Credit Risk Glossory

Imagine you are manaing credit portfolio of 10 million, which has 100 bonds, each of them worth 100,000 dollar across 100 diffferent firms. And we and to estimate the loss under systemetic shock.

Assumptions:

• Each bond has:

  • Baseline default probability of $p_{k}^{0} = 1\%$.
  • Recovery rate of $40\%$.
  • Correlation to system factor $\rho_{k} = 0.2$.

• System factor realization (for example, recession) = $z = 1$.

Default?

A default occurs when a borrower fails to meet debt obligations, such as interest or principal payments on a bond or loan.

• In a portfolio, if a bond issuer goes bankrupt, it defaults.
• The loss is calculated as:

Loss = (1 - Recovery Rate) × Exposure

For example, with a 40% recovery on a $100,000 bond:

Loss = (1 - 0.4) × 100,000 = 60,000

Credit Portfolio?

A credit portfolio consists of credit-risky financial assets, including:

• Corporate bonds
• Bank loans
• Credit derivatives (e.g. CDS)

Such portfolios are exposed to the risk of borrower default.

Baseline Default Probability $p_k^0$?

This is the unconditional probability that a firm will default, assuming normal market conditions.

• Estimated from credit ratings, CDS spreads, or historical default rates.
• Example: A BB-rated firm may have $p_k^0 = 0.01$ (1% default probability).

Systemic Factor Correlation $\rho_k$?

$\rho_k$ measures how strongly asset $k$'s default risk is correlated with the macroeconomy.

• If $\rho_k$ is close to 0 → risk is mostly firm-specific.
• If $\rho_k$ is close to 1 → risk is strongly tied to macro conditions.

Typical values:

• $\rho_k = 0.1–0.2$: Low to moderate sensitivity.
• $\rho_k = 0.3–0.5$: High sensitivity to economic downturns.

Systemic Factor Realization $z$?

The variable $z ~ N(0,1)$ models the macroeconomic environment:

• $z = 0$: Normal conditions
• $z = -1$: Recession (1 standard deviation below mean)
• $z = -2$: Financial crisis or deep recession

This is used to simulate default probabilities under stress.

Why Gaussian Conditional Independence?

The Gaussian Conditional Independence (GCI) model allows:

• Simulation of losses across many economic scenarios.
• Estimation of expected loss, VaR, and CVaR.
• Modeling of systemic vs. idiosyncratic risks.
• Regulatory capital and stress-testing applications.

CDF (Cumulatuve Distribution Function)?

The cumulative distribution function of a real-valued random variable $X$ is the function given by

$$F(x) = \mathbb{P}[X\leq x].$$

This means

The probability that the random variable $X$ is less than or equal to some value $x$. It's the area under the probability curve to the left of $x$.

Let's say $X \sim \mathcal{N}(0,1)$, where 0 is the mean and 1 is the variance ,i.e., standard normal distribution where mean $\mu = 0$, symmetrical bell curve. We wonder what's the chance a standard normal vairable is $\leq 0$? That is, $F(1) = \mathbb{P}[X\leq 0]$? The answer is $50%$ since the curve is symmetric around 0. $$ F(0) = \Phi(0) = 0.5 $$

A simple example that we can analogy is the exam scores. Let's say your exam score $X$, which is a random variable, you don't know yet. And we have $X \sim \mathcal{N}(70,10^{2})$, which means that the average score $\mu = 70$ and standard deviation $\sigma = 10$. Now, you want to know that

What's the chance your score is less than 85?

That will be $$ \mathbb{P}[X\leq x] = \mathbb{P}[X\leq 85], $$ where $X$ is your future score which you don't know yet, $x= 85$ is the specific score you are comparing to. And the result will be $$ Z = \frac{X - \mu}{\sigma} = \frac{85 - 70}{10} = 1.5 $$ thus, we plug back to our CDF function

$$\Phi(z) = \frac{1}{\sqrt{2\pi}}\int_{\infty}^{z} e^{-\frac{t^{2}}{2}} dt = \Phi(1.5) \approx 0.9332$$

This means there's a $93.3\%$ chance your score $\leq 85$.

