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CFA Level I Exam · The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Par Curve and Bootstrapping Spot Rates Explained

Updated 7 October 2026 · Fact-checked

The par curve shows the coupon rates at which bonds of each maturity would price exactly at par. To bootstrap spot rates, start with the shortest maturity, whose par rate equals its spot rate. Then solve each longer par bond for its final discount factor using earlier ones, and convert that factor to a spot rate.

Understand Par Curve and Bootstrapping Spot Rates

A par rate is the coupon rate that makes a bond's price equal to its face value. For a bond priced at par, the coupon rate equals the yield to maturity. The par curve plots these par rates against maturity: the 1-year par rate, the 2-year par rate, and so on.

A par rate is a single yield applied to every cash flow in the bond. A spot rate is different. It is the yield on a zero-coupon bond for one specific maturity, and it discounts only the cash flow that arrives at that date. A par rate is a kind of average of the spot rates up to that maturity, weighted by the discount factors. When the curve slopes upward, spot rates at longer maturities usually sit above par rates at the same maturity.

Bootstrapping turns par rates into spot rates one maturity at a time. The 1-year par rate is also the 1-year spot rate, because there is only one cash flow. For the 2-year par bond, you know the coupon, you know the 1-year discount factor, and the only unknown is the 2-year discount factor. Solve for it. Then use both known factors to solve for the 3-year factor, and keep going.

Work in discount factors, not rates. A par bond has a price of 1 per 1 of face value, so the equation is simple: 1 = coupon × (sum of earlier discount factors) + (1 + coupon) × final discount factor. Rearrange and you have the new factor. The last step converts the factor to a spot rate.

The exam tests this with annual-pay bonds. You may be asked for a discount factor, a spot rate, or a forward rate that depends on the spot rates you just found.

Key formulas to remember

Par bond pricing identity (annual pay)
1 = c × (DF₁ + DF₂ + … + DFₙ₋₁) + (1 + c) × DFₙ
c is the par rate for maturity n, written as a decimal. DF is the discount factor per 1 of face value.
Bootstrapped discount factor
DFₙ = [1 − c × (DF₁ + … + DFₙ₋₁)] ÷ (1 + c)
Use the par rate cₙ for maturity n and every earlier discount factor.
Spot rate from discount factor
zₙ = (1 ÷ DFₙ)^(1/n) − 1
On the exam this is usually annual compounding.
Discount factor from spot rate
DFₙ = 1 ÷ (1 + zₙ)ⁿ
Use this to go back and check a result.
First step
z₁ = c₁
The 1-year par rate equals the 1-year spot rate when the bond pays once a year.

How to solve Par Curve and Bootstrapping Spot Rates questions

Use this method for any bootstrapping question with annual-pay par bonds. Keep discount factors at six decimals and round only the final rate.

  1. 1List the par rates by maturity and note the payment frequency. Assume annual pay unless the question says otherwise.
  2. 2Set the 1-year spot rate equal to the 1-year par rate. Compute DF₁ = 1 ÷ (1 + z₁).
  3. 3For the next maturity n, write the sum of all earlier discount factors.
  4. 4Compute DFₙ = [1 − c × (sum of earlier DFs)] ÷ (1 + c), where c is the par rate for maturity n.
  5. 5Convert to a spot rate: zₙ = (1 ÷ DFₙ)^(1/n) − 1.
  6. 6Repeat steps 3 to 5 for each longer maturity until you reach the one asked for.
  7. 7Check by repricing: coupon × sum of DFs + 1 × DFₙ should equal 1 for a 1-unit par bond. Then pick the matching option.

Quickest way: Discount-factor shortcut

When to use it: Use it whenever the question gives par rates and asks for a spot rate or discount factor. Skip the spot rates of the middle years unless they are asked for.

  1. Compute only the discount factors, not the spot rates, for the intermediate years.
  2. Keep a running total of the discount factors. It is the sum you need at each step.
  3. Apply DFₙ = [1 − c × running sum] ÷ (1 + c).
  4. Convert to a spot rate only once, for the maturity asked.
  5. If options are close, sanity-check: for an upward-sloping par curve, the spot rate should be a bit above the par rate at that maturity.

Common mistakes in Par Curve and Bootstrapping Spot Rates

  • Treating the par rate as the spot rate for every maturity

    Both are quoted as annual yields, so they look the same.

    Fix: Only the 1-year (single cash flow) par rate equals the spot rate. Longer par rates discount all coupons at one rate, so bootstrap them.

  • Discounting the coupons at the new par rate

    Students apply the old YTM habit to every cash flow.

    Fix: Use the already-found discount factors for the earlier coupons. The par rate only sets the coupon size.

