Copulas for Credit

Credit traders, quants, and risk managers use copulas to see how the value changes with the assumption about default correlation.

Separating each name's distribution from the dependence

A CDS spread gives the distribution of the time to default of one name. How to derive the hazard rate from the spread is covered in Pricing a CDS. Whether five names tend to default in the same period cannot be read from any single CDS. The value of a tranche of a synthetic CDO or of a first-to-default swap depends on this clustering. The clustering is expressed through an assumption about default correlation, which cannot be observed directly.

A copula is a function that sets the dependence between names without changing each name's own distribution. For $m$ variables, given each variable's distribution function $F_i$ and a copula $C$ that links them, the joint distribution function is determined as follows¹.

$$F(x_1,\dots,x_m) = C\bigl(F_1(x_1),\dots,F_m(x_m)\bigr)$$

A default indicator takes only the value 0 or 1, so the meaning of its correlation is not intuitive. Credit models therefore use the time to default $\tau_i$ as the variable. Let $p_i(T)$ be the probability that name $i$ defaults by year $T$. Then $p_i(\tau_i)$ is a uniform value between 0 and 1, and the copula is specified as the joint distribution of this set of uniform values².

The default probabilities used for each name's own distribution are backed out from CDS prices, so they are values under the risk-neutral probability. Hull's presentation says that risk-neutral estimates should be used to value credit derivatives, and real-world estimates (derived from historical data) for credit VaR and scenario analysis³.

The one-factor Gaussian copula

The Gaussian copula builds the dependence from a set of standard normal variables. The form used to value tranches is the one-factor form, in which each name's variable is built from a common factor $M$ and a name-specific factor $Z_i$. In Hull's presentation, as of 2005 the market valued tranches with a Gaussian copula in which the correlation was the same for every pair of names⁴.

$$X_i = \sqrt{\rho}\, M + \sqrt{1-\rho}\, Z_i$$

$M$ and $Z_i$ are independent standard normal variables. The correlation between $X_i$ and $X_j$ is $\rho$⁵⁴. Name $i$ is read as defaulting by year $T$ when $\Phi(X_i) \le p_i(T)$, where $\Phi$ is the standard normal distribution function. The lower $X_i$ is, the more likely name $i$ is to default, and the lower $M$ is (the worse the common factor), the more likely all names are to default together⁶.

Suppose all names have the same probability and recovery rate $R$ and the same correlation (a homogeneous portfolio), and there are $N$ names. The default probability of each name, given the value of $M$, is as follows.

$$p(T \mid M) = \Phi\!\left(\frac{\Phi^{-1}\bigl(p(T)\bigr) - \sqrt{\rho}\, M}{\sqrt{1-\rho}}\right)$$

Once $M$ is fixed, the names default independently, so the number of defaults $K$ by year $T$ follows a binomial distribution with probability $p(T \mid M)$. Averaging over the distribution of $M$ gives the distribution of the number of defaults. Here $\varphi$ is the standard normal density⁷.

$$\Pr(K=k) = \int_{-\infty}^{\infty} \binom{N}{k}\, p(T \mid M)^{k}\,\bigl(1-p(T \mid M)\bigr)^{N-k}\, \varphi(M)\, dM$$

This integral has no closed form, so it is approximated numerically. Because everything up to the integral can be written as a formula, Brigo et al. call the method semi-analytical⁸. The procedure computes the cash flows given the common factor, then integrates over the distribution of the common factor. This can be done accurately and efficiently with Gaussian quadrature⁹. The loss is $(1-R)K/N$.

Worked example (100 hypothetical names)

Consider a hypothetical example in which a portfolio has 100 names with equal notional amounts. The hazard rate is a constant 2% a year for every name, the recovery rate is 40%, and the horizon is 5 years. The default probability over 5 years is $p = 1 - e^{-0.02 \times 5} = 9.516\%$, and $\Phi^{-1}(p) = -1.3096$. Each default raises the portfolio loss by $(1-40\%)/100 = 0.6\%$.