Here, we define a standardized variable from a normal distribution: $$ Z = \frac{X - \mu}{\sigma} $$ where $X$ is the raw value (this can be return, credit score), $\mu$ is the mean of the distrubtion, $\sigma$ is the standard deviation, $Z \sim \mathcal{N}(0,1)$: always stardard normal

Loss given default

A Loss Given Default (LGD) for asset $k$ means the fraction of value you loss if asset $k$ defaults. $$ \lambda_{k} = 1 - \text{Recovery Rate}. $$ For example, if a bond pays back $40\%$ when defaulting, we have a $\text{Recovery Rate}$ of 0.4. Thus, $\lambda_{k} = 1 - 0.4 = 0.6$. This means that you lose $60\%$ of the value upon default.

Why use CDF for Risk Estimation?

Because Value-at-Risk (VaR) is defined as a quantile, which is the inverse of the CDF.

"Find the smallest $x$ such that at least $\alpha %$ of losses are below it"

$$\text{VaR}_{\alpha} = \text{inf}\{x | F_{L}(x)\geq \alpha \}$$

Therefore, if you already know $F_{L}(x)$, you can

find VaR by solving:

$$F_{L}(x^{*}) = \alpha \rightarrow x^{*} = \text{VaR}_{\alpha}$$

and find CVaR:

$$\text{CVaR}_{\alpha} = \mathbb{E}[L|L\leq\text{VaR}_{\alpha}]$$

Then you average all loeese up to the VaR threshold.

Recover Rate

The recovery rate is the percentages of defaulted debt that a lender can recover.[2]

Bernoulli Variable

A Bernoulli variable is a random variable with only two outcomes:

• 1 if event happens, e.g. default.
• 0 if event does not happen

$$D_k \sim \text{Bernoulli}(p_{k}(z))$$

This means that:

• $D_k = 1$: asset $k$ defaults with probability $p_{k}(z)$
• $D_k = 0$: asset $k$ survivies with probability $1 - p_{k}(z)$

Uncentainty Models

In finance, an uncertainty model referes to any mathematical or probabilistic framework used to quantify and manage unknowns espeically future outcomes that are not deterministic.

We know that Finance is full of randomness, such as returns, defaults, interest rate, asset prices, and uncertainty models helps you describe, simulate, and price that radomness.

Below are some common types of uncertainty models:

• Probabilistic models: This model uses distributuions to describe outcomes. For example: Value-at-Risk (VaR), default models.
• Stochastic models: This model uses random variables/ processes to model time evolution. For example: Black–Scholes model for option pricing.
• Scenario-based model: This model define possible future paths or economic regiimes. For example, stress testing.
• Bayuesian models: This model combines prior beliefs with observed data. For example: Portfolio updating, risk estimation.

Latent Variables and Observables

Latent variables in finance refer to hidden forces—such as investor sentiment, creditworthiness, or real earnings quality—that influence market behavior but cannot be directly measured or retrieved from any financial terminal. Unlike observable quantities like stock prices, volumes, or P/E ratios, latent variables are not directly recorded. Instead, they must be inferred from patterns in observable data, which makes them essential yet elusive components in financial modeling.

• Observable variable: You see it directly, i,e,. stock price, trading volume, P/E ratio.
• Latent variable: Sits in the background, driving what you see, but you cannot measure it with a single gauge.

These variables are typically abstract, multidimensional, and expressed only indirectly through their impact on observable metrics. For instance, a rise in market fear may cause VIX to spike, bond yields to fall, and equity prices to decline. None of these alone fully captures fear, but together they imply its presence. The problem is worsened by noise—macroeconomic surprises, news shocks, and irregular market reactions—making latent signals hard to isolate or measure cleanly.

To extract latent variables, finance relies on statistical inference tools. Principal Component Analysis (PCA) can uncover dominant hidden drivers behind correlated movements. Factor models help decompose asset returns into exposure to unobserved risks. Kalman filters track time-varying latent states under noisy conditions. These techniques don’t let us see the latent variables directly—but they allow us to reconstruct and quantify them, enabling more robust understanding of markets and better decision-making under uncertainty.

Reference

[1]. https://en.wikipedia.org/wiki/Cumulative_distribution_function

[2]. https://www.investopedia.com/terms/r/recovery-rate.asp

[3]. https://en.wikipedia.org/wiki/Bernoulli_distribution