Actuarial Mathematics for Modelling · Term structure of interest rates

Theories of the Term Structure of Interest Rates

Updated 11 October 2026 · Fact-checked

Term structure theories explain why yields differ by term. Expectations theory says long yields reflect expected future short rates. Liquidity preference adds a premium for longer terms. Market segmentation says each maturity has its own supply and demand. To answer, name the theory, state its assumption, and link it to the curve's shape.

Understand Theories of the Term Structure

The term structure of interest rates describes how yields vary with the term to maturity. Plotted for bonds of the same credit quality, it gives the yield curve. The curve is usually described by its shape: normal (upward sloping), inverted (downward sloping), flat, or humped.

Three theories explain the shape. The pure expectations theory says the long-term yield is set by what the market expects short-term rates to be over that period. A rising curve means the market expects short rates to rise. A falling curve means it expects them to fall. Forward rates are then unbiased predictors of future spot rates.

The liquidity preference theory says investors prefer short-dated bonds. Long bonds have more price risk, so investors need extra yield to hold them. Borrowers, in contrast, like to borrow long. The extra yield is the liquidity premium, and it rises with term. This is why the curve is often upward sloping even when no rise in rates is expected. Under this theory, forward rates are higher than expected future spot rates.

The market segmentation theory says the market is split into separate maturity segments. Each group has its own preferred term, for example life insurers and pension funds at long terms, banks at short terms. Yields at each term are set by supply and demand in that segment alone. So the curve's shape reflects relative demand across segments, not expected future rates. A related view, the preferred habitat theory, allows investors to move out of their segment if paid enough.

None of the theories is fully right. In practice the curve reflects a mix: expected rates, risk premiums and institutional demand. Exams ask you to describe each theory, say what shape it implies and judge its limits.

Key rules to remember

Forward rate from spot rates (annual compounding): (1 + s_t)^t = (1 + s_(t-1))^(t-1) × (1 + f_(t-1,t)); Here s_t is the t-year spot rate and f_(t-1,t) is the one-year forward rate from t-1 to t. Use it to test what the curve implies about expected rates.
Pure expectations theory: f_(t-1,t) = E[ future one-year spot rate at time t-1 ]; Forward rates are unbiased estimates of future spot rates. No risk premium.
Liquidity preference theory: f_(t-1,t) = E[ future spot rate ] + liquidity premium, with premium > 0 and increasing with term; Forward rates overstate expected future spot rates. This gives an upward bias to the curve.
Curve shape under expectations theory: Rising curve ⇒ expected rise in short rates; falling curve ⇒ expected fall; Holds only for the pure theory. Under liquidity preference, a rising curve does not by itself prove rates are expected to rise.

How to solve Theories of the Term Structure questions

Use this method for both descriptive and numerical questions on term structure theories.

1Read what the question asks: describe a theory, explain a shape, or interpret given yields.
2If yields are given, state the shape: upward sloping, inverted, flat or humped.
3If needed, calculate implied forward rates from the spot rates using the compounding relationship.
4Name the theory and state its key assumption in one sentence.
5Explain what that theory says about the given shape, and what it says about forward rates versus expected spot rates.
6Compare with at least one other theory if the question says compare, contrast or discuss.
7Add a limitation or a practical point, such as that real curves mix several effects.
8Finish with a clear conclusion that answers the exact question.

Quickest way: One-line test for each theory

When to use it: Use in multiple-choice questions and when you need a fast outline for a written answer.

Expectations: look for the phrase 'expected future short rates' and no premium.
Liquidity preference: look for 'premium for longer terms' and a bias towards an upward slope.
Market segmentation: look for 'separate markets' and 'supply and demand at each term'.
For numbers, compute the implied forward rate first, then compare it to the expected rate given.
If forward rate exceeds the expected rate, the gap is the liquidity premium.

