5. Default Intensity Models
5. Default Intensity Models
1. Poisson/Exponential Distribution
1.1 Bernoulli Trail
A Bernoulli trial is an experiment or process where the outcome can take on only two values: success or failure (ie., a binomial distribution). Success and failure are relative terms that denote that either the event happens (“success”) or does not happen (“failure”). The obvious connection to our discussion is that a firm does or does not default during a particular time period.
Let us define the relevant time period as T 2 − T 1 = τ T_2 - T_1 =\tau T2−T1=τ where the firm will default with probability π \pi π and remains solvent with probability 1 − π 1- \pi 1−π.
The mean and variance of a Bernoulli distribution is equal to π \pi π and π ( 1 − π ) \pi(1 - \pi) π(1−π), respectively.
An important property of the Bernoulli distribution is that each trial is conditionally independent. That is, the probability of default in the next period is independent of default in any previous period. Hence, if a firm has survived until the current period, the probability of default in the next period is the same as in its first year of existence. This memoryless property is exactly the same as studying a series of coin flips.
1.2 Poisson Distribution
Poisson distribution is used to model number of default events over time.
f ( x ) = P ( X = x ) = λ x x ! e − λ f(x)=P(X=x)=\cfrac{\lambda^x}{x!}e^{-\lambda} f(x)=P(X=x)=x!λxe−λ
- X X X refers to the number of success per unit.
- λ \lambda λ refers to the average or expected number of successes per unit.
The mean and variance of a Poisson distributed random variable is equal to λ \lambda λ.
1.3 Exponential Distribution
The exponential distribution is often used to model the time it takes a company to default. The probability density function for this distribution is as follows:
f ( x ) = 1 β × e − x / β , x ≥ 0 f(x)=\cfrac{1}{\beta}\times e^{-x/\beta}, x \geq 0 f(x)=β1×e−x/β,x≥0
The scale parameter, β \beta β, is greater than zero and is the reciprocal of the “rate” parameter λ \lambda λ (i.e., λ = 1 / β \lambda=1/\beta λ=1/β).
The rate parameter measures the rate at which it will take an event to occur. In the context of waiting for a company to default, the rate parameter is known as the hazard rate and indicates the rate at which default will arrive.
2. Hazard Rate
2.1 Definition
The hazard rate (ie, default intensity) is represented by the (constant) parameter λ \lambda λ and the probability of default over the next, small time interval, d t dt dt, is λ d t \lambda dt λdt.
Cumulative probability of default (cumulative PD \text{PD} PD)
If the time of the default event is denoted t ∗ t^* t∗, the cumulative default time distribution, F(t), represents the probability of default over ( 0 , t ) (0, t) (0,t):
P ( t ∗ < t ) = F ( t ) = 1 − e − λ t P(t^*
This equation calculates the cumulative probability of default (cumulative PD \text{PD} PD), which is an unconditional default probability.
Survival distribution
P ( t ∗ > t ) = 1 − F ( t ) = e − λ t P(t^*>t)=1-F(t)=e^{-\lambda t} P(t∗>t)=1−F(t)=e−λt
Both survival and default probabilities sum to 1 1 1 at each point in time. In other words, if you have not defaulted by time t t t, then you have survived until this point.
As t t t increases, the cumulative default probability approaches 1 1 1 and the survival probability approaches 0 0 0.
Marginal default probability
P ( t < t ∗ < t + τ ) = P ( t ∗ < t + τ ) − P ( t ∗ < t ) = 1 − e − λ ( t + τ ) − ( 1 − e − λ t ) = e − λ t ( 1 − e − λ τ ) P(t
It is evident that this quantity is always positive indicating that the probability of default increases over time related to the intensity parameter λ \lambda λ.
Conditional default probability
If we examine the probability of default over ( t , t + τ ) (t, t+\tau) (t,t+τ) given survival up to time t t t, the function is a conditional default probability. The instantaneous conditional default probability (for small τ \tau τ) is equal to λ τ λ\tau λτ.
P [ t ∗ < t + τ ∣ t ∗ > t ] = P [ t < t ∗ < t + τ ] P [ t ∗ > t ] = e − λ t ( 1 − e − λ τ ) e − λ t = 1 − e − λ τ P[t^*
Therefore, the conditional probability of default over ( t , t + τ ) (t, t+\tau) (t,t+τ) , assuming survival over ( 0 , t ) (0, t) (0,t) is equal to the difference between the unconditional probability of default over ( 0 , τ ) (0, \tau) (0,τ). This is memoryless property .
2.2 Risk-Neutral Hazard Rate
Comparing zero-coupon corporate bonds to maturity-matched default- free government bonds. Since the only cash flows occur at maturity, the current prices differ based on their yields.
