15 Credit Risk Models

15.1 Introduction

Two questions define the study of credit risk models in this chapter. First, how can the probability that a borrower defaults — and the timing of that default — be modelled in a way that is mathematically tractable even though default times do not follow a normal distribution? Second, how does correlation in default probabilities across borrowers affect the pricing and risk of structured credit products, such as the tranched securities that pool many individual loans? These questions are of fundamental importance to financial markets. Credit risk is the defining feature of corporate bonds, loans, and mortgage-backed securities, and the failure to model it correctly — particularly the second question about correlation — was a central cause of the 2007–2009 global financial crisis. Understanding these models is therefore both an intellectual necessity and a lesson drawn from one of the most consequential episodes in modern financial history.

The first question motivates the Gaussian copula approach to modelling default times. Default times are non-negative and typically right-skewed random variables; they cannot be directly plugged into standard multivariate normal models. The Gaussian copula resolves this by applying a transformation: each entity’s default time is first mapped through its own marginal cumulative distribution function, producing a uniform random variable, and then through the inverse of the standard normal CDF, producing a standard normal variable that preserves the original marginal default probability. This transformation makes it possible to impose a correlation structure using familiar normal distribution tools. The key structural assumption is that each entity’s transformed default indicator can be decomposed into a common systematic factor — shared by all borrowers — and an idiosyncratic component specific to that borrower, with the parameter $a_i$ controlling the relative weight of each. Conditional on the common factor, defaults become independent, which makes computation of joint default probabilities tractable.

The second question is explored through a stylized two-loan example that builds direct intuition for the effects of correlation on tranche values. When the two loans default independently, the senior tranche — which collects the first payment regardless of which borrower makes it — defaults only if both loans fail simultaneously, an event with probability $\phi^2$. As positive correlation is introduced through a common factor, this probability rises: because defaults tend to cluster together in bad states of the world, the protection that diversification provides to the senior tranche evaporates. The junior tranche, which collects only when both loans perform, correspondingly increases in expected value as correlation rises. This result captures in miniature the essential fragility of AAA-rated tranches of mortgage-backed securities: when a common economic shock can drive many borrowers into default simultaneously, the senior position is far less safe than its rating under an independence assumption would suggest.

The chapter begins by developing the Gaussian copula model formally, deriving the conditional default probability given the common factor and discussing the role of the loading parameter in governing default correlation. It then turns to the two-loan tranche example, computing expected payoffs under both independence and correlated defaults and showing analytically how correlation redistributes value between the senior and junior tranches. Together, these two parts illuminate both the power and the limitations of factor-based credit models.

In the news

Pricing the risk that a borrower fails to repay is the business of a fast-growing — and increasingly scrutinized — corner of finance. Global Finance reported that the roughly $2 trillion private-credit market is reaching a “crossroads,” with defaults rising and regulators warning about thin oversight even as spreads remain tight. Estimating a borrower’s probability of default and the loss it would cause, and reading those probabilities out of market spreads, is precisely what the credit models in this chapter do. Read it at Global Finance.

15.2 The Gaussian Copula Model of Time to Default

Let $t$ be the random variable denoting the time to default of a particular entity (mortgage, bond payment, etc.). In general $t$ will not be normally distributed. Suppose that the CDF of $t$ is given by $F(t)$. So $F(t_0)$ is the probability that the entity defaults up to the time $t_0$. Let $N(x)$ be the CDF of the standard normal distribution (i.e. the normal distribution with mean 0 and variance 1). One can transform the non-normal random variable $t$ into a normally distributed random variable $x$. To do so, let $x(t)$ be the solution to the equation

\[N(x(t)) = F(t)\]

Note that by definition, the probability that $t \le t_0$ is equal to the probability that $x \le x(t_0)$. Thus, by letting $x(t) = N^{-1}(F(t))$ we have transformed $t$ into a normally distributed random variable.

Consider now a set of entities indexed by $\{1,\ldots, n\}$ with times to default given by $\{t_1,t_2,t_3,\ldots,t_n\}$. We will assume that the transformation of these times to default can be modelled by the function

\[x_i = a_i Y + \sqrt{1-a_i^2}\epsilon_i\]

where $-1 < a_i < 1$. The variables $\epsilon_i$ represent the individual factors that affect default so it is assumed that $cov(\epsilon_i,\epsilon_j) = 0$ for all $i$ and $j$. $Y$ is a common factor affecting default amongst all entities. It is also assumed that $Y$ and $\epsilon_i$ have independent standard normal distributions for all $i$. Note that

\[cov(x_i,x_j) = Ex_i x_j = a_i a_j\]

because of the independent standard normal assumption.

