将次贷打包为MBS，评级机构给予AAA评级（依赖历史数据，忽略系统性相关）
CDO分层 CDO²进一步分层，底层资产的违约风险被放大和隐匿
评级失真 高斯Copula模型低估了住房价格同时下跌的概率
系统性传染 所有层级几乎同步亏损，分散化假象破灭

关键数据：

  • 2006年美国次贷发放量占全部住房贷款的约20%
  • ABX指数（次贷CDS指数）从2007年1月的100跌至2009年初的不到5
  • 全球银行业次贷相关损失超过$1万亿

根本教训：

  • 证券化链条中的道德风险和激励扭曲
  • 评级方法对系统性风险的忽视
  • 复杂分层产品的不透明性——投资者无法穿透理解底层资产质量
  • 流动性风险与信用风险的交互放大

Inverse Floaters

To illustrate the concept of tranching, we consider a simple example with a two-tranche structure that splits up interest rate risk. The collateral consists of a regular five-year, 6% coupon $100 million note. This can be split up into a floating-rate note (FRN) that pays LIBOR on a notional of $50 million, and an inverse floater that pays 12% − LIBOR on a notional of $50 million. Because the coupon on the inverse floater cannot go below zero, this imposes another condition on the inverse floater coupon . The exact formulas are:

  • Try to verify that the outgoing cash flows exactly add up to the incoming cash flows
  • Suppose the dollar duration of the original five-year note is 4.5 years, try to analyze the duration of the floater and the inverse floater just before a reset
  • What if the coupon is tied to twice LIBOR, i.e. ?

CMOs

  • Sequential-pay tranches: defined by prioritizing the payment of principal into different tranches

  • Planned amortization class (PAC)

    • PAC bonds offer a fixed redemption schedule as long as prepayments on the collateral stays within a specified PSA range (say 100 to 250 PSA), called the PSA collar
    • the principal payment is set at the minimum payment of these two extreme values for every month of its life
    • all prepayment risk is transferred to other bonds in the CMO structure, called support bonds.

Managing Credit Risk

Measuring the Distribution of Credit Losses

Default mode (DM): considering only losses due to defaults instead of changes in market values

For a portfolio of counterparties, the credit loss (CL) is

The net replacement value (NRV)（净重置价值）

  • A typical distribution of credit profits & losses (P&L)

  • The distribution of P&L is highly skewed to the left

    • similar to a short position in an option
    • Merton model:

  • Major features
    • The Expected credit loss (ECL) represents the average credit loss. The pricing of the portfolio should be such that it covers the expected loss
    • The Unexpected credit loss (UCL) represents the loss that will not be exceeded at some level of confidence, typically 99.9%. Taking the deviation from the expected loss gives the unexpected credit loss. The institution should have enough equity capital to cover the unexpected loss.

The effect of correlations

  • Correlations across default event

  • Correlations across default event and exposure

    • wrong-way trades（逆向风险交易/错向风险交易）: exposure is positively correlated with the probability of default

    • right-way trades（正向风险交易/顺向风险交易） occurs when the transaction is a hedge for the counterparty

      Credit risk is lowered for right-way trades, where the counterparty is using the trade as a hedge. Conversely, wrong-way trades create a positive correlation between the credit exposure and the probability of default.

Measuring Expected Credit Loss

Expected Loss over a Target Horizon

Assuming independency,

The Time Profile of Expected Loss

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 only, using the cumulative default rate and discount factor :

Measuring Credit VaR

  • Credit VaR over a Target Horizon
    • At a given confidence level , the worst credit loss (WCL) is defined such that

  • Credit VaR is defined as the unexpected credit loss at some confidence level, which is measured as the deviation from ECL:

    • It should be viewed as the economic capital to be held as a buffer against unexpected losses
  • Using Credit VaR to Manage the Portfolio
    • rationales on trades: profitability vs. credit risk (credit VaR)
    • the marginal contribution to risk can be used to analyze the incremental effect of a proposed trade on the total portfolio risk
    • the marginal analysis can also help to establish the remuneration of capital（资本报酬率） required to support the position

