Economic Modelling · Models of the term structure of interest rates

Term Structure and Yield Curve Basics: Spot and Forward Rates

Updated 11 October 2026 · Fact-checked

The term structure of interest rates shows how yields vary with the time to maturity. A spot rate y_t is the annual rate for a single payment at time t. A forward rate f_{t,r} is the rate agreed now for borrowing from t to t+r. You derive forward rates by equating accumulations: (1+y_{t+r})^(t+r) = (1+y_t)^t × (1+f_{t,r})^r.

Understand Term Structure and Yield Curve Basics

Interest rates are not the same for every term. A 1-year bond and a 10-year bond usually pay different annual yields. The term structure of interest rates describes this link between yield and term. Plotting yield against term gives the yield curve.

The spot rate y_t is the annual effective yield on a zero-coupon bond bought now and maturing at time t. It is the rate that applies to one single payment at t. The discount function is the price now of ₹1 payable at t: v(t) = (1 + y_t)^(-t). Spot rates and discount factors carry the same information.

A forward rate f_{t,r} is the annual rate, fixed today, for money lent from time t to time t+r. The special case of one year is f_t = f_{t,1}. Forward rates come from no-arbitrage. Investing for t+r years at the spot rate must give the same result as investing for t years and then rolling into the forward contract for r years. If not, you could make a risk-free profit.

The par yield c_t is the coupon rate at which a t-year coupon bond, redeemed at par, is priced at par. It is found from the spot rates (or discount factors). Par yields differ from spot rates because a coupon bond pays cash flows at many dates, so its yield is a blend of several spot rates.

The curve shape tells you about the forward rates. If spot rates rise with term (an upward-sloping curve), forward rates lie above the spot rates. If spot rates fall (downward-sloping or inverted), forward rates lie below them. A flat curve gives forward rates equal to the spot rate. A humped curve rises then falls. The shape is explained by theories such as expectations, liquidity preference and market segmentation. Those are covered in a separate topic.

Key rules to remember

Discount function from spot rate: v(t) = (1 + y_t)^(-t); y_t is the annual effective spot rate for term t. Price today of ₹1 at time t.
Forward rate from spot rates (general): (1 + y_{t+r})^(t+r) = (1 + y_t)^t × (1 + f_{t,r})^r; Rearrange for f_{t,r}. Rates are annual effective and the same compounding basis is used throughout.
One-year forward rate: 1 + f_t = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t = v(t) ÷ v(t+1); f_t is the rate from time t to t+1. Here f_0 = y_1.
Spot rate from forward rates: (1 + y_t)^t = (1 + f_0)(1 + f_1)...(1 + f_{t-1}); The spot rate is a geometric average of the one-year forward rates.
Discount factor from forward rates: v(t) = [(1 + f_0)(1 + f_1)...(1 + f_{t-1})]^(-1); Same relation as above, written as a price.
Par yield: c_n = (1 − v(n)) ÷ (v(1) + v(2) + ... + v(n)); Coupons paid annually, bond redeemed at par, priced at par. Denominator is the sum of discount factors.
Price of a coupon bond from spot rates: P = Σ C_t × v(t); Discount each cash flow at the spot rate for its own term.

How to solve Term Structure and Yield Curve Basics questions

Most questions give you one set of rates (spot, forward or par) and ask for another. Work through discount factors, as they link everything.

1Identify what you are given (spot rates, forward rates, par yields or bond prices) and what you must find. Note whether rates are annual effective.
2Convert the given rates into discount factors v(t) = (1 + y_t)^(-t), or v(t+1) = v(t) ÷ (1 + f_t) if you have forward rates.
3For a forward rate, use 1 + f_t = v(t) ÷ v(t+1) for one year, or [v(t) ÷ v(t+r)]^(1/r) = 1 + f_{t,r} for r years.
4For a par yield, use c_n = (1 − v(n)) ÷ Σ v(k), summing k from 1 to n.
5For a bond price, discount each coupon and the redemption payment using its own spot rate.
6Check the result against the curve shape. On a rising curve, forward rates should exceed spot rates.
7State the answer with the correct unit (annual effective rate, as a percentage) and the time period it applies to.

Quickest way: Work in discount factors, not rates

When to use it: Use it for any numerical question that links spot rates, forward rates and par yields, especially under time pressure.

Write v(1), v(2), v(3)... in a short list from the spot rates. Keep 5 or 6 decimals.
For a one-year forward rate, divide: f_t = v(t) ÷ v(t+1) − 1.
For a multi-year forward rate, use f_{t,r} = [v(t) ÷ v(t+r)]^(1/r) − 1.
For a par yield, compute (1 − v(n)) ÷ Σ v(k) directly from the list.
Quick check: if the spot curve is rising, your forward rate must be above the later spot rate.

