Lesson 2 of 4 11–14% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Par rates from spot rates

A par rate is the coupon rate (and yield) that makes a bond priced with today's spot rates worth exactly 100. Set price equal to 100 in the spot-pricing equation and solve for the coupon. Par curves strip out the tax and trading quirks of discount and premium bonds.

6 min cram, 28 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 9. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Spot rates and pricing a bond with the spot curve

On the exam

Define par and forward rates, and calculate par rates, forward rates from spot rates, spot rates from forward rates, and the price of a bond using forward rates

Define par rates and calculate them from spot rates. Expect a 3- or 4-year par rate from discount factors, a trap that averages the spots or uses the final spot, and a question on how the par curve sits against an upward spot curve.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What is a par rate?

The par rate for a maturity is the coupon rate that makes a bond, valued with today’s spot rates, worth exactly 100. A bond priced at par has a yield equal to its coupon, so the same number is also the bond’s yield. Plotted against maturity, par rates form the par curve .

Analysts use par rates because real bonds trade at a discount or premium, and that brings tax and trading distortions. A hypothetical bond at par avoids them. Official government yield curves are often published as par curves for this reason. Actual on-the-run bonds have yields close to, but not exactly equal to, par rates.

Three rates that can describe one bond

Rate for a bond or a maturitythey agree only in special cases

Spot rate Memorize yield on a zero-coupon bond; one cash flow, one date
Par rate Memorize coupon and yield of a hypothetical bond priced at 100 off the spot curve
- 1-year par rate equals the 1-year spot rate
- Longer par rates a blend of the spot rates up to maturity
Yield to maturity Memorize internal rate of return (IRR) of a real bond at whatever price it trades

InteractivePar rates from the spot curve

Try this 0 of 3

Make the par curve sit exactly on the spot curve
Invert the spot curve and compare par with spot
From the worked example, raise only the 4-year spot rate by 0.50

Spot rates (zero-coupon yields), annual:

z₁%z₂%z₃%z₄%

Show the working for:

2-year par rate3.691%

2-year spot rate3.70%

Sum of d₁ to d₂1.89891

From spot rates to par rates (d = 1 ÷ (1 + z)ᵗ)
Year	Spot	d	Sum of d	Par
1	3.20%	0.9690	0.9690	3.200%
2	3.70%	0.9299	1.8989	3.691%
3	4.10%	0.8864	2.7853	4.077%
4	4.40%	0.8418	3.6271	4.362%

Spot curve
Par curve

The 4-year par rate (4.362%) is 0.038% below the 4-year spot rate (4.40%): par rates blend all the spot rates up to maturity, so they sit on the side of the earlier ones, which are lower on a rising curve.

Show working

Price = 100 means 100 = PMT × (d₁ + … + d₂) + 100 × d₂, so PMT = 100 × (1 − d₂) ÷ (d₁ + … + d₂).

d₁ = 1 ÷ (1 + 3.20%) = 0.96899

d₂ = 1 ÷ (1 + 3.70%)² = 0.92991

2-year par rate = (1 − 0.92991) ÷ (0.96899 + 0.92991) = 0.07009 ÷ 1.89891 = 3.691%.

Check: a 2-year bond with a 3.691% coupon priced off the spot curve: 3.576 + 96.424 = 100.000, exactly par.

The 2-year par rate on your calculator

1.0321/xSTO1d₁ = 0.96899
1.037x²1/xSTO2d₂ = 0.92991
1−RCL2=÷(RCL1+RCL2)=×100=3.6909, the 2-year par rate in %

1/x reciprocal · STO store · x² square · RCL recall a stored number

1.0321/xSTO1d₁ = 0.96899
1.037ENTER2y^x1/xSTO2d₂ = 0.92991
1RCL2−RCL1RCL2+÷100×3.6909, the 2-year par rate in %

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

Display: 3.6909 = 3.691%

The par rate is the coupon that makes the spot curve price a bond at exactly 100.

How do you calculate a par rate from spot rates?

Memorize

Start from the spot-pricing equation and set the price to 100. A discount factor turns each spot rate into the present value of one unit paid at that date.

Discount factor Use it

d_t = \frac{1}{(1 + z_t)^{t}}

With coupon $c$ per 100, the price equals 100 when

100 = c\,(d_1 + d_2 + \cdots + d_N) + 100\,d_N

Solve for $c$ :

Par rate from spot rates Memorize

c_N = \frac{1 - d_N}{d_1 + d_2 + \cdots + d_N}

The result is a decimal per unit of par. Multiply by 100 for a percent. The numerator is the part of the 100 that the coupons must supply, because the principal alone is worth $d_N$ .

This works on a coupon date. Between coupon dates, it is the flat price (without accrued interest), not the full price, that is set equal to 100.

Worked example The 2-year par rate

Spot rates are 3.20% (1 year), 3.70% (2 years), 4.10% (3 years) and 4.40% (4 years). Find the 2-year par rate.

d₁ = 1 ÷ 1.032 = 0.96899.
d₂ = 1 ÷ 1.037² = 0.92991.
Par rate = (1 − 0.92991) ÷ (0.96899 + 0.92991) = 0.07009 ÷ 1.89890 = 3.691%.

