FRM Exam Part I · Pricing Conventions, Discounting, and Arbitrage

Forward Rates and Par Yields: How to Calculate Them

Updated 11 October 2026 · Fact-checked

A forward rate is the interest rate for a future period implied by today's spot rates. Compute it by dividing the growth factor of the longer spot rate by that of the shorter, then taking the root for the forward period. A par yield is the coupon that prices a bond at par using the spot curve.

Understand Forward Rates and Par Yields

A spot rate (zero rate) is the annualised rate on a single payment received at a future date T with no coupons before it. The set of spot rates across maturities is the term structure.

A forward rate is the rate for a future period, such as years 1 to 2, that you can lock in today. It follows from no-arbitrage. Investing for 2 years at the 2-year spot rate must give the same result as investing for 1 year at the 1-year spot rate and then rolling over at the forward rate. If the two results differ, you could borrow in one and lend in the other for a risk-free profit.

This links forwards to the shape of the curve. When the spot curve slopes upward, the forward rate for a period lies above the spot rate for the same end date. When the curve is inverted, forwards lie below spot rates. The forward rate is the marginal rate of extending the investment by one more period.

A par yield is the coupon rate at which a bond with that maturity prices exactly at its face value, when each cash flow is discounted at its own spot rate. Yield to maturity (YTM) is different. It is the single discount rate that makes the present value of one bond's cash flows equal its market price. A bond priced at par has a YTM equal to its coupon rate. So the par yield is the YTM of a hypothetical par bond built from the spot curve.

For an upward-sloping spot curve, the par yield for a given maturity is usually below the spot rate for that maturity. The reason is that coupons arrive early and are discounted at lower rates.

Key formulas to remember

Forward rate, annual compounding: (1 + z₂)^T₂ = (1 + z₁)^T₁ × (1 + f)^(T₂ − T₁), so f = [(1 + z₂)^T₂ ÷ (1 + z₁)^T₁]^(1 ÷ (T₂ − T₁)) − 1; z₁ and z₂ are spot rates for maturities T₁ and T₂. f is the forward rate between T₁ and T₂.
Forward rate, continuous compounding: f = (z₂ × T₂ − z₁ × T₁) ÷ (T₂ − T₁); Valid only when all rates are continuously compounded. Here the forward is a weighted difference of the spot rates.
Forward rate from discount factors: (1 + f) = [d(T₁) ÷ d(T₂)]^(1 ÷ (T₂ − T₁)); d(T) = 1 ÷ (1 + z)^T is the discount factor. For a one-year forward period, 1 + f = d(T₁) ÷ d(T₂).
Par yield, annual coupons: c = (1 − d(N)) ÷ Σ d(t), for t = 1 to N; Per 1 of face value. Gives the annual coupon rate that prices the bond at par.
Par yield, m coupons per year: c = m × (1 − d(N)) ÷ Σ d(tᵢ); The sum runs over all coupon dates, spaced 1/m years apart. d(N) is the discount factor at final maturity.
Yield to maturity: Price = Σ C ÷ (1 + y)^t + Face ÷ (1 + y)^N; Solve for y by trial or with a financial calculator (N, PV, PMT, FV, then CPT I/Y).

How to solve Forward Rates and Par Yields questions

Use this method for any question on forward rates, par yields or YTM. First decide which quantity you are asked for, then keep the compounding convention consistent.

1Identify the target: forward rate, par yield or YTM. Note the periods involved, such as 1y into 2y, or a 3-year par bond.
2Write down the compounding convention (annual, semiannual or continuous) and keep it for every rate in the question.
3For a forward rate, compute the growth factor of each spot rate: (1 + z)^T for discrete compounding, or e^(zT) for continuous.
4Divide the longer growth factor by the shorter. Raise the result to 1 ÷ (T₂ − T₁) and subtract 1. For continuous rates use the weighted-difference formula instead.
5For a par yield, convert the spot rates to discount factors, add the coupon-date factors, then apply c = m × (1 − d(N)) ÷ Σ d.
6For YTM, set the bond price equal to the discounted cash flows at one rate y. Solve with the calculator, or check candidate answers by pricing the bond at each.
7Sanity check: an upward-sloping curve should give a forward above the spot rate. A par bond's YTM should equal its coupon.

Quickest way: Growth-factor ratio and calculator shortcuts

When to use it: Use it when the question gives spot rates and asks for a single forward rate, par yield or YTM, and you have the exam calculator.

For a one-year forward, compute (1 + z₂)^T₂ ÷ (1 + z₁)^T₁ in a single calculator line and subtract 1. For example (1.04)² ÷ 1.03 − 1.
For a rough check, the forward is about [z₂T₂ − z₁T₁] ÷ (T₂ − T₁). The estimate is close but not exact for annual compounding.
For a par yield, store the discount factors in memory and sum them. Then compute (1 − d(N)) ÷ sum.
For YTM, enter N, PV (negative price), PMT and FV in the TVM keys, then compute I/Y. Check that the compounding frequency matches the question.
If the answer options are far apart, estimate first and pick the nearest. Do a full calculation only when options are close.

