6 Fixed income

CFA二级固定收益科目，其中利率二叉树为重点内容，需要掌握计算，其他部分重点理解相关概念。

Fixed income

• The Term Structure and Interest Rate Dynamics
  • Yield curve and spread
    • 一些概念
      • Interest rates
        • Spot rates (Sj)
        • Forward rates: f(j , k); maturing at time j+k and delivered at time j
      • Discount factors
        • Spot
        • Forward
    • Forward rate model
    • Interest curves
      • Yield curves
        • 图形
        • 特点
          • When the spot curve slopes upward, the forward curve will lie above the spot curve; conversely, , the forward curve will lie below
          • An upward-sloping yield curve is reflecting a market expectation of rising or at least stable future inflation (associated with relatively strong economic growth)
          • An inverted yield curve may reflect a market expectation of declining future inflation rates
      • Par curve
        • Definition: represents the yields to maturity on coupon-paying government bonds, priced at par over a range of maturities
        • Recently issued ("on the run") bonds are most often used to create the curve
        • Bootstrapping: The zero-coupon rates are determined by using the par yields and solving for the zero-coupon rates one by one, from the shortest to longest maturities (forward substitution)
      • Swap rate curve
        • Swap rate: the fixed rate in a plain vanilla interest rate swap
        • The reasons to use swap rate curve as a benchmark interest rate curve rather than a government bond yield curve
          • The swaps are more liquid than Treasury bonds
          • The swap market is not regulated by any government, making swap rates in different countries more comparable
          • The swap curve typically has yield quotes at more maturities
        • The pricing of swap rate
    • Active Bond Portfolio Management
      • Expected future spot rates ≠ quoted forward rate
        • The investor's expected future spot rates <(>) a quoted forward rate for the same maturity
          • The bond to be undervalued (overvalued)
            • The market is effectively discounting the bond's payments at a higher(lower) rate
            • The bond's market price is below(above) the intrinsic value perceived by the investor
        • Bond return = Receipt of promised coupons (and principal) + Reinvestment of coupon payments +/– Capital gain/Loss on sale prior to maturity
      • Rolling down the yield curve（riding the yield curve）
        • When a yield curve is upward sloping, the forward curve is always above the current spot curve
        • The trader expects the yield curve to remain static over an investment horizon
          • Buying bonds with a maturity longer than the investment horizon would provide a total return greater than the return on a maturity-matching strategy
    • The Relation Between Yield-to-Maturity and Spot Rate
      • For zero coupon bonds: YTM = spot rates
      • For non zero coupon bonds: unless the spot rate curve is horizontal, YTM does not equal to spot rates
      • YTM of the bond should be some weighted average of spot rates used in the valuation of the bond
      • YTM= expected rate of return if
        • A bond is held to maturity
        • All promised coupon and principal payments are made in full when due
        • The coupons are reinvested at the original YTM
      • The YTM can provide a poor estimate of expected return if
        • Interest rates are volatile
        • The yield curve is sloped either upward or downward
        • There is significant risk of default
        • The bond has one or more embedded options. (e.g., put, call, or conversion)
    • Spread
      • Swap spread
        • 公式: Swap spread = Swap rate – government bond interest rate
        • 风险
          • The swap spread is a barometer of the market's perceived credit risk relative to default-risk-free rates, roughly reflects the default risk of a commercial bank
          • The spread typically widens countercyclically, exhibiting greater values during recessions
        • 图形
      • I-spread
        • 公式: I-Spread= bond rates - swap rates (of the same maturities)
        • 风险: only reflects compensation for credit and liquidity risks
      • Z-spread
        • 定义: when the spread added to the benchmark spot curve, produces a value equal to the market price of the bond
        • 评价
          • Z-spread provide a more accurate measure of credit and liquidity than swap spreads
          • Z-spread will be more accurate than a linearly interpolated yield (I-spread)
      • TED spread
        • 公式: TED spread = LIBOR – T-bill rate
        • 风险
          • A key indicator of perceived credit and liquidity risks
          • Ted spread more accurately reflects risk in the banking system
      • LIBOR-OIS spread
        • 公式: LFBOR - overnight indexed swap (OIS) rate
        • 风险: LIBOR-OIS spread is a useful measure of credit risk and an indication of liquidity risk of money market
  • Theories of the term structure of interest rates
    • Expectations Theory
      • Unbiased(pure) expectations theory
        • The forward rate is an unbiased predictor of the future spot rate
