Updated for the 2026-2027 CFA® Level I curriculum.
Par rates, spot rates, and forward rates describe the same no-arbitrage term structure from different angles. CFA Level I questions test how to convert among them.
Quick Answer
A par rate is the coupon rate that makes a bond price equal par. A forward rate is the future-period rate implied by current spot rates. Use discount factors to calculate par rates, compound spot rates to isolate forward rates, and chain forward rates to recover spot rates or price cash flows.
Key Takeaways
A par rate sets a coupon bond's price to par.
A forward rate applies to a future interval.
Spot and forward rates are linked by no-arbitrage compounding.
Forward rates are implied rates, not guaranteed forecasts.
All rates must use the same compounding convention.
What You Need to Know for CFA Level I
Calculate a par rate from discount factors.
Calculate a forward rate from spot rates.
Recover a spot rate from forward rates.
Price a bond using forward-derived discount factors.
This page focuses on calculations. The separate curve-comparison note explains how spot, par, yield, and forward curves differ.
What Are Par and Forward Rates?
The par rate is the coupon rate that makes a newly priced bond equal face value. A forward rate is the rate for a future borrowing or investment period implied by today's term structure.
How to Calculate a Par Rate
A par rate is calculated from the discount factors for each coupon-payment period. For a bond with annual payments:
where:
= par coupon rate per period
= discount factor for period t
= discount factor for the final payment period
= payment-period index
= number of payment periods
Adjust the coupon and rate periodicity when payments occur more than once per year.
How to Calculate Forward Rates From Spot Rates
To isolate the forward rate between periods m and n, equate the compounded return over the full horizon with the return earned through period m and then through the forward interval. With annual compounding:
where:
= m-period spot rate
= n-period spot rate
= forward rate from period m to period n
= number of periods to the start of the forward interval
= number of periods to the end of the forward interval
How to Recover Spot Rates From Forward Rates
You can recover an n-period spot rate by compounding the sequence of one-period forward rates through period n:
where:
= n-period spot rate
= one-period forward rate from period to period t
= period index in the forward-rate sequence
= number of periods to the spot-rate maturity
After compounding the forward rates, take the root and subtract 1 to solve for .
How to Price a Bond Using Forward Rates
Forward rates can also be chained to discount each bond cash flow back to today. For each cash flow at time t, use all one-period forward rates from today through that payment date:
where:
= current bond price
= cash flow in period t
= one-period forward rate from period to period j
= index for each forward-rate interval
= cash-flow period
= total number of payment periods
The product inside brackets compounds the forward rates through each maturity; raising it to converts that accumulation factor into the discount factor.
Worked Par and Forward Rate Example
With a one-year spot rate of 4% and a two-year spot rate of 5%:
Discount factors are .
A two-year par coupon rate is , or about 4.98%.
Illustrative Example
The 6.01% forward rate is the one-year rate beginning one year from today that makes a two-year investment consistent with the observed one-year and two-year spot rates. It is an implied no-arbitrage rate, not a promise about the future.
Common Exam Traps
Treating an implied forward rate as a guaranteed forecast
It is a rate implied by today's spot-rate relationship under the stated conventions, not a promise about the future realized short rate.
Using the wrong maturities in a forward-rate exponent
A one-year rate beginning two years from now spans years two to three; derive it from the two- and three-year discount factors or spot rates.
Mixing rates and discount factors without conversion
A discount factor is a present-value multiplier, while a spot rate is a yield. Convert consistently before solving for a forward rate.
Forgetting coupon frequency in a par-rate calculation
A semiannual-pay bond has half-year cash flow dates and periodic rates. Match the coupon and discount periods before solving for the coupon that makes price equal par.
Confusing a par yield with a spot rate
The par yield is the coupon rate that prices a bond at par using the relevant spot curve; each cash flow is discounted at its own maturity-specific spot rate.
Practice Question
The one-year spot rate is 4% and the two-year spot rate is 5%, with annual compounding. The one-year forward rate one year from now is closest to:
4.00%
5.00%
6.01%
Correct Answer: Option C
Using annual compounding, .
Option A: The 4.00% figure is the one-year spot rate, not the one-year forward rate beginning one year from now.
Option B: The 5.00% figure is the two-year spot rate, not the implied forward rate for the second year.
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FAQs About Par and Forward Rates
What is a par rate?
It is the coupon rate that makes a bond's price equal its face value.
How do you calculate a forward rate from spot rates?
Equate compounded returns over the same horizon and solve for the missing future-period rate.
Is a forward rate a forecast of a future spot rate?
Not by definition. It is the rate implied by current spot rates under no-arbitrage.