Common mistakes in Forward Rates and Par Yields

Taking the simple difference of spot rates as the forward rate
It is quick and gives roughly the right size, so it feels correct.
Fix: Use the growth-factor ratio for discrete compounding. The simple difference works exactly only with continuous compounding, and then only after weighting by maturity.
Using the wrong exponent when taking the root
Students forget that the forward period is T₂ − T₁, not T₂.
Fix: Write the exponent 1 ÷ (T₂ − T₁) before you compute. For a one-year forward period it is simply 1.
Mixing compounding conventions
Some rates in a question are quoted continuous and others annual, or semiannual rates are treated as annual.
Fix: Convert every rate to one convention first. Check the question wording for how each rate is compounded.
Confusing par yield with YTM
Both are called yields and both can equal the coupon of a par bond.
Fix: Remember that par yield is derived from the whole spot curve and is the coupon that gives a price of par. YTM is one discount rate for one bond with a given price.
Forgetting the factor m in the par yield formula for semiannual coupons
The annual formula is memorised and applied to half-yearly discount factors.
Fix: Sum the half-yearly discount factors, then multiply (1 − d(N)) ÷ Σ d by m = 2 to get the annualised coupon rate.
Discounting coupons at the YTM when asked to price from the spot curve
YTM is the familiar pricing rate.
Fix: If spot rates are given, discount each cash flow at its own spot rate. Use YTM only when asked for the bond's single yield.

Worked examples

Example 1

The 1-year spot rate is 3% and the 2-year spot rate is 4%, both annually compounded. Find the 1-year forward rate starting in one year.

Show the solution

Use (1 + z₂)² = (1 + z₁) × (1 + f).
Growth factor for 2 years: 1.04² = 1.0816.
Growth factor for 1 year: 1.03.
1 + f = 1.0816 ÷ 1.03 = 1.05010.
f = 1.05010 − 1 = 0.05010, so about 5.01%.

Answer: The 1-year forward rate one year from now is about 5.01%.

Example 2

Annually compounded spot rates are 3% (1 year), 4% (2 years) and 5% (3 years). Find the par yield for a 3-year bond paying annual coupons.

Show the solution

Discount factors: d₁ = 1 ÷ 1.03 = 0.970874; d₂ = 1 ÷ 1.0816 = 0.924556; d₃ = 1 ÷ 1.157625 = 0.863838.
Sum of discount factors: 0.970874 + 0.924556 + 0.863838 = 2.759268.
Numerator: 1 − d₃ = 1 − 0.863838 = 0.136162.
c = 0.136162 ÷ 2.759268 = 0.049346, so about 4.93%.
Check: a 3-year bond with a 4.93% coupon priced at the spot curve is worth about 100, so its YTM also equals 4.93%.

Answer: The 3-year par yield is about 4.93%, which is below the 3-year spot rate of 5%.

Exam tips

Read the compounding convention in the stem first. Most lost marks on this topic come from a mismatch between annual, semiannual and continuous rates.
Use the sanity checks. With an upward-sloping curve the forward should sit above the later spot rate, and the par yield below it.
If a question gives a bond price and asks for YTM, use the calculator TVM keys. Do not iterate by hand.
Expect discount factors as inputs in some questions. Know the formula that goes straight from d(T₁) and d(T₂) to the forward rate.
In multiple-choice options, the wrong answers often come from the simple-difference shortcut or a wrong exponent. Compute the correct value before comparing options.

Practice questions from Pricing Conventions, Discounting, and Arbitrage

Forward Rates and Par Yields in other exams

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

Forward Rates and Par Yields: frequently asked questions

How do I calculate an implied forward rate from spot rates?

Divide the growth factor of the longer spot rate, (1 + z₂)^T₂, by that of the shorter, (1 + z₁)^T₁. Raise the ratio to 1 ÷ (T₂ − T₁) and subtract 1. With continuous compounding use (z₂T₂ − z₁T₁) ÷ (T₂ − T₁).

What is the difference between par yield and yield to maturity?

Par yield is the coupon rate that makes a bond price at par when each cash flow is discounted at its own spot rate. YTM is the single rate that equates a specific bond's cash flows to its market price. For a bond priced at par, the two are the same number.

Why is the forward rate above the spot rate when the curve slopes upward?

The longer spot rate is an average of rates over all periods up to that date. If that average is rising, the rate for the last period must be higher than the average. So the forward exceeds the spot rate for the same end date.

Do I need a financial calculator for forward rates and par yields?

A standard calculator with power and root keys is enough for forward rates and par yields. A financial calculator helps most for YTM, where you enter N, PV, PMT and FV and compute I/Y.