        • The predictions of the unbiased expectations theory are consistent with the assumption of risk neutrality
      • Local expectations theory
        • The expected return for every bond over short periods is the risk-free
        • Although the theory requires that risk premiums be nonexistent for very short holding periods, no such restrictions are placed on longer-term investments
    • Liquidity Preference Theory
      • The liquidity premiums exist to compensate investors for the added interest rate risk they face when lending long term
      • Yield curve is usually upward-sloping, but a downward-sloping yield curve could still be consistent with the existence of liquidity premiums
      • The liquidity premium rises as maturities rise
    • Segmented Markets Theory
      • The yields are not a reflection of expected spot rates or liquidity premiums, but just solely a function of the supply and demand for funds of a particular maturity
      • Assumes that market participants are either unwilling or unable to invest in anything other than securities of their preferred maturity
    • Preferred Habitat Theory
      • If the expected additional returns to be gained become large enough, institutions will be willing to deviate from their preferred maturities or habitats
      • The theory moves closer to explaining real-world phenomena
  • Yield curve models
    • Yield curve Movement
      • Shaping risk: the sensitivity of a bond's price to the changing shape of the yield curve
    • Factors affecting the shape of the yield curve
      • Level factor: a reflection of parallel yield curve moves in which rates move in the same direction
      • Steepness factor: refers to a non-parallel shift in the yield curve when either short-term rates change more than long-term rates or long-term rates change more than short-term rates
      • Curvature factor: This variable explaining the "twist" in the yield curve has the smallest impact of the three
    • Yield curve risk models
      • Yield curve risk: the risk to portfolio value arising from unanticipated changes in the yield curve
      • Key rate duration (KRD)
      • Level, steepness, and curvature
        • Level (ΔxL): A parallel increase or decrease of interest rates
        • Steepness (ΔxS): Long-term interest rates increase while short-term rates decrease
        • Curvature (ΔxC): Increasing curvature means short- and long-term interest rates increase while intermediate rates do not change
        • The proportional change in portfolio value resulted from yield curve movement
      • Other factors
        • Developing interest rate views using macroeconomic variables
          • During economic expansions
            • Monetary authorities raise benchmark rates
              • Bearish flattening
                • Flatter yield curve
          • During economic recessions
            • Monetary authorities cut benchmark rates
              • Bullish steepening
                • Steeper yield curve
          • 概要
        • Yield Volatility
          • The reasons why quantifying interest rate volatilities is important
            • Most fixed-income instruments and derivatives have embedded options and option valuescrucially depend on the level of interest rate volatilities
            • Fixed-income interest rate risk management is clearly an important part of any management process
          • Term structure of interest rate volatility
            • The term structure is a representation of the yield volatility of a zero-coupon bond
            • The volatility term structure typically shows that short-term rates are more volatile than long-term rates
              • Short-term volatility is most strongly linked to uncertainty regarding monetary policy
              • Long-term volatility is most strongly linked to uncertainty regarding the real economy and inflation
          • The square root rule of interest volatility
• The Arbitrage-Free Valuation Framework
  • Arbitrage-free valuation methods
    • Arbitrage opportunities
      • 定义: an arbitrage transaction involves no initial cash outlay but a positive riskless profit (cash flow) at some point in the future; basic principle of the "law of one price"
      • 类型
        • Value additivity: the value of the whole must equal the sum of the values of the parts
        • Dominance: A financial asset with a risk-free payoff in the future must have a positive price today
    • 过程
      • Stripping: if the portfolio of strips is trading for less than an intact bond, one can purchase the strips, combine them (reconstituting), and sell them as a bond
      • Reconstitution: if the bond is worth less than its component parts, one could purchase the bond, break it into a portfolio of strips (stripping), and sell those components
      • The arbitrage-free valuation approach does not allow a market participant to realize an arbitrage profit through stripping and reconstitution
    • Arbitrage-free valuation
      • A fixed-rate, option-free bond
      • Bonds with embedded options: binomial interest rate tree
  • Binomial interest rate tree framework