Portfolio Credit Risk Models

Approaches to Portfolio Credit Risk Models

  • Model Type
    • Top-down models group credit risks using single statistics. They aggregate many sources of risk viewed as homogeneous into an overall portfolio risk, without going into the details of individual transactions. It is appropriate for retail portfolios with large numbers of credits, but less so for corporate or sovereign loans.
    • Bottom-up models account for futures of each instrument. It is most similar to the structural decomposition of positions that characterizes market VaR systems. It is appropriate for corporate and capital market portfolios. It is also most useful for taking corrective action, because the risk structure can be reverse-engineered to modify the risk profile
  • Risk Definitions
    • Default-mode models consider only outright default as a credit event
    • Mark-to-market (MTM) models consider changes in market values and ratings changes, including defaults
  • Models of Default Probability
    • Conditional models incorporate changing macroeconomic factors into the default probability through a functional relationship
    • Unconditional models have fixed default probabilities and tend to focus on borrower-or factor-specific information
  • Models of Default Correlations
    • Structural Models explain correlations by the joint movements of assets
    • Reduced-form models explain correlations by assuming a particular functional relationship between the default probability and background factors

  • Comparison of Credit Risk Models

    CreditMetrics CreditRisk+ Moody's KMV Credit Portfolio View
    Originator JPMorgan Credit Suisse KMV McKinsey
    Model type Bottom-up Bottom-up Bottom-up Top-down
    Risk definition Market value (MTM) Default losses (DM) Default losses (MTM/DM) Market value (MTM)
    Risk drivers Asset values Default rates Asset values Macro factors
    Credit events Rating change/default Default Continuous default probability Rating change/default
    Probability Unconditional Unconditional Conditional Conditional
    Volatility Constant Variable Variable Variable
    Correlation From equities (structural) Default process (reduced-form) From equities (structural) From macro factors
    Recovery rates Random Constant within band Random Random
    Solution Simulation/analytic Analytic Simulation Simulation

Credit VaR 算例

算例设定

目标资产：某银行对公司A的贷款

  • 本金 (Principal): ¥10,000,000
  • 当前评级: BBB
  • 剩余期限: 1年
  • 违约损失率 (LGD): 45%
  • 置信水平: 99%
  • 无风险利率: 3%

方法一：蒙特卡罗模拟法 (Monte Carlo Simulation)


背景与基本思想
蒙特卡罗方法通过大量随机模拟来估计损失分布，是最直观和灵活的Credit VaR计算方法。

核心思想：

  • 模拟大量可能的违约情景
  • 统计损失分布
  • 计算指定置信水平下的VaR

数学模型

  1. 违约概率建模
    使用Logistic回归模型：

    违约

    其中：
    • (截距项)
    • (系数)
    • (标准化风险因子)

  1. 损失计算公式

    损失违约

    其中：

    • 违约 为违约指示函数

  2. 详细计算步骤

    • 步骤1：参数设定
      • 模拟次数: N = 100,000
      • EAD = ¥10,000,000
      • LGD = 45%
      • BBB级1年期违约率 = 0.18%
  • 步骤2：生成随机违约情景
# 伪代码示例
for i in range(100000):
    X_i = random.normal(0, 1)
    PD_i = 1 / (1 + exp(-(-2.5 + 0.8 * X_i)))
    U_i = random.uniform(0, 1)
    Default_i = 1 if U_i <= PD_i else 0
    Loss_i = 10000000 * 0.45 * Default_i
  • 步骤3：损失分布统计
    模拟结果（部分）：
情景1: X=-0.52, PD=0.16%, U=0.9995, 违约=否, 损失=0
情景2: X=1.83, PD=0.48%, U=0.0032, 违约=是, 损失=¥4,500,000
情景3: X=-1.21, PD=0.11%, U=0.8821, 违约=否, 损失=0
...
  • 步骤4：Credit VaR计算
    违约次数: 181次 (在100,000次模拟中)
实际违约率: 0.181% ≈ 0.18% ✓

预期损失 (EL):
EL = 10,000,000 × 0.18% × 45% = ¥8,100

99% VaR计算：
├─ 99%分位数对应第99,000个损失值
├─ 排序后第99,000个值 = 0
└─ 99% VaR = ¥4,500,000 - ¥8,100 = ¥4,491,900

扩展到投资组合

  • 相关性建模

    # 多因子模型
X_i = √ρ × Z + √(1-ρ) × ε_i

其中：
├─ Z: 系统性风险因子
├─ ε_i: 资产特有风险因子
├─ ρ: 资产相关系数
└─ 所有因子服从N(0,1)

  • 组合损失计算
    组合损失 = Σ(i=1 to n) EAD_i × LGD_i × I_i(违约)

