FRM Exam Part I · Bond Yields and Return Calculations

Spot, Forward and Par Rates: How to Bootstrap the Curve

Updated 11 October 2026 · Fact-checked

A spot rate is the yield on a zero-coupon bond to a given maturity. A forward rate is the rate implied today for a future period. A par rate is the coupon that prices a bond at par. Bootstrap spot rates from par yields one maturity at a time, then derive forwards from discount factors.

Understand Spot, Forward and Par Rates

A spot rate (zero rate) is the annualised yield on a zero-coupon bond that pays one cash flow at maturity. The 3-year spot rate s3 is the return you lock in today for lending money for exactly 3 years. Every cash flow is discounted at the spot rate for its own maturity. The discount factor is DF(t) = 1 ÷ (1 + s_t)^t, using annual compounding.

A forward rate is the rate for a future period that is implied by today's spot curve. The 2y1y forward is the rate for a one-year loan starting in 2 years. It follows from no-arbitrage: lending for 3 years must give the same result as lending for 2 years and then rolling into the implied 1-year forward. If spot rates rise with maturity, forwards sit above spots.

A par rate (par yield) is the coupon rate that makes a bond with that maturity price at exactly 100% of face value. Coupon bonds mix several cash flows, so a par yield is a kind of average of spot rates. For an upward-sloping curve, the par yield is below the spot rate of the same maturity, because the early coupons are discounted at lower spot rates.

Bootstrapping turns par yields into spot rates. You start with the shortest maturity, where par yield equals spot rate. For each longer maturity you set the price of the par bond equal to 1, plug in the discount factors you already know, and solve for the one unknown discount factor. Then you convert the discount factor to a spot rate. The par yield curve, the spot curve and the forward curve contain the same information. You can move between them.

In this guide, rates use annual compounding unless the question says otherwise. If a question uses semiannual par yields, the same method applies with half-year periods and coupons of c ÷ 2.

Key formulas to remember

Discount factor from spot rate: DF(t) = 1 ÷ (1 + s_t)^t; Annual compounding. s_t is the t-year spot rate as a decimal.
Spot rate from discount factor: s_t = (1 ÷ DF(t))^(1/t) − 1; Use the y^x key on your calculator with 1/t as the exponent.
Par bond condition: 1 = c × [DF(1) + DF(2) + … + DF(n)] + DF(n); c is the annual par coupon rate per 1 of face value. The bond prices at par.
Bootstrapping the last discount factor: DF(n) = [1 − c_n × (DF(1) + … + DF(n−1))] ÷ (1 + c_n); Needs the earlier discount factors. Work from the shortest maturity upward.
Forward rate from spot rates: (1 + s_T)^T = (1 + s_t)^t × (1 + f(t,T))^(T−t); f(t,T) is the forward rate from time t to T. Rearrange to solve for f.
One-period forward from discount factors: f(t, t+1) = DF(t) ÷ DF(t+1) − 1; Fastest route once you have discount factors.
Par yield from discount factors: c_n = (1 − DF(n)) ÷ [DF(1) + … + DF(n)]; Annual coupons. The reverse of bootstrapping.

How to solve Spot, Forward and Par Rates questions

Convert everything to discount factors first. Discount factors are the common language of spot, forward and par rates.

1Identify what you are given (par yields, spot rates or forwards) and what is asked. Note the compounding and the coupon frequency.
2If par yields are given, set the 1-year spot equal to the 1-year par yield (for an annual-pay curve).
3Compute DF(1) = 1 ÷ (1 + s1). Keep at least six decimal places.
4For each next maturity n, solve the par condition: DF(n) = [1 − c_n × sum of earlier DFs] ÷ (1 + c_n).
5Convert to a spot rate with s_n = (1 ÷ DF(n))^(1/n) − 1, if the question asks for spot rates.
6For a forward rate between t and T, use f = (DF(t) ÷ DF(T))^(1/(T−t)) − 1. For a one-year gap this is DF(t) ÷ DF(t+1) − 1.
7Check the answer: with an upward curve, forwards should be above spots, and spots above par yields at the same maturity.

Quickest way: Discount-factor shortcut

When to use it: