In this chapter, the differences between risk-free and default risky interest rates are discussed together with credit spreads and default probability approximations with respect to credit spreads. However, for the most part of the chapter, we seek to unravel the link between credit spreads and default probabilities.

# Credit Spreads

To better understand credit spread, we look at the following subdivisions:

**Yield Spread:** Often applied in price quotes than in fixed-income analysis, it is the distinction between a credit-risky bond’s yield to maturity and a benchmark government bond with similar maturity period.

**i-spread:** The interpolated spread is the difference between a credit-risky bond’s yield and the linearly populated yield between the two benchmark government bonds or swap rates having maturities flanking that of the credit-risky bond.

**Z-spread:** To arrive at a bond’s market price, a spread, that may be measured based on a government bond curve, has to be added to the Libor spot curve. This spread is called the Z-spread.

The Z-spread has to satisfy the following equation:

$$ { p }_{ \tau ,h }=ch\sum _{ i=1 }^{ \frac { \tau }{ h } }{ { e }^{ -\left( { r }_{ ih }+z \right) ih }+ } { e }^{ -\left( { r }_{ \tau }+z \right) \tau } $$

Where \({ p }_{ \tau ,h }\) is the \(\tau\)-year credit-risky bond’s price having a frequency \(h\) payment and coupon \(c\).

**Asset-Swap Spread:** A bond’s asset swap floating leg has quoted margin called the asset swap rate.

**Credit Default Swap Spread:** Similar bonds of the same issuer have basis points of a CDS as an expression market premium. This market premium is the credit default swap spread.

**Option-Adjusted Spread:** This is a z-spread version accounting for options embedded in the bonds and is identical only to the Z-spread if no options are contained in the bond.

**Discount Margin:** The current Libor rate has a fixed spread precisely pricing the bond and is known as the discount margin.

# Spread Mark-to-Market

The market value change corresponding to one change in the z-spread basis point can be determined for a credit-risky bond. This change is called the spread 01 and can be calculated by making a 0.5 basis points increase or decrease then repricing the bonds of each of the credit shocks and calculating the difference.

A reduction in the discount factor applied to cash flows further in the future is due to an increase in spread and a corresponding decline in bond prices. The proportional impact of a spread change on a credit-risky bond price is determined by the spread duration in a similar manner as the duration measure exists for interest rates providing a rate change in a bond value’s proportional impact.

# Default Curve Analytics

Analytics of default timing is a focal point in reduced form models of credit risk. The models typically operate in default mode with rating migrations being disregarded.

Similar to single factor credit risk model, estimates from default probabilities, derived from internal or rating agencies’ ratings or structural credit models, are relied on by reduced form models.

A Bernoulli trial can represent a single company’s default risk with \(\tau ={ T }_{ 2 }-{ T }_{ 1 }\) over a fixed time period. Hence, the probability of default occurring is \(\pi \) and \(1- \pi \) for the firm to remain solvent.

If default and solvency outcomes are assigned values of \(0\) and \(1\) over time (\({ T }_{ 1 }\), \({ T }_{ 2 }\)], then a random variable following a Bernoulli distribution is defined.

$$ E\left\{ ({ T }_{ 1 },{ T }_{ 2 }] \right\} =\pi $$

The variance of default is \(\pi \left( 1-\pi \right) \) with successive time intervals (\({ T }_{ 2 }\), \({ T }_{ 3 }\)], (\({ T }_{ 3 }\), \({ T }_{ 4 }\)], with length \(\tau\) the Bernoulli trial can be repeated while \(\pi \) remains constant.

The implication of this model is that Bernoulli trials are conditionally independent such that for each future interval,(\({ T }_{ j }\), \({ T }_{ j+1 }\)] is independent of any earlier default event interval (\({ T }_{ j }\), \({ T }_{ j+1 }\)].

$$ \forall \left( j>i \right) $$

# The Hazard Rate

The hazard rate is the parameter driving default, denoted by \( \lambda \), with a time dimension taken as annual. It is also referred to as the default intensity. Over a tiny time interval \(dt\), then the default probability is:

$$ \lambda dt $$

And the consequent no default probability is:

$$ 1-\lambda dt $$

To define default concepts, the hazard rate is taken as constant.

# Default Time Distribution Function

The probability of default sometimes between time \(t\) and now is the default time distribution \(F\left( \tau \right) \) and is given by the following relation:

$$ P\left[ { t }^{ \ast }<t \right] \equiv F\left( t \right) =1-{ e }^{ -\lambda t } $$

The survival time distribution is the probability of no default between time \(t\) and the current time and must sum exactly to \(1\). It is given by:

$$ P\left[ { t }^{ \ast }\ge t \right] =1-P\left[ { t }^{ \ast }<t \right] =1-F\left( t \right) ={ e }^{ -\lambda t } $$

# Default Time Density Function

The default time density function is the derivative of the default time distribution w.r.t \(t\) and is sometimes called the marginal default probability and is always positive; given as:

$$ \frac { \partial }{ \partial t } P\left[ { t }^{ \ast }<t \right] ={ F }^{ \prime \left( t \right) }=\lambda { e }^{ -\lambda t } $$

With a small \(\lambda\), its increase will be quite slow and is consequently followed by a decline in the survival probability:

$$ \frac { \partial }{ \partial t } P\left[ { t }^{ \ast }\ge t \right] ={ -F }^{ \prime \left( t \right) }=-\lambda { e }^{ -\lambda t }<0 $$

The default time density is also always positive, but with a fast enough rise in hazard rate with the time horizon, the incline in cumulative default probability may be at an increasing pace.

# Conditional Default Probability

It can be defined as:

$$ P\left( { t }^{ \ast }<t+\tau |{ t }^{ \ast }>t \right) =\frac { P\left[ { t }^{ \ast }>t\cap { t }^{ \ast }<t+\tau \right] }{ P\left[ { t }^{ \ast }<t \right] } $$

The likelihood of surviving to time \(t\) then defaulting between time \(t\) and \(t+\tau \) in the constant hazard rate model is:

$$ P\left[ { t }^{ \ast }>t\cap { t }^{ \ast }<t+\tau \right] =F\left( t+\tau \right) -F\left( t \right) $$

$$ ={ e }^{ -\lambda }\left( 1-{ e }^{ -\lambda t } \right) $$

$$ =\left[ 1-F\left( t \right) F\left( \tau \right) \right] $$

$$ =P\left[ { t }^{ \ast }>t \right] P\left( { t }^{ \ast }<t+\tau |{ t }^{ \ast }>t \right) $$

Then:

$$ F\left( \tau \right) =P\left( { t }^{ \ast }<t+\tau |{ t }^{ \ast }>t \right) $$

The likelihood of a security to default in a short time interval \(\left( t,t+\tau \right) \) if until time \(t\) it has not yet defaulted, will be:

$$ \begin{matrix} & lim & \\ \tau & \rightarrow & 0 \\ t & > & 0 \end{matrix}P\left( t<{ t }^{ \ast }<t+\tau |{ t }^{ \ast }>t \right) =\frac { { F }^{ \prime }\left( t \right) \tau }{ 1-F\left( t \right) } =\lambda t $$

# Risk-Neutral Estimates of Default Probabilities

Our aim here is to determine how market prices extract default probabilities with the assistance of the health algebra, The spread over the risk-free rate on a bond that is defaul table with maturity \(T\) is denoted by \({ z }_{ t }\) and the constant risk-neutral hazard rate at time \(T\) is \({ \lambda }_{ T }^{ \ast }\).

## Basic Analytics of Risk-Neutral Default rates

Bonds and credit default swaps (CDSs) are the two main security types used in estimating the likelihood of defaults. The following are the notations applied in this section:

\({ p }_{ T }\) is the current default-free price of \( \tau \)-year zero-coupon bond, \({ p }_{ T }^{ corp }\) is a defaultable \(\tau\)-year zero-coupon bond’s current price, \({ r }_{ T }\) is a default-free bond’s continuously compounded discount rate, \({ z }_{ T }\) is the defaultable bond’s continuously compounded spread, \(R\) is the recovery rate and \({ \lambda }_{ T }^{ \ast }\) is the \( \tau \)-year risk neutral hazard rate, and finally \(1-{ e }^{ -{ \lambda }_{ T }^{ \ast } }\) is the annualized risk neutral default probability.

