Strategic Financial Management · Equity and Bond Valuation and Evaluation of Performance

Term Structure of Interest Rates and Yield Curve Theories

Updated 11 October 2026 · Fact-checked

The term structure of interest rates shows how yields differ across maturities for bonds of similar credit risk. Plotted, it is the yield curve. To solve questions, read spot rates, compute implied forward rates using (1+s₂)² = (1+s₁)(1+f), and match the curve shape to a theory: expectations, liquidity preference or market segmentation.

Understand Term Structure of Interest Rates

The term structure of interest rates is the relationship between the yield on a bond and its time to maturity, holding credit risk the same. Usually you compare government securities, so default risk is not a factor. A graph of yield against maturity is the yield curve.

A spot rate is today's yield on a zero-coupon bond for a given maturity. A forward rate is the rate for a future period that is implied by today's spot rates. If a 1-year spot is 6% and a 2-year spot is 7%, the market is implicitly pricing a 1-year rate, one year from now, of about 8.01%.

The yield curve has four common shapes. Upward sloping (normal): long rates are above short rates. Downward sloping (inverted): short rates are above long rates. Flat: rates are similar across maturities. Humped: rates rise, peak at medium maturity, then fall. Upward curves are often linked to expected growth or rising rates. Inverted curves are often read as a signal of expected falling rates or slowdown, but this is a market reading, not a certainty.

Three theories explain the shape. The pure expectations theory says long rates are an average of current and expected future short rates, so forward rates are unbiased predictions of future spot rates. The liquidity preference theory says investors want a premium for holding longer bonds because of higher price risk, so forward rates include a liquidity premium and the curve slopes up more than expectations alone imply. The market segmentation theory says investors and borrowers stick to preferred maturities (for example banks prefer short, insurers and pension funds prefer long), so each maturity has its own supply and demand and the rate is set separately in each segment.

A related view is the preferred habitat theory: investors prefer a maturity but will move if the yield premium is enough. It sits between expectations and segmentation.

Key rules to remember

Implied forward rate (one period, annual compounding): (1 + s₂)² = (1 + s₁) × (1 + ₁f₂), so ₁f₂ = (1 + s₂)² ÷ (1 + s₁) − 1; s₁ and s₂ are 1-year and 2-year spot rates. ₁f₂ is the 1-year rate starting at the end of year 1.
General forward rate: (1 + sₙ)ⁿ = (1 + sₘ)ᵐ × (1 + ₘfₙ)ⁿ⁻ᵐ, so ₘfₙ = [(1 + sₙ)ⁿ ÷ (1 + sₘ)ᵐ]^(1 ÷ (n−m)) − 1; Gives the annualised forward rate between year m and year n.
Spot rate from a bond price: Price of zero-coupon bond = Face value ÷ (1 + sₙ)ⁿ; Rearrange to find sₙ = (Face value ÷ Price)^(1/n) − 1.
Pure expectations theory: (1 + sₙ)ⁿ = (1 + s₁)(1 + E[₁f₂])(1 + E[₂f₃])...; Forward rate equals expected future spot rate. Long rate is a geometric average of expected short rates.
Liquidity preference theory: Forward rate = Expected future spot rate + Liquidity premium; The premium is positive and generally rises with maturity. So the curve is higher than under pure expectations.

How to solve Term Structure of Interest Rates questions

Questions on this topic are either numerical (forward rates from spot rates) or descriptive (shape and theory). Use this method for both.

1Write down the given rates and label them clearly as spot rates, yields to maturity or forward rates, with the maturity of each.
2Identify the curve shape by comparing short and long rates: rising, falling, flat or humped.
3For a forward rate, set up the no-arbitrage equation: growth over the longer period equals growth over the shorter period times growth over the forward period.
4Solve for the forward rate and convert it to a percentage. Keep at least four decimals in intermediate steps.
5If the question gives an expected future spot rate, compare it with the forward rate. The difference is the liquidity premium (or an arbitrage signal).
6Name the theory that fits the explanation asked for, and give its reason in one or two sentences.
7State the conclusion in words: what the market expects and what an investor or borrower should do.

Quickest way: Forward rate shortcut using growth factors

When to use it: When a numerical MCQ gives two spot rates and asks for the forward rate.

Compute the growth factor of the longer spot: (1 + s)ⁿ.
Compute the growth factor of the shorter spot: (1 + s)ᵐ.
Divide long by short.
Take the root for the forward period if it is more than one year, then subtract 1.
Check with approximation: for one year, f ≈ 2 × s₂ − s₁. If your answer is far from this, recheck.

