spread risk and default intensity models - bionic turtle · p2.t6. malzchapter 7 spread risk and...

P2.T6. Malz Chapter 7

Spread Risk and Default Intensity Models

Bionic Turtle FRM Video TutorialsBy: David Harper CFA, FRM, CIPM

Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also violates GARP’s ethical standards. 1

Agenda

Malz, Financial Risk Management: Models, History and Institutions

• Chapter 7: Spread Risk and Default Intensity Models

P2.T6. Malz, Chapter 7: Spread Risk and Default Intensity Models

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P2.T6. Malz Chapter 7


WorkbookExam Relevance(XLS not topic) Worksheet

T6.Malz.7 Credit Spreads Medium 7. Credit Spreads

Medium 7. Poisson and Exponential

Low 7. Hazard rate

3

Chapter 7: Spread Risk and Default Intensity Models

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Define the different ways of representing spreads.

Each of the credit spreads attempt to decompose bond interest into

i. a component that is compensation for credit and liquidity risk and ii. a component that is compensation for the time value of money (TVM)


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TVM

Credit,Liquidity

Risk


• Yield spread is the difference between yield to maturity (YTM) of a risky bond and YTM of a benchmark government bond with the ~ same maturity

Yield spread = YTM[risky bond] – YTM[riskless government bond ~maturity].

Yield spread is commonly used in price quotes, but less so for fixed-income analysis.

• Interpolated spread (i-spread): o The benchmark government bond (or new plain vanilla interest-rate swap) almost

never has the same maturity as a particular credit-risky bond; the maturities can be very different.

o The i-spread is the difference between the yield of the credit-risky bond and the linearly interpolated yield between the two benchmark government bonds or swap rates with maturities flanking that of the credit-risky bond.

Commonly used (like yield spread) for price quotes.


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• Zero-coupon (z-spread) is the spread that must be added to the LIBOR spot curve to arrive at the market price of the bond; however, it may also be measured relative to a government bond curve. Therefore, it is good practice to specify the reference risk-free curve being used.

• Asset-swap spread is the spread or quoted margin on the floating leg of an asset swap on a bond.

• Credit default swap spread is the market premium, expressed in basis points, of a CDS on similar bonds of the same issuer.


,

1

( ) ih

hr z ih r z

hi

p c ch e e

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• Option-adjusted spread (OAS) is a version of the z-spread that takes account of options embedded in the bonds. o If the bond contains no options, OAS is identical to the z-spread.

• Discount margin (a.k.a., quoted margin) is a spread concept applied to floating rate notes. It is the fixed spread over the current (one-or three-month) LIBOR rate that prices the bond precisely.

The discount margin is the floating-rate note analogue of the yield spread for fixed-rate bonds.


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Face 100 2-year maturityCoupon 6.0% semiannual coupon

0.5 1.0 1.5 2.0 Swap 0.400% 0.800% 1.200% 1.600%Spot 0.400% 0.799% 1.200% 1.602%

$3.00 $3.00 $3.00 $103.00DF 0.9980 0.9920 0.9822 0.9685

$2.99 $2.98 $2.95 $99.75 $108.67

0.0%

1.0%

2.0%

3.0%

4.0%

0.5 1.0 1.5 2.0



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Face 100 2-year maturity

Coupon 6.0% semiannual coupon

0.5 1.0 1.5 2.0 $3.00 $3.00 $3.00 $103.00

Yield (YTM) 4.00% 4.00% 4.00% 4.00%$2.94 $2.88 $2.83 $95.08 $103.73

Rf Spot 0.40% 0.80% 1.20% 1.60%

Z-spread 2.43% 2.43% 2.43% 2.43%Rf spot + Z 2.83% 3.23% 3.63% 4.03%

$2.96 $2.90 $2.84 $95.02 $103.72 0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

0.5 1.0 1.5 2.0

Swap

Spot

Define and compute the Spread ‘01.

• The Spread ’01 is analogous to the DV01. o Recall the DV01 is the price increase (“mark-to-market gain”) implied by with a one

basis point decline in interest rate

(in the FRM, we follow Tuckman by specifically employing a yield-based DV01 where the interest rate factor is the yield to maturity, YTM).

• The “spread01” is also called the DVCS and it measures the price change implied by a one basis point change in the spread.

o We (Malz) calculate the DVCS by re-pricing the bond after shocking the z-spread.


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Define and compute the Spread ‘01.