The spot rate \({ r }_{ T }\) is defined by the following relation:

$$ { p }_{ T }={ e }^{ -{ r }_{ T }\tau } $$

Where:

$$ { p }_{ \tau }^{ corp }\le { p }_{ \tau }\quad and\quad { z }_{ \tau }\ge 0 $$

The following assumptions on default and recovery should be made for the calculation of hazard rates: in the next \(\tau\) years,the issuer can default any time. At the maturity date, a deterministic and known recovery payment will be made to creditors in case of default regardless of when the default happens.

In a defaultable bond, the expected value of investors receiving either $1 or 0 in \(\tau\) years is:

$$ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }\times 1+\left( 1-{ e }^{ -{ \lambda }_{ T }^{ \ast }\tau } \right) \times 0 $$

The payoff’s expected present value is:

$$ { e }^{ -{ r }_{ \tau }^{ \ast }\tau }\left[ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }\times 1+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) \times 0 \right] $$

If the present expected values of the payments are set to be equal to the price of the bond,and the recovery rate \(R\) is a non-negative value on \(\left( 0,1 \right) \), we have:

$$ { e }^{ -\left( { r }_{ \tau }+{ z }_{ \tau } \right) \tau }={ e }^{ -{ r }_{ \tau }^{ }\tau }\left[ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) R \right] $$

or

$$ { e }^{ -{ z }_{ \tau }\tau }={ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) R=1-\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) \left( 1-R \right) $$

The \( \tau \)-year default probability is equal to the additional credit-risk discount on the defaultable loan divided by the LGD:

$$ 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }=\frac { 1-{ e }^{ -{ z }_{ \tau }\tau } }{ 1-R } $$

and:

$$ -\left( { r }_{ \tau }+{ z }_{ \tau } \right) \tau =-{ r }_{ \tau }\tau +log\left[ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) R \right] $$

or

$$ { z }_{ \tau }\tau =-log\left[ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) R \right] $$

To solve the above expression numerically for \({ \lambda }_{ \tau }^{ \ast }\tau \), we can apply estimations.

$$ { e }^{ x }\approx 1+x\quad and\quad log\left( 1+x \right) \approx x $$

So:

$$ { e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau }+\left( 1-{ e }^{ -{ \lambda }_{ \tau }^{ \ast }\tau } \right) R\approx 1-{ \lambda }_{ \tau }^{ \ast }\tau +{ \lambda }_{ \tau }^{ \ast }\tau =1-{ \lambda }_{ \tau }^{ \ast }\tau \left( 1-R \right) $$

Thus:

$$ log\left[ 1-{ \lambda }_{ \tau }^{ \ast }\tau \left( 1-R \right) \right] \approx -{ \lambda }_{ \tau }^{ \ast }\tau \left( 1-R \right) $$

Combining these results we get:

$$ { z }_{ \tau }\tau \approx { \lambda }_{ \tau }^{ \ast }\tau \left( 1-R \right) \Rightarrow { \lambda }_{ \tau }^{ \ast }\tau \approx \frac { { z }_{ \tau } }{ 1-R } $$

# Time Scaling of Default Probabilities

To start our analysis, we approximate the default probability \(\pi \) over a specified timeframe on the basis of default probability given by a rating agency or a model or on a market credit spread. The likelihood of default can always be varied from timeframe to timeframe by using the hazard rates algebra.

Assuming that an approximation of annual default likelihood is set as \({ \pi }_{ 1 }\), from hazard rate definition, we have:

$$ { \pi }_{ 1 }=1-{ e }^{ -\lambda } $$

Then:

$$ \lambda =log\left[ 1-{ \pi }_{ 1 } \right] $$

$$ \Rightarrow { \pi }_{ 1 }=1-{ e }^{ -log\left( 1-{ \pi }_{ 1 } \right) } $$

$$ \Rightarrow { \pi }_{ t }=1-{ \left( 1-{ \pi }_{ 1 } \right) }^{ t } $$

# Credit Default Swaps

Practically, the approximation of hazard rates from the CDS prices has the following merits:

**Standardization:**CDS trading for most firms generally happens in 1,3,5,7 and 10 standardized maturity years with the most liquid point being the five-year.**Coverage:**The largest CDS data collectors and purveyor gives curves on about 2000 corporate issuers worldwide with 800 being in the US.**Liquidity:**CDS liquidity with different maturities is usually less distinct than a specific issuer’s bond.

The expected spread payments’ present value is set by the protected buyer equal to the protected seller’s payment’s expected present value in case of a default, to determine the default probability function using CDSs.

