Advanced Financial Management · Security Valuation

Term Structure of Interest Rates and Spot Rates for CA Final AFM

Updated 5 October 2026 · Fact-checked

The term structure of interest rates shows how yield changes with maturity. A spot rate is the yield on a zero-coupon bond for a given term. Forward rates are implied by spot rates: (1+S₂)² = (1+S₁)(1+f₁,₂). To value a bond, discount each cash flow at the spot rate of its own year.

Understand Term Structure of Interest Rates and Spot Rates

A spot rate is the annual yield today on a zero-coupon bond that matures at a given date. The 1-year spot rate applies to a cash flow in year 1. The 3-year spot rate applies to a cash flow in year 3. Different maturities usually have different spot rates.

The term structure of interest rates is the relationship between yield and time to maturity, for bonds of similar credit risk. When you plot it on a graph, you get the yield curve. It can be upward sloping (longer maturity, higher yield), downward sloping (inverted) or flat. It can also be humped.

A forward rate is the rate for a future period that is implied by today's spot rates. If you can earn S₁ for one year and S₂ per year for two years, the second year must be earning an implied rate. That rate is the forward rate f₁,₂. It makes you indifferent between investing for two years directly and investing for one year then rolling over, if the forward rate is realised.

Three theories explain the shape of the curve. The pure expectations theory says forward rates are the market's expected future spot rates, so an upward curve means rising rates are expected. The liquidity preference theory says investors want a premium for locking money for longer, so forward rate = expected spot rate + liquidity premium. The curve then tends to slope upward even when rates are expected to stay flat. The market segmentation theory says investors and borrowers stick to preferred maturities, so each segment has its own supply and demand and sets its own rate. A related view, the preferred habitat theory, lets investors move out of their segment if the premium is large enough.

For bond valuation, the right method is to discount each coupon and the principal at the spot rate for that year. This is more accurate than using a single yield to maturity for all cash flows. The yield to maturity is a blended rate that depends on the coupon.

Key rules to remember

Spot rate to forward rate (one period ahead): (1 + Sₙ)ⁿ = (1 + Sₙ₋₁)ⁿ⁻¹ × (1 + fₙ₋₁,ₙ); Gives the one-year forward rate for year n. So fₙ₋₁,ₙ = (1 + Sₙ)ⁿ ÷ (1 + Sₙ₋₁)ⁿ⁻¹ − 1.
General forward rate between years m and n: (1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + fₘ,ₙ)ⁿ⁻ᵐ; Use for a multi-year forward rate, such as the 2-year rate starting after year 1. Annual compounding assumed.
Bond value using spot rates: P = Σ [Cₜ ÷ (1 + Sₜ)ᵗ] + M ÷ (1 + Sₙ)ⁿ; Cₜ is coupon in year t, M is redemption value, Sₜ is the spot rate for year t.
Liquidity preference theory: Forward rate = Expected future spot rate + Liquidity premium; Premium is positive and usually grows with maturity. Under pure expectations the premium is zero.
Spot rate from discount factors: Sₙ = (1 ÷ DFₙ)^(1/n) − 1; DFₙ is the price today of ₹1 received in year n.

How to solve Term Structure of Interest Rates and Spot Rates questions

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

1Read what is given: spot rates, forward rates, bond prices or discount factors. Note whether compounding is annual.
2Identify what is asked: a forward rate, a spot rate, a bond price, or an explanation of a theory.
3For a forward rate, write the identity (1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + f)ⁿ⁻ᵐ and solve for f. Keep four decimal places in the factors.
4For bond valuation, list each year's cash flow and the spot rate for that year. Discount each at its own rate and add.
5If the question asks for the expected spot rate under liquidity preference, subtract the liquidity premium from the forward rate.
6Check the answer: an upward-sloping spot curve should give forward rates above the spot rates.
7Interpret in a line: say what the curve shape suggests, and name the theory that explains it.

Quickest way: Factor-ratio shortcut for forward rates

When to use it: Use when you must find a one-year forward rate from two consecutive spot rates under annual compounding.

Compute (1 + Sₙ)ⁿ for the longer term.
Compute (1 + Sₙ₋₁)ⁿ⁻¹ for the shorter term.
Divide the first by the second.
Subtract 1 to get the forward rate.
For bond pricing, write a table with columns: year, cash flow, spot rate, discount factor, present value. Add the last column.