方法二：CreditMetrics方法

背景与基本思想
由JP Morgan开发，基于信用评级转移和市值重估的方法。

核心思想：

  • 基于历史评级转移矩阵
  • 考虑评级变化对资产价值的影响
  • 通过正态分布门槛模型模拟评级转移

数学模型

  1. 评级转移矩阵（BBB级1年期，%）
AAA AA A BBB BB B CCC 违约
0.03 0.31 5.95 86.93 5.30 1.17 0.12 0.18
  1. 门槛模型：资产价值变化
  1. 详细计算步骤
    • 步骤1：评级转移概率校准
      门槛值计算：
      ├─ P(违约) = 0.18% → Z_违约 = Φ^(-1)(0.0018) = -2.88
      ├─ P(≤CCC) = 0.30% → Z_CCC = Φ^(-1)(0.0030) = -2.75
      ├─ P(≤B) = 1.47% → Z_B = Φ^(-1)(0.0147) = -2.17
      ├─ P(≤BB) = 6.77% → Z_BB = Φ^(-1)(0.0677) = -1.49
      └─ P(≤A) = 92.70% → Z_A = Φ^(-1)(0.9270) = 1.46
    • 步骤2：资产价值重估
      各评级对应的贷款价值：
      ├─ A级: PV = ¥10,339,806
      ├─ BBB级: PV = ¥10,407,767
      ├─ BB级: PV = ¥10,563,107
      ├─ B级: PV = ¥10,825,243
      ├─ CCC级: PV = ¥11,378,641
      └─ 违约: PV = ¥5,500,000

  • 步骤3：损失分布计算

      预期损失：
  EL = 0.0147×67,961 + 0.8522×0 + 0.0553×155,340
  + 0.0117×417,476 + 0.0012×970,874 + 0.0018×4,907,767
  = ¥24,472

  损失分布排序（从小到大）：
  ├─ 前1.47%: 收益¥67,961
  ├─ 1.47%-86.69%: 损失¥0
  ├─ 86.69%-92.22%: 损失¥155,340
  ├─ 92.22%-93.39%: 损失¥417,476
  ├─ 93.39%-99.51%: 损失¥970,874
  └─ 99.51%-100%: 损失¥4,907,767

  99% VaR计算：
  ├─ 99%分位数落在违约区间
  └─ Credit VaR = ¥4,907,767 - ¥24,472 = ¥4,883,295

方法三：CreditRisk+方法

背景与基本思想
由Credit Suisse开发，基于保险精算思想的解析方法。

核心思想：

  • 违约事件服从泊松分布
  • 通过概率生成函数求解损失分布
  • 计算速度快，适合大型组合

数学模型

  1. 基本假设
    ├─ 各资产违约相互独立
    ├─ 违约次数服从泊松分布: N ~ Poisson(λ)
    ├─ 损失率固定或已知分布
    └─ 可处理大量小额敞口
  1. 损失分布推导

单一资产违约概率: p_i
敞口大小: EAD_i
损失严重度: LGD_i

期望违约次数: λ = Σp_i
概率生成函数

  1. VaR计算
    λ = 0.0018
    P(L = 0) = e^(-0.0018) = 0.9982
    P(L = 45单位) = 0.0018

99.9% VaR = 45单位 = ¥4,500,000
Credit VaR = ¥4,500,000 - ¥8,100 = ¥4,491,900

方法四：结构化方法 (Structural Approach)

背景与基本思想
基于Merton模型，将违约视为公司价值低于债务价值的事件。

核心思想：

  • 公司价值服从几何布朗运动
  • 违约发生在公司价值触及违约边界时
  • 通过期权定价理论计算违约概率

数学模型
公司价值动态:

参数: , , ,

违约概率计算:

预期损失:

99% VaR = ¥5,200,000 - ¥180,440 = ¥5,019,560

各方法比较总结

计算结果对比

方法 Credit VaR 预期损失 计算复杂度 适用场景
蒙特卡罗 ¥4,491,900 ¥8,100 高 灵活性要求高
CreditMetrics ¥4,883,295 ¥24,472 中等 评级数据充足
CreditRisk+ ¥4,491,900 ¥8,100 低 大型组合
结构化方法 ¥5,019,560 ¥180,440 高 上市公司

