CFA Level I Exam · Credit Risk

Structural vs Reduced-Form Credit Models Explained

Q: What is the main difference between structural and reduced-form credit models?

Structural models link default to the firm's asset value falling below its debt, so they use balance sheet logic and option theory. Reduced-form models treat default as a random event with an intensity estimated from market data. Structural models explain default; reduced-form models fit market prices.

Q: Why is equity a call option on the firm's assets?

Equity holders receive whatever is left after debt is paid, which is max(A − K, 0) at maturity. Their loss is limited to zero because of limited liability. That matches the payoff of a call with strike equal to the debt face value.

Q: What does CVA measure?

CVA is the present value of the expected loss from a counterparty defaulting. It combines expected exposure, probability of default, loss given default and discounting. Subtracting it from the risk-free value gives the value that reflects credit risk.

Updated 7 October 2026 · Fact-checked

Structural models treat equity as a call option on the firm's assets, with the debt face value as the strike, so default happens when assets fall below debt. Reduced-form models skip the balance sheet and model default as a random event with a default intensity. Both estimate default probability, loss and credit spreads.

Understand Structural and Reduced-Form Credit Models

A credit model tries to answer one question: how likely is default, and how much do you lose if it happens? The two model families answer it in different ways.

Structural models start from the firm's balance sheet. Shareholders have limited liability. At debt maturity, if asset value A is above the debt face value K, equity holders pay the debt and keep A − K. If A is below K, they walk away and bondholders take the assets. So equity payoff = max(A − K, 0). That is a call option on the firm's assets with strike K. Bondholders hold the equivalent of a risk-free bond and are short a put option on the assets with the same strike. Default is likely when asset value falls below the debt level. Higher asset volatility makes both options worth more. That raises the value of the put that bondholders are short, so risky debt is worth less.

Reduced-form models do not model why the firm defaults. Default is treated as an unexpected event that arrives randomly, with a default intensity (hazard rate) that can change with the economy and firm data. The inputs are observable, such as bond spreads, CDS spreads and macro variables. Because they use market data, they fit market prices well and are widely used for pricing and risk.

Credit valuation adjustment (CVA) is the present value of expected credit loss on a derivative or exposure. It uses expected exposure, probability of default and loss given default. The value of a risky position is the risk-free value minus CVA.

The term structure of credit spreads shows credit spread against maturity. For high-quality issuers it usually slopes upward, because default risk is low now and could rise later. For weak or distressed issuers it can be downward sloping, because surviving the near term lowers later risk.

Key formulas to remember

Equity as a call option: Equity at maturity = max(A − K, 0); A = asset value, K = face value of debt (strike). Equity holders keep the upside above debt.
Debt as risk-free bond minus a put: Debt at maturity = min(A, K) = K − max(K − A, 0); Today: risky debt value = PV of K − value of put on assets. Equity + debt = assets.
Approximate credit spread: Credit spread ≈ PD × LGD; Annual probability of default times loss given default. A rough link, not exact.
Expected loss: Expected loss = PD × LGD × exposure; LGD = 1 − recovery rate.
Credit valuation adjustment: CVA = Σ [expected exposure_t × PD_t × LGD × discount factor_t]; PD_t is the marginal (period-specific) probability of default in period t, not the cumulative probability of default up to t. Sum across all periods.
Risky value: Value with credit risk = risk-free value − CVA; CVA is always a reduction in value for the party holding the exposure.
Constant hazard rate: Survival to t = e^(−λt); PD over t = 1 − e^(−λt); λ is the default intensity. Approximately λ ≈ spread ÷ LGD.

How to solve Structural and Reduced-Form Credit Models questions

Use this method for any question on credit models, CVA or credit spreads.

1Identify the model type from the wording: assets, strike or option language means structural; intensity, hazard rate or observable market inputs means reduced-form.
2For structural questions, set K as the debt face value and compare A with K at maturity.
3Compute the payoffs: equity = max(A − K, 0); debt = min(A, K); check that they add up to A.
4For reduced-form or spread questions, use spread ≈ PD × LGD, and remember LGD = 1 − recovery rate.
5For CVA, build a table by period: expected exposure × PD × LGD × discount factor, then add the rows.
6For term structure questions, match the issuer quality to the slope: high quality upward, weak or distressed can be downward sloping.
7Check the sign and units, then eliminate the two wrong options.