With correlation $\rho = 0.3$, the default probability of each name for each value of the common factor $M$ is as follows. For example, for $M=-2$, $(-1.3096 - 0.5477 \times (-2)) / 0.8367 = -0.2560$, and $\Phi(-0.2560) = 39.90\%$ ($\sqrt{0.3} = 0.5477$, $\sqrt{0.7} = 0.8367$).

When the common factor is bad ($M$ is low), close to 40 of the 100 names default, and when it is good, about 1 does. The overall average is 100 × 9.516% = 9.5 names regardless of the correlation, and the expected loss of the portfolio is 5.71%.

The next table changes only the correlation, to 0.1, 0.3, and 0.5, and shows the result of computing the integral above numerically. The tranches are 0–3% (equity), 3–7% (mezzanine) and 7–15% (senior), and the loss fractions are found from the loss at year 5 only. Discounting and the timing of losses by payment date are not included¹⁰.

Losses start to hit the senior tranche at 12 or more defaults, and at 25 or more (a loss of 15%) it is wiped out. Expected losses are fractions of each tranche's notional amount.

A high correlation raises the probability of no defaults at all and the probability of many defaults at the same time. From the protection seller's side, the equity tranche gains value because its expected loss falls from 89.7% to 55.8%, and the senior tranche loses value because its expected loss rises from 11.7% to 17.8%. The mezzanine tranche's expected loss fell from 50.4% to 32.5% in this example. The mezzanine is affected by both the low-default and high-default sides, so the direction depends on the tranche's position and terms. From the buyer's side, gains and losses are reversed.

Steps for valuing a tranche

Valuing a tranche with the one-factor Gaussian copula follows the steps below. The correlation $\rho$ is a single value, set separately from the default probabilities and the recovery rate.

1. Set each name's default probability from CDS prices, and set a recovery rate.
2. Choose a single correlation $\rho$.
3. Find the loss distribution for each payment date by integrating over the common factor¹¹.
4. From the distribution, find the expected loss of each tranche, compute the values of the premium leg and the default leg the way a CDS is valued, and set the spread¹².

The distribution in step 3 can also be found with Monte Carlo simulation, which generates losses with random numbers instead of integrating (see Monte Carlo Simulation)¹². Assuming a homogeneous portfolio gives the loss distribution in a semi-analytical formula⁵. In practice, however, you need a hedge ratio for each name, so the heterogeneous form, in which each name has a different spread, is used¹³.

Fitting market prices with base correlation

Hold each name's default probability and recovery rate fixed, and change only the correlation until the model price matches the market quote. The correlation that does this is called the implied correlation¹⁴. It is the same idea as implied volatility¹⁵.

There are two ways to find a correlation for each tranche. Compound correlation sets one correlation for each tranche and values the whole tranche with a copula that has that single correlation. For some tranche quotes no correlation can be found, and for others several can be found¹⁶. Base correlation, by contrast, applies the correlation to the equity tranche from zero loss up to the upper end $B$. For a 3–7% quote, you first find $\rho_3$ from the 0–3% quote, then use that value to find the 0–7% correlation $\rho_7$¹⁷.

Here $A$ is the attachment point and $B$ the detachment point of a tranche. The expected loss of tranche $A$ to $B$ is a fraction of the tranche's notional amount. It can be written as the difference between the expected losses of the equity tranches from zero loss to $A$ and from zero to $B$¹⁸. $E[L_{0,K}]$ is the expected loss of the tranche from zero loss to $K$, as a fraction of its notional amount.

$$E[L_{A,B}] = \frac{B \cdot E[L_{0,B}](\rho_B) - A \cdot E[L_{0,A}](\rho_A)}{B-A}$$

Consider a hypothetical example in which the base correlations in the example above are $\rho_3 = 0.1$ and $\rho_7 = 0.3$. At correlation 0.1, the expected loss of 0–3% is $E[L_{0,3\%}] = 89.7\%$. The expected loss of 0–7% comes from the 0–3% (71.7%) and 3–7% (39.8%) figures in the correlation 0.3 row: $(3 \times 71.7\% + 4 \times 39.8\%)/7 = 53.5\%$. The expected loss of the mezzanine tranche (3–7%), using unrounded values, is $(7 \times 0.53484 - 3 \times 0.89737)/4 \approx 0.2629$, or about 26.3%¹⁹. This differs from the 39.8% obtained by valuing the same tranche with a single copula at correlation 0.3. Base correlation values the two ends of the tranche with copulas of different correlations.