CDSs are traded in terms of spread; the expression of price when two traders make a deal is in terms of the spread premium paid by the counterparty purchasing the protection to the seller.

# Building Default Probability Curves

The conditional default likelihood at a time \(t\) is denoted by \( \lambda \left( t \right) ,t\epsilon [0,\infty )\).The default likelihood over interval \([0,t)\) is:

$$ { \pi }_{ t }=1-{ e }^{ -\int _{ 0 }^{ t }{ \lambda \left( s \right) ds } } $$

For a constant hazard rate, by \( \lambda \left( t \right) = \lambda ,t\epsilon [0,\infty )\), then the above equation reduces to:

$$ { \pi }_{ t }=1-{ e }^{ -\lambda t } $$

As the generalized CDS standard maturities are 1, 3, 5, 7 and 10 years, 5-piecewise constant hazard rates can be extracted from data:

$$ \lambda \left( t \right) =\begin{Bmatrix} { \lambda }_{ 1 } \\ { \lambda }_{ 2 } \\ { \lambda }_{ 3 } \\ { \lambda }_{ 4 } \\ { \lambda }_{ 5 } \end{Bmatrix}for\begin{Bmatrix} 0<t\le 1 \\ 1<t\le 3 \\ 3<t\le 5 \\ 5<t\le 7 \\ 7<t \end{Bmatrix} $$

Then:

$$ \int _{ 0 }^{ t }{ \lambda \left( s \right) ds } =\begin{Bmatrix} { \lambda }_{ 1 }t \\ { { \lambda }_{ 1 }+\left( t-1 \right) \lambda }_{ 2 } \\ { { \lambda }_{ 1 }+2{ \lambda }_{ 2 }+\left( t-3 \right) \lambda }_{ 3 } \\ { { \lambda }_{ 1 }+2{ \lambda }_{ 2 }+2\lambda }_{ 3 }+\left( t-5 \right) { \lambda }_{ 4 } \\ { { { \lambda }_{ 1 }+2{ \lambda }_{ 2 }+2\lambda }_{ 3 }+3\lambda }_{ 4 }+\left( t-7 \right) { \lambda }_{ 5 } \end{Bmatrix}for\begin{Bmatrix} 0<t\le 1 \\ 1<t\le 3 \\ 3<t\le 5 \\ 5<t\le 7 \\ 7<t \end{Bmatrix} $$

If \({ s }_{ \tau }\) isthe premium a particular company’s \(\tau\)-year CDS, with \(1-{ \pi }_{ t }\) being the likelihood to survive up to date, then this expected value can be expressed as:

$$ \frac { 1 }{ 4\times { 10 }^{ 4 } } { s }_{ \tau }\sum _{ u=1 }^{ 4 }{ { p }_{ 0.25u }\left( 1-{ \pi }_{ 0.25u } \right) } $$

Whereby \({ p }_{ t }\) is the risk-free zero-coupon bond price maturing at time \(t\). The likelihood of paying on date \(t\) is equal to \({ \pi }_{ t }-{ \pi }_{ t-0.25 }\), and the likelihood of defaulting during the time interval \((t-\frac { 1 }{ 4 } ]\) is equivalent to the likelihood of surviving to time \((t-\frac { 1 }{ 4 } ]\) less the small likelihood of surviving to time \(t\). The expected present value of the free leg is:

$$ \frac { 1 }{ 4\times { 10 }^{ 4 } } { s }_{ \tau }\sum _{ u=1 }^{ 4\tau }{ { p }_{ 0.25u }\left[ \left( 1-{ \pi }_{ 0.25u } \right) +0.5\left( { \pi }_{ 0.25u }-{ \pi }_{ 0.25\left( u-1 \right) } \right) \right] } $$

In the event of a default occurring in the quarter ending at time \(t\), the contingent payment’s present value is \(\left( 1-R \right) { p }_{ t }\) per dollar of national, whose expected present value is:

$$ \left( 1-R \right) { p }_{ t }\left( { \pi }_{ t }-\pi _{ t-0.25 } \right) $$

Therefore, the contingent leg’s expected present value is:

$$ \left( 1-R \right) \sum _{ u=1 }^{ 4\tau }{ { p }_{ 0.25u }\left( { \pi }_{ 0.25u }-{ \pi }_{ 0.25\left( u-1 \right) } \right) } $$

