Default dependence and credit portfolios, financial engineering for quants lesson

A lender holds two loans, and each borrower defaults within the horizon with probability $0.1$ . What is the probability that both default? Under independence it is $0.01$ , if the two defaults always happen together it is $0.1$ , and both answers fit the same two marginal probabilities. This lesson writes defaults as indicators, finds which joint default probabilities a set of marginals allows, builds a joint law from a latent Gaussian factor and its Gaussian copula, separates tail dependence from dependence at a fixed threshold, and shows in an exact two-name example that dependence reshapes the loss distribution while the expected loss stays fixed.

Key terms

Default indicator

Definition Fix a horizon $T$ and one probability measure for all statements, a pricing measure $\mathbb{Q}$ when the law is used for prices or the physical measure $\mathbb{P}$ for risk statements, and write $P$ and $E$ for probability and expectation under it. For names $i = 1, \dots, n$ with default times $\tau_i$ , the default indicator of name $i$ is

I_i = \mathbf{1}_{\{\tau_i \le T\}},

equal to $1$ if name $i$ defaults by $T$ and $0$ otherwise. Its marginal default probability is $p_i = P(\tau_i \le T)$ . Because $I_i^2 = I_i$ , the indicator has mean $E[I_i] = p_i$ and variance $\operatorname{Var}(I_i) = p_i(1 - p_i)$ , and for two names

\begin{aligned} &\operatorname{Cov}(I_i, I_j) \\[0.4em] &= P(I_i = 1,\, I_j = 1) - p_i p_j . \end{aligned}

With notional $w_i$ and a deterministic recovery fraction $R_i$ in $[0, 1]$ , the loss on name $i$ at the horizon is $w_i(1 - R_i)\, I_i$ currency units. This page counts losses at the horizon and ignores when they occur and how they are discounted.

In plain words The indicator turns a default event into a number that is $1$ or $0$ . Its average is the default probability, and the loss on a name is a fixed amount times its indicator, so the expected loss of one name needs nothing beyond its default probability.

Example Take a constructed name with notional $100$ , recovery fraction $0.5$ and default probability $0.1$ , so a default costs $50$ currency units. The indicator has mean $0.1$ and variance $0.1 \cdot 0.9 = 0.09$ . The loss has mean $50 \cdot 0.1 = 5$ and variance $50^2 \cdot 0.09 = 225$ squared currency units, a standard deviation of $15$ .

Joint default law

Definition The joint default law of $n$ names is the distribution of the vector $(I_1, \dots, I_n)$ , that is, the probabilities of the $2^n$ default patterns in $\{0, 1\}^n$ . They are nonnegative, add to $1$ , and each marginal $p_i$ is the sum over the patterns with $I_i = 1$ . The marginals fix $n$ of the $2^n - 1$ free numbers, so for $n \ge 2$ they determine the law only when at most one $p_i$ lies strictly between $0$ and $1$ . For two names with marginals $p_1$ and $p_2$ , let $q = P(I_1 = 1,\, I_2 = 1)$ be the joint default probability. The four states have the probabilities below.

State	Probability
both default	$q$
only name 1 defaults	$p_1 - q$
only name 2 defaults	$p_2 - q$
neither defaults	$1 - p_1 - p_2 + q$

They are all nonnegative exactly when

\begin{aligned} q &\ge \max(0,\, p_1 + p_2 - 1), \\[0.4em] q &\le \min(p_1, p_2), \end{aligned}

the two-name Fréchet bounds. Every $q$ in this interval gives a valid joint law, and outside it no joint law with these marginals exists. For $0 < p_1, p_2 < 1$ the indicator correlation is

\begin{aligned} \rho_I &= \frac{q - p_1 p_2}{s_1 s_2}, \\[0.4em] s_i &= \sqrt{p_i(1 - p_i)} . \end{aligned}

The Fréchet bounds confine this correlation to a subinterval of $[-1, 1]$ . It reaches $1$ exactly when $p_1 = p_2$ , at $q = p_1$ , and $-1$ exactly when $p_1 + p_2 = 1$ , at $q = 0$ .

In plain words The marginals say how often each name defaults. The joint law says how the defaults line up across scenarios, and many different alignments are consistent with the same marginals. A dependence input is admissible only if it produces nonnegative probabilities for every default pattern.

Example Let both names have $p = 0.1$ . Independence gives $q = 0.01$ , defaults that always happen together give $q = 0.1$ , and defaults that never happen together give $q = 0$ . The constructed value $q = 0.03$ gives the states $0.03$ , $0.07$ , $0.07$ and $0.83$ , which are nonnegative, add to $1$ and reproduce both marginals. Its covariance is $0.03 - 0.01 = 0.02$ and its indicator correlation is $0.02/0.09 = 2/9 \approx 0.222$ . The smallest admissible value $q = 0$ gives $\rho_I = -0.01/0.09 = -1/9$ , so an indicator correlation of $-0.5$ is impossible for these marginals.

Latent Gaussian factor

Definition Let $Y, \epsilon_1, \dots, \epsilon_n$ be independent standard normal random variables and $0 \le \rho \le 1$ . The one-factor latent variables are

Z_i = \sqrt{\rho}\, Y + \sqrt{1 - \rho}\, \epsilon_i .

Each $Z_i$ is standard normal, because the variances add to $\rho + (1 - \rho) = 1$ , and $\operatorname{Cov}(Z_i, Z_j) = \rho$ for $i \ne j$ , because two names share only the $Y$ term. For marginals $0 < p_i < 1$ , set the threshold $c_i = \Phi^{-1}(p_i)$ and declare default when $Z_i \le c_i$ . Then $P(I_i = 1) = \Phi(c_i) = p_i$ exactly, whatever $\rho$ is. For $0 \le \rho < 1$ , the event $Z_i \le c_i$ is the event $\epsilon_i \le (c_i - \sqrt{\rho}\, Y)/\sqrt{1 - \rho}$ , and $\epsilon_i$ is independent of $Y$ , so $P(I_i = 1 \mid Y) = p_i(Y)$ with

\begin{aligned} p_i(y) &= \Phi\left(\frac{c_i - \sqrt{\rho}\, y}{\sqrt{1 - \rho}}\right) . \end{aligned}

Given $Y$ , the indicators are independent, since they are functions of the independent $\epsilon_i$ . Averaging over the factor gives every joint probability, for example

\begin{aligned} &P(I_i = 1,\, I_j = 1) \\[0.4em] &\quad = \int_{-\infty}^{\infty} p_i(y)\, p_j(y)\, \varphi(y)\, dy \end{aligned}

for $i \ne j$ , which equals the bivariate standard normal probability $\Phi_2(c_i, c_j; \rho)$ . For $0 < \rho < 1$ each $p_i(y)$ is strictly decreasing in $y$ , and $\rho = 0$ gives $p_i(y) = p_i$ and independent names. The case $\rho = 1$ is separate, because the formula divides by $\sqrt{1 - \rho} = 0$ . Then $Z_i = Y$ for every $i$ , so $I_i = \mathbf{1}_{\{Y \le c_i\}}$ , each default is certain or impossible once $Y$ is known, and names with equal marginals default together. The factor is a modeling device for dependence, not an observed quantity, and $\rho$ is a correlation of latent variables, not of default indicators.

In plain words One common random number plays the role of the state of the economy. A low value raises every name’s default probability at once, a high value lowers them all, and once that number is fixed the names default independently. The thresholds are set so that each name keeps exactly its own default probability.

Example With $p = 0.1$ and $\rho = 0.3$ , the threshold is $c = \Phi^{-1}(0.1) \approx -1.2816$ . The conditional default probability is about $0.412$ at $y = -2$ , about $0.0628$ at $y = 0$ and about $0.00225$ at $y = 2$ , and averaging $p(Y)$ over the standard normal law returns $0.1$ . The value at $y = 0$ , the median of $p(Y)$ , lies below the mean $0.1$ because the rare bad states, where $p(y)$ is large, pull the mean up. By numerical integration the joint default probability is $\Phi_2(c, c; 0.3) \approx 0.0216$ , more than twice the independent $0.01$ . A numerical root search shows that $\rho \approx 0.46$ reproduces the constructed $q = 0.03$ of the previous card, whose indicator correlation is only $2/9 \approx 0.22$ .

Gaussian copula

Definition An $n$ -dimensional copula is the joint CDF $C$ of a random vector $(U_1, \dots, U_n)$ whose coordinates are each uniform on $[0, 1]$ . For a correlation matrix $R$ , symmetric and positive semidefinite with unit diagonal, the Gaussian copula is

\begin{aligned} &C_R(u_1, \dots, u_n) \\[0.4em] &= \Phi_R\bigl(\Phi^{-1}(u_1), \dots, \\[0.4em] &\qquad \Phi^{-1}(u_n)\bigr) \end{aligned}

for $u$ in $(0, 1)^n$ , where $\Phi_R$ is the CDF of $N(0, R)$ . If $Z \sim N(0, R)$ , then $U_i = \Phi(Z_i)$ is uniform by the probability integral transform, and $U$ has the copula $C_R$ . The one-factor model is the case in which every off-diagonal entry of $R$ equals $\rho$ . Let $F_i$ be the CDF of the default time $\tau_i$ , for example $F_i(t) = 1 - \exp\bigl(-\int_0^t \lambda_i(s)\, ds\bigr)$ for an intensity $\lambda_i$ , and let $F_i^{-1}(u) = \inf\{t \ge 0 : F_i(t) \ge u\}$ be its generalized inverse, equal to $+\infty$ when no such $t$ exists. Setting

\begin{gathered} \tau_i = F_i^{-1}(U_i), \\[0.4em] \{\tau_i \le T\} = \{U_i \le F_i(T)\} \end{gathered}

gives default times with the marginals $F_i$ and the dependence of $C_R$ , and the event at the horizon is $Z_i \le \Phi^{-1}(F_i(T))$ , the threshold model with $p_i = F_i(T)$ . Sklar’s theorem, stated here without proof, says that when the $F_i$ are continuous, the joint CDF of the default times is

\begin{aligned} &P(\tau_1 \le t_1, \dots, \tau_n \le t_n) \\[0.4em] &\quad = C\bigl(F_1(t_1), \dots, F_n(t_n)\bigr) \end{aligned}

for exactly one copula $C$ . Prices of single-name contracts depend only on the marginals $F_i$ , so they can calibrate the marginals but carry no information about $C$ , which is a separate model input.

In plain words A copula is the dependence that remains after each variable is expressed as its own percentile. The Gaussian copula takes that dependence from a correlated normal vector and attaches it to any marginal default time curves, so the curves and the dependence can be chosen separately.

Example Take a constructed name with constant intensity $\lambda = 0.02$ per year and horizon $T = 5$ years. Then $F(5) = 1 - e^{-0.1} \approx 0.0952$ and the latent threshold is $\Phi^{-1}(0.0952) \approx -1.3096$ . A draw $Z \approx -1.645$ gives $U = 0.05$ and the default time

\tau = \frac{-\ln(0.95)}{0.02} \approx 2.565

years, a default before the horizon, consistent with $Z$ lying below the threshold. Replacing the Gaussian copula by another copula with the same $F_i$ leaves every single-name default probability unchanged and in general changes the joint law.

Tail dependence

Definition For a random vector $(U_1, U_2)$ with uniform marginals and copula $C$ , the lower tail dependence coefficient is

\lambda_L = \lim_{u \downarrow 0} \frac{C(u, u)}{u},

when the limit exists. It equals $\lim_{u \downarrow 0} P(U_2 \le u \mid U_1 \le u)$ , the limiting probability that the second variable is in its lowest fraction $u$ given that the first is. Independence, with $C(u, u) = u^2$ , gives $\lambda_L = 0$ , and the comonotone copula $\min(u_1, u_2)$ gives $\lambda_L = 1$ . Two results are stated here without proof. A bivariate Gaussian copula with correlation $\rho < 1$ has $\lambda_L = 0$ . A bivariate Student $t$ copula with $\nu > 0$ degrees of freedom and correlation $\rho > -1$ has

\begin{aligned} \lambda_L &= 2\, t_{\nu + 1}(-k) > 0, \\[0.4em] k &= \sqrt{\frac{(\nu + 1)(1 - \rho)}{1 + \rho}}, \end{aligned}

where $t_{\nu + 1}$ is the Student $t$ CDF with $\nu + 1$ degrees of freedom. The coefficient is a limit and says nothing directly about the ratio $C(u, u)/u$ at a fixed threshold $u$ , which is the conditional probability that matters for two names with default probability $u$ at a given horizon.

