NISM-Series-X-A: Investment Adviser (Level 1) · Investing in Fixed Income Securities

Yield Curve and Term Structure of Interest Rates Explained

Q: What is an inverted yield curve?

It is a yield curve where short-term yields are higher than long-term yields, so the curve slopes downward. It is often read as a sign that the market expects interest rates to fall, sometimes because of a slowdown. It is not a guaranteed forecast.

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

A spot rate is the yield on a zero-coupon bond for one maturity. Yield to maturity is a single rate that equates a bond's price to all its cash flows. Spot rates discount each cash flow separately.

Q: Which theories explain the term structure of interest rates?

The main ones are pure expectations theory, liquidity preference theory and market segmentation theory. Preferred habitat theory is a related extension. Each explains the curve shape through a different cause.

Q: Why does the yield curve usually slope upward?

Long bonds carry more price risk and tie up money for longer. Investors ask for a liquidity premium, and rates may also be expected to rise. Both push long-term yields above short-term yields.

Updated 11 October 2026 · Fact-checked

A yield curve plots yields of similar-quality bonds against their maturities. The term structure of interest rates is the relationship between yield and maturity. To solve questions, identify the curve's shape (normal, inverted, flat, humped), then match it to a theory: expectations, liquidity preference, or market segmentation. Spot rates discount single cash flows.

Understand Yield Curve and Term Structure of Interest Rates

A yield curve is a graph. The x-axis shows time to maturity. The y-axis shows yield. You plot bonds of the same credit quality, such as government securities, so that only maturity differs. The relationship between yield and maturity is called the term structure of interest rates.

A spot rate is the yield on a zero-coupon bond for a given maturity. It is the rate for a single cash flow received at one date. A coupon bond is a bundle of cash flows, so you can value it by discounting each cash flow at the spot rate for its own date. A curve built from spot rates is the spot rate curve or zero curve.

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. It is often read as a sign that the market expects rates to fall, and sometimes a slowdown.
Flat: yields are about the same across maturities.
Humped: yields rise for medium maturities, then fall for longer ones.

Three theories explain the shapes. The pure expectations theory says long-term rates are an average of expected future short-term rates. An upward curve means rates are expected to rise. An inverted curve means they are expected to fall. The liquidity preference theory says investors want a premium for holding long bonds, because long bonds carry more price risk. So the curve tends to slope up even when rates are expected to stay flat. The market segmentation theory says investors and borrowers stick to preferred maturities, so each maturity segment has its own supply and demand, and yields in each segment are set separately. A related idea, the preferred habitat theory, says investors will move out of their preferred maturity if paid enough.

A shift in the curve means all yields move. A parallel shift moves all maturities by the same amount. A steepening widens the gap between long and short yields. A flattening narrows it.

Key formulas to remember

Bond price using spot rates: P = C₁ ÷ (1 + s₁)¹ + C₂ ÷ (1 + s₂)² + … + (Cₙ + F) ÷ (1 + sₙ)ⁿ; Each cash flow is discounted at the spot rate for its own year. s is the spot rate, C the coupon, F the face value.
Forward rate from spot rates (annual compounding): (1 + s₂)² = (1 + s₁) × (1 + f); f is the one-year rate one year from now. Rearranged: f = (1 + s₂)² ÷ (1 + s₁) − 1.
Pure expectations idea: Long-term rate ≈ average of expected future short-term rates; Under this theory forward rates are unbiased predictors of future spot rates.
Liquidity preference idea: Long-term yield = expected average short rate + liquidity premium; The premium is positive and tends to rise with maturity.
Curve shape summary: Normal: long > short; Inverted: long < short; Flat: long ≈ short; Compare yields at the short and long ends of the curve.

How to solve Yield Curve and Term Structure of Interest Rates questions

Use this method for any question on yield curves, spot rates or term structure theories.

1Read what is asked: a shape, a theory, a price, or a forward rate.
2If a table of yields is given, compare short-maturity and long-maturity yields to name the shape.
3If the question gives a theory name, recall its single core claim: expectations, liquidity premium, or separate maturity segments.
4For price questions, discount each cash flow at the spot rate for its own year. Do not use one rate for all.
5For forward rate questions, set (1 + longer spot)ⁿ equal to (1 + shorter spot)ᵐ × (1 + f)ᵏ and solve for f.
6Check the answer: forward rate should be above both spot rates when the curve is upward sloping.
7Eliminate options that mix up theories, such as attributing a liquidity premium to pure expectations.