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Spread ’01: change in value (price) due to a one basis point change in z-spread

$94.85

$94.90

$94.95

$95.00

$95.05

$95.10

$95.15

459 460 460 461 461 462 462

Spread01

Explain how default risk for a single company can be modeled as a Bernoulli trial.

• Default risk for a single company can be represented as a Bernoulli trial. Over some fixed time horizon (τ) =T2−T1, there are just two outcomes for the firm: o Default occurs with probability (π), oro The firm remains solvent with probability (1−π).

• If we assign the values 1 and 0 to the default and solvency outcomes over the time interval (T1,T2], we define a random variable that follows a Bernoulli distribution. o The time interval (T1,T2] is important: The Bernoulli trial does not ask “does the firm

ever default?,” but rather, “does the firm default over the next year?”

• The mean and variance of a Bernoulli-distributed variable are easy to compute:o The expected value of default on (T1,T2] is equal to the default probability π, ando The variance of default is π * (1−π).


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Explain how default risk for a single company can be modeled as a Bernoulli trial.


Binomial is a series of Bernoulli trials• If Bernoulli trials can be repeated during successive time intervals, of identical

length τ, and where the probability of default during each interval is a constant value π, then we can say the trials are conditionally independent (a.k.a., the model has the property of memorylessness). o This series of independent and identically (i.e., same probability of default)

distributed Bernoulli trials is characterized by a binomial distribution.

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Explain the relationship between exponential and Poisson distributions.

• The binomial distribution is discrete but it can be inconvenient to model default over time with a discrete distribution.

• An alternative is to model the random time at which a default occurs as the first arrival time—the time at which the modeled event occurs—of a Poisson process. In a Poisson process, the number of events in any time interval is Poisson-distributed.

• The time to the next arrival of a Poisson-distributed event is described by the exponential distribution. In this way, we can model the “time to default” as an exponentially distributed random variable.


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Define the hazard rate and use it to define probability functions for default time and conditional default probabilities.

The hazard rate (a.k.a., default intensity) is denoted by lambda, λ.

• Can interpret hazard rate as the instantaneous conditional default probability.

Default time distribution function• The (cumulative) default time distribution function is the probability of default

sometime between now and (τ) and is given by:

• The survival and default probabilities must sum to exactly 1.0 at every instant (t), so the probability of no default sometime between now and time t, called the survival time distribution, is given by:


[ * ] ( ) 1 tP t t F t e

[ * ] 1 [ * ] 1 ( ) tP t t P t t F t e

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Default time density function• The default time density function or marginal default probability is the

derivative of the default time distribution w.r.t. t:

This is always a positive number, since default risk “accumulates;” i.e., the probability of default increases for longer horizons. If lambda, λ, is small, it will increase at a very slow pace. With a constant hazard rate, the marginal default probability is positive but declining.

• The survival probability, in contrast, is declining over time:


[ * ] ( ) tP t t F t et

[ * ] ( ) 0tP t t F t et

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Conditional default probability• If we ask, what is the probability of default over some horizon (t, t + τ) given

that there has been no default prior to time (t), we are asking about a conditional default probability.

By the definition of conditional probability, it can be expressed as the ratio of the probability of the joint event of survival up to time (t) and default over some horizon (t, t + τ), to the probability of survival up to time (t):

* *

* | **

P t t t tP t t t t

P t t

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Cumulative default time distribution and marginal default probability

0%

20%

40%

60%

80%

100%

0 5 10 15

Cumulative default time distribution1 - exp(-λt)

15.00%

5.00%

2.50%

1.00%

0.10%0%

5%

10%

15%

20%

0 5 10 15

Marginal default probabilityλ*exp(-λt)

15.00%

5.00%

2.50%

1.00%

0.10%

Calculate risk-neutral default rates from spreads.

• The spread is approximately equal to the default probability multiplied by the loss given default (LGD), such that the hazard rate is approximated by the spread divided by (1-R):


* *(1 )1

zz R

R

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Describe advantages of using the CDS market to estimate hazard rates.

• Corporations do not issue many zero-coupon bonds; most corporate zero-coupon issues are commercial paper, which have a typical maturity of less than one year, and are issued by only a small number of highly-rated “blue chip” companies. Commercial paper even has a distinct rating system.