    • 定义: Assumes that interest rates have an equal probability of taking one of two possible values in the next period , Over multiple periods, the set of possible interest rate paths that are used to value bonds with a binomial model is called a binomial interest rate tree
    • 假设条件
      • The interest rate tree is constructed using the lognormal random walk model with two desirable properties
        • Higher volatility at higher rates
        • Non-negative interest rates
      • Volatility estimates
        • Historical data
        • Implied volatility
    • 图形
      • The interest rate at each node is the forward rate from the current period to the next period
      • A node is a point in time when interest rates can take one of two possible paths, an upper path, H, or a lower path, L
      • 参数说明
        • σ = assumed volatility of the one-year rate
        • i1,L = the lower one-year forward rate one year from now at Time 1
        • i 1,H = the higher one-year forward rate one year from now at Time 1
      • If the interest rate volatility increases ,the forward rates shown in the binomial interest rate tree will spread out
      • The middle forward rate in a period is close to the implied one-year forward rate for that period
    • Valuation methods
      • Backward induction: process of valuing a bond using a binomial interest rate tree
      • Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths
  • Monte Carlo Method
    • 作用: often used when a security's cash flows are path dependent
    • 步骤
      • Simulate numerous (say, 500) paths of one-month interest rates under a volatility assumption and probability distribution
        • Generate spot rates from the simulated future one-month interest rates
          • Determine the cash flow along each interest rate path
            • Calculate the present value for each path
              • Calculate the average present value across all interest rate paths
    • Note
      • A constant (drift term) is added to all interest rates on all paths such that the average present value for each benchmark bond equals its market value. When this technique is used, the model is said to be drift adjusted
      • Increasing the number of paths increases the accuracy of the estimate in the statistical sense, but does not mean the model is closer to the true fundamental value of the security
      • The Monte Carlo estimation is mean reversion
  • Modern Term Structure Models
    • Equilibrium Term Structure Models
      • 特点
        • Attempt to describe changes in the term structure through the use of fundamental economic variables that are assumed affect interest rates
        • Require the specification of a drift term and the assumption of a functional form for interest rate volatility
        • They are one factor or multifactor models and both the Vasicek and CIR models assume a single factor——the short term interest rate
      • 模型
        • Cox-Ingersoll-Ross model
          • 假设条件
            • Interest rate movements are driven by individuals choosing between consumption today versus investing and consuming at a later time
            • If interest rates rise, then investors will increase their investment (delay consumption), thereby increasing the supply of long-term funds, which will inevitably lead to a decline in long-term interest rates and vice versa
          • 公式
            • 组成部分
              • dr = change in the short- term interest rate
              • k = speed of mean reversion parameter
              • θ is the long run mean rate
              • rt= the current interest rate(the short term interest rate)
              • dt = a small increase in time
              • σ = volatility
              • dz = a small random walk movement
            • 说明
              • The model has two parts
                • A deterministic part (sometimes called a "drift term") , the expression in dt
                • A stochastic ( i.e., random ) part , the expression in dz , which models risk
              • The deterministic part , k(θ -r )dt , ensures mean reversion with the speed of adjustment governed by the strictly positive parameter k
              • The standard deviation factor makes volatility proportional to the square root of the short-term rate , which allows for volatility to increase with the level of interest rates
        • Vasicek model
          • 公式
          • 和Cox-Ingersoll-Ross模型比较
            • The Vasicek model has the same drift term as the CIR model and thus tends toward mean reversion in the short rate
            • Non interest rate(r) term appears in the second term
            • Assume constant volatility over the period of analysis and does not increase with the level of interest rates
          • Disadvantage: It is theoretically possible for the interest rate to become negative
    • Arbitrage-Free Models
      • Advantage over equilibrium model: the ability to calibrate arbitrage- free models to match current market prices
      • 模型
        • Ho–Lee model
          • 公式
          • 描述
            • The model can be calibrated to market data by inferring the form of the time-dependent drift term, θt, from market prices , which means the model can precisely generate the current term structure
            • No mean reversion
        • Kalotay–Williams–Fabozzi model