Quickest way: Shape-to-theory shortcut

When to use it: Use for conceptual MCQs that ask what a given curve shape implies or which theory explains it.

Upward curve with no premium: pure expectations says rates are expected to rise.
Upward curve even if rates are expected flat: liquidity preference.
Downward curve: expectations says rates will fall. Liquidity preference needs falling expectations large enough to beat the premium.
Segments with unrelated yields or kinks: market segmentation.
For forward rates, remember the two-year spot is roughly the average of the one-year spot and the forward rate.

Common mistakes in Yield Curve and Term Structure of Interest Rates

Calling a curve inverted when long yields are higher.
The word inverted sounds like the opposite of what students picture.
Fix: Inverted means short-term yields exceed long-term yields. The curve slopes down.
Discounting all cash flows at the bond's single yield when asked to use spot rates.
Students are used to the yield to maturity method.
Fix: When spot rates are given, discount each cash flow at the spot rate for its own maturity.
Saying that pure expectations theory includes a liquidity premium.
The theories are learned together and blur.
Fix: Pure expectations has no premium. Liquidity preference adds a premium for longer maturities.
Treating the yield curve as showing bonds of mixed credit quality.
Students plot any bonds they know.
Fix: A proper yield curve uses bonds of the same credit quality, so only maturity differs.
Computing the forward rate as the simple difference of two spot rates.
It looks quicker than compounding.
Fix: Use the compounding relationship. Simple subtraction is only a rough approximation and may not match an option.
Assuming an inverted curve always means a recession will follow.
News headlines overstate it.
Fix: Say it is often read as a signal of expected rate falls or slowdown, not a certain outcome.

Worked examples

Example 1

The one-year spot rate is 6% and the two-year spot rate is 7% (annual compounding). What is the implied one-year forward rate one year from now, to two decimal places?

Show the solution

Write the relationship: (1 + s₂)² = (1 + s₁) × (1 + f).
Compute (1.07)² = 1.1449.
Divide by (1.06): 1.1449 ÷ 1.06 = 1.08009.
Subtract 1: f = 0.08009, or about 8.01%.

Answer: About 8.01%

Example 2

A two-year government bond has face value ₹1,000 and pays an annual coupon of ₹80. The one-year spot rate is 6% and the two-year spot rate is 7%. What is its price using spot rates, to the nearest rupee?

Show the solution

Year 1 cash flow is ₹80. PV = 80 ÷ 1.06 = ₹75.47.
Year 2 cash flow is ₹80 + ₹1,000 = ₹1,080.
Discount at the two-year spot: 1,080 ÷ (1.07)² = 1,080 ÷ 1.1449 = ₹943.34.
Add the present values: 75.47 + 943.34 = ₹1,018.81.
Round to the nearest rupee.

Answer: ₹1,019

Exam tips

Learn the four shapes by comparing short and long yields. Many questions are one-line definitions.
Match each theory to its single key idea. Trap options swap them.
For numerical questions, check whether spot rates or one yield is to be used. Spot rates mean year-by-year discounting.
Do not guess on a 2-mark caselet question. A wrong answer costs 25% of the marks assigned to it, so a wrong answer on 2 marks costs 0.5 mark.
In caselets, read the table of yields first and name the shape before looking at the options.

Practice questions from Investing in Fixed Income Securities

Yield Curve and 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.

Yield Curve and Term Structure of Interest Rates: frequently asked questions

What is an inverted yield curve?

It is a yield curve where short-term yields are higher than long-term yields, so the curve slopes downward. It is often read as a sign that the market expects interest rates to fall, sometimes because of a slowdown. It is not a guaranteed forecast.

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

A spot rate is the yield on a zero-coupon bond for one maturity. Yield to maturity is a single rate that equates a bond's price to all its cash flows. Spot rates discount each cash flow separately.

Which theories explain the term structure of interest rates?

The main ones are pure expectations theory, liquidity preference theory and market segmentation theory. Preferred habitat theory is a related extension. Each explains the curve shape through a different cause.

Why does the yield curve usually slope upward?

Long bonds carry more price risk and tie up money for longer. Investors ask for a liquidity premium, and rates may also be expected to rise. Both push long-term yields above short-term yields.