Business Economics · Role of money and interest rates in the economy

Term Structure of Interest Rates and the Yield Curve

Updated 11 October 2026 · Fact-checked

The term structure of interest rates shows how yields differ by maturity for similar bonds. The yield curve plots those yields against time to maturity. To answer exam questions, name the shape (normal, inverted, flat), then explain it using expectations, liquidity preference or market segmentation theory, and state what it implies for future rates.

Understand Term Structure and Yield Curve

A yield is the annual return you earn if you buy a bond and hold it to maturity. A 1-year government bond and a 10-year government bond rarely give the same yield. The pattern of yields across maturities is the term structure of interest rates.

The yield curve is a graph of this pattern. Maturity is on the horizontal axis and yield is on the vertical axis. To compare like with like, the bonds should have the same credit risk, usually government securities. Differences in yield for the same maturity come from risk, such as credit risk and liquidity risk. Differences across maturity come from the term structure.

There are three common shapes:

Normal (upward sloping): long-term yields are higher than short-term yields.
Inverted (downward sloping): short-term yields are higher than long-term yields. Markets often read this as a sign that rates are expected to fall, for example when a slowdown is expected.
Flat: yields are similar at all maturities. This often appears when the curve is moving between normal and inverted.

Three theories explain the shape. The pure expectations theory says a long-term yield is an average of the current short rate and expected future short rates. An upward curve means short rates are expected to rise. The liquidity preference theory says investors prefer short-term assets because they are less exposed to price changes and can be turned into cash sooner. So they demand a term premium to hold long bonds. This makes the curve slope up more than expectations alone would suggest. The market segmentation theory says investors and borrowers have preferred maturities, such as pension funds and life insurers for long bonds. Each maturity then has its own supply and demand, and yields are set separately. A related idea, the preferred habitat theory, allows investors to move away from their preferred maturity if the yield compensates them.

An important link for actuaries: a yield is not the same as a forward rate. The yield curve tells you the return from today to a given date. Forward rates tell you the rate implied for a future period. Under pure expectations, forward rates equal expected future spot rates. Under liquidity preference, forward rates are higher than expected future spot rates by the term premium.

Key rules to remember

Two-year yield from a forward rate (annual compounding): (1 + y₂)² = (1 + y₁) × (1 + f₁,₂); y₁ and y₂ are the 1-year and 2-year spot yields. f₁,₂ is the forward rate for year 2, agreed today.
Forward rate from spot yields: f₁,₂ = (1 + y₂)² ÷ (1 + y₁) − 1; Rearranged form of the rule above. Use decimals in the working.
General n-year spot yield: (1 + yₙ)ⁿ = (1 + y₁)(1 + f₁,₂)(1 + f₂,₃)…(1 + fₙ₋₁,ₙ); Holds by no-arbitrage. Gives the long yield as a geometric average of forward rates.
Pure expectations theory: f₁,₂ = E[ y₁ in one year ]; The forward rate equals the expected future short rate. The long yield is an average of expected short rates.
Liquidity preference theory: f₁,₂ = E[ y₁ in one year ] + L, where L > 0; L is the liquidity or term premium. Forward rates overstate expected future short rates.
Shape rule: Normal: yₙ rises with n. Inverted: yₙ falls with n. Flat: yₙ is about constant.; Describe the shape using the yields given, not just the label.

How to solve Term Structure and Yield Curve questions

Use this method for both calculation and discussion questions on the term structure.

1Read what is asked: describe a shape, explain it with a theory, or calculate a forward rate or yield.
2List the data given: maturities, yields, and whether they are spot yields or forward rates. Note the compounding basis.
3Identify the shape by comparing short and long yields. Say clearly if it is normal, inverted or flat, and quote the numbers.
4For a calculation, use the no-arbitrage link (1 + y₂)² = (1 + y₁)(1 + f₁,₂). Work in decimals and keep at least four decimal places until the end.
5For an explanation, choose the theory asked for. State its main claim, then apply it to the given shape.
6Add the contrast if relevant. For example, pure expectations needs no term premium, while liquidity preference adds a positive one.
7State the implication: what the curve suggests about expected short rates, risk or the economy, and any limits of the theory.
8Check the answer: units in percent, sensible size, and the shape matches your conclusion.

Quickest way: Shape, theory, implication

When to use it: Use this for MCQs and short written parts when time is tight.

Compare the shortest and longest yield. Higher long means normal, lower long means inverted, similar means flat.
For a forward rate, use f = (1 + y₂)² ÷ (1 + y₁) − 1. If y₂ > y₁, f is above y₂.
Link theory to words: expectations means expected future short rates, liquidity preference means a term premium, segmentation means separate markets by maturity.
If asked what a pure expectations theory implies for an inverted curve, answer that short rates are expected to fall.
Eliminate options that say a theory ignores risk or uses the wrong direction.

