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)