Economic Modelling · Models of the term structure of interest rates

Theories of the Term Structure of Interest Rates

Updated 11 October 2026 · Fact-checked

Theories of the term structure explain why yields differ by maturity. Expectations theory says long yields reflect expected future short rates. Liquidity preference adds a premium for long bonds. Market segmentation says supply and demand set yields in separate maturity markets. Preferred habitat lets investors leave their segment for a premium.

Understand Theories of the Term Structure

The term structure of interest rates describes how yields vary with the term of a bond. Plot yield against term and you get the yield curve. It can be upward sloping (normal), downward sloping (inverted) or humped. The theories try to explain why the curve takes a given shape.

Pure expectations theory says the forward rates implied by today's curve are unbiased predictions of future spot rates. A long bond is then just a chain of short investments, so investors are indifferent between them. A rising curve means the market expects short rates to rise. A falling curve means it expects them to fall. Under this theory the curve is flat only if short rates are expected to stay constant.

Liquidity preference theory says investors dislike risk from price changes. Long bonds have greater price volatility when yields move. So investors demand extra yield, a liquidity premium, for holding longer bonds. Forward rates then equal expected future spot rates plus a premium that tends to grow with term. This theory gives an upward bias to the curve. It can still invert if expected falls in short rates outweigh the premium.

Market segmentation theory says investors and borrowers have strong preferences for particular maturities, often because of the matching of their liabilities. A life insurer with long liabilities buys long bonds. A bank with short deposits buys short bonds. Yields at each term are set by supply and demand in that segment alone. Expectations of future rates play no direct role, and the curve can take any shape.

Preferred habitat theory sits between the others. Investors prefer a maturity range but will move outside it if the extra yield is high enough. So yields reflect expected short rates, plus premiums that can be positive or negative and vary with term, depending on supply and demand in each habitat. It explains why curves are not always upward sloping, and why supply of government bonds at a given term can change yields.

Key rules to remember

Forward rate from spot rates (annual compounding): (1 + y_t)^t = (1 + y_{t-1})^(t-1) × (1 + f_{t-1,t}); y_t is the t-year spot rate and f_{t-1,t} is the one-year forward rate from t-1 to t. Needs a consistent compounding basis.
Pure expectations theory: f_{t-1,t} = E[ one-year spot rate at time t-1 ]; Forward rates are unbiased predictors of future spot rates. No risk premium.
Liquidity preference theory: f_{t-1,t} = E[ spot rate at t-1 ] + L_t, with L_t > 0; L_t is the liquidity premium, usually increasing with term. Forwards overstate expected future spot rates.
Preferred habitat theory: f_{t-1,t} = E[ spot rate at t-1 ] + P_t; P_t can be positive or negative, depending on supply and demand in each maturity habitat.

How to solve Theories of the Term Structure questions

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

1Read what is asked: calculate a forward rate or expected rate, or explain a curve shape using a named theory.
2Write down the given yields and state the compounding basis and whether they are spot rates or par yields.
3If a calculation is needed, derive forward rates from spot rates using the accumulation relationship.
4Choose the theory named in the question. Pure expectations: expected spot rate equals forward rate. Liquidity preference: subtract the premium. Preferred habitat: adjust for the stated premium.
5State what the result says about market expectations, for example rising or falling short rates.
6For discussion questions, give the core idea of the theory, the shape it predicts, and one limitation.
7Compare theories where asked: say what drives the curve in each case, and whether expectations matter.

Quickest way: Match the shape to the theory

When to use it: Use this for multiple-choice questions and short explanations of yield curve shapes.

Ask first: does the theory use expected future short rates? Expectations, liquidity preference and preferred habitat do. Pure market segmentation does not.
Ask: is there a risk premium? Liquidity preference has a positive premium rising with term. Preferred habitat has premiums of either sign. Pure expectations has none.
Ask: can investors switch maturities freely? Market segmentation says no. Preferred habitat says yes, for a price.
For a calculation, compute forwards from spots and then remove any stated premium to get expected spot rates.

Common mistakes in Theories of the Term Structure

Treating liquidity preference as a theory that the curve must always slope upward.
The premium is positive, so students assume the curve can never be flat or inverted.
Fix: Say the premium biases the curve upward. If expected short rates fall enough, the curve can still be flat or inverted.
Saying expectations play a role in market segmentation theory.
Students blend the theories and assume expectations always matter.
Fix: State that in pure market segmentation, yields come from supply and demand within each maturity segment, not from expected future rates.
Confusing preferred habitat with market segmentation.
Both mention maturity preferences.
Fix: Remember that in preferred habitat investors will leave their habitat if compensated. In segmentation they will not.
Using the forward rate as the expected spot rate under liquidity preference.
This is correct only for pure expectations theory.
Fix: Under liquidity preference, expected spot rate = forward rate minus the premium.
Mixing up compounding bases when deriving forward rates.
Students rush and combine annual and continuously compounded rates.
Fix: Write the basis next to every rate. Under continuous compounding use y_t × t = y_{t-1} × (t-1) + f_{t-1,t}.

Worked examples

Example 1

One-year and two-year spot rates are 4% and 5% per year, annual compounding. Under pure expectations theory, find the expected one-year spot rate in one year's time.

Show the solution

Use (1 + y_2)^2 = (1 + y_1) × (1 + f_{1,2}).
Compute (1.05)^2 = 1.1025.
Then 1 + f_{1,2} = 1.1025 ÷ 1.04 = 1.060096.
So f_{1,2} = 6.0096%, about 6.01%.
Under pure expectations theory, the expected spot rate equals the forward rate.

Answer: The expected one-year spot rate in one year is about 6.01%.

Example 2

Using the data above, suppose liquidity preference theory applies with a liquidity premium of 0.5% in the one-year forward rate from time 1 to 2. Find the expected one-year spot rate in one year and compare it with pure expectations.

Show the solution

The forward rate is f_{1,2} ≈ 6.01%, from the same calculation.
Under liquidity preference, f = E[spot] + L, so E[spot] = f − L.
E[spot] = 6.01% − 0.50% = 5.51%.
Compare with the pure expectations result of 6.01%.
The premium means the market expects a smaller rise in short rates than the pure expectations reading suggests.

Answer: The expected spot rate is about 5.51%, which is 0.5 percentage points lower than under pure expectations.

Exam tips

For a discussion question, name the theory, say what drives yields, state the curve shape it predicts, and give one weakness.
In calculations, state the compounding basis and show the forward rate formula before substituting.
Link each theory to a real investor, such as a life insurer with long liabilities, to support market segmentation and preferred habitat.
Write the effect of a premium in words: forward rates exceed expected spot rates when the premium is positive.

Practice questions from Models of the 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

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

Pure expectations theory says forward rates equal expected future spot rates. Liquidity preference adds a positive premium to long-term yields, so forward rates are higher than expected spot rates. This is why it gives an upward bias to the curve.

How does market segmentation theory explain the yield curve?

It says each maturity has its own market with its own supply and demand, driven by investor preferences such as liability matching. Yields at each term are set separately. The curve can therefore take any shape, and expectations of future rates are not needed.

How is preferred habitat theory different from market segmentation?

Both say investors prefer certain maturities. In preferred habitat, investors will move to other maturities if the extra yield compensates them. In market segmentation they stay within their segment.

Can the yield curve be inverted under liquidity preference theory?

Yes. The liquidity premium pushes the curve up, but if the market expects short rates to fall by more than the premium, the curve can be flat or downward sloping.