Understand the principles and application of simple models for credit risk

Welcome to Credit Risk Modelling!

Hello there! Welcome to one of the most practical chapters in the CM2 curriculum. In this section, we are moving into the world of Asset Valuations by looking at Credit Risk. If you have ever lent money to a friend and worried they might not pay you back, you’ve already experienced the core of credit risk! For actuaries, we need to put numbers to that worry so we can price bonds and manage portfolios effectively.

Don't worry if the math looks a bit scary at first—we'll break it down step-by-step. By the end of these notes, you'll understand why a company is like a "call option" and how we can predict the chance of a business "going bust."

1. What is Credit Risk?

At its simplest, credit risk is the risk that a borrower (a person, a company, or a government) fails to meet their obligations to pay back interest or principal on a debt. In the world of asset valuation, this means the bond we bought might end up being worth much less than we hoped.

There are two main ways we model this in CM2:
1. Structural Models: We look inside the company to see if they have enough assets to cover their debts.
2. Reduced-Form Models: We treat default like a random "bolt from the blue" that could happen at any time.

Quick Review: Credit risk isn't just about total failure (default); it also includes downgrade risk (the risk that a company's credit rating drops from, say, AA to BBB).

2. Structural Models: The Merton Model

The Merton Model is the "grandaddy" of structural models. It uses the principles of Option Pricing (which you’ve seen in earlier chapters) to value a company’s debt and equity.

The Big Idea: The Company as an Option

Imagine a company has a total value \(V_t\) and a single debt \(D\) that must be paid at time \(T\).
• If at time \(T\), the company is worth more than the debt (\(V_T > D\)), the shareholders pay the debt and keep the rest. Their payoff is \(V_T - D\).
• If at time \(T\), the company is worth less than the debt (\(V_T < D\)), the shareholders walk away, give the company to the lenders, and get nothing. Their payoff is \(0\).

Does this look familiar? This is exactly the payoff of a European Call Option! Equity is a call option on the assets of the firm with a strike price equal to the debt.

The Formulas You Need

Using the Black-Scholes framework, the value of the Equity (\(E_t\)) is:
\( E_t = V_t \Phi(d_1) - D e^{-r(T-t)} \Phi(d_2) \)

Where:
\( d_1 = \frac{\ln(V_t / D) + (r + 0.5\sigma^2)(T-t)}{\sigma \sqrt{T-t}} \)
\( d_2 = d_1 - \sigma \sqrt{T-t} \)

Did you know? In this model, the Probability of Default is simply the probability that the company value is less than the debt at time \(T\). Under the real-world measure (not the risk-neutral one), this is \(1 - \Phi(d_2)\) (adjusted for the real growth rate \(\mu\) instead of \(r\)).

Key Takeaway:

Structural models assume we have perfect information about a company's assets and volatility. Default happens endogenously—meaning it is caused by the internal value of the firm falling too low.

3. Reduced-Form Models (Intensity Models)

Sometimes, we don't know the exact value of a company's assets. Reduced-form models ignore the balance sheet and instead treat default as a random event governed by a "hazard rate" or intensity, denoted as \(\lambda\).

The "Bolt from the Blue" Analogy

Think of it like a lightbulb. You don't know exactly when it will pop, but you know the average rate at which bulbs fail. The intensity \(\lambda\) is the instantaneous probability of default. If \(\lambda\) is constant, the time to default follows an Exponential Distribution.

Key Concepts:

• Survival Probability: The chance the firm survives until time \(t\) is \(P(T > t) = e^{-\lambda t}\).
• Default Probability: The chance the firm fails before time \(t\) is \(1 - e^{-\lambda t}\).
• Recovery Rate (\(R\)): If the company defaults, you don't always lose everything. You might get \(R\) cents on the dollar.

Common Mistake: Students often confuse these with structural models. Remember: If the question mentions a "hazard rate," "intensity," or "Poisson process," it's a Reduced-Form Model.

4. Credit Ratings and Transition Matrices

Another way to model credit risk is to look at Credit Ratings (like those from Moody's or S&P). Companies move between ratings (e.g., from A to BBB) over time. We model this using a Markov Chain.

The Transition Matrix

A transition matrix \(P\) shows the probability of moving from one state to another over a set period (usually one year).
• Rows represent the current rating.
• Columns represent the rating at the end of the period.
• One state is always Default, which is an "absorbing state" (once you're in default, you usually stay there!).

Memory Aid: "Rows are where you start, Columns are where you land."

Example: If you want the probability of defaulting over two years, you square the matrix (\(P^2\)) and look at the "Default" column for your starting rating.

5. Valuing Risky Bonds

How do we actually price a bond that might default? We discount the expected cash flows. For a simple zero-coupon bond with face value \(1\):

Price = (Probability of Survival \(\times\) Value if no default) + (Probability of Default \(\times\) Value if default)

Using the intensity \(\lambda\) and recovery rate \(R\), the value of a risky zero-coupon bond is:
\( B(t, T) = e^{-r(T-t)} [ e^{-\lambda(T-t)} + R(1 - e^{-\lambda(T-t)}) ] \)

Simplified Logic:
If there is no recovery (\(R=0\)), the price is just \( e^{-(r+\lambda)(T-t)} \). The intensity \(\lambda\) acts just like an extra bit of interest rate—we call this the credit spread.

6. Summary and Comparison

To wrap up, let's compare our two main modeling families:

Structural Models (Merton):
• Pros: Connects equity and debt; provides an economic reason for default.
• Cons: Assets (\(V_t\)) aren't directly observable; hard to calibrate; default is "predictable" in the short term (which isn't realistic).

Reduced-Form Models:
• Pros: Easier to calibrate to market prices; reflects the "surprise" nature of default.
• Cons: Doesn't explain why a default happened; treats default as a purely mathematical event.

Final Tip for the Exam: If you see a question asking for the "Value of the Firm" or "Volatility of Assets," head straight for the Merton Model formulas. If you see "Hazard Rate" or "Recovery Rate," use the Reduced-Form approach!

Don't worry if this seems tricky at first! Credit risk is a deep subject, but if you master the link between call options and company equity, you've already climbed the steepest part of the hill. Keep practicing those transition matrices!