Common mistakes in Term Structure of Interest Rates and Spot Rates

Taking the forward rate as the simple difference of spot rates, such as S₂ − S₁.
Students treat rates as additive and ignore compounding.
Fix: Always use the ratio of compounded factors, (1 + S₂)² ÷ (1 + S₁) − 1.
Discounting all bond cash flows at one rate, usually the YTM, when the question gives spot rates.
Habit from basic bond valuation.
Fix: If spot rates are given, discount each year's cash flow at that year's spot rate.
Using the wrong exponent, for example squaring when the term is three years.
Rushing and mixing up the year number with the period of the forward rate.
Fix: The exponent equals the number of years in that spot rate. Write the exponent beside each rate before calculating.
Mixing up the theories: saying liquidity preference predicts a downward curve by default.
Memorising names without the logic.
Fix: Remember that liquidity preference adds a premium for longer maturities, so the curve tends to slope upward. Pure expectations has no premium, and segmentation depends on demand and supply in each segment.
Treating forward rates as certain future rates.
The word 'implied' is read as 'guaranteed'.
Fix: State that forward rates are implied by today's curve. They are the expected spot rate only under pure expectations, and include a premium under liquidity preference.

Worked examples

Example 1

The 1-year spot rate is 6% and the 2-year spot rate is 7% per annum, with annual compounding. The 3-year spot rate is 8%. Find the one-year forward rate for year 2 and for year 3.

Show the solution

Forward rate for year 2: f₁,₂ = (1.07)² ÷ 1.06 − 1.
(1.07)² = 1.1449.
1.1449 ÷ 1.06 = 1.08009, so f₁,₂ = 8.01% approximately.
Forward rate for year 3: f₂,₃ = (1.08)³ ÷ (1.07)² − 1.
(1.08)³ = 1.259712.
1.259712 ÷ 1.1449 = 1.10028, so f₂,₃ = 10.03% approximately.
Check: the spot curve is upward sloping, so both forward rates are above the spot rates.

Answer: The forward rate for year 2 is about 8.01% and for year 3 about 10.03%.

Example 2

A 3-year bond with face value ₹1,000 pays an annual coupon of 10%. Spot rates are: 1-year 6%, 2-year 7%, 3-year 8%. Find the bond's value using spot rates.

Show the solution

Cash flows: year 1 ₹100, year 2 ₹100, year 3 ₹1,100.
PV of year 1 = 100 ÷ 1.06 = ₹94.34.
PV of year 2 = 100 ÷ 1.1449 = ₹87.34.
PV of year 3 = 1,100 ÷ 1.259712 = ₹873.22.
Add: 94.34 + 87.34 + 873.22 = ₹1,054.90.
Interpretation: the price is above face value because the coupon rate of 10% is above all the spot rates.

Answer: The value of the bond is about ₹1,054.90.

Exam tips

If a question gives spot rates, use them to discount each cash flow. Do not switch to YTM unless asked.
Show the formula line (1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + f)ⁿ⁻ᵐ before the working. Method marks depend on it.
In theory questions, answer in a structure: the idea, what it says about the curve, and one limitation.
In case-scenario MCQs, first decide whether the question asks for a forward rate, a spot rate or a theory, and then apply the right identity. There is no negative marking, so always attempt.
Round factors to at least four decimals, and state that annual compounding is assumed.

Practice questions from Security Valuation

Term Structure of Interest Rates and Spot 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 and Spot Rates: frequently asked questions

How do I calculate a forward rate from spot rates?

Divide the compounded factor of the longer spot rate by that of the shorter one, then subtract 1. For the year-2 rate, f₁,₂ = (1 + S₂)² ÷ (1 + S₁) − 1. For a multi-year forward rate, take the root of the ratio equal to the number of forward years.

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

Under pure expectations, the forward rate equals the expected future spot rate, so the curve shape reflects only rate expectations. Under liquidity preference, the forward rate also includes a premium for holding longer bonds. So the curve tends to slope upward even if rates are expected to stay flat.

What does an inverted yield curve mean?

It means short-term yields are higher than long-term yields. Under pure expectations, this suggests that the market expects interest rates to fall. Other theories add that demand in different maturity segments can also cause it.

Why use spot rates instead of YTM to value a bond?

Each cash flow is received at a different time, and the market gives a different rate for each maturity. Discounting each flow at its own spot rate prices the bond consistently with the zero-coupon market. YTM is a single blended rate that depends on the coupon.