Common mistakes in Term Structure of Interest Rates

Taking the forward rate as the simple average or the difference of spot rates, for example 7% − 6% = 1%.
Rates feel additive, and compounding is ignored.
Fix: Always divide growth factors: (1 + s₂)² ÷ (1 + s₁) − 1.
Using yield to maturity of coupon bonds as if it were the spot rate.
The terms are used loosely in questions and books.
Fix: Use spot rates for forward calculations. If only coupon bond data is given, derive spot rates first by discounting each cash flow, unless the question says to use the yields directly.
Mixing up liquidity preference and market segmentation.
Both explain why long and short rates differ.
Fix: Remember: liquidity preference says maturities are substitutes, but long ones need a premium. Segmentation says maturities are not substitutes at all, so each market clears separately.
Saying an inverted curve always means a recession.
It is a popular statement.
Fix: Say it is often read as a market expectation of falling short rates or weaker growth. Under expectations theory it simply means expected future rates are lower.
Forgetting the exponent when the forward period is more than one year.
The one-year formula is memorised without its logic.
Fix: For an n−m year forward period, take the (n−m)th root of the ratio before subtracting 1 to get an annual rate.

Worked examples

Example 1

The 1-year spot rate is 6% and the 2-year spot rate is 7% (annual compounding). (a) Find the implied 1-year forward rate one year from now. (b) Under the pure expectations theory, what is the expected 1-year spot rate next year?

Show the solution

Write (1.07)² = (1.06) × (1 + f).
(1.07)² = 1.1449.
1 + f = 1.1449 ÷ 1.06 = 1.08009.
f = 0.08009, about 8.01%.
Under pure expectations, the forward rate equals the expected future spot rate, so expected spot = 8.01%.

Answer: (a) Implied forward rate ≈ 8.01%. (b) Expected 1-year spot rate next year ≈ 8.01%.

Example 2

Spot rates are: 1-year 5%, 2-year 6%, 3-year 7% (annual compounding). (a) Find the 1-year forward rate between year 2 and year 3. (b) If the market expects the 1-year rate in two years to be 7.5%, what liquidity premium is implied under liquidity preference theory? (c) Which shape is this curve?

Show the solution

(a) (1.07)³ = (1.06)² × (1 + f).
(1.07)³ = 1.225043. (1.06)² = 1.1236.
1 + f = 1.225043 ÷ 1.1236 = 1.09029.
f ≈ 9.03%.
(b) Liquidity premium = forward rate − expected spot = 9.03% − 7.5% = 1.53%.
(c) Spot rates rise with maturity, so the curve is upward sloping (normal).

Answer: (a) Forward rate ≈ 9.03%. (b) Implied liquidity premium ≈ 1.53%. (c) Upward sloping curve.

Exam tips

Practise the forward rate calculation until it takes under a minute. It is the most likely numerical question.
In descriptive answers, give each theory its core claim, what it says about forward rates, and what shape it can explain. Expectations can explain any shape, liquidity preference explains why upward slopes are common, segmentation explains kinks at specific maturities.
If asked to compare theories, use a short two-column list in your answer with clear headings such as assumption, forward rate meaning and criticism.
Always end a case answer with an interpretation, such as what the curve implies for borrowing or investing decisions.
Check whether the question gives spot rates, YTMs or forward rates before you start. Using the wrong one costs the whole answer.

Practice questions from Equity and Bond Valuation and Evaluation of Performance

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

What is the difference between liquidity preference theory and market segmentation theory?

Liquidity preference theory treats different maturities as substitutes but says investors demand a premium for longer maturities because of higher price risk. Market segmentation theory says investors and borrowers stay in their preferred maturity, so rates in each segment are set by separate supply and demand. Segmentation does not need expected future rates to explain the curve.

How do I calculate a forward rate from spot rates?

Equate the growth of a longer investment to a shorter one rolled into the forward period. For one year forward after one year: f = (1 + s₂)² ÷ (1 + s₁) − 1. For longer forward periods, divide the growth factors and take the root for the number of forward years.

What does an inverted yield curve mean?

It means short-term yields are above long-term yields. Under expectations theory, it signals that the market expects short rates to fall. It is often watched as a sign of slowing growth, but it is an indicator, not a guarantee.

Is the forward rate a prediction of the future spot rate?

Under pure expectations theory, yes. Under liquidity preference theory, the forward rate is higher than the expected spot rate by a liquidity premium. So it is a biased predictor in that case.