The fair market CDS spread solves:

$$ \frac { 1 }{ 4\times { 10 }^{ 4 } } { s }_{ \tau }{ \Sigma }_{ u=1 }^{ 4\tau }{ p }_{ 0.25u }\left[ \left( 1-\pi _{ 0.25u } \right) +0.5\left( { \pi }_{ 0.25u }-{ \pi }_{ 0.25\left( u-1 \right) } \right) \right] =\left( 1-R \right) { \Sigma }_{ u=1 }^{ 4\tau }{ p }_{ 0.25u }\left( { \pi }_{ 0.25u }-{ \pi }_{ 0.25\left( u-1 \right) } \right) $$

To find the default curve for a firm for which only a single CDS spread for a term like 5 years, \(t=0.25,0.50,\dots 5,\) and \(\lambda :{ \pi }_{ t }=1-{ e }^{ -\lambda \ast t },t>0\). Substituting this, we have:

$$ \frac { { s }_{ \tau } }{ 4\times { 10 }^{ 4 } } { \Sigma }_{ u=1 }^{ 4\tau }{ p }_{ 0.25u }\left[ { e }^{ -\lambda \frac { u }{ 4 } }+0.5\left( { e }^{ -\lambda \frac { u-1 }{ 4 } }-{ e }^{ -\lambda \frac { u }{ 4 } } \right) \right] =\left( 1-R \right) { \Sigma }_{ u=1 }^{ 4\tau }{ p }_{ 0.25u }\left( { e }^{ -\lambda \frac { u-1 }{ 4 } }-{ e }^{ -\lambda \frac { u }{ 4 } } \right) $$

Where \(\tau=5\),then the equation remains to be solved for one unknown variable.

# Spread Risk

Spread risk is the risk of loss due to credit-risky securities pricing changes and only affects credit portfolios. It is generated by price changes as opposed to credit state of securities changes.

## Mark-to-Market of a CDS

If the mark-to-market value of a CDS is zero, the implication is that neither counterparty owes the other anything. It is computed in a similar manner as the spread 01, but a parallel shift up and down is carried for the entire CDS curve by 0.5 bps instead of increasing and decreasing the spread by 0.5 bps.

The hazard rate curve is recomputed for each shift of the CDS curve away from its initial level, and with the shocked hazard rate curve, we recomputed the CDS value. The difference between the two shocked values is, therefore, the spread 01 of the CDS.

# Spread Volatility

Prices fluctuations in credit risky-bonds due to the market assessment of the value of default and credit transition risk contrary to changes in risk-free rates credit spreads are expressed in changes in credit spreads.

# Practice Questions

1) Suppose that \(\lambda =0.455\), then determine the conditional one-year default probability using the survival through the first year.

- 0.3655
- 0.6343
- 0.5975
- 0.2319

The correct answer is **A**.

To calculate the conditional one-year default probability, we must first determine:

The unconditional one-year default probability given as:

$$ 1-{ e }^{ -\lambda } $$

$$ =1-{ e }^{ -0.455 }=0.3656 $$

The survival probability given by:

$$ { e }^{ -\lambda } $$

$$ { e }^{ -0.455 }=0.6344 $$

The unconditional two-year default probability given by:

$$ 1-{ e }^{ -2 \lambda } $$

$$ =1-{ e }^{ -2\times 0.455 }=0.5975 $$

We then take the difference between the two probabilities as:

$$ 0.5975-0.3656= 0.2319 $$

Finally, we divide the difference by the one-year survival probability to get:

$$ \frac { 0.2319 }{ 0.6344 } =0.3655 $$

2) Given that \(\lambda =0.822\) and \(t=2\) years,calculate the marginal default probability.

- 0.2754
- 0.3802
- 0.1588
- 0.3243

The correct answer is **C**.

Recall that the default time density function is the derivative of the default time distribution with respect to \(t\):

$$ \frac { \partial }{ \partial t } P\left[ { t }^{ \ast }<t \right] ={ F }^{ \prime \left( t \right) }=\lambda { e }^{ -\lambda t } $$

$$ \frac { \partial }{ \partial t } P\left[ { t }^{ \ast }<t \right] ={ F }^{ \prime \left( t \right) }=0.822{ e }^{ -0.822×2 }=0.1588 $$