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IAI Actuarial Core Principles · Economic Modelling

Models of the Term Structure of Interest Rates for CM2

The term structure shows how yields vary with maturity. You start with spot rates, forward rates and discount factors, then learn the theories of the curve and no-arbitrage pricing. Models of how the curve moves, such as Vasicek and CIR, are background only unless the current IAI syllabus lists them. You solve questions by stating assumptions and applying formulas step by step.

What this chapter covers

This chapter is about one question: how do interest rates of different maturities relate to each other? You begin with the yield curve, spot rates, forward rates and discount factors. You then meet the economic theories that explain why the curve slopes up, down or humps. These are expectations, liquidity preference and market segmentation.

The second half is more mathematical. You price bonds using no-arbitrage arguments and a risk-neutral measure. You then see how one-factor short rate models, multifactor models and forward rate models describe the movement of the curve. Treat these models as background. The 2026 CM2 topic list is Rational economic theory, Measures of investment risk, Asset valuations, Liability valuations and Option theory. It has no separate term structure or short rate model topic. Models such as Vasicek and CIR matter only to the extent the current IAI syllabus specifies, so check the syllabus before you spend time on them.

The core ideas link to CM1 and CM2. They build on the theory of interest from CM1. They feed into asset and liability valuation, where values depend on the discount curve. They also support option theory, where the behaviour of interest rates matters. Paper B is a computer-based paper, so practise applying the methods on a computer as well as learning the formulas.

The 2026 syllabus weights asset valuations at 30% and liability valuations at 20%, and both depend on a sound view of discount rates. A student who is weak on yield curves and discounting loses marks in several topics, not one. The ideas suit both parts of the exam. Paper A tests definitions, derivations and interpretation in MCQs and written answers. Paper B is a 1 hour 45 minute computer-based exam, so practise doing the working on a computer. Check the current IAI syllabus to see how far the model material goes. Clear working and stated assumptions earn marks even when a number is slightly off, so steady practice here pays back.

Models of the term structure of interest rates: topics in the order to study them

  1. 1Term Structure and Yield Curve BasicsSpot rates, forward rates and discount factors are the language of every later topic, so master the conversions between them first.
  2. 2Theories of the Term StructureThese are descriptive and need no heavy maths. They explain curve shapes and give you the context for model choices.
  3. 3No-Arbitrage and Risk-Neutral Pricing of BondsThis sets the pricing framework that any model must satisfy. It also supports asset valuation and option theory, so it must come before any specific model.
  4. 4One-Factor Short Rate ModelsStudy these, such as Vasicek and CIR, only to the depth the current IAI syllabus requires. Check first whether they are listed. If they are, apply the no-arbitrage framework and compare their features.
  5. 5Multifactor and Forward Rate ModelsThese extend the one-factor ideas. Treat them as background reading unless the current IAI syllabus lists them.
  6. 6Calibration and Practical Use of Interest Rate ModelsCalibration needs you to know the models first. Study it only as far as the syllabus requires. It helps you practise computer-based work for Paper B.

How to prepare Models of the term structure of interest rates

Build the chapter in layers. Get the definitions right, then the pricing logic, then the models. Keep a one-page sheet for each model and practise on a phone in short sessions.

  1. Practise converting between spot rates, forward rates, par yields and discount factors until it is automatic. Check each answer by converting back.
  2. Write each theory of the term structure in two lines: what it assumes and what curve shape it predicts. Add one weakness for each.
  3. Learn the no-arbitrage idea in words first. A price is consistent if no trade earns a riskless profit. Then link it to discounting expected payoffs under the risk-neutral measure.
  4. If the current syllabus includes short rate models, list for each one the dynamics, whether rates can go negative, whether mean reversion is present, and whether the curve fits the market exactly. Compare models in a small grid.
  5. Work through at least a few written questions that ask you to compare models, if the syllabus covers them, since explaining strengths and weaknesses is a common task.
  6. Do a simulation or fitting exercise on a computer for the parts the syllabus covers. Write down your method, notation, assumptions and result, as the exam expects.
  7. Revise by redoing past questions under time. Review every lost mark and note whether it was a concept, a calculation or a presentation error.

Common mistakes in Models of the term structure of interest rates

  • Mixing up spot rates and forward rates in a calculation.

    Fix: Write the time periods on each rate before using it. Check by rebuilding the spot rate from your forward rates.

  • Using the real-world probability measure when pricing under no-arbitrage.

    Fix: State the measure you are using. Pricing uses the risk-neutral measure, while forecasting uses real-world assumptions.

  • Describing a term structure theory without saying what curve shape it implies.

    Fix: For each theory, write the assumption, the predicted shape and one criticism.

  • Claiming a model is better without a reason tied to its features.

    Fix: Compare models on mean reversion, negative rates, fit to the market curve, and ease of use. Use these four points.

  • Giving a Paper B answer with a result but no method or assumptions.

    Fix: Write the method, the formula in standard notation, the working and the result in words. Mention any parameter assumed.

  • Leaving calibration until the last week.

    Fix: Practise a small fit early. It reinforces model features and prepares you for the computer-based paper.

Last-day revision: Models of the term structure of interest rates

  • Spot rate: yield on a zero-coupon bond from now to time t.
  • Forward rate: rate agreed today for borrowing over a future period, implied by spot rates.
  • Discount factor from spot rates: v(t) = (1 + s_t)^(−t), when rates are annual effective.
  • Expectations theory: long rates reflect the average of expected future short rates.
  • Liquidity preference theory: investors want a premium for longer maturities, so the curve tends to slope up.
  • Market segmentation: different maturities are driven by separate supply and demand.
  • No-arbitrage: a model must not allow a riskless profit, so prices are consistent across bonds.
  • Risk-neutral pricing of a zero-coupon bond: P(t,T) = E_Q[exp(−∫ r_s ds from t to T) | F_t]. The expectation is under the risk-neutral measure Q, and the discount factor depends on the whole short rate path.
  • If your syllabus covers them: Vasicek has mean reversion and normally distributed rates, so negative rates are possible.
  • If your syllabus covers them: CIR has mean reversion and a square-root term that keeps rates non-negative under its usual parameter condition.
  • Calibration means choosing parameters so model prices match observed market prices.
  • Always state your assumptions and notation before calculating.

Models of the term structure of interest rates practice questions

Models of the term structure of interest rates in other exams

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

Models of the term structure of interest rates: frequently asked questions

Which subject covers the term structure of interest rates?

The basics build on the theory of interest rates in CM1. In CM2, yield curves and discounting matter for asset valuations, liability valuations and option theory. The 2026 CM2 topic list has no separate term structure topic, so check the current IAI syllabus for the exact scope.

Do I need to memorise the Vasicek and CIR formulas?

First check whether the current IAI syllabus lists them. If it does, know the model dynamics and the features each one gives, such as mean reversion and the treatment of negative rates. For derivations or bond price formulas, follow the notation in the IAI study material and practise using them.

How is this chapter tested in Paper B?

Paper B is a 1 hour 45 minute computer-based exam. Tasks may involve working with rates or prices on a computer. Show your method, assumptions and result clearly, and check the syllabus and past papers for what is actually asked.

Where should I start if my maths is weak?

Start with yield curve basics and the theories of the term structure, since they need little advanced maths. Then move to no-arbitrage ideas in words before the model equations.