Quickest way: Keyword and payoff shortcut

When to use it: Conceptual comparison questions and short payoff questions where you have about 90 seconds.

Spot the keyword: option on assets, strike = debt means structural. Default intensity or observable inputs means reduced-form.
Remember the contrast: structural explains why default occurs; reduced-form treats default as a surprise.
For payoffs, picture A against K: above K equity gets the excess; below K bondholders get all of A.
For CVA, multiply row by row and add. Do not discount twice.
Cross out any option that reverses the roles, such as equity as a put or debt as a long put.

Common mistakes in Structural and Reduced-Form Credit Models

Calling equity a put option on the firm's assets.
Students link default with puts and forget who holds the option.
Fix: Equity holders own the upside above K, so equity is a call. Bondholders are short the put.
Saying reduced-form models need the firm's asset value and its volatility.
Mixing the two model types.
Fix: Reduced-form models use observable inputs and a default intensity. Asset value and volatility belong to structural models.
Forgetting that LGD is 1 minus the recovery rate.
Spread and loss questions give recovery, not LGD.
Fix: Convert first: a 40% recovery rate gives LGD of 60%.
Leaving out the discount factor or the PD in a CVA sum.
CVA has four inputs and one is easily dropped.
Fix: Write the four columns in a table before multiplying.
Assuming every credit spread curve slopes upward.
Applying the usual yield curve shape to all issuers.
Fix: Upward for high-quality issuers; weak or distressed issuers with high near-term risk can show a downward-sloping curve.
Treating spread = PD × LGD as exact.
Exam notes show it as a simple formula.
Fix: Use it as an approximation. Spreads also reflect liquidity and risk premiums.

Worked examples

Example 1

A firm has zero-coupon debt with a face value of €100 million due in one year. At maturity, the firm's assets are worth €85 million. In a structural model, how much do bondholders receive? A. €0 B. €15 million C. €85 million

Show the solution

Debt face value K = €100 million; asset value A = €85 million.
Since A < K, equity holders do not exercise the call: equity = max(85 − 100, 0) = €0.
Bondholders receive min(A, K) = min(85, 100) = €85 million.
Check: equity 0 + debt 85 = assets 85. The loss to bondholders is €15 million, but they receive €85 million.

Answer: C. €85 million

Example 2

A bank has a two-year derivative exposure to a counterparty. Year 1: expected exposure $10 million, PD 2%, discount factor 0.97. Year 2: expected exposure $8 million, PD 3%, discount factor 0.94. LGD is 60%. What is the CVA? A. $0.12 million B. $0.25 million C. $0.40 million

Show the solution

Year 1: 10 × 0.02 × 0.60 × 0.97 = 0.1164 million.
Year 2: 8 × 0.03 × 0.60 × 0.94 = 0.13536 million.
CVA = 0.1164 + 0.13536 = 0.25176 million, about $0.25 million.
Option A is roughly year 1 alone. Option C overstates the loss.

Answer: B. $0.25 million

Exam tips

Read the first noun: option or asset language signals structural; intensity or hazard language signals reduced-form.
Know who holds which option: equity is a long call, bondholders are short a put on the assets.
Write LGD = 1 − recovery before any spread or CVA calculation.
For spread term structure items, link the slope to credit quality and match it to the option.
CVA questions are plug-in arithmetic; use a table and keep four decimals until the last step.

Practice questions from Credit Risk

Structural and Reduced-Form Credit Models in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Structural and Reduced-Form Credit Models: frequently asked questions

What is the main difference between structural and reduced-form credit models?

Structural models link default to the firm's asset value falling below its debt, so they use balance sheet logic and option theory. Reduced-form models treat default as a random event with an intensity estimated from market data. Structural models explain default; reduced-form models fit market prices.

Why is equity a call option on the firm's assets?

Equity holders receive whatever is left after debt is paid, which is max(A − K, 0) at maturity. Their loss is limited to zero because of limited liability. That matches the payoff of a call with strike equal to the debt face value.

What does CVA measure?

CVA is the present value of the expected loss from a counterparty defaulting. It combines expected exposure, probability of default, loss given default and discounting. Subtracting it from the risk-free value gives the value that reflects credit risk.

What shape is the term structure of credit spreads?

For high-quality issuers it usually slopes upward, because default risk is low in the near term and could rise later. For weak or distressed issuers it can be downward sloping, because surviving the near term lowers later risk. Exam questions test this link to credit quality.