If the upper-end correlation $\rho_7$ is set to 0.5, the expected loss of 0–7% found by numerical integration is 42.5%. With the same formula, the mezzanine expected loss falls to 7.0%.

Base correlation is used because the correlation moves smoothly with the upper end, which makes it easier to value tranches with non-standard upper ends by interpolation²⁰. Christoffersen et al.'s 2008 presentation says that a non-standard (bespoke) tranche is usually valued using market information on the index tranches²¹.

Hull's presentation includes an example for August 4, 2004, for the DJ iTraxx EUR (a European CDS index). The implied correlations found tranche by tranche (compound correlations) were widely scattered, from 0.055 to 0.312, and base correlation rose with the upper end²².

This table shows that a copula with a single correlation cannot explain all the tranches with the same correlation, and that a different value is needed for each upper end.

Default correlation and hedging

The correlation in a copula is backed out from market tranche prices. It reflects the market's view rather than being an observed value. Christoffersen et al.'s 2008 presentation reports that the correlation backed out from CDOs moves in the same direction as the correlation estimated from CDS and equity data²³.

Hedging uses single-name CDS. The heterogeneous form is used to obtain the hedge ratio for each name¹³. Fermanian and Vigneron find that the profit and loss of a hedge with single-name CDS is flat only under narrow conditions: no default occurs, the CDS have the same maturity as the position, and spreads move in a particular relationship. To prepare for default as well, you use two CDS of different maturities on one name and pay the carry (the cost of that protection), so the profit and loss is not flat²⁴.

Dynamic loss models, which model the movement of the loss on the tranche as a whole, do not model each name's default, so they cannot do partial hedging name by name. This is one reason the market has stayed with base correlation²⁵.

Assumptions, limits and the problems that were pointed out

The 2010 paper by Brigo et al. lists the problems of the methods of the time, including base correlation, as criticisms that had been published before the financial crisis²⁶.

• Collapsing the correlations into one value (for an index of 125 names, 7,750 pairwise correlations)
• Getting a different correlation for each tranche, which fits contradictory loss distributions to the same portfolio
• With compound correlation, being unable in some cases to find a correlation from the quote
• With base correlation, using a different correlation at each end of a tranche, so the expected loss can turn negative, which violates the no-arbitrage conditions
• Having no dynamics over time (the models are static) and being unable to explain quotes of different maturities at the same time

Deep diveWhy base correlation can give a negative expected loss

As the formula above shows, the expected loss of tranche $A$ to $B$ is the difference between a term computed under a copula with correlation $\rho_B$ and a term computed under a copula with correlation $\rho_A$. The two terms come from different distributions, so if the upper-end correlation is much higher than the lower-end correlation, the difference can turn negative. A loss must be zero or more, so a negative expected loss violates the no-arbitrage conditions. Brigo et al. show an example in which the expected loss is initially negative for the 6–9% tranche of the DJ iTraxx in 2005²⁷.

Hull's presentation shows that using a t distribution with 4 degrees of freedom for both the common factor and the name-specific factor reproduces the iTraxx quotes of the same day well, and that base correlation is then almost constant across upper ends (0.241 to 0.266)²⁸. However, Brigo et al. judge that, because it has few parameters and has numerical problems, it is unlikely to solve the problem of fitting the quotes²⁹. Hull's presentation also says that implied correlation (base correlation in particular) should be interpreted with care³⁰.

Brigo et al. also write that the mathematical model alone was not the cause of the crisis. Despite the criticism, the base correlation form of the Gaussian copula was still the main valuation method as of 2010²⁵.