The 1-year par rate is 3.200%, the same as the 1-year spot rate, because a one-year bond has a single cash flow.

On your calculator

1.0321/xSTO1
1.037y^x2=1/xSTO2
1−RCL2=÷(RCL1+RCL2)=×100= 3.6909

1/x reciprocal · STO store · y^x power · RCL recall a stored number

1.0321/xSTO1
1.037ENTER2y^x1/xSTO2
1RCL2−RCL1RCL2+÷100× 3.6909

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

Worked example The 4-year par rate, and a check

Same spot rates. d₃ = 0.88644 and d₄ = 0.84178. The sum of the four discount factors is 3.62712.

Par rate = (1 − 0.84178) ÷ 3.62712 = 4.362%.
The 3-year par rate works out the same way: 4.077%.
Check: price a 4-year bond with a 4.362% coupon off the spot curve. 4.227 + 4.056 + 3.867 + 87.850 = 100.000.

The 4-year par rate on your calculator

1.0321/xSTO1
1.037y^x2=1/xSTO2
1.041y^x3=1/xSTO3
1.044y^x4=1/xSTO4
1−RCL4=÷(RCL1+RCL2+RCL3+RCL4)=×100= 4.3622

1/x reciprocal · STO store · y^x power · RCL recall a stored number

1.0321/xSTO1
1.037ENTER2y^x1/xSTO2
1.041ENTER3y^x1/xSTO3
1.044ENTER4y^x1/xSTO4
1RCL4−RCL1RCL2+RCL3+RCL4+÷100× 4.3622

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

Rearrange itPar rate: solve for the par rate or the last spot rate

The question asks for

Spot rates are 3.20% (1 year), 3.70% (2 years), 4.10% (3 years) and 4.40% (4 years). The 4-year par rate is closest to:

1-year spot rate%2-year spot rate%3-year spot rate%4-year spot rate%

c_{4} = \frac{1 - d _{4}}{d _{1} + d _{2} + d _{3} + d _{4}}, d_{t} = \frac{1}{( 1 + z _{t} ) ^{t}}

(1 − 0.84178) ÷ 3.62712 = 4.362%

4-year par rate4.36%

On your calculator

1.0321/xSTO1
1.037y^x2=1/xSTO2
1.041y^x3=1/xSTO3
1.044y^x4=1/xSTO4
1−RCL4=÷(RCL1+RCL2+RCL3+RCL4)=×100=4.3622

1/x reciprocal · STO store · y^x power · RCL recall a stored number

1.0321/xSTO1
1.037ENTER2y^x1/xSTO2
1.041ENTER3y^x1/xSTO3
1.044ENTER4y^x1/xSTO4
1RCL4−RCL1RCL2+RCL3+RCL4+÷100×4.3622

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

How is a par rate different from a yield to maturity?

Both are yields, but they answer different questions. A par rate assumes the bond is worth 100 and discounts each cash flow at its spot rate. A yield to maturity is the single rate that equates a real bond’s cash flows with whatever price it trades at. A bond priced at par off the spot curve has a YTM equal to its par rate, by construction.

Exam tip Spot rates, par rates and YTMs for the same maturity are three different numbers. Read the question for which one is asked, and check the 1-year case: there, spot and par match.

Go deeper Why official yield curves are par curves

A published curve is meant to be a clean benchmark for pricing and for setting coupons on new bonds. A par curve answers directly: what coupon would a new bond at par need? It also avoids the bumps that come from older bonds trading at a discount or premium.

Go deeper Par rates and swap fixed rates

The fixed rate on a plain interest rate swap is found the same way: the fixed payment that makes the swap worth zero at the start, with floating payments worth par. The calculation is the par-rate formula with discount factors from the relevant curve. The Derivatives topic develops it.

Memorize this

MemorizePar rate from spot rates

Memory hook Price 100 = coupons plus principal. Principal is worth 100 × d_N, so the coupons must be worth 100 × (1 − d_N).

Know it cold $c_N = \dfrac{1 - d_N}{\sum_{t=1}^{N} d_t}$ with $d_t = 1/(1 + z_t)^t$ . Multiply by 100 for a percent. Never forget the final principal.

MemorizePar rate versus spot rate

Memory hook The first year has no coupons before it, so 1-year par equals 1-year spot.

Know it cold With positive rates: upward spot curve, par curve slightly below it. Flat: equal. Inverted: slightly above. (Negative spot rates flip the par side.) Par rate = coupon = yield for a bond at par.

Three rates that can describe one bond

Rate for a bond or a maturitythey agree only in special cases

Spot rate Memorize yield on a zero-coupon bond; one cash flow, one date
Par rate Memorize coupon and yield of a hypothetical bond priced at 100 off the spot curve
- 1-year par rate equals the 1-year spot rate
- Longer par rates a blend of the spot rates up to maturity
Yield to maturity Memorize internal rate of return (IRR) of a real bond at whatever price it trades

Discount factor

d_t = \dfrac{1}{(1 + z_t)^{t}}

Quick example

The 3-year spot rate is 3.6%. d₃ = 1 ÷ 1.036³ = 0.8993: one unit paid in 3 years is worth 0.8993 today.