The price of a default-free bond maturing in τ \tau τ is: P τ r f = e − r f τ × τ P^{rf}_{\tau}=e^{-rf_{\tau}\times \tau} Pτrf=e−rfτ×τ
The price of a risky(corporate) bond maturing in τ \tau τ is P τ c o r p = e − ( r f τ + z τ ) × τ = P τ r f × e − z τ × τ P^{corp}_{\tau}=e^{-(rf_{\tau}+z_{\tau})\times \tau} =P_{\tau}^{rf}\times e^{-z_{\tau}\times \tau} Pτcorp=e−(rfτ+zτ)×τ=Pτrf×e−zτ×τ
If there is no default, the price between the corporate and default-free bond converge to par over τ \tau τ.
In case of default, creditors will recover a fraction of par, which is the recovery rate denoted as RR \text{RR} RR ( 0 ≤ RR ≤ 1 0\leq \text{RR}\leq1 0≤RR≤1).
Risk-neural default probabilities with RR = 0 \text{RR}=0 RR=0
The risky(corporate) bond investor receives $ 1 1 1 (par) if no default and $ 0 0 0 if there is a default. On average, the expected value is:
e − λ τ ∗ × τ × 1 + ( 1 − e − λ τ ∗ × τ ) × 0 = e − λ τ ∗ × τ e^{-\lambda_{\tau}^* \times \tau} \times 1 + (1-e^{-\lambda_{\tau}^* \times \tau}) \times 0=e^{-\lambda_{\tau}^* \times \tau} e−λτ∗×τ×1+(1−e−λτ∗×τ)×0=e−λτ∗×τ
On a present value basis discounting at risk-free rate generates:
e − r f τ × τ × e − λ τ ∗ × τ = e − ( − r f τ + λ τ ∗ ) × τ e^{-rf_{\tau}\times \tau} \times e^{-\lambda_{\tau}^* \times \tau}=e^{-(-rf_{\tau}+\lambda_{\tau}^*) \times \tau} e−rfτ×τ×e−λτ∗×τ=e−(−rfτ+λτ∗)×τ
The final step is to equate this present value expression to the risky bond price;
e − ( − r f τ + z τ ) × τ = e − ( − r f τ + λ τ ∗ ) × τ → λ τ ∗ = z τ e^{-(-rf_{\tau}+z_{\tau}) \times \tau}=e^{-(-rf_{\tau}+\lambda_{\tau}^*) \times \tau}\to \lambda_{\tau}^*=z_{\tau} e−(−rfτ+zτ)×τ=e−(−rfτ+λτ∗)×τ→λτ∗=zτ
When the recovery rate is zero, the risk-neutral hazard rate λ τ ∗ \lambda^*_{\tau} λτ∗ equals to credit spread (z-spread, z τ z_{\tau} zτ).
Risk-neutral default probabilities with RR ≠ 0 \text{RR}\neq0 RR=0
When we introduce a positive recovery rate, the present value changes slightly and s to the risky bond price:
e − ( − r f τ + z τ ) × τ = e − r f τ × τ × [ e − λ τ ∗ × τ × 1 + ( 1 − e − λ τ ∗ × τ ) × RR ] e^{-(-rf_{\tau}+z_{\tau}) \times \tau}=e^{-rf_{\tau}\times \tau}\times \left[e^{-\lambda_{\tau}^* \times \tau}\times 1+(1-e^{-\lambda_{\tau}^* \times \tau}) \times \text{RR}\right] e−(−rfτ+zτ)×τ=e−rfτ×τ×[e−λτ∗×τ×1+(1−e−λτ∗×τ)×RR]
→ 1 − e − λ τ ∗ × τ = 1 − e − z τ × τ 1 − RR \to1-e^{-\lambda_{\tau}^*\times \tau}=\cfrac{1-e^{-z_{\tau}\times \tau}}{1-\text{RR}} →1−e−λτ∗×τ=1−RR1−e−zτ×τ
Assuming e x = 1 + x e^x=1+x ex=1+x, than the risk-neutral hazard rate λ τ ∗ = z τ 1 − RR \lambda_{\tau}^*=\cfrac{z_{\tau}}{1-\text{RR}} λτ∗=1−RRzτ
Stated differently, the loss given default ( 1 − RR 1-\text{RR} 1−RR) times the default probability (hazard rate) is approximately equal to the credit spread (z-spread, z τ z_{\tau} zτ).
2.3 Using CDS market to estimate hazard rate
The primary advantage of using CDS to estimate hazard rates is that CDS spreads are observable. Although we can create a model for the hazard rate (the probability of default in the next period conditional on surviving until the current period), the estimated value would inherently be a guess. Instead, we can draw on the logic of a reduced form model to use the observable, liquid CDS to infer the estimates of the hazard rate.
