## **Learning Outcomes**

This article explains how to distinguish between structural and reduced-form credit risk models and apply each framework to typical CFA Level 2-style problems. It defines the economic intuition behind structural models, including the role of firm value, capital structure, and the default barrier, and shows how option-pricing techniques are used to derive model-implied default probabilities and credit spreads. It also explains the statistical foundations of reduced-form, or intensity-based, models, emphasizing hazard rates, survival probabilities, and recovery assumptions, and how these inputs are extracted from bond prices and CDS spreads. The article examines how both approaches are used to value risky debt, compute expected loss, and interpret credit spreads as compensation for default risk and risk premia. It highlights the endogenous nature of default in structural models versus the exogenous treatment in reduced-form models, and summarizes the practical advantages, limitations, and typical use cases of each. Throughout, the focus remains on concepts, formulas, and interpretations that are most likely to be tested in the CFA Level 2 exam.

## **CFA Level 2 Syllabus**

For the CFA Level 2 exam, you are expected to understand the theoretical models for credit risk assessment and credit portfolio strategies, with a focus on the following syllabus points:

- Explaining expected exposure, loss given default, probability of default, and credit valuation adjustment (CVA)
- Describing and contrasting structural (balance sheet-based) and reduced-form (statistical) credit models, including their assumptions and key features
- Calculating the value of risky debt using credit risk parameters
- Interpreting how market-implied and structural default probabilities are used in credit risk management and bond valuation

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. According to structural credit models, what financial metric is compared to the face value of debt to determine default?
2. What is the main difference between how structural and reduced-form models estimate default probability?
3. If a credit-risky bond is priced below an otherwise identical risk-free bond, which risk parameter can be directly inferred from this price difference?
4. True or false? Reduced-form models provide an economic explanation for the cause of default.

## **Introduction**

Credit risk is the possibility of loss due to a borrower's failure to meet contractual debt obligations. Quantifying this risk is critical in valuing corporate bonds, credit derivatives, and managing portfolios exposed to default risk. Two main types of credit risk models are widely used: structural models, which use economic reasoning based on a firm's balance sheet, and reduced-form models, which take a statistical approach based on observable credit market data. Understanding the mechanics, assumptions, and practical use of both methods is essential for effective credit analysis.

## STRUCTURAL CREDIT MODELS

Structural models, sometimes called firm-value models, tie a company's default risk to its capital structure and asset value. The classic example is the Merton model, which treats a firm's equity as a call option on the assets, with a strike price equal to the face value of debt.

> **Key Term: structural model**
> A credit risk model in which default is predicted to occur when a company's asset value falls below a threshold (typically the value of its liabilities).

In the structural approach, the probability of default is the probability that the firm's asset value will be less than the default barrier (usually total debt) at the debt's maturity. The model uses option pricing theory to estimate this probability.

> **Key Term: default barrier**
> The value of liabilities or debt that serves as the trigger for default in a structural model, often set at the total face value of outstanding debt.

A structural model provides economic intuition by modeling default as a rational decision: if assets are worth less than the firm's obligations at maturity, equity holders will default and hand over the assets to debt holders.

### **Worked Example 1.1**

**A company has assets currently valued at \$120 million and a single zero-coupon bond with a face value of \$100 million maturing in one year. If the standard deviation of asset returns is 35%, and the risk-free rate is 4%, estimate the probability that the company defaults in one year.**

> **Answer:**
> Model the equity as a call option on assets with a strike price of \$100 million. Using the Black-Scholes formula for a call option, calculate the probability that asset value at maturity falls below \$100 million. The resulting cumulative normal probability gives the model-implied probability of default.

Structural models can generate estimates for default probability, expected loss, credit spreads, and option-like features in corporate debt. However, they rely on the ability to observe or estimate the firm's asset values and volatilities, which are not directly traded.

### **Exam Warning**

> On the exam, ensure you recognize that structural models produce _endogenous_ default probabilities derived from firm balance sheet information, not simply observed from market prices.

## REDUCED-FORM CREDIT MODELS

Reduced-form, or intensity-based, models treat default as a random event driven by observable market variables, with probabilities estimated statistically (often called the hazard rate).

> **Key Term: reduced-form model**
> A credit risk model in which default is assumed to occur randomly, with probability based on a statistically estimated hazard rate using market data.

These models do not rely on a specific economic explanation for default, nor do they require explicit modeling of a firm's asset values. Instead, default intensity (hazard rate) is estimated from historical data, bond prices, or credit default swap (CDS) spreads.

> **Key Term: hazard rate**
> The conditional probability that a default occurs in a given short time interval, assuming that default has not occurred so far.

Reduced-form models allow default probability and recovery rates to change with observable variables (interest rates, economic indicators, market factors). Unlike structural models, defaults in reduced-form frameworks are _exogenous_ and may arrive unexpectedly.

### **Worked Example 1.2**

**Suppose a bond has a flat hazard rate of 2% per year and a recovery rate of 40%. What is the 3-year survival probability for this bond, and what is the expected loss if the notional is \$100?**

> **Answer:**
> Survival over 3 years = (1 – hazard rate)³ = (0.98)³ ≈ 94.1%.  
> Expected loss on notional = [1 – survival probability] × (1 – recovery rate) × notional = (1 – 0.941) × 0.6 × \$100 ≈ \$3.54.

Reduced-form models are especially useful for calibrating to observed market prices and for products (like CDS) where default parameters are implied from pricing rather than firm fundamentals.

## COMPARISON: STRUCTURAL VS. REDUCED-FORM MODELS

Structural models:

- Use capital structure and firm value to explain why default occurs
- Require detailed balance sheet information and assumptions about asset volatility
- Provide economic intuition and link credit risk to business fundamentals

Reduced-form models:

- Do not rely on firm-level fundamentals; estimate default probability from observed prices and statistical methods
- Treat default as a stochastic, sometimes unpredictable event
- Are flexible for changing market conditions and can be quickly calibrated to market data

Both frameworks are used to price risky bonds and credit derivatives, and to estimate credit spreads:

> **Key Term: credit spread**
> The yield difference between a risky bond and a risk-free benchmark, compensating investors for default risk and other risk premiums.

### **Worked Example 1.3**

**A 5-year zero-coupon risky bond is priced to yield 5%, while a risk-free bond of the same maturity yields 3.5%. Calculate the credit spread and discuss what it represents.**

> **Answer:**
> Credit spread = 5% – 3.5% = 1.5%.  
> This spread reflects compensation for expected loss from default (probability × loss severity) and risk premium for bearing credit risk.

## PRACTICAL LIMITATIONS

While structural models offer a rationale for default and allow linkage to firm-specific news, they struggle with complex capital structures and off-balance-sheet liabilities, and their default barriers may be uncertain. Reduced-form models rely on the accuracy and relevance of statistical estimates, and do not explain the root cause of default. Both frameworks simplify a complex reality and should be complemented by qualitative analysis.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Structural models relate default to a firm's asset value and liabilities using option theory
- Reduced-form models estimate default as a random, time-dependent process via hazard rates
- Structural models provide economic visibility into default; reduced-form models focus on statistical calibration to prices
- Both approaches are used to estimate default probability, expected loss, and credit spreads
- Limitations include unobservable inputs (structural model) and limited economic explanation (reduced-form model)

## **Key Terms and Concepts**

- structural model
- default barrier
- reduced-form model
- hazard rate
- credit spread

Credit risk and portfolio strategies - Structural and reduced-form credit models