Common mistakes in Theories of the Term Structure

Saying that under every theory an upward curve means rates will rise.
Students learn the expectations result and apply it to all theories.
Fix: Restrict that statement to pure expectations theory. Under liquidity preference, part of the slope is a premium.
Treating the liquidity premium as decreasing with term.
Confusing liquidity with marketability.
Fix: Longer bonds carry more price risk, so the premium increases with term.
Claiming market segmentation explains the curve using expected future rates.
Mixing it up with expectations theory.
Fix: In segmentation, yields come from supply and demand within each maturity band, independent of expectations.
Mixing up spot rates and forward rates in calculations.
Both are quoted as annual percentages.
Fix: Label each rate with its term. Use (1 + s_t)^t on the left and build the forward from the ratio of accumulations.
Describing only the shape and not giving a reason.
Students treat the question as a definition.
Fix: Always link the shape to a theory's mechanism, then comment on its limits.

Worked examples

Example 1

The one-year spot rate is 6% and the two-year spot rate is 7% per annum effective. (a) Calculate the implied one-year forward rate from time 1 to 2. (b) Under pure expectations theory, what does this say about the expected one-year rate in a year's time? (c) Under liquidity preference theory with a liquidity premium of 0.5%, what is the expected one-year rate?

Show the solution

(a) Use (1 + s_2)^2 = (1 + s_1)(1 + f).
1.07^2 = 1.1449.
f = 1.1449 ÷ 1.06 − 1 = 1.080094 − 1 = 0.080094, about 8.01%.
(b) Under pure expectations, forward rate = expected future spot rate, so the expected one-year rate in a year is about 8.01%.
(c) Under liquidity preference, forward = expected spot + premium.
Expected spot = 8.01% − 0.5% = 7.51% (approximately).

Answer: (a) about 8.01%; (b) expected one-year rate about 8.01%; (c) expected one-year rate about 7.51%.

Example 2

The yield curve is steeply upward sloping. Explain how each of the three theories could account for this.

Show the solution

Expectations theory: the market expects short-term rates to rise, so long yields, which average expected short rates, are above current short yields.
Liquidity preference theory: investors demand a premium for holding longer bonds because of greater price risk. Yields rise with term even if short rates are expected to stay flat.
Market segmentation theory: demand is strong at short terms (for example from banks) pushing short yields down, or supply of long bonds is heavy relative to long-term demand, pushing long yields up.
Conclusion: the slope may come from expected rate rises, a risk premium, or segment imbalances. In practice, a mix is likely, so the slope alone does not prove expected rises.

Answer: Upward slope is explained by expected rate rises (expectations), a rising liquidity premium (liquidity preference), or relative supply and demand across maturities (segmentation). Real curves usually reflect a combination.

Exam tips

For 'compare' questions, set out each theory's driver, forward rate meaning and implied shape in parallel.
In calculations, show the compounding equation before substituting numbers so you earn method marks.
Always state assumptions, such as annual effective rates and risk-free bonds of equal credit quality.
Give one limitation per theory. For example, pure expectations ignores risk, and segmentation ignores arbitrage across terms.
In MCQs, watch for options that attach a premium to expectations theory or expected rates to segmentation theory.

Practice questions from Term structure of interest rates

Theories of the Term Structure in other exams

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

Theories of the Term Structure: frequently asked questions

Why is the yield curve usually upward sloping?

Liquidity preference theory says investors want a premium for holding longer bonds, which have more price risk. Expectations of rising rates can add to the slope. Heavy demand for short-term bonds can also contribute.

What is the difference between expectations theory and liquidity preference theory?

Expectations theory says forward rates equal expected future spot rates. Liquidity preference adds a positive premium that grows with term, so forward rates exceed expected spot rates.

What does market segmentation theory say about the yield curve?

It says investors and borrowers stick to their preferred maturities. Yields at each term are set by supply and demand in that segment. The curve's shape therefore reflects relative demand across segments.

What does an inverted yield curve mean?

Short-term yields are higher than long-term yields. Under pure expectations theory it signals that short rates are expected to fall. Under other theories, it may also reflect strong demand for long bonds.