方法特点

  • 蒙特卡罗方法

    • 优点：最灵活，可处理复杂相关性
    • 缺点：计算量大，收敛速度慢
    • 适用：研究分析，模型验证

  • CreditMetrics

    • 优点：基于市场数据，考虑评级变化
    • 缺点：依赖评级转移矩阵的稳定性
    • 适用：评级数据可靠的组合

  • CreditRisk+

    • 优点：计算速度快，解析解
    • 缺点：假设限制较多，难处理相关性
    • 适用：大型零售贷款组合

  • 结构化方法

    • 优点：理论基础牢固，前瞻性强
    • 缺点：参数估计困难，模型复杂
    • 适用：上市公司，公司债券

实务应用建议

  • 监管资本计算：优先使用CreditRisk+或简化CreditMetrics
  • 风险管理决策：推荐蒙特卡罗方法
  • 组合优化：结合多种方法进行敏感性分析
  • 模型验证：使用历史数据回测各方法准确性

课堂练习

某银行持有一个由5笔贷款组成的投资组合，各贷款参数如下：

贷款 评级 EAD (万元) PD (%) LGD (%)
1 AAA 500 0.02 30
2 A 800 0.10 35
3 BBB 1,000 0.30 40
4 BB 600 1.50 45
5 B 400 5.00 50

(a) 计算该组合的预期信用损失（ECL）。
(b) 假设各贷款之间违约相互独立，计算信用损失的标准差。
(c) 如果资产相关系数为0.15，信用损失的标准差如何变化？

课堂练习

Consider a 3-year CDS with a notional of $20 million. The risk-neutral annual default probability is 2.5%, the recovery rate is 35%, and the risk-free rate is 4% (continuous).

(a) Calculate the fair CDS spread.
(b) If the market spread is 180bp, what is the mark-to-market value of the CDS for the protection buyer?
(c) Explain how the CDS spread would change if the counterparty's credit rating deteriorated from A to BBB.

End of L04

Key takeaways:

  • Credit risk is driven by three factors: PD, CE, and LGD
  • Default probabilities can be estimated from credit ratings (actuarial) or market prices (risk-neutral)
  • The Merton model provides a structural framework linking equity to default risk
  • Credit exposures vary by instrument type and can be modified by margins, collateral, and netting
  • Credit derivatives (CDS, TRS, CLN) allow efficient transfer and trading of credit risk
  • Portfolio credit risk models (CreditMetrics, CreditRisk+, KMV) differ in their treatment of correlations and credit events
  • 2008 crisis and Archegos event illustrate the importance of model risk, counterparty risk, and transparency

#### A Detailed Example from the Text <font size=5> It is instructive to work through a simplified example. Consider a firm with assets worth $V = \$100$ and with volatility $\sigma_V = 20\%$. In practice, one would have to start from the observed stock price and volatility and iterate to find $\sigma_V$. The horizon is $\tau = 1$ year. The risk-free rate is $r = 10\%$ using continuous compounding. The debt face value is $K = \$99.46$, which implies a risk-free current value of $Ke^{-r\tau} = \$90$ and a leverage factor of $x = 0.9$. Working through the Merton analysis, one finds that the current stock price should be $S = \$13.59$. Hence the current bond price is $$B = V - S = \$100 - \$13.59 = \$86.41$$ which implies a yield of $\ln(K/B)/\tau = \ln(99.46/86.41) = 14.07\%$, or a yield spread of $4.07\%$. The current value of the credit put is then $$P = Ke^{-r\tau} - B = \$90 - \$86.41 = \$3.59$$ </font> --- <font size=5> The analysis also generates values for $N(d_2) = 0.6653$ and $N(d_1) = 0.7347$. Thus the risk-neutral probability of default is $\text{EDF} = N(-d_2) = 1 - N(d_2) = 33.47\%$. Finally, let us decompose the expected loss at expiration: $$N(-d_2)[K - Ve^{r\tau}N(-d_1)/N(-d_2)] = 0.3347 \times [\$99.46 - \$110.56 \times 0.2653/0.3347]$$ $$= 0.3347 \times [\$11.85] = \$3.96$$ This combines the probability of default with the expected loss upon default, which is $\$11.85$. </font> --- <font size=5> Note that the model needs very high leverage, here $x = 90\%$, to generate a reasonable credit spread of $4.07\%$. This implies a debt-to-equity ratio of $0.9/0.1 = 900\%$, which is unrealistically high for this type of spread. With lower leverage, say $x = 0.7$, the credit spread shrinks rapidly, to $0.36\%$. At $x = 50\%$ or below, the predicted spread goes to zero. As this leverage would be considered normal, the model fails to reproduce the size of observed credit spreads. Perhaps it is most useful for tracking time variation in estimated default frequencies. </font> ---