If the probability that entity $i$ will default by time $T$ is $F_i (T)$, then this occurs when $x_i \le N^{-1}(F_i (T))$. From our model of $x_i$, this will happen when

\[a_i Y + \sqrt{1-a_i^2}\epsilon_i < N^{-1}(F_i(T))\]

or rearranging, when

\[\epsilon_i \le \frac{N^{-1}(F_i(T)) - a_i Y}{\sqrt{1-a_i^2}}.\]

Therefore, if we knew the value of the factor $Y$, then we would know the probability of default which is

\[F_i (T|Y) = N \left ( \frac{N^{-1}(F_i(T)) - a_i Y}{\sqrt{1-a_i^2}} \right )\]

If we assume a common loading $a_i = \rho$ for all $i$ and a common marginal default probability $F_i(T) = F(T)$, then this becomes

\[F(T|Y) = N \left ( \frac{N^{-1}(F(T)) - \rho Y}{\sqrt{1-\rho^2}} \right ).\]

This is the key result of the model: conditional on the common factor $Y$, every entity defaults with this same probability $F(T|Y)$ and — because the idiosyncratic shocks $\epsilon_i$ are mutually independent — the defaults are conditionally independent. The joint probability that a group of entities all default by $T$ is therefore the product of their conditional probabilities, integrated over the distribution of $Y$:

\[\Pr(\text{all default}) = \int_{-\infty}^{\infty} \Big[F(T|Y)\Big]^{k} \, n(Y)\,dY,\]

where $k$ is the number of entities and $n(\cdot)$ is the standard normal density. Correlation enters entirely through the shared factor $Y$: a bad draw of $Y$ raises $F(T|Y)$ for every entity at once, which is what makes many defaults cluster in bad states.

15.3 A simple example

Consider a one year loan that will make a payment of $M$ if not in default and 0 under default. The probability of the loan defaulting is $\phi$. This asset has an expected value of $(1-\phi)M$ and a variance of $\phi(1-\phi)M^2$. For simplicity, suppose that the covariance of this asset with the market is 0. Thus, the CAPM says that the price of this asset would satisfy

\[E \left (\frac{M - P}{P} \right ) - r_f = 0\]

which implies that its price is given by

\[P = \frac{EM}{1+r_f} = \frac{(1 - \phi)M}{1 + r_f}\]

Now assume that there are two such loans, each identical and each having independent probabilities of defaulting. The likelihood that both will default is $\phi^2$, while the likelihood that one of these loans defaults is $2\phi(1-\phi)$ while the probability that none will default is $(1-\phi)^2$.

Consider constructing a security that will receive the first mortgage payment, no matter who it comes from. Such a security will default only if both of the loans default. As such, it has an expected payoff of $(1-\phi^2)M$, a variance of $\phi^2 (1-\phi^2) M^2$ and a market price of

\[P = \frac{EM}{1+r_f} = \frac{(1 - \phi^2)M}{1 + r_f}\]

A second security will receive the second payoff regardless of from whom it comes. Such a security will payoff as long as both mortgages don't default. This security has expected payoff $(1-\phi)^2 M$, variance $(1-\phi)^2(1-(1-\phi)^2) M^2$ and market price

\[P = \frac{(1 - \phi)^2 M}{1 + r_f}\]

Now suppose that the probabilities of each of these loans defaulting are not independent. Specifically, suppose that the probability of each loan defaulting is given by $\phi_i = a y + \phi$ where $y$ is a random, common factor that affects default probabilities for all of the loans. Notice that if $a$ is 0 then this model becomes the old model. Suppose that conditional on a particular $y$, the default probabilities are still independent, so the way that dependence is introduced into the model is through the random factor $y$. Assume that $y \sim N(0,\sigma^2)$, and, to keep the results below in the same units as the independent case, adopt the normalization $\sigma^2 = \phi^2$ (the common factor has standard deviation equal to the baseline default probability). The two facts we need about $y$ are $E_y[y] = 0$ and $E_y[y^2] = \sigma^2$.

A security based on the first loan payment received will default only if both loans default. Conditional on $y$ the two defaults are independent, so that conditional probability is $\phi_i^2 = (ay+\phi)^2$, and the payoff probability is $1 - (ay+\phi)^2$. Taking the expectation over $y$, and being careful to keep the $a^2 y^2$ term,

\[\begin{aligned} E_y\!\left[1 - (ay+\phi)^2\right] &= 1 - E_y\!\left[a^2 y^2 + 2a\phi\, y + \phi^2\right] \\ &= 1 - a^2 E_y[y^2] - 2a\phi\, E_y[y] - \phi^2 \\ &= 1 - a^2\sigma^2 - \phi^2 = 1 - \phi^2 - a^2\phi^2, \end{aligned}\]

where the last step uses $E_y[y]=0$, $E_y[(ay)^2] = a^2 E_y[y^2] = a^2\sigma^2$, and the normalization $\sigma^2 = \phi^2$. (The earlier version dropped the $a^2 y^2$ contribution; it is precisely this term that produces the $a^2\phi^2$ correction.) This implies a new price of

\[P = \frac{(1 - \phi^2 -a^2 \phi^2)M}{1 + r_f}\]

Thus, when correlation increases, the relatively risk-free asset becomes less valuable.

The second asset pays off only when both assets do not default. Conditional on $y$ the probability that neither defaults is $(1-\phi_i)^2 = (1 - ay - \phi)^2$. Expanding fully — again keeping the $a^2 y^2$ term — and taking the expectation,

\[\begin{aligned} E_y\!\left[(1 - \phi - ay)^2\right] &= E_y\!\left[(1-\phi)^2 - 2(1-\phi)a\,y + a^2 y^2\right] \\ &= (1-\phi)^2 - 2(1-\phi)a\,E_y[y] + a^2 E_y[y^2] \\ &= (1-\phi)^2 + a^2\sigma^2 = (1-\phi)^2 + a^2\phi^2, \end{aligned}\]

using $E_y[y]=0$, $E_y[y^2]=\sigma^2$, and the normalization $\sigma^2 = \phi^2$. Note that the expected value of the second asset increases. It now has value given by

\[P = \frac{[(1-\phi)^2 + a^2 \phi^2]M}{1 + r_f}\]

Comparing the two tranches, the common factor moves value from the senior security to the junior one. The senior tranche loses $a^2\phi^2 M/(1+r_f)$ and the junior tranche gains exactly the same amount, so the two effects offset: the combined value of the two tranches is still $\frac{[2(1-\phi)]M}{1+r_f}$, equal to the value of simply holding both loans. Correlation does not create or destroy value; it redistributes it. The intuition is that positive correlation makes joint outcomes more likely — both loans defaulting together, or both surviving together — which thins out the “one defaults, one survives” middle case. That hurts the senior tranche, whose safety relied on it being unlikely that both borrowers fail at once, and helps the junior tranche, whose payoff relied on it being likely that both survive.

This is the essential lesson of the chapter. A senior (“AAA”) tranche rated on the assumption of independent defaults ($a = 0$, default probability $\phi^2$) is materially riskier once a common factor is present, because its default probability rises to $\phi^2 + a^2\phi^2$. In the language of the 2007–2009 crisis, the common factor $y$ is a nationwide shock to house prices: when it turns bad, many mortgages default together, and the diversification that supposedly protected the senior tranche evaporates precisely when it is needed most.

15.4 Application: An analyst underwriting a loan

An analyst at a private-credit fund must decide whether to extend a loan to a mid-sized company and what interest rate would compensate for the risk of not being repaid. The models in this chapter are the analyst’s underwriting toolkit. By estimating the borrower’s probability of default, the loss given default, and the exposure at default, the analyst can compute the expected loss the loan carries and judge whether the offered yield is adequate. Structural models tie the chance of default to the firm’s assets and leverage, while reduced-form models read default intensity from market prices; together they let the analyst translate a credit spread into an implied default probability and ask whether the market — currently pricing very little default risk into a fast-growing, opaque asset class — is paying enough for the danger being taken on.

Further listening

Odd Lots (Bloomberg) — “What’s Actually Going On With Private Credit”: inside the $2-trillion private-credit boom and the rising default risk these credit models are built to price.

15.5 Homework problems

15.5.1 Conceptual

CR-C1. A junior analyst argues that because a borrower’s time to default $t$ is non-negative and right-skewed, “credit risk simply cannot be handled with normal-distribution machinery.” Explain how the transformation $x(t) = N^{-1}(F(t))$ addresses this objection. In your answer, state precisely what is preserved by the transformation and why the probability that $t \le t_0$ equals the probability that $x \le x(t_0)$.

CR-C2. A modeler applies the transformation $x(t) = N^{-1}(F(t))$ to two entities whose marginal default distributions $F_1(t)$ and $F_2(t)$ are very different — one a sub-prime mortgage that defaults quickly, the other an investment-grade bond that defaults slowly. Explain why, after transformation, both $x_1$ and $x_2$ are standard normal even though $t_1$ and $t_2$ are not identically distributed, and explain what this common normal scale buys the modeler when he wants to impose correlation across the two entities.

CR-C3. In the factor model $x_i = a_i Y + \sqrt{1-a_i^2}\,\epsilon_i$, describe the economic meaning of the common factor $Y$ and the idiosyncratic term $\epsilon_i$. Contrast a scenario in which every $a_i$ is close to 1 with one in which every $a_i$ is close to 0, explaining what each implies for how defaults cluster, and give a real-world example of an economic force that would be captured by $Y$.

CR-C4. Explain why the decomposition $x_i = a_i Y + \sqrt{1-a_i^2}\,\epsilon_i$ is written with the coefficient $\sqrt{1-a_i^2}$ on the idiosyncratic term rather than an arbitrary constant. Show that this choice keeps each $x_i$ standard normal, and explain how it makes $a_i$ interpretable directly as the correlation of $x_i$ with the common factor $Y$. Why is preserving the marginal distribution of $x_i$ essential to the copula construction?

CR-C5. The chapter shows $F_i(T|Y) = N\!\left(\frac{N^{-1}(F_i(T)) - a_i Y}{\sqrt{1-a_i^2}}\right)$. Explain why conditioning on the common factor $Y$ makes the defaults of different entities independent of one another, and why this conditional-independence property is what makes joint default probabilities computationally tractable. What is lost if one instead tries to model the joint default distribution directly?

CR-C6. Interpret the conditional default probability $F(T|Y) = N\!\left(\frac{N^{-1}(F(T)) - \rho Y}{\sqrt{1-\rho^2}}\right)$ as $Y$ moves. Explain why a low (unfavorable) draw of $Y$ raises the conditional default probability for every entity at once, and describe the limiting behavior as $\rho \to 0$ and as $\rho \to 1$. Why does this single expression capture the entire correlation structure of the model?

CR-C7. In the two-loan example, the senior security (first payment received) has default probability $\phi^2$ under independence but $\phi^2 + a^2\phi^2$ once the common factor is introduced, while the junior security’s survival probability rises from $(1-\phi)^2$ to $(1-\phi)^2 + a^2\phi^2$. Explain in words why positive correlation makes the senior tranche riskier while making the junior tranche more valuable, and relate this to why diversification “evaporates” for the senior position in bad states of the world.

CR-C8. The chapter argues that when the common factor is introduced, the senior tranche loses exactly $a^2\phi^2 M/(1+r_f)$ in value and the junior tranche gains exactly the same amount, so that the combined value of the two tranches is unchanged. Explain why correlation redistributes value rather than creating or destroying it, and why this conservation is what one should expect given that the two tranches together simply reproduce holding both loans outright.

CR-C9. Explain how the two-loan example illuminates the 2007–2009 financial crisis. In particular, discuss why a AAA rating assigned to a senior mortgage-backed tranche under an independence assumption ($a=0$) could be dangerously misleading, and what feature of the housing market corresponds to the common factor $y$ in the model.

CR-C10. A ratings analyst insists that because each individual mortgage in a pool has only a small default probability $\phi$, the senior tranche built on that pool must be nearly riskless. Using the result that the senior tranche’s default probability is $\phi^2 + a^2\phi^2$ rather than $\phi^2$, explain the flaw in this reasoning and identify the single modeling assumption whose misjudgment does the most damage to the senior tranche’s safety.

15.5.2 Quantitative

CR-Q1. An entity has a constant hazard (default intensity) of $\lambda = 0.03$ per year, so its default-time CDF is $F(t) = 1 - e^{-\lambda t}$. (a) Compute the probability the entity defaults within $T = 5$ years, $F(5)$. (b) Using $N^{-1}$, find the transformed threshold $x(5) = N^{-1}(F(5))$. Report both numbers.