  • Forgetting to add the coupon to the final payment

    The last cash flow is coupon plus principal, and the denominator becomes (1 + c).

    Fix: Divide by (1 + c), not by c or 1.

  • Leaving the answer as a discount factor

    The algebra stops one step early.

    Fix: Take (1 ÷ DF)^(1/n) − 1 and read the question to see whether it wants DF or a rate.

  • Rounding discount factors too early

    Rounding to two or three decimals feels neat.

    Fix: Keep at least six decimals until the final spot rate. Options can differ by only a few basis points.

  • Using the wrong root

    Students take the square root for a 3-year rate.

    Fix: The root equals the maturity in years: 1/2 for year 2, 1/3 for year 3.

Worked examples

Example 1

Annual-pay par rates are 2.00% for 1 year, 3.00% for 2 years and 4.00% for 3 years. What is the 3-year spot rate? A) 3.02% B) 4.06% C) 4.50%

Show the solution
  1. z₁ = 2.00%, so DF₁ = 1 ÷ 1.02 = 0.980392.
  2. DF₂ = [1 − 0.03 × 0.980392] ÷ 1.03 = 0.970588 ÷ 1.03 = 0.942318.
  3. Sum of DF₁ and DF₂ = 1.922710.
  4. DF₃ = [1 − 0.04 × 1.922710] ÷ 1.04 = (1 − 0.076908) ÷ 1.04 = 0.923092 ÷ 1.04 = 0.887588.
  5. z₃ = (1 ÷ 0.887588)^(1/3) − 1 = (1.126649)^(1/3) − 1 ≈ 1.04056 − 1.
  6. Calculator (BA II Plus): 1.126649 [yˣ] 0.333333 [=] then subtract 1. On the HP 12C: 1.126649 [ENTER] 3 [1/x] [yˣ].

Answer: B) about 4.06%. It is above the 4.00% par rate, as expected on an upward-sloping curve.

Example 2

The 1-year spot rate is 3.00% and the 2-year annual-pay par rate is 5.00%. What is the 2-year spot rate? A) 4.00% B) 5.05% C) 6.10%

Show the solution
  1. DF₁ = 1 ÷ 1.03 = 0.970874.
  2. DF₂ = [1 − 0.05 × 0.970874] ÷ 1.05 = (1 − 0.048544) ÷ 1.05 = 0.951456 ÷ 1.05 = 0.906149.
  3. z₂ = (1 ÷ 0.906149)^(1/2) − 1 = (1.103573)^0.5 − 1.
  4. √1.103573 ≈ 1.05051, so z₂ ≈ 5.05%.
  5. Check: 5 ÷ 1.03 + 105 ÷ 1.0505² ≈ 4.854 + 95.146 = 100.00.

Answer: B) about 5.05%. The 2-year spot rate is slightly above the 5.00% par rate.

Exam tips

  • Questions are standalone with three options, so you can often drop one option fast: on an upward-sloping par curve, the spot rate should exceed the par rate at that maturity.
  • Do not spend time computing spot rates for years the question does not ask about. Compute discount factors and convert once.
  • If the question gives spot rates and asks for a par rate, reverse the identity: par rate = (1 − DFₙ) ÷ (sum of DF₁ to DFₙ).
  • Read whether the question wants a discount factor, a spot rate or a forward rate. Forward rates come from the discount factors you just found.
  • Store DF values in calculator memory or write them down. There is no penalty for a wrong answer, so never leave a question blank.

Practice questions from The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Par Curve and Bootstrapping 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.

Par Curve and Bootstrapping Spot Rates: frequently asked questions

What is the par curve in CFA Level I?

The par curve is the set of par rates for bonds of different maturities. A par rate is the coupon rate that makes a bond price equal to its face value. It is a yield-to-maturity-style measure, not a rate for a single cash flow.

What is the difference between a par rate and a spot rate?

A par rate is one yield that fits a whole coupon bond priced at par. A spot rate is the yield on a zero-coupon bond for one maturity and discounts a single cash flow. Par rates are blends of spot rates, so the two differ unless the curve is flat.

How do you bootstrap spot rates step by step?

Set the 1-year spot rate equal to the 1-year par rate. For each longer maturity, solve the par bond identity for the final discount factor using the earlier ones, then convert it to a spot rate with (1 ÷ DF)^(1/n) − 1. Repeat until you reach the maturity you need.

Why is the spot rate above the par rate on an upward-sloping curve?

The par rate averages the spot rates for all the bond's cash flows, and the early ones are lower. The final payment carries the most weight and is discounted at the highest spot rate. So the longest spot rate must be above the par rate to balance the price at par.