In plain words Tail dependence asks whether joint extremes stay comparable to single extremes as the extremes become rarer. Under a Gaussian copula with correlation below $1$ they do not, since the conditional probability of a joint extreme eventually vanishes, yet at any practical threshold joint defaults can still be far more likely than under independence.

Example For the Gaussian copula with $\rho = 0.3$ , numerical integration gives $C(u, u)/u \approx 0.216$ at $u = 0.1$ , $0.056$ at $u = 0.01$ , $0.015$ at $u = 0.001$ and $0.0003$ at $u = 10^{-6}$ . Each value exceeds the independent value $u$ , by a factor of about $2$ at $u = 0.1$ and about $300$ at $u = 10^{-6}$ , yet the sequence tends to $0$ . For a Student $t$ copula with $\nu = 4$ and the same $\rho = 0.3$ , the formula gives $\lambda_L \approx 0.162$ . Even with $\rho = 0$ it gives $\lambda_L \approx 0.076$ , so a $t$ copula with zero correlation does not describe independent names.

Portfolio loss distribution

Definition With fixed notionals $w_i$ , deterministic recovery fractions $R_i$ and the indicators of the first card, the portfolio loss at the horizon is

L = \sum_{i=1}^{n} w_i (1 - R_i)\, I_i ,

in currency units, and the portfolio loss distribution is the law of $L$ , with CDF $F_L(x) = P(L \le x)$ . Linearity of expectation gives

E[L] = \sum_{i=1}^{n} w_i (1 - R_i)\, p_i

for every joint law, with no independence assumption. The variance needs the pairwise joint default probabilities,

\begin{aligned} \operatorname{Var}(L) &= \sum_{i, j} w_i w_j (1 - R_i) \\[0.4em] &\quad \times (1 - R_j) \operatorname{Cov}(I_i, I_j), \end{aligned}

and tail probabilities $P(L > x)$ and quantiles $\inf\{x : F_L(x) \ge \alpha\}$ for $0 < \alpha < 1$ need the full joint law. In the one-factor model with $0 \le \rho < 1$ and homogeneous names, each with default probability $p$ and loss $\ell$ per default, the number of defaults $N$ is binomial with parameters $n$ and $p(Y)$ given $Y$ , so $L = \ell N$ and

\begin{aligned} &P(N = k) \\[0.4em] &= \int_{-\infty}^{\infty} \binom{n}{k} p(y)^{k} \\[0.4em] &\quad \times \bigl(1 - p(y)\bigr)^{n - k} \varphi(y)\, dy . \end{aligned}

In plain words The mean of the loss sees only the marginals. Its spread also sees how pairs of names move together, and its shape, which governs severe-loss probabilities and quantiles, sees the whole joint law.

Example Two constructed names each have $p = 0.1$ and lose $50$ currency units at default. With $q = 0.03$ the loss is $0$ , $50$ or $100$ with probabilities $0.83$ , $0.14$ and $0.03$ . Its mean is $50 \cdot 0.14 + 100 \cdot 0.03 = 10$ , its second moment is $2500 \cdot 0.14 + 10000 \cdot 0.03 = 650$ , and its variance is $650 - 100 = 550$ squared currency units, which agrees with $2 \cdot 2500 \cdot 0.09 \cdot (1 + 2/9)$ from the covariance formula. Under independence the probabilities are $0.81$ , $0.18$ and $0.01$ , the mean is still $10$ and the variance is $450$ . Dependence moved probability out of the one-default state to both ends. The probability of the largest loss triples, while the probability of no loss rises from $0.81$ to $0.83$ . With the lower quantile convention, the $0.98$ quantile is $100$ under dependence and $50$ under independence, the $0.95$ quantile is $50$ for both, and the $0.82$ quantile is $0$ under dependence and $50$ under independence. So not every risk statistic rises with dependence.