Common mistakes in Term Structure and Yield Curve

Confusing the yield curve with the forward rate curve.
Both are plotted against time and both come from bond prices.
Fix: A yield covers the whole period from today to maturity. A forward rate covers only one future period. Forward rates are higher than yields when the yield curve is upward sloping.
Saying liquidity preference theory means long rates always exceed short rates.
Students remember that a term premium is positive and stop there.
Fix: The premium pushes the curve up relative to expectations. If short rates are expected to fall sharply, the curve can still be flat or inverted.
Stating that expectations theory includes a risk premium.
The theories are mixed up when revising.
Fix: Pure expectations has no premium. Liquidity preference and preferred habitat add one.
Calculating a forward rate by simple subtraction, such as 6% − 5%.
Compounding is ignored.
Fix: Use f₁,₂ = (1 + y₂)² ÷ (1 + y₁) − 1. Subtraction is only a rough guide.
Treating an inverted curve as a certain predictor of recession.
News headlines present it as a rule.
Fix: Say it is often followed by slowdowns but is not a guarantee. It reflects expectations and may also reflect policy or demand for long bonds.
Comparing yields on bonds with different credit risk and calling the gap a term effect.
The condition of equal risk is overlooked.
Fix: Separate maturity effects from credit and liquidity spreads. Compare government bonds of different maturities for the term structure.

Worked examples

Example 1

The 1-year spot yield is 5.0% a year and the 2-year spot yield is 6.0% a year, both annual effective. (a) Describe the shape of the curve. (b) Calculate the one-year forward rate for year 2. (c) Under pure expectations, what does this imply?

Show the solution

(a) The 2-year yield (6.0%) is higher than the 1-year yield (5.0%). The curve slopes upward, so it is normal.
(b) Use (1 + y₂)² = (1 + y₁)(1 + f₁,₂).
(1.06)² = 1.1236.
1.1236 ÷ 1.05 = 1.070095.
f₁,₂ = 1.070095 − 1 = 0.070095, or about 7.01%.
(c) Under pure expectations, the forward rate equals the expected 1-year rate in one year. So the market expects the 1-year rate to rise from 5.0% to about 7.01%.

Answer: (a) Normal, upward sloping. (b) f₁,₂ ≈ 7.01%. (c) Short rates are expected to rise to about 7.01% next year.

Example 2

Explain how the expectations theory and the liquidity preference theory differ in explaining an upward sloping yield curve. Say what each implies about the 1-year rate expected in one year if the one-year forward rate for year 2 is 7.0%.

Show the solution

Expectations theory: the long yield is an average of expected future short rates. An upward slope means the market expects short rates to rise.
Liquidity preference theory: investors prefer short assets and require a term premium to hold long bonds. Part of the upward slope is compensation for this, not expected rises in short rates.
Apply to the forward rate of 7.0%.
Under pure expectations, f₁,₂ = E[ y₁ in one year ], so the expected 1-year rate in one year is 7.0%.
Under liquidity preference, f₁,₂ = E[ y₁ in one year ] + L with L > 0. So the expected rate is less than 7.0%.
For example, if L = 0.5%, the expected rate is 6.5%. The value of L is not given, so only the direction can be stated.

Answer: Expectations theory explains the slope entirely by expected rises in short rates, so the expected rate is 7.0%. Liquidity preference explains part of it by a positive term premium, so the expected rate is below 7.0%.

Exam tips

In written answers, quote the yields given when you name the shape. Marks are usually for evidence, not just the label.
Always state the theory's claim, then link it to the shape. A definition alone rarely earns full marks.
For forward rate questions, show the equation, the substitution and the result. Keep four decimal places in the working.
In MCQs, look for options that swap the theories, such as a term premium under pure expectations.
State assumptions, such as annual compounding and equal credit risk, to protect marks.

Practice questions from Role of money and interest rates in the economy

Term Structure and Yield Curve 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 and Yield Curve: frequently asked questions

What does an inverted yield curve mean?

It means short-term yields are above long-term yields. Under expectations theory, the market expects short rates to fall. It is often linked to expected economic slowdown, but it is not a certain forecast.

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

Expectations theory says long yields reflect only expected future short rates. Liquidity preference adds a positive term premium because investors prefer short maturities. So forward rates exceed expected future short rates under liquidity preference.

What is market segmentation theory?

It says that investors and borrowers prefer certain maturities, so each maturity has its own supply and demand. Yields at different maturities are then set largely separately. Preferred habitat theory softens this by letting investors switch if the yield is attractive.

How do I calculate a forward rate from spot yields?

Use (1 + y₂)² = (1 + y₁)(1 + f₁,₂), so f₁,₂ = (1 + y₂)² ÷ (1 + y₁) − 1. Use decimals and annual compounding unless told otherwise. The result is the rate for year 2 fixed today.