Our previous analysis on estimating hazard rates did not fully capture the complexities of the bond market. First, published estimates of default probabilities are insufficient as they are typically provided for a one-year horizon which may not match the duration of the analysis. Second, few corporations issue zero-coupon bonds. One can view commercial paper as de facto zero-coupon bonds but the issuing universe is restricted to large, highly-rated corporations. CDS can overcome these difficulties because liquid contracts exist for several maturities (e.g, 1,3, 5,7, and 10 years are common).
Furthermore, a large number of liquid CDS curves are available (800 in U.S. markets, 1,200 in international markets) and the contracts are more liquid than the underlying cash bonds (i.e., narrower spreads and more volume).
2.4 Hazard Rate Curves
3. Spread and Spread Risk
3.1 Spread Conventions
Credit spread represents the difference in yields between the security of interest (e.g, corporate bond) and a reference security (typically a higher rated instrument). Ideally, these two securities would have the same maturity, so the difference in yields represents the difference in risk premiums, not compensation for the time value of money.
| Spread Measure | Definition |
|---|---|
| Yield spread | YTM risky bond - YTM benchmark government bond |
| i-spread | YTM risky bond-linearly interpolated YTM on benchmark government bond |
| z-spread | Basis points added to each spot rate on a benchmark curve |
| Asset-swap spread | Spread on floating leg of asset swap on a bond |
| CDS spread | Market premium of CDS of issuer bond |
| Option adjusted spread (OAS) |
z-spread adjusted for optionality of embedded options z-spread=OAS if no option |
| Discount margin | Fixed spread above current LIBOR needed to price bond correctly |
The more common spread definitions (yield spread, i-spread) are demonstrated in the following examples.
Example 1: Assume the following information regarding XYZ Company and U.S. Treasury yields.
| XYZ | U.S. Treasury | |
|---|---|---|
| Coupon rate | 6 % 6\% 6% semi-annual coupon | 4 % 4\% 4% semi-annual coupon |
| Time to maturity | 20 20 20 years ( 7.25 % 7.25\% 7.25% YTM) | 20 20 20 years ( 4.0 % 4.0\% 4.0% YTM) |
yield spread = 7.25 % − 4 % = 3.25 % = 7.25\% - 4\% = 3.25\% =7.25%−4%=3.25% ( 325 325 325 basis points)
Example 2: Assume the following information regarding XYZ Company and U.S. Treasury yields.
| XYZ | U.S. Treasury | |
|---|---|---|
| Coupon rate | 6 % 6\% 6% semi-annual coupon | 4 % 4\% 4% semi-annual coupon |
| Time to maturity | 19 19 19 years ( 7.25 % 7.25\% 7.25% YTM) | 20 20 20 years ( 4.0 % 4.0\% 4.0% YTM) 18 18 18 years ( 3.6 % 3.6\% 3.6% YTM) |
Because the maturity of the XYZ bond does not match exactly with the maturity of the quoted Treasury bonds, the i-spread will be computed as:
i-spread = 7.25 % − ( 4.0 % + 3.6 % ) / 2 = 3.45 % = 7.25\% - (4.0\% + 3.6\%)/2=3.45\% =7.25%−(4.0%+3.6%)/2=3.45%
3.2 Spread’01
DV01 \text{DV01} DV01 captures the dollar price change from a one basis point change in the current yield.
DVCS \text{DVCS} DVCS (i.e, spread ′ 01 \text{spread} '01 spread′01) captures the potential change in the bond price from a one basis point change in the z-spread.
The z-spread \text{z-spread} z-spread is shocked 0.5 0.5 0.5 basis points up and 0.5 0.5 0.5 basis points down and the difference is computed. If the current z-spread \text{z-spread} z-spread is 207 207 207 bps and the bond is priced at $ 92 92 92, we could consider incremental 0.5 0.5 0.5 basis point changes to compute the spread ′ 01 \text{spread} '01 spread′01.
- When the z-spread \text{z-spread} z-spread increased by 0.5 0.5 0.5 basis points to 207.5 207.5 207.5 bps, the new bond price is $ 91.93 91.93 91.93.
- When the z-spread \text{z-spread} z-spread tightens by 0.5 0.5 0.5 basis points to 206.5 206.5 206.5 bps, the bond price is $ 92.14 92.14 92.14.
Hence, given a $ 100 100 100 par value, the spread ′ 01 \text{spread} '01 spread′01 : 92.14 − 91.93 = 0.21 92.14-91.93 = 0.21 92.14−91.93=0.21 dollars per basis point.