CR-Q2. An entity’s default-time CDF is $F(t) = 1 - e^{-\lambda t}$ with $\lambda = 0.05$. (a) Find the time-to-default value $t^{*}$ that transforms to the standard-normal threshold $x = 0$, i.e. solve $N^{-1}(F(t^{*})) = 0$. (b) Verify by computing $F(t^{*})$ and confirming it equals $N(0)$. Report $t^{*}$ and $F(t^{*})$.

CR-Q3. Using the conditional default probability $F_i(T|Y) = N\!\left(\frac{N^{-1}(F_i(T)) - a_i Y}{\sqrt{1-a_i^2}}\right)$, take $F_i(T) = 0.05$ and loading $a_i = 0.5$. (a) Compute $F_i(T|Y)$ when the common factor takes a “good” value $Y = +1$. (b) Compute it when the common factor takes a “bad” value $Y = -1$. Comment on which state produces more defaults.

CR-Q4. Using $F(T|Y) = N\!\left(\frac{N^{-1}(F(T)) - \rho Y}{\sqrt{1-\rho^2}}\right)$, take an unconditional default probability $F(T) = 0.10$ and a bad common-factor draw $Y = -1.5$. Compute the conditional default probability for (a) $\rho = 0.2$ and (b) $\rho = 0.7$. Report both and explain why the higher loading produces a larger conditional default probability in the bad state.

CR-Q5. The pairwise correlation of the default drivers is $cov(x_i,x_j) = a_i a_j$. (a) If two entities have loadings $a_i = 0.6$ and $a_j = 0.8$, compute $cov(x_i,x_j)$. (b) State what single common loading $a$ (assumed equal across both entities) would be required to produce the same pairwise correlation. Report both.

CR-Q6. A pool has three entities with loadings $a_1 = 0.5$, $a_2 = 0.5$, $a_3 = 0.9$ on the common factor $Y$. (a) Compute the pairwise default-driver correlation $cov(x_i,x_j) = a_i a_j$ for each of the three pairs $(1,2)$, $(1,3)$, $(2,3)$. (b) Which pair is most correlated, and explain in one sentence why the entity with the largest loading dominates the correlation structure. Report all three correlations.

CR-Q7. Two identical one-year loans each pay $M = \$1{,}000$ if not in default and 0 otherwise, with default probability $\phi = 0.10$ and risk-free rate $r_f = 0.04$. Assuming independence, compute the market price of (a) the senior security (receives the first payment, defaults only if both default) and (b) the junior security (receives the second payment, pays only if neither defaults). Use $P_{\text{senior}} = \frac{(1-\phi^2)M}{1+r_f}$ and $P_{\text{junior}} = \frac{(1-\phi)^2 M}{1+r_f}$.

CR-Q8. Now introduce the common factor with $\phi_i = ay + \phi$, keeping $\phi = 0.10$, $M = \$1{,}000$, $r_f = 0.04$, and loading $a = 0.20$. Using the correlated-default results $P_{\text{senior}} = \frac{(1-\phi^2 - a^2\phi^2)M}{1+r_f}$ and $P_{\text{junior}} = \frac{[(1-\phi)^2 + a^2\phi^2]M}{1+r_f}$, compute both prices and report the change in each relative to the independent-default prices from CR-Q7.

CR-Q9. For the correlated two-loan model with $\phi = 0.10$, $M = \$1{,}000$, $r_f = 0.04$, and loading $a = 0.30$: (a) compute the dollar amount $\frac{a^2\phi^2 M}{1+r_f}$ that is transferred from the senior tranche to the junior tranche. (b) Confirm value conservation by showing that $P_{\text{senior}} + P_{\text{junior}}$ equals $\frac{2(1-\phi)M}{1+r_f}$, the value of holding both loans outright. Report the transfer amount and the combined value.

CR-Q10. Under the common-factor model the senior tranche’s default probability is $\phi^2 + a^2\phi^2$. With $\phi = 0.10$: (a) compute the senior tranche’s default probability under independence ($a = 0$) and under a loading of $a = 0.5$. (b) Compute the percentage increase in the senior tranche’s default probability caused by moving from $a=0$ to $a=0.5$. Report both probabilities and the percentage increase.