CFA Level I Exam · Yield and Yield Spread Measures for Fixed-Rate Bonds

Spot Rates, Forward Rates and the Yield Curve

Updated 7 October 2026 · Fact-checked

A spot (zero) rate is the yield on a single cash flow paid at one future date. You bootstrap spot rates from par yields, one maturity at a time. A forward rate is the rate between two future dates implied by spot rates. Price each cash flow with its own spot rate.

Understand Spot Rates, Forward Rates and the Yield Curve

A bond is a bundle of cash flows paid on different dates. A spot rate (zero rate) is the yield for one cash flow received at one date, with no coupons in between. The 3-year spot rate is the annual return for money locked up for exactly 3 years.

Yield to maturity (YTM) is different. It is one single rate that discounts all of a bond's cash flows to its price. It is a kind of average of the spot rates, weighted by when the cash flows arrive. Two bonds with the same maturity but different coupons can have different YTMs. Both bonds still use the same spot rates. That is why pricing with spot rates is more precise than pricing with YTM.

The par curve shows the coupon rate (equal to the YTM) at which a bond of each maturity would price at exactly 100. The spot curve shows zero rates by maturity. When the curve slopes upward, the par yields sit below the spot rates at longer maturities. This is because coupons paid early are discounted at lower rates.

Bootstrapping turns par yields into spot rates. Start with the 1-year par yield, which equals the 1-year spot rate. For the 2-year bond, discount the first coupon at the 1-year spot rate and solve for the 2-year spot rate so the price equals 100. Then repeat for 3 years, and so on.

A forward rate is the rate for a loan that starts at a future date. It is the rate that makes you indifferent between investing for the long period, or investing for the short period and then reinvesting at the forward rate. The notation f(A,B) means the B-year rate starting A years from now. For example, f(2,1) is the 1-year rate starting in 2 years.

Key formulas to remember

Bond price using spot rates: PV = CF₁ ÷ (1 + S₁) + CF₂ ÷ (1 + S₂)² + … + (CFₙ + Par) ÷ (1 + Sₙ)ⁿ; Each cash flow uses the spot rate for its own date. Annual coupons assumed here.
Discount factor: DFₜ = 1 ÷ (1 + Sₜ)ᵗ; Price = Σ (cash flow × discount factor).
Par bond bootstrapping (2-year example): 100 = C₂ ÷ (1 + S₁) + (100 + C₂) ÷ (1 + S₂)²; Use the par yield as the coupon rate C₂. Known spots go in first. Solve for the one unknown spot.
Implied forward rate: (1 + S_(A+B))^(A+B) = (1 + S_A)^A × (1 + f(A,B))^B; f(A,B) is the B-year rate starting A years ahead.
Forward rate solved: f(A,B) = [ (1 + S_(A+B))^(A+B) ÷ (1 + S_A)^A ]^(1/B) − 1; Take the B-th root. For B = 1 there is no root.
Forward rate from discount factors: 1 + f(A,1) = DF_A ÷ DF_(A+1); A quick route when the question gives discount factors.

How to solve Spot Rates, Forward Rates and the Yield Curve questions

Most questions ask you to price a bond, bootstrap a spot rate, or find a forward rate. This method works for all three.

1Identify what you are given: spot rates, par yields, discount factors, or forward rates. Note the compounding (assume annual unless told otherwise).
2If you are given par yields, bootstrap. Set the 1-year spot equal to the 1-year par yield. Then work forward one maturity at a time.
3For each new maturity, write the price as 100 for a par bond. Discount every coupon you already know at its own spot rate.
4Move the known present values to the other side. Divide the final cash flow (100 + coupon) by the remaining amount, then take the n-th root and subtract 1.
5For a forward rate, write (1 + long spot)^long = (1 + short spot)^short × (1 + f)^B. Compute the long and short growth factors and divide.
6For bond pricing, discount each cash flow with its own spot rate and add them. Compare with the YTM approach only if asked.
7Sanity check. With an upward-sloping curve, the forward rate should exceed the longer spot rate. Every spot rate should be close to its par yield.

Quickest way: Growth-factor shortcut for forwards and bootstrapping

When to use it: Use it under time pressure when options are spread apart, since you only need to pick the correct option.

For forwards, compute (1 + S_long)^long ÷ (1 + S_short)^short. For a 1-year forward the result minus 1 is the answer. No root is needed.
Estimate first: f(2,1) ≈ 3 × S₃ − 2 × S₂. With S₂ = 3% and S₃ = 4% this gives 6%. The exact figure is slightly higher.
If the curve is upward sloping, discard any option below the longer spot rate for a forward.
For bootstrapping, the new spot rate is always slightly above the par yield on an upward-sloping curve. Eliminate options below the par yield.
Calculator: use 1.04 [yˣ] 3 [=] for the powers on the BA II Plus. On the HP 12C, use 1.04 [ENTER] 3 [yˣ]. Keep full precision and round only at the end.