Check yourself

Two names have default probabilities $0.2$ and $0.3$ by the same horizon. Which joint default probabilities are possible, and which indicator correlations do they allow? A model input asks for an indicator correlation of $0.9$ . Is that possible?

Show the answer

The four state probabilities $q$ , $0.2 - q$ , $0.3 - q$ and $0.5 + q$ must be nonnegative, so the Fréchet bounds give $0 \le q \le 0.2$ . The indicator standard deviations multiply to $\sqrt{0.2 \cdot 0.8 \cdot 0.3 \cdot 0.7} = \sqrt{0.0336} \approx 0.1833$ , so the correlation $(q - 0.06)/0.1833$ ranges from about $-0.327$ at $q = 0$ to about $0.764$ at $q = 0.2$ . A correlation of $0.9$ would need

q = 0.06 + 0.9 \cdot 0.1833 \approx 0.225,

and then the state in which only the first name defaults would have probability $0.2 - 0.225 < 0$ . No joint law has these marginals and that correlation. The mistake is to treat every number in $[-1, 1]$ as an admissible correlation of default indicators.

In the one-factor Gaussian model every name has $p = 0.02$ and $\rho = 0.25$ . Find the conditional default probability given $Y = -2$ . A colleague wants the case $\rho = 1$ and substitutes it into the same formula. What goes wrong, and what are the conditional default probabilities at $Y = -2$ and at $Y = -2.1$ when $\rho = 1$ ?

Show the answer

The threshold is $c = \Phi^{-1}(0.02) \approx -2.0537$ and $\sqrt{\rho}\, y = 0.5 \cdot (-2) = -1$ , so

\begin{aligned} p(-2) &= \Phi\left(\frac{-2.0537 + 1}{\sqrt{0.75}}\right) \\[0.4em] &\approx \Phi(-1.2168) \approx 0.112, \end{aligned}

more than five times the marginal $0.02$ . The formula was derived by dividing by $\sqrt{1 - \rho}$ , which needs $\rho < 1$ , and at $\rho = 1$ the denominator is $0$ . In that case $Z_i = Y$ for every name, so a name defaults exactly when $Y \le c$ . At $Y = -2$ , which is above $-2.0537$ , no name defaults and the conditional probability is $0$ . At $Y = -2.1$ every name defaults and it is $1$ . The marginals are still right, since $P(Y \le c) = 0.02$ , but all names now default together.

A constructed model has a common factor with two states. In the good state, which has probability $0.8$ , each name defaults independently with probability $0.05$ . In the bad state, which has probability $0.2$ , each defaults independently with probability $0.3$ . For three names, find the marginal default probability, the joint default probability and the indicator correlation of two names, and the probability that all three default. Compare with independent names that have the same marginal.

Show the answer

The marginal is $0.8 \cdot 0.05 + 0.2 \cdot 0.3 = 0.1$ . Given the state the names are independent, so the joint default probability of two names is $0.8 \cdot 0.05^2 + 0.2 \cdot 0.3^2 = 0.002 + 0.018 = 0.02$ . The covariance $0.02 - 0.01 = 0.01$ equals the variance of the conditional default probability, which is why a common factor of this kind can only add nonnegative covariance, and the indicator correlation is $0.01/0.09 = 1/9 \approx 0.111$ . All three default with probability

\begin{aligned} &0.8 \cdot 0.05^3 + 0.2 \cdot 0.3^3 \\[0.4em] &= 0.0001 + 0.0054 = 0.0055, \end{aligned}

five and a half times the independent value $0.1^3 = 0.001$ . No name defaults with probability $0.8 \cdot 0.95^3 + 0.2 \cdot 0.7^3 = 0.7545$ , against $0.9^3 = 0.729$ under independence, while the expected number of defaults is $0.3$ in both models.

Default dependence and credit portfolios

Key terms

Default indicator

Joint default law

Latent Gaussian factor

Gaussian copula

Tail dependence

Portfolio loss distribution

Check yourself

How this lesson connects

Builds on

Used later in