IAI Actuarial Core Principles · Economic Modelling

Simple Models for Credit Risk: Merton, Hazard Rates and Spreads

Simple models for credit risk estimate the chance a borrower defaults and the loss if it does. You study structural models (Merton), reduced form models (hazard rates), Markov rating transitions and credit spreads. Solve questions by stating assumptions, writing the model formula, computing step by step, and interpreting the result.

What this chapter covers

This chapter covers how to model the risk that a borrower fails to pay. It starts with the basic language: default, recovery, credit ratings and expected loss. Then it gives you three families of models. Structural models link default to the value of a firm's assets. Reduced form models treat default as a random event with an intensity, called the hazard rate. Markov rating models describe how ratings move over time using a transition matrix.

The chapter ends by tying these models to market prices. A risky bond pays less than a risk-free bond, so its price is lower and its yield is higher. The gap is the credit spread. You learn to split the spread into expected loss and a premium for risk, and to price a risky bond from default probabilities and recovery.

The chapter connects to the rest of CM2. It uses the discounting and risk-neutral ideas from asset valuation, and the Merton model uses the same option logic as the option theory chapters. The Markov and hazard rate ideas also link back to survival and multiple state models from your earlier subjects. Expect both MCQs and written questions, and some computation that you can practise in Excel or R for Paper B.

Credit risk questions are very testable because they mix a clear model with short calculations. A Merton question can be solved with an option formula, a hazard rate question with an exponential survival probability, and a transition matrix question with matrix multiplication. These are marks you can secure with practice. The chapter also feeds into asset valuation and option theory, so time spent here repays you elsewhere in CM2. Remember that CM2 has a Paper A and a computer-based Paper B, and you must meet the pass rules for both, so being able to code or build these calculations in a spreadsheet is useful.

Simple models for credit risk: topics in the order to study them

1Credit Risk Basics and Credit RatingsIt defines default, recovery, expected loss and ratings, which every later model uses.
2Structural Models of Credit Risk (Merton Model)It builds on option ideas you already know and shows why firm value drives default.
3Reduced Form Models and Hazard RatesIt gives the intensity-based view of default, which is simpler to calculate and leads straight to pricing.
4Markov Rating Transition ModelsIt extends default to rating changes over many periods and uses matrix methods.
5Credit Spreads and Pricing Credit Risky BondsIt brings all models together to price bonds and explain spreads, so it comes last.

How to prepare Simple models for credit risk

Treat this chapter as one idea seen through four lenses. Learn the definitions first, then each model, then use them together on pricing.

Write a one-page glossary of default probability, recovery rate, loss given default, exposure and expected loss. Learn how they combine.
Learn the Merton set-up: assets, debt as a zero-coupon liability, default at maturity if assets fall below debt. Be able to state each assumption and what the equity and debt payoffs look like.
Practise hazard rate questions. Link the hazard rate to survival probability by S(t) = exp(−∫ λ(s) ds) and, for a constant λ, S(t) = exp(−λt).
Work through transition matrices by hand. Multiply matrices for multi-period probabilities and check that each row sums to 1.
Practise pricing a risky bond: discount expected cash flows, then compute the yield and the spread. Compare with the risk-free bond.
Do past-paper style questions with timing. For each, write assumptions, formula, working and a one-line interpretation.
Rebuild two or three calculations in Excel or R so you are ready for the computer-based paper.

Common mistakes in Simple models for credit risk

Mixing up real-world and risk-neutral default probabilities.
Fix: Note which measure each question uses. Use risk-neutral probabilities for pricing and real-world ones for expected loss estimates.
Treating the spread as equal to expected loss.
Fix: Say it is an approximation and mention that spreads also reflect risk premium and liquidity.
Forgetting the assumptions of the Merton model.
Fix: Write the assumptions first: single zero-coupon debt, default only at maturity, lognormal asset value and constant interest rate.
Errors in transition matrix multiplication or using the wrong row.
Fix: Read the starting rating along the row, multiply row by column, and check each result row sums to 1.
Confusing the hazard rate with the probability of default.
Fix: Remember the hazard rate is an instantaneous intensity. Convert it to a probability using 1 − exp(−λt) for a constant rate.
Giving a number without interpretation in written answers.
Fix: Add one sentence saying what the result means for the lender or investor.

Last-day revision: Simple models for credit risk

Expected loss = probability of default × loss given default × exposure.
Loss given default = 1 − recovery rate, when recovery is a fraction of exposure.
Structural models link default to firm asset value falling below the debt level.
In the Merton model, equity behaves like a call option on the firm's assets.
In the Merton model, risky debt equals risk-free debt minus a put option on the assets.
Reduced form models treat default time as random, driven by a hazard rate.
For a constant hazard rate λ, survival to time t is exp(−λt).
In a transition matrix, each row sums to 1 and default is usually an absorbing state.
Multi-period transition probabilities come from powers of the one-period matrix, under the time-homogeneous Markov assumption.
Credit spread = risky yield − risk-free yield.
Spread is not only expected loss; it also includes compensation for risk and other factors.
State your assumptions before computing, and interpret the answer at the end.

Simple models for credit risk practice questions

Simple models for credit risk in other exams

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

Simple models for credit risk: frequently asked questions

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

Structural models explain default from the firm's balance sheet, usually asset value against debt. Reduced form models do not explain why default happens; they model it as a random event with an intensity. Structural models give insight, while reduced form models are often easier to calibrate to market prices.

How much of the Merton model do I need to know?

You should know the set-up, assumptions, payoffs to equity and debt, and how default probability follows from the model. Be ready to interpret results and to compute with given values. Check the latest IAI syllabus for the exact depth.

Is this chapter calculation-heavy?

Yes, in parts. Hazard rates, transition matrices and bond pricing need steady arithmetic. Practise by hand for Paper A and in a spreadsheet or R for the computer-based paper.

How should I revise this chapter in the last week?

Go through the formula list, redo one worked question for each model, and write short explanations of the assumptions. Then attempt timed questions that mix models, such as pricing a bond using a hazard rate.