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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Term Structure of Interest Rates: Spot Rates, Forward Rates and Pricing

The term structure of interest rates describes how yields vary with the term of a cash flow. You solve questions by building spot rates from bond prices, deriving forward rates from spot rates using no-arbitrage, then pricing cash flows with them. Duration and convexity measure sensitivity to yield changes and support immunisation.

What this chapter covers

This chapter shows that one interest rate is not enough. A cash flow due in 1 year and one due in 10 years can be discounted at different rates. You learn three linked ways to describe the same curve: spot rates (yield on a zero-coupon bond), forward rates (rates agreed now for a future period) and par yields or yield to maturity (a single rate for a coupon bond).

The link between them is no-arbitrage. If spot rates are y_t, then (1 + y_t)^t = (1 + y_s)^s × (1 + f_{s,t-s})^(t-s) for s < t, where f_{s,t-s} is the forward rate for the period from s to t, in annual effective terms. Almost every numerical question uses this one idea. You then add economic theories that explain why the curve slopes up, down or humps, and finally the risk tools: duration, convexity and immunisation.

This chapter sits between the basic interest-rate and equation-of-value work and the later pricing and reserving work in CM1. Theory of interest rates carries a large share of the 2026 syllabus (25%), and the curve ideas return when you value annuities, bonds and liabilities with time-varying rates. The same ideas also feed into CM2 on asset and liability valuation, so the effort pays off twice.

The chapter is calculation-heavy but follows a small set of rules, so it rewards practice more than memory. It gives short MCQ marks (convert a spot rate to a forward rate, find a bond price) and longer written parts (derive a curve from bond prices, explain a theory, show Redington's conditions). Duration and immunisation also appear in Paper B style computer work, where you build a curve or test a portfolio in R or Excel. A student who is fluent here gains speed and accuracy for the rest of the paper.

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

  1. 1Spot Rates and Zero-Coupon Bond PricingSpot rates are the base of the curve; every other rate in the chapter is defined from them.
  2. 2Forward Rates and Implied Forward RatesForward rates come directly from spot rates by no-arbitrage, so learn them once spot rates feel automatic.
  3. 3Par Yields and Yield to MaturityThese describe coupon bonds, and you need spot and forward rates to see why a coupon bond's yield is a blend of them.
  4. 4Theories of the Term StructureOnce you know the shapes a curve can take, you can explain them with expectations, liquidity preference and market segmentation.
  5. 5Duration, Convexity and ImmunisationThis needs bond pricing and yield ideas first, and it applies them to managing interest-rate risk.

How to prepare Term structure of interest rates

Treat this chapter as one curve seen from several sides. Practise moving between spot rates, forward rates, discount factors and prices until each conversion takes under a minute.

  1. Write the notation page first: y_t for the t-year spot rate, f_{t,r} for the forward rate starting at t for r years, v(t) for the discount factor. State the conventions (annual effective unless told otherwise) at the start of every answer.
  2. Drill the no-arbitrage identity in both directions. Given spot rates, find forwards. Given forwards, rebuild spot rates. Always check by recomputing the discount factor.
  3. Practise bootstrapping: use the prices of coupon bonds with successive maturities to solve for spot rates one term at a time, starting from the shortest.
  4. Compute par yields and yield to maturity for the same bond and see why they differ from the spot rates. Use a table of cash flows and discount factors so the working is clear.
  5. Learn each term structure theory in three lines: the main idea, the curve shape it predicts, and one weakness. Practise writing a short comparison, since written questions often ask for it.
  6. For duration and convexity, practise the formulas, then Redington's three conditions for immunisation: equal present values of assets and liabilities, equal discounted mean terms, and asset convexity (spread of cash flows) greater than liability convexity. Finish with timed mixed questions and one computer-based exercise.

Common mistakes in Term structure of interest rates

  • Mixing up spot rates, forward rates and yields to maturity in the same calculation.

    Fix: Label every rate with its type and term before you calculate. Discount each cash flow with the spot rate for its own term, not with the bond's yield.

  • Forgetting the exponents in the forward rate formula, or using the wrong number of years.

    Fix: Rebuild it each time from the idea that investing to time t equals investing to time s and then rolling forward. Check that the powers add up to the total term.

  • Treating a theory of the term structure as a proven fact.

    Fix: Describe each theory as an explanation, state the curve it implies, and mention at least one weakness, such as the lack of a risk premium in pure expectations theory.

  • Bootstrapping in the wrong order and using unknown longer spot rates.

    Fix: Always begin with the shortest maturity. Each new bond should add exactly one unknown spot rate.

  • Stating Redington's conditions incompletely, for example leaving out the convexity condition.

    Fix: Write all three conditions every time. Explain that the convexity condition makes the surplus rise for small yield changes in either direction.

  • Skipping the assumptions on compounding and cash flow timing.

    Fix: Open each written answer with one line stating the compounding basis and timing. This protects method marks even if you slip numerically.

Last-day revision: Term structure of interest rates

  • Spot rate y_t is the yield on a zero-coupon bond paying 1 at time t: price = (1 + y_t)^(-t).
  • Discount factor v(t) = (1 + y_t)^(-t).
  • Forward rate from s to t: (1 + f_{s,t-s})^(t-s) = (1 + y_t)^t ÷ (1 + y_s)^s.
  • For a one-year forward rate: 1 + f_{t,1} = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t.
  • Bootstrapping: solve spot rates in order of maturity, using the shorter spot rates already found.
  • Yield to maturity is the single rate that makes the bond's price equal to the present value of its cash flows.
  • A par yield is the coupon rate at which a bond prices at par.
  • On a rising curve, the yield to maturity of a coupon bond is below the spot rate for its final maturity.
  • Expectations theory: forward rates equal expected future spot rates; liquidity preference adds a premium for longer terms; market segmentation treats maturities as separate markets.
  • Macaulay duration = Σ t × PV(cash flow at t) ÷ Σ PV(cash flow); modified duration = Macaulay duration ÷ (1 + i).
  • Price change ≈ −(modified duration) × (change in yield) × price, and convexity improves the estimate for larger yield moves.
  • Redington immunisation: PV assets = PV liabilities, equal duration, and asset convexity greater than liability convexity.

Term structure of interest rates practice questions

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.

Term structure of interest rates: frequently asked questions

How do I get a forward rate from spot rates?

Equate two ways of investing to time t. The accumulated value at the t-year spot rate must equal the one at the s-year spot rate, followed by the forward rate from s to t. Rearrange to get (1 + f)^(t-s) = (1 + y_t)^t ÷ (1 + y_s)^s.

What is the difference between spot rate and yield to maturity?

A spot rate applies to a single cash flow at one date, as on a zero-coupon bond. Yield to maturity is one rate that discounts all of a coupon bond's cash flows to its price. Because the coupon bond mixes several spot rates, its yield is a kind of average.

Do I need to memorise all the theories of the term structure?

You need the main idea of each, the curve shape it suggests and one criticism. Written questions often ask you to compare or comment, so short structured answers earn the marks.

Is this chapter tested in the computer-based paper?

It can be, because building a curve from bond prices and computing duration and convexity are well suited to R or Excel. Practise setting out cash flows and discount factors in a clear layout so you can check your work.