• In practice, hazard rates are usually estimated from the prices of CDS. Advantages include:o Standardization. In contrast to most developed-country central governments,

private companies do not issue bonds with the same cash flow structure and the same seniority in the firm’s capital structure at fixed calendar intervals. For many companies, however, CDS trading occurs regularly in standardized maturities of 1, 3, 5, 7, and 10 years, with the five-year point generally the most liquid.

o Coverage. The universe of firms on which CDS are issued is large. Markit Partners, the largest collector and purveyor of CDS data, provides curves on about 2,000 corporate issuers globally, of which about 800 are domiciled in the United States.

o Liquidity. When CDS on a company’s bonds exist, they generally trade more heavily and with a tighter bid-offer spread than bond issues. The liquidity of CDS with different maturities usually differs less than that of bonds of a given issuer.


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The following AIMS are not explained in this (version of this) video

• Explain how a CDS spread can be used to derive a hazard rate curve.• Construct a hazard rate curve from a CDS spread curve.• Construct a default distribution curve from a hazard rate curve.


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Explain how the default distribution is affected by the sloping of the spread curve.

Spread curves, and thus hazard curves, may be upward-or downward-sloping.

• An upward-sloping spread curve leads to a default distribution that has a relatively flat slope for shorter horizons, but a steeper slope for more distant ones. The intuition is that the credit has a better risk-neutral chance of surviving the next few years, since its hazard rate and thus unconditional default probability has a relatively low starting point. But even so, its marginal default probability, that is, the conditional probability of defaulting in future years, will fall less quickly or even rise for some horizons.

• A downward-sloping curve, in contrast, has a relatively steep slope at short horizons, but flattens out more quickly at longer horizons. The intuition here is that, if the firm survives the early, “dangerous” years, it has a good chance of surviving for a long time.


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Explain how the default distribution is affected by the sloping of the spread curve.

• Spread curves are typically gently upward sloping. If the market believes that a firm has a stable, low default probability that is unlikely to change for the foreseeable future, the firm’s spread curve would be flat if it reflected default expectations only. However, spreads also reflect some compensation for risk. For longer horizons, there is a greater likelihood of an unforeseen and unforeseeable change in the firm’s situation and a rise in its default probability. The increased spread for longer horizons is in part a risk premium that compensates for this possibility.

• Downward-sloping spread curves are unusual, a sign that the market views a credit as distressed, but became prevalent during the subprime crisis.


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Define spread risk and its measurement using the mark-to-market and spread volatility.

• Spread risk is the risk of loss from changes in the pricing of credit-risky securities. Although it only affects credit portfolios, it is closer in nature to market than to credit risk, since it is generated by changes in prices rather than changes in the credit state of the securities.

Mark-to-Market of a CDS • Like many derivatives (e.g., interest rate swap), at inception the mark-to-

market value of the CDS is zero: neither counterparty owes the other anything. o If the spread increases, the premium paid by the fixed-leg counterparty increases.

This causes a gain to existing fixed-leg payers, who in retrospect got into their position cheaply [and are synthetically short the reference], and a loss to the contingent-leg parties [i.e., protection-sellers who are synthetically long the reference], who are receiving less premium than if they had entered the position after the spread widening. This mark-to-market effect is the spread01 of the CDS.


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• To compute the mark-to-market, we carry out the same steps needed to compute the spread01 of a fixed-rate bond. However, rather than shocking the z-spread (a single number), by +/- one-half a basis point (0.5 bps), we execute a parallel shift up and down of the entire CDS curve by 0.5 bps. This is similar to the procedure we carried out in computing DV01 for a default-free bond, in which we shifted the entire spot curve up or down by 0.5 bps.

• For each shift of the CDS curve away from its initial level, we re-compute the hazard rate curve, and with the shocked hazard rate curve we then re-compute the value of the CDS. The difference between the two shocked values is the spread01 of the CDS.


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Spread Volatility • Fluctuations in the prices of credit-risky bonds due to the market assessment

of the value of default and credit transition risk, as opposed to changes in risk-free rates, are expressed in changes in credit spreads. Spread risk therefore encompasses both the market’s expectations of credit risk events and the credit spread it requires in equilibrium to put up with credit risk. The most common way of measuring spread risk is via the spread volatility or “spread vol,” the degree to which spreads fluctuate over time. Spread vol is the standard deviation—historical or expected—of changes in spread, generally measured in basis points per day.


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End of P2.T6. Malz, Chapter 7:Spread Risk and Default Intensity Models

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