Even for the same tranche, the price changes with the choice of correlation and model, so you bear model risk. The procedure for fitting a model to market quotes is covered in Model Calibration.

Notes

1. David X. Li, "The Valuation of Basket Credit Derivatives: A Copula Function Approach" (Fields Institute presentation, December 8, 2000), slides 17–18. The form of the formula in the text. ↩

2. Damiano Brigo, Andrea Pallavicini, and Roberto Torresetti, "Credit Models and the Crisis" (arXiv:0912.5427v3, February 2010), section 3 (page 20). ↩

3. John Hull, "Credit Derivatives Markets" (Fields Institute presentation, February 7, 2005; hereafter Hull's presentation), slides 6 and 11. The distinction between risk-neutral and real-world estimates, and Hull's recommendation on when to use each. ↩

4. Hull's presentation (2005), slide 12. As of February 2005, the correlation model the market used for index tranches was a Gaussian copula (of the time to default) with the same correlation between any two names, with $x_i = aM + \sqrt{1-a^2} Z_i$, where $M$ and $Z_i$ are independent with mean 0 and variance 1. The correlation between $x_i$ and $x_j$ is $a^2$, and $x_i$ is mapped to the default time percentile to percentile. ($\rho$ in the text corresponds to $a^2$ in the presentation.) ↩ ↩²

5. Brigo et al., section 3.1 (page 22). $X_i = \sqrt{\rho_i} S + \sqrt{1-\rho_i} Y_i$, where $S$ is the common factor and $Y_i$ is the name-specific factor, both independent standard normal. If the correlation is constant and homogeneous, the overall correlation is a single value $\rho$, and assuming homogeneity gives the loss distribution in a semi-analytical formula. ($M$ and $Z_i$ in the text are $S$ and $Y_i$ in the paper.) ↩ ↩²

6. Brigo et al., section 3 (page 21), equation (4). The loss is $\bar{L}_t = \sum_i \frac{1}{M}(1-R_i) 1_{\{\Phi(X_i) \le p_i(t)\}}$. ($M$ in the paper is the number of names, which is $N$ in the text.) ↩

7. Brigo et al., section 3.1 (pages 22–23), equations (5) and (6). Given the common factor, the defaults of the names are independent, with $Q\{\tau_i < T \mid S\} = \Phi((\Phi^{-1}(1-\exp(-\Lambda(T))) - \sqrt{\rho} S)/\sqrt{1-\rho})$. The number of defaults follows a binomial distribution, and integrating over the common factor gives the loss distribution. ↩

8. Brigo et al., section 3.1 (page 23). The integral in equation (6) has no closed form and the other quantities can be found analytically, so the paper calls the method semi-analytical. ↩

9. Hull's presentation (2005), slide 13. In the procedure of computing the cash flows given the common factor and integrating over the distribution of the factor, the integration can be done accurately and efficiently with Gaussian quadrature. ↩

10. This site's own calculation. A homogeneous portfolio of 100 names (a constant hazard rate of 2% a year, a recovery rate of 40%, a horizon of 5 years, and each default raising the loss by 0.6%) was integrated numerically with the one-factor Gaussian copula. Discounting and the timing by payment date are not included, and the values in the table are rounded. ↩

11. Brigo et al., section 3.2 (page 23). For given dates, the loss distribution is found, the expected losses are computed, and the fair tranche spreads are obtained. ↩

12. Brigo et al., section 3.1 (page 22). From Gaussian variables that include the common factor, the portfolio loss at any time can be generated with random numbers, and the present values of the premium leg and the default leg of a tranche can be found. ↩ ↩²

13. Brigo et al., section 3, Remark 3.1 (page 21). In practice, because the hedge ratio for each name must be computed, a heterogeneous form with different values for each name is used. The dispersion of spreads complicates the valuation. ↩ ↩²

14. Brigo et al., section 3 (page 21). The implied correlation is the $\rho$ that is consistent with the tranche quote, found by backing it out with the other model parameters (default probabilities and recovery rate) held fixed. ↩

