Welcome to Spread Risk and Default Intensity Models!
Welcome, future FRM charterholders! If you’ve made it to Part II, you already know that credit risk isn't just about whether someone defaults—it's also about how the market's perception of that risk changes over time. In this chapter, we are going to dive into Spread Risk (the risk that the "gap" between risky and safe bonds changes) and Default Intensity Models (the math we use to predict the "instantaneous" chance of default).
Don't worry if these terms sound a bit intimidating. Think of it this way: if Structural Models (like Merton’s) are like looking under the hood of a car to see if the engine will fail, Intensity Models are like looking at the historical traffic data to see how often crashes happen on this road. Let's break it down step-by-step!
1. Understanding Credit Spreads
Before we model anything, we need to understand what we are measuring. A Credit Spread is simply the extra yield an investor demands for taking on credit risk compared to a risk-free benchmark (like a Government Treasury bond).
The Basic Formula:
\( Spread = Yield_{Risky} - Yield_{Risk-free} \)
Analogy: Imagine you lend \$100 to the government, and they pay you 3% interest. Now imagine your neighbor asks for \$100. Because your neighbor is riskier than the government, you might charge them 8%. That 5% difference (8% - 3%) is the spread. It compensates you for the risk that your neighbor might move away and never pay you back!
Why do Spreads Change?
Spreads don't stay still. They move because of:
1. Credit Quality Changes: The borrower's financial health gets better or worse.
2. Liquidity: It becomes harder or easier to sell the bond in the market.
3. Market Sentiment: Investors become "scared" and demand more premium for any kind of risk.
Quick Review: When credit spreads widen (increase), the price of the bond falls. When spreads narrow (tighten), the price of the bond rises.
2. Default Intensity (The Hazard Rate)
The "heart" of reduced-form models is the Default Intensity, also known as the Hazard Rate, denoted by the Greek letter \( \lambda \) (lambda).
Default intensity is the instantaneous probability of default. It represents the probability that a borrower defaults in the next very small increment of time, given that they haven't defaulted yet.
Key Properties of Hazard Rates:
- It is conditional: It assumes the borrower is still "alive" (hasn't defaulted) right now.
- It is exogenous: Unlike the Merton model, we don't look at the company's balance sheet. We just assume default is a "random bolt from the blue."
Survival Probability:
If the hazard rate \( \lambda \) is constant, the probability that a company will survive until time \( t \) is:
\( P(Survive) = e^{-\lambda t} \)
Probability of Default (PD):
Since you either survive or you don't, the probability of default is simply:
\( PD = 1 - e^{-\lambda t} \)
Memory Aid: Think of \( \lambda \) as the "speed" of default. If the speed is high, the survival probability drops very quickly!
3. Reduced-Form Models vs. Structural Models
It is crucial for the FRM exam to know the difference between these two approaches. This chapter focuses on Reduced-Form Models.
Structural Models (The Merton Model):
- Default happens when Assets < Debt.
- Requires knowing the value of the firm's assets (which is hard to see).
- Weakness: Cannot easily explain why spreads exist for very short-term bonds.
Reduced-Form Models (Intensity Models):
- Default is a "surprise" event (a Poisson process).
- Uses market data (bond prices and spreads) as inputs.
- Strength: Very good at matching actual market prices and spreads.
Key Takeaway: Reduced-form models are preferred by traders because they can be "calibrated" to match the prices we see on our screens today.
4. Estimating Hazard Rates from Market Spreads
How do we find \( \lambda \) in the real world? We look at the credit spread! For a risk-neutral world with a constant hazard rate and a known Recovery Rate (RR), there is a famous "rule of thumb" equation:
\( Spread \approx \lambda \times (1 - RR) \)
Where:
- \( (1 - RR) \) is the Loss Given Default (LGD).
Step-by-Step Example:
Suppose a 1-year corporate bond has a spread of 200 basis points (0.02) over Treasuries. The estimated Recovery Rate is 40%. What is the Hazard Rate (\( \lambda \))?
1. Identify the LGD: \( 1 - 0.40 = 0.60 \).
2. Set up the equation: \( 0.02 = \lambda \times 0.60 \).
3. Solve for \( \lambda \): \( \lambda = 0.02 / 0.60 = 0.0333 \) or 3.33%.
Common Mistake: Don't forget to convert basis points to decimals! 100 bps = 0.01.
5. Spread Risk and Spread Volatility
Even if a bond never defaults, an investor can still lose money if the Spread Volatility is high. Spread risk is the risk of financial loss due to changes in the level of credit spreads.
Spread Duration
Similar to interest rate duration, Spread Duration measures how much a bond's price will change for a 1% change in its credit spread.
\( \Delta Price \approx -Spread\ Duration \times \Delta Spread \times Price \)
Did you know? For most corporate bonds, the Spread Duration is roughly equal to the Modified Duration. This means if spreads widen by 1%, the price impact is almost the same as if interest rates had risen by 1%.
6. Summary of Key Concepts
Quick Review Box:
- Spread: The "extra" yield for taking credit risk.
- Hazard Rate (\( \lambda \)): The instantaneous probability of default.
- Survival Probability: \( e^{-\lambda t} \).
- Reduced-Form Models: Use market prices; default is a random surprise.
- Spread-Hazard Relationship: \( Spread \approx \lambda \times LGD \).
Final Encouragement: You've got this! The math in intensity models can look scary with integrals and exponents, but if you remember that Spread is just a reflection of Probability (\( \lambda \)) times Loss (LGD), you will be able to answer most conceptual questions on the FRM exam. Keep practicing those calculations!