Common mistakes in Term Structure and Yield Curve Basics

Averaging spot rates to get a forward rate or taking the difference y_{t+1} − y_t.
Students treat rates as additive, but interest compounds.
Fix: Always equate accumulation factors: 1 + f_t = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t.
Using the wrong exponent, for example (1 + y_2)^1 instead of (1 + y_2)^2.
Rushing and forgetting that the exponent is the term in years.
Fix: Write the accumulation factor as (1 + y_t)^t for each term before dividing.
For a multi-year forward rate f_{t,r}, forgetting to take the r-th root.
The ratio gives a total r-year accumulation, not an annual rate.
Fix: After dividing, raise to the power 1/r and subtract 1.
Confusing par yield with spot rate, or discounting a coupon bond at its par yield.
Both are called yields and both can be quoted for the same term.
Fix: Remember that the spot rate is for one payment at t. The par yield is the coupon rate that prices a coupon bond at par. Discount each cash flow at its own spot rate.
Leaving out the redemption payment when pricing a bond or finding a par yield.
Attention goes to the coupons only.
Fix: List all cash flows in a line, including the final coupon plus redemption, before discounting.
Stating that an upward-sloping curve always means higher interest rates will occur.
Forward rates are read as forecasts.
Fix: Say that forward rates are implied by today's prices. Whether they predict future spot rates depends on the theory of the term structure.

Worked examples

Example 1

Annual effective spot rates are y_1 = 4%, y_2 = 5% and y_3 = 6%. Calculate (a) the one-year forward rate f_1 from time 1 to 2, (b) the two-year forward rate f_{1,2} per annum from time 1 to 3, and (c) the 3-year par yield.

Show the solution

Discount factors: v(1) = 1/1.04 = 0.961538. v(2) = 1/1.05² = 1/1.1025 = 0.907029. v(3) = 1/1.06³ = 1/1.191016 = 0.839619.
(a) 1 + f_1 = v(1) ÷ v(2) = 0.961538 ÷ 0.907029 = 1.060096. Using accumulation factors, this is 1.05² ÷ 1.04 = 1.1025 ÷ 1.04 = 1.060096. So f_1 = 6.01%.
(b) (1 + f_{1,2})² = v(1) ÷ v(3) = 1.06³ ÷ 1.04 = 1.191016 ÷ 1.04 = 1.145208. Square root = 1.070144. So f_{1,2} = 7.01%.
(c) Σ v(k) = 0.961538 + 0.907029 + 0.839619 = 2.708186. 1 − v(3) = 0.160381.
Par yield = 0.160381 ÷ 2.708186 = 0.059221, which is 5.92%.

Answer: (a) f_1 ≈ 6.01%. (b) f_{1,2} ≈ 7.01% per annum. (c) c_3 ≈ 5.92%. The par yield lies below the 3-year spot rate of 6%, as expected on a rising curve.

Example 2

The one-year forward rates are f_0 = 3%, f_1 = 4% and f_2 = 5%. A bond pays ₹100 at the end of each of the next 3 years (no redemption payment). Find the 3-year spot rate and the price of the bond.

Show the solution

Accumulation to time 3: (1 + y_3)³ = 1.03 × 1.04 × 1.05 = 1.12476.
So (1 + y_3) = 1.12476^(1/3) ≈ 1.03997, so y_3 ≈ 4.00%.
Discount factors from forward rates: v(1) = 1/1.03 = 0.970874.
v(2) = 1/(1.03 × 1.04) = 1/1.0712 = 0.933532.
v(3) = 1/1.12476 = 0.889079.
Price = 100 × (0.970874 + 0.933532 + 0.889079) = 100 × 2.793485 = ₹279.35.

Answer: The 3-year spot rate is about 4.00%. The bond price is about ₹279.35.

Exam tips

Show the accumulation equation first, then the rearranged result. Method marks are given for the setup even if arithmetic slips.
Keep at least 5 or 6 decimal places in discount factors. Rounding early is a common way to lose accuracy in the final rate.
Read the question for the time convention. f_{t,r} is a rate from t to t+r. Make sure you give it per annum, not as a total over r years.
In written answers, interpret the result. Say whether the curve is rising, flat or inverted, and link it to the forward rates.
In the computer-based paper, build discount factors in a column, then compute forward rates and par yields from that column so you can check each step.

Practice questions from Models of the term structure of interest rates

Term Structure and Yield Curve Basics 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 Basics: frequently asked questions

What is the difference between spot rates and forward rates?

A spot rate applies from now to a future date t. A forward rate applies to a period starting at a future date t, and is fixed today. Forward rates are implied by spot rates through no-arbitrage.

How do I derive forward rates from spot rates?

Equate two ways of investing to time t+r. One is at the spot rate y_{t+r}. The other is at y_t for t years and then at the forward rate for r years. Solve (1 + y_{t+r})^(t+r) = (1 + y_t)^t (1 + f_{t,r})^r for f_{t,r}.

What is a par yield and how is it different from a spot rate?

A par yield is the coupon rate that makes a coupon-paying bond, redeemed at par, worth its par value. A spot rate is the yield on a single payment at one date. The par yield is a weighted blend of spot rates.

What does an inverted yield curve mean?

It means spot rates fall as term increases, so long-term yields are below short-term yields. The implied forward rates are then below the spot rates. Different theories of the term structure explain why this happens.