Common mistakes in Spot Rates, Forward Rates and the Yield Curve

Using YTM to discount every cash flow when asked to price with spot rates.
YTM is the rate you meet first, and it feels like one rate fits the whole bond.
Fix: Each cash flow uses the spot rate for its own date. Write the rate beside each cash flow before computing.
Treating a par yield as a spot rate for maturities beyond one year.
The 1-year par yield equals the 1-year spot, so students assume the same holds for 2 years and more.
Fix: Only the first maturity is identical. Bootstrap every later maturity.
Forgetting to discount the first coupon at the earlier spot rate when bootstrapping.
Students solve the 2-year bond with only one discount rate.
Fix: Put each known coupon at its known spot rate, then solve for the single unknown spot.
Using the wrong power in a forward rate calculation.
The notation f(A,B) is easy to misread, as the periods and exponents get swapped.
Fix: Write the exponents as years. Long rate: (A+B). Short rate: A. Take the B-th root for the result.
Averaging spot rates instead of using growth factors.
Averaging seems natural. For example, (3% + 4%) ÷ 2 = 3.5%.
Fix: Rates compound. Always divide growth factors, never subtract or average rates.
Taking the bond price at 100 but forgetting the principal in the last cash flow.
Students discount only the coupon at the final date.
Fix: The final cash flow is 100 + coupon.

Worked examples

Example 1

A 1-year annual-pay par bond yields 2.00% and a 2-year annual-pay par bond yields 3.00%. The 2-year spot rate is closest to: A. 2.50%, B. 3.02%, C. 3.50%.

Show the solution

The 1-year spot rate equals the 1-year par yield, so S₁ = 2.00%.
A 2-year par bond has a 3% coupon and a price of 100. Write: 100 = 3 ÷ 1.02 + 103 ÷ (1 + S₂)².
3 ÷ 1.02 = 2.9412.
100 − 2.9412 = 97.0588.
(1 + S₂)² = 103 ÷ 97.0588 = 1.06121.
1 + S₂ = √1.06121 = 1.03015.
S₂ = 3.015%, which rounds to 3.02%. It is slightly above the 3.00% par yield, as expected on an upward-sloping curve.

Answer: B. 3.02%

Example 2

The 2-year spot rate is 3.00% and the 3-year spot rate is 4.00% (annual compounding). The 1-year forward rate two years from now, f(2,1), is closest to: A. 3.50%, B. 6.03%, C. 7.00%.

Show the solution

Use (1 + S₃)³ = (1 + S₂)² × (1 + f(2,1)).
(1.04)³ = 1.124864.
(1.03)² = 1.0609.
1 + f(2,1) = 1.124864 ÷ 1.0609 = 1.06029.
f(2,1) = 6.03%.
Check: the quick estimate 3 × 4% − 2 × 3% = 6% is close. A 3.50% answer would be the simple average, which ignores compounding.

Answer: B. 6.03%

Exam tips

Read whether the question gives spot rates, par yields or forward rates. The method depends on it, and the wrong starting point loses the item.
Questions often include distractors from the quick estimate or simple averages. Compute with growth factors to avoid them.
If the curve slopes upward, forward rates exceed spot rates, and spot rates exceed par yields at longer maturities. Use this to eliminate options quickly.
With 90 seconds per item, do the bootstrap only up to the maturity asked. Do not compute all earlier spots if you only need S₂.
Keep full calculator precision between steps. Rounding S₂ early can push the answer to the wrong option.

Practice questions from Yield and Yield Spread Measures for Fixed-Rate Bonds

Spot Rates, Forward Rates and the Yield Curve in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Spot Rates, Forward Rates and the Yield Curve: frequently asked questions

What is the difference between a spot rate and YTM?

A spot rate is the yield for one cash flow at one date. YTM is a single rate that discounts all of a bond's cash flows to its price. Spot rates price each cash flow separately, so they are more precise for bonds of different coupons.

How do I bootstrap spot rates from par yields?

Set the 1-year spot equal to the 1-year par yield. For each longer maturity, write the price as 100, discount the known coupons with earlier spot rates, and solve for the one unknown spot rate. Repeat for each maturity.

How do I calculate an implied forward rate?

Use (1 + S_long)^long = (1 + S_short)^short × (1 + f)^B. Divide the long growth factor by the short growth factor. If the forward period is more than one year, take the root equal to the number of forward years.

What is the difference between the par curve and the spot curve?

The par curve shows the coupon rate (equal to YTM) at which bonds of each maturity price at par. The spot curve shows zero-coupon rates. On an upward-sloping curve, spot rates are above par yields at longer maturities.