On your calculator

1.036y^x3=1/x 0.8993

y^x power · 1/x reciprocal

1.036ENTER3y^x1/x 0.8993

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Par rate from spot rates

c_N = \dfrac{1 - d_N}{d_1 + d_2 + \cdots + d_N}

Quick example

Spot rates 2.0% (1 year) and 2.6% (2 years). d₁ = 0.98039, d₂ = 0.94996. Par rate = (1 − 0.94996) ÷ (0.98039 + 0.94996) = 2.592%.

On your calculator

1.021/xSTO1
1.026y^x2=1/xSTO2
1−RCL2=÷(RCL1+RCL2)=×100= 2.5923

1/x reciprocal · STO store · y^x power · RCL recall a stored number

1.021/xSTO1
1.026ENTER2y^x1/xSTO2
1RCL2−RCL1RCL2+÷100× 2.5923

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

How it is asked

3-year par rate from three spot rates

Spot rates 2.4%, 2.9%, 3.3%. 3-year par rate?

d = 0.97656, 0.94443, 0.90719; (1 − 0.90719) ÷ 2.82818 = 3.28%. Averaging the spots (2.87%) or using 3.3% are the traps.
Where is the par curve?

The spot curve slopes upward. Par curve relative to spot?

Slightly below spot.
Par rate or YTM?

Which rate is the IRR of a bond at its market price?

The YTM. A par rate assumes a price of 100 and spot-rate discounting.

Exam traps

Reporting the N-year spot rate as the N-year par rate. They are equal only at one year, or on a flat curve.
Averaging the spot rates.
Forgetting the final principal's discount factor in the numerator, 1 − d_N.
Confusing the par rate with the YTM. The YTM is the IRR at the market price, whatever it is.

Checkpoint

4 exam-style questions. About 90 seconds each.

1Spot rates are 3.0% (1 year), 3.5% (2 years) and 3.9% (3 years). The 3-year par rate is closest to:
2When spot rates are positive and the spot curve slopes upward, the par curve most likely lies:
3A par rate is best described as the:
4Compared with the 1-year spot rate, the 1-year par rate is:

Key takeaways

Par rate: the coupon that makes the spot-curve price equal 100.
Par = (1 − d_N) ÷ sum of d_t, with d_t = 1 ÷ (1 + z_t)^t.
Upward spot curve with positive rates: par slightly below spot. The 1-year par rate equals the 1-year spot rate.

Discount factor

d_t = \dfrac{1}{(1 + z_t)^{t}}

Quick example

The 3-year spot rate is 3.6%. d₃ = 1 ÷ 1.036³ = 0.8993: one unit paid in 3 years is worth 0.8993 today.

On your calculator

1.036y^x3=1/x 0.8993

y^x power · 1/x reciprocal

1.036ENTER3y^x1/x 0.8993

ENTER enter (on the HP, puts the number on the stack) · y^x power · 1/x reciprocal

Par rate from spot rates

c_N = \dfrac{1 - d_N}{d_1 + d_2 + \cdots + d_N}

Quick example

Spot rates 2.0% (1 year) and 2.6% (2 years). d₁ = 0.98039, d₂ = 0.94996. Par rate = (1 − 0.94996) ÷ (0.98039 + 0.94996) = 2.592%.

On your calculator

1.021/xSTO1
1.026y^x2=1/xSTO2
1−RCL2=÷(RCL1+RCL2)=×100= 2.5923

1/x reciprocal · STO store · y^x power · RCL recall a stored number

1.021/xSTO1
1.026ENTER2y^x1/xSTO2
1RCL2−RCL1RCL2+÷100× 2.5923

1/x reciprocal · STO store · ENTER enter (on the HP, puts the number on the stack) · y^x power · RCL recall a stored number

All Fixed Income formulas, every lesson

Practice exam-style questions on Term Structure: Spot, Par and Forward Curves

Next lesson Implied forward rates: from spot to forward and back

Questions about this lesson?

Where this comes back

Implied forward rates: from spot to forward and back Term Structure: Spot, Par and Forward Curves
Pricing and Valuing Interest Rate Swaps Derivatives

Questions candidates ask

What is a par rate?

The coupon rate, equal to the yield, at which a bond priced from the spot curve is worth exactly 100. It is a hypothetical benchmark bond, not a quoted market bond.

How do you calculate a par rate from spot rates?

Compute each discount factor, 1 ÷ (1 + spot)^t. Then divide one minus the final discount factor by the sum of all the discount factors.

Why is the par curve below the spot curve?

When rates are positive and the curve slopes up, earlier coupons are discounted at lower rates, so a smaller coupon is enough to reach 100. The gap is small because the final payment dominates. With negative spot rates the par curve sits slightly above an upward spot curve instead.

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