15. Christoffersen et al., "Measuring and Modeling Default Correlation" (Bank of Canada presentation, September 13, 2008; hereafter Christoffersen et al.'s presentation), slide 7. The definition of implied correlation and the analogy with implied volatility. ↩

16. Brigo et al., section 3 (page 21) and section 3.2 (pages 24–25). Compound correlation is consistent for a single tranche, but depending on the quote no correlation is found or several are found. ↩

17. Brigo et al., section 3.2 (pages 23–24). Base correlation finds $\rho_3$ from the 0–3% equity tranche, then finds the next correlation (such as $\rho_6$) for the next tranche while keeping $\rho_3$. Finding a new correlation for each tranche is compound correlation. ↩

18. Brigo et al., equation (2) in section 2 (page 18) and equation (7) in section 3.2 (page 23). $\bar{L}^{A,B}_t = \frac{1}{B-A}[B \bar{L}^{0,B}_t - A \bar{L}^{0,A}_t]$. Equation (7) is the expected loss with each term valued at a different correlation. ↩

19. This site's own calculation. With the same numerical integration as the table above, the expected loss of 0–3% at correlation 0.1 (0.89737) and that of 0–7% at correlation 0.3 (0.53484) give 0.26294. With the upper-end correlation at 0.5, the 0–7% expected loss of 0.42480 gives 0.0704. ↩

20. Brigo et al., section 3.3 (pages 26–27). Base correlation is much smoother in the upper end than compound correlation, and for non-standard tranches such as 4–15%, the attachment point and the detachment point can each be interpolated. ↩

21. Christoffersen et al.'s presentation (2008), slide 4. Non-standard (bespoke) CDOs are usually valued using market information on standard (index) products. ↩

22. Hull's presentation (2005), slide 14. The implied correlations and base correlations for the DJ iTraxx EUR tranches (0–3%, 3–6%, 6–9%, 9–12%, and 12–22%) on August 4, 2004. ↩

23. Christoffersen et al.'s presentation (2008), slide 21. The implied correlation backed out from CDOs moves in the same direction as the time series of correlations derived from CDS and equity data, the correlation varies a great deal over time, and the Gaussian copula model can have value. ↩

24. Jean-David Fermanian and Olivier Vigneron, "On break-even correlation" (arXiv:1204.2251), page 7 (section 2). The condition under which the profit and loss is flat under assumption (A), and the case of two CDS of different maturities. ↩

25. Brigo et al., section 8 (page 57). The criticism had been made before the crisis, the Gaussian copula continues to be used in the form of base correlation, and dynamic loss models do not model each name's default and so cannot hedge partially name by name. The model was one of many factors in the crisis and not the main one. Section 1 (page 12) also says that base correlation is still the main valuation method for synthetic CDOs. ↩ ↩²

26. Brigo et al., section 1 (pages 10–12) and section 3.3 (pages 27–28). The paper lists the five problems of the methods of the time given in the text and writes that the criticism had been published in 2006. The 125 names come from slide 8 of Christoffersen et al.'s presentation. ↩

27. Brigo et al., section 3.3 (pages 26–28). The expected loss is determined by two terms that use two different correlations, and it turns negative when the upper-end correlation is much higher than the lower-end correlation. Figure 7, the 6–9% tranche (the 10-year DJ iTraxx on August 3, 2005, with the upper-end correlation set to 48%). A loss must be zero or more, so this violates the no-arbitrage constraint. ↩

28. Hull's presentation (2005), slides 17–18. In a double t copula that uses a t distribution with 4 degrees of freedom for $M$ and $Z_i$, the base correlations for the August 4, 2004 iTraxx quotes are 0.266, 0.266, 0.260, 0.253, and 0.241. ↩

29. Brigo et al., section 3.4 (page 28). The double t copula of Hull and White (2004) can reproduce the quotes of one day without a correlation skew, but it has few parameters and numerical problems, so it is judged unlikely to solve the problem of fitting the quotes. ↩

30. Hull's presentation (2005), slide 22. Implied correlation (base correlation in particular) should be interpreted with care. ↩