• Valuation and Analysis of Bonds with Embedded Options
  • Overview of Embedded Options
    • The term"embedded bond options" or embedded options refers to contingency provisions found in the bond's indenture or offering circular
    • Callable bonds
      • Types
        • European-style option
        • American-style option
        • Bermudan-style: option can be exercised at fixed dates after the lockout period
      • 特点: Most callable bonds include a call protection period during which the issuer cannot call the bond
    • Putable bonds
      • The put provision allows the bondholders to put back the bonds to the issuer prior to maturity, usually at par. This usually happens when interest rates have risen and higher-yielding bonds are available
      • Similar to callable bonds, most putable bonds include protection periods
    • Extendible bond: At maturity, the holder of an extendible bond has the right to keep the bond for a number of years after maturity, possibly with a different coupon
    • Complex Embedded Options
      • Convertible bonds : another type of bond with an embedded option. The conversion option allows bondholders to convert their bonds into the issuer's common stock
      • Estate put bonds: In the event of the holder’s death, this bond can be put at par by the heir(s)
      • Sinking fund bond: Sinking fund bonds (sinkers): require the issuer to set aside funds periodically to retire the bond (a sinking fund)
  • Relationships between the Values of a Callable or Putable Bond, Straight Bond, and Embedded Option
    • Callable bond
      • Value of callable bond = Value of straight bond – Value of issuer call option
      • Value of issuer call option = Value of straight bond – Value of callable bond
    • Putable bond
      • Value of putable bond = Value of straight bond + Value of investor put option
      • Value of investor put option = Value of putable bond – Value of straight bond
  • Basic process for valuing callable (or putable) bond: Apply backward induction valuation methodology
    • Call rule: When valuing a callable bond, the value at any node where the bond is callable must be the call price or the computed value if the bond is not called, whichever is lower
    • Put rule: for a putable bond, the value used at any node corresponding to a put date must be either the put price or the computed value if the bond is not put, whichever is higher
  • Factors affecting the value of embedded bonds
    • Interest Rate Volatility
      • Option values are positively related to the volatility of their underlying
      • The value but isof a straight bond is unaffected by changes in the volatility of interest rates
      • 变化过程: Volatility change →option value→ change bond price/value change
    • Option-adjusted spreads (OAS)
      • 定义: If risk-free rates are used to discount cash flows of a credit risky corporate bond, the calculated value will be too high. To correct for this, a constant spread must be added to all one-period rates in the tree such that the calculated value equals the market price of the risky bond. This constant spread is called the option adjusted spread (OAS)
      • 评价
        • OAS is used by analysts in relative valuation; bonds with similar credit risk should have the same OAS
        • Bonds with low OAS (relative to peers) are considered to be overvalued
      • How interest rate volatility affects option-adjusted
        • As the assumed level of volatility used in an interest rate tree increases, the computed OAS (for a given market price) for a callable bond decreases
        • The computed OAS of a puttable bond increases as the assumed level of volatility in the binomial tree increases
        • OAS计算
    • Relationship Between Volatility and OAS
  • The interest rates risk of embedded bond
    • Effective duration (ED)
      • 公式
      • 不同工具Effective Duration
        • Effective duration (zero-coupon) ≈ maturity of the bond
        • Effective duration of fixed-rate bond < maturity of the bond
        • Effective duration of Cash = 0
        • Effective duration of floater ≈ time (years) to next reset
        • Effective duration (callable) ≤ effective duration (straight)
        • When interest rates fall (rise), the call (put) option gives issuer the right to retire the bond at the call price/ the investor can put the bond and reinvest the proceeds of the retired bond at a higher yield. Thus, the call (put) option reduces the effective duration of the putable bond relative to that of the straight bond
      • Compares the effective durations of option-free, callable, and putable bonds
        • Straight bonds have positive effective convexity
        • Callable bonds are unlikely to be called and will exhibit positive convexity when rates are high
        • Putable bonds exhibit positive convexity throughout
      • One-Sided Durations
        • 公式
        • 比较
          • When the underlying option is at-the-money (or near-the-money), callable bonds will have lower one-sided down duration than one-side up duration
          • A near-the-money putable bond will have larger one-sided down duration than one-sided up-duration
      • Key rate durations(partial durations)
        • Key rate durations can sometimes be negative for maturity points that are shorter than the maturity of the bond being analyzed if the bond is a zero-coupon bond or has a very low coupon
        • Option-free bond
          • Trading at par, the bonds maturity matched rate is the only rate that affects the bonds value
          • Not trading at par, shift the maturity-matched par rate has the greatest effect
        • Callable bonds
          • With low coupon rate are unlikely to be called , hence , the rate that has the highest effect on the value of the callable bond is the maturity-matched rate
          • As the bond's coupon increase , however so does the likelihood of the bond being called. The rate that has the highest effect on the callable bond's value is the time-to-exercise rate
        • Putable bonds
          • With high coupon rates are unlikely to be put, their prices are most sensitive to their maturity-matched rates
          • A low-coupon bond is more likely to be put, its price is most sensitive to the time-to-exercise rate
    • Effective convexities (EC)
      • Straight bonds have positive effective convexity
      • Callable bonds are unlikely to be called and will exhibit positive convexity when rates are high
        • The effective convexity turns negative when the underlying call option is near the money
        • The upside potential of the bond's price is limited due to the call(while the downside is not protected)
      • Putable bonds exhibit positive convexity throughout
  • Value Of A Capped Or Floored Floating-rate Bond
    • Value of capped floater = Value of straight bond 一 Value of embedded cap (protects the issuer)
    • Value of floored floater= Value of straight bond + Value of embedded floor (protects the investor)
  • Convertible bond
    • 定义: The owner of the convert bond has the right to convert the bond into a fixed number of common shares of the issuer during a specified timeframe (conversion period) and at a fixed amount of money (conversion price)
    • A convertible bond includes an embedded call option
    • 一些概念
      • Conversion ratio: the number of common shares for which a convertible bond can be exchanged
      • Conversion price: the bond issue price divided by the conversion ratio
      • Market conversion price (conversion parity price)
        • 定义: the price that the convertible bondholder would effectively pay for the stock if she bought the bond and immediately converted it
        • Market conversion price=market price of convertible bond/Conversion ratio
      • Conversion value = market price of stock × conversion ratio
      • Minimum value of a convertible bond = max (straight value, conversion value)
      • Market conversion premium per share
        • Market conversion premium per share = market conversion price − stock's market price
        • Market conversion premium ratio = Market conversion premium per share / market price of common stock
        • Premium over straight value= (market price of convertible bond/straight value)-1
    • Effects of convertible bond
      • Volatility
        • The value of a noncallable / nonputable convertible bond = straight value + value of call option on stock
        • Stock price volatility ↓=>Value of the call option on the stock ↓ =>convertible bond value ↓
      • Embedded option
        • Callable convertible bond value = straight value of bond+ value of call option on stock − value of call option on bond
        • Callable and putable convertible bond value = straight value of bond+ value of call option on stock − value of call option on bond+ value of put option on bond
  • Comparison of the Risk–Return Characteristics
    • Fixed-income equivalent (busted convertible): the price of the common stock associated with a convertible issue is so low that it has little or no effect on the convertibles market price and the bond trades as though it is a straight bond
    • Common stock equity: the price of the stock may be high enough that the price of the convertible behaves as though it is an equity security
    • Hybrid security: most of the time , convertible bond is a hybrid security with the characteristics of equity and a fixed-income security
• Credit Analysis Models
  • Modeling credit risk and the credit valuation adjustment
    • Expected exposure: the amount of money a bond investor in a credit risky bond stands to lose at a point in time before any recovery is factored in, which changes over time
    • Recovery rate: the percentage recovered in the event of a default Recovery rate=1- loss severity
    • Loss given default (LGD)= loss severity * exposure
    • Expected loss = Default probability × Loss severity given default(LGD)
    • Credit valuation adjustment (CVA)
      • CVA: the sum of the present value of the expected loss for each period
      • CVA = price of risk-free bond – price of risky bond
    • Fair value of corporate bond = VND (the value for the corporate bond assuming no default) – CVA
  • Credit scores and credit ratings
    • Credit scoring
      • 作用: used primarily in the retail lending market for small businesses and individuals
      • 判断标准: A higher credit score indicates better credit quality
      • 特点: Credit ratings do not adjust dynamically to changes in the economic environment
      • Five primary factors in the algorithm
        • 35% for the payment history
        • 30% for the debt burden
        • 15% for the length of credit history
        • 10% for the types of credit used
        • 10% for recent searches for credit
    • Credit rating
      • 作用: Credit ratings are widely used in corporate and sovereign bond markets
      • The three major global credit rating agencies are Moody's Investors Service, Standard & Poor's, and Fitch Ratings
      • Notching
        • 定义: an adjustment to the issuer rating to reflect the priority ofclaim for specific debt issues of that issuer and to reflect any subordination
        • The issuer rating for a company is typically for its senior unsecured debt
        • The rating on subordinated debt is adjusted or "notched" by lowering it one or two levels
    • Credit Transition Matrix
      • 应用: the expected percentage price change = modified duration * the change in the spread
    • The credit migration reduces the expected return
      • The probabilities for change are not symmetrically distributed around the current rating, which are skewed toward a downgrade rather than an upgrade
      • The increase in the credit spread is much larger for downgrades than the decrease in the spread for upgrades
  • Structural and reduced-form credit models
    • Structural models
      • 内容
        • Structural models are based on the structure of a company's balance sheet and are originated to understand the economics of a company's liabilities and build on the insights of option pricing theory
        • Their key insights were that a company defaults on its debt if the value of its assets falls below the amount of its liabilities and that the probability of that event has the features of an option
      • Assumptions
        • Company's assets (A) are traded in a frictionless arbitrage-free market;
        • The value of the company's assets has a lognormal distribution;
        • The company has a simple balance sheet structure with only one class of simple zero-coupon debt D(K,T).
        • The probability of default is endogenous(internal)
      • Analogy
        • Call option analogy for equity: holding the company's equity is economically equivalent to owning a European call option on the company's assets (A) with strike price K (face value of zero-coupon debt) and maturity T (maturity of debt), because they have the same payoff at maturity T
        • Debt option analogy: owning the company's debt is economically equivalent to owning a riskless bond, and simultaneously selling a European put option on the assets (A) of the company with strike price K (face value of zero-coupon debt) and maturity T
      • 优缺点
        • 优点
          • Provide insight into the nature of credit risk
          • Provide an option analogy for understanding a company's default probability and recovery rate
        • 缺点
          • Model assumptions of simple balance sheet and traded assets are not realistic
          • Estimation procedures do not consider business cycle
          • Limitations on available data
    • Reduced models
      • 内容
        • Do not rely on the structure of a company's balance sheet and therefore do not assume that the assets of the company trade
        • Unlike structural models that aim to explain why default occurs (i.e., when the asset value falls below the amount of liabilities), reduced-form models aim to explain statistically when
      • 假设: Default is exogenous(external) variable that occurs randomly
      • 优缺点
        • 优点
          • Inputs are observable variables, including historical data
          • The default intensity is estimated using regression analysis on company specific variables and macroeconomic variables. This flexibility allows the model to directly reflect the business cycle in the credit risk measure
        • 缺点
          • Do not explain the economic reasons for default
          • Assume that default comes as a “surprise” and can occur at any time
  • Credit spreads
    • Interpreting changes in credit spreads
      • Credit spread on a risky bond = YTM of risky bond – YTM of benchmark
      • The value of a risky bond, assuming it does not default, is its value given no default (VND) → CVA = VND – value of risky debt
    • The determinants of the term structure of credit spreads
      • Credit quality
      • Financial conditions affect the credit spread curve
      • Market demand and supply influence the shape of the spread curve
      • Equity market volatility
  • Credit analysis for securitized debt
    • Collateral pool
    • Homogeneity
    • Servicer quality
    • Structure determines the tranching or other management of credit and other risks in a collateral pool
• Credit Default Swaps
  • Credit default swaps (CDS) fundamentals
    • 定义: A credit default swap (CDS) is a contract between two parties in which one party purchases protection from another party against losses from the default of a borrower. CDS is essentially an insurance contract. CDS have emerged as the primary type of credit derivative
    • 组成部分
      • The underlying is the credit quality of a borrower
      • The protection buyer pays the seller a premium
      • The protection buyer receives in return a promise that if default occurs
    • 过程
  • Basic features of CDS
    • Notional amount/principal: the amount of protection being purchased
    • CDS spread (%): the periodic premium that the buyer of a CDS pays to the seller for protection against credit risk
    • CDS coupon rate (%): the periodic premium that the buyer actually pays to the seller. Typically, for standardization
      • 1% for a CDS on an investment-grade company or index
      • 5% for a CDS on a high-yield company or index
    • Upfront payment/upfront premium: the differential between the credit spread and the standard coupon rate that converted to a present value basis
  • Types of credit default swaps
    • Single name CDS
      • A CDS on one specific borrower (reference entity)
        • Reference obligation: a particular debt instrument issued by the borrower that is the designated instrument being covered by CDS
        • Usually a senior unsecured obligation (senior CDS)
        • Any debt obligation issued by the borrower that is pari passu (ranked equivalently in priority of claims) or higher relative to the reference obligation is covered
      • The payoff of the CDS is determined by the cheapest-to-deliver obligation
    • Index CDS: A CDS that allows participants to take positions on the credit risk of a combination of borrowers
      • The notional principle is the sum of the protection on all the borrowers
      • Credit correlation is a key determinant of its value.The more correlated the defaults, the more costly it is to purchase the index CDS
  • Common types of credit events
    • Bankruptcy: allows the defaulting party to work with creditors under the supervision of the court so as to avoid full liquidation
    • Failure to pay: a borrower does not make a scheduled payment of principal or interest on any outstanding obligations after a grace period, without a formal bankruptcy filing
    • Restructuring: the issuer forces its creditors to accept terms that are different than those specified in the original issue
      • Reduction or deferral of principal or interest
      • Change in seniority or priority of an obligation
      • Change in the currency in which principal or interest is scheduled to be paid
  • Settlement protocols
    • Physical settlement: actual delivery of the debt instrument in exchange for a payment by the credit protection seller of the notional amount of the contract
    • Cash settlement: the credit protection seller pays cash to the credit protection buyer
      • Payout ratio (%) = 1 - recovery rate (%)
      • Payout amount = Payout ratio (%) × Notional amount
  • Pricing and Application of CDS
    • Pricing of CDS
      • Protection leg: the contingent payment that the credit protection seller may have to make to the credit protection buyer
      • Premium leg: the series of payments the credit protection buyer promises to make to the credit protection seller
      • 公式
        • Upfront premium (%) by buyer ≈ (CDS spread - CDS coupon) × Duration of CDS
        • Upfront payment = PV (Protection leg) – PV (Premium leg)
        • The lower the CDS coupon rate, the higher the upfront premium
      • Factors that influence the market's pricing of CDS: The higher the probability of default or the loss given default, the higher the CDS spread
    • Profit of CDS
      • Profit for protection buyer (%) ≈ Change in spread (%) × Duration of CDS
      • Profit for protection buyer ($) ≈ Change in spread (%) × Duration of CDS × Notional amount($)
    • Application
      • Credit curve: the term structure of credit spread or the credit spreads for a range of maturities of a company's debt. Upward-sloping credit curves imply a greater likelihood of default in later years
      • Managing credit exposure: the taking on or shedding of credit risk in light of changing expectations and/or valuation disparities
        • Adjustment of credit exposure: increasing/decreasing credit exposure by selling/buying CDS if having assumed too little/much credit risk
        • Naked CDS: buying or selling credit protection without credit exposure to the reference entity
        • Long/short trade: taking a long position in one CDS and a short position in another
        • Curve trade: buying a CDS of one maturity and selling a CDS on the same reference entity with a different maturity
          • If an investor believes that the credit curve will become steeper, he can buy a long-term CDS and sell a short-term CDS
      • Valuation disparity: the focus is on differences in the pricing of credit risk in the CDS market relative to that of the underlying bonds
        • Basis trade: exploit the difference of credit spread between bond market and CDS market
        • Arbitrage trade: buy the cheaper and sell the more expensive
          • The cost of the index is not equivalent to the aggregate cost of the index components
          • If a synthetic CDO is not equivalent to the actual CDO
            • Synthetic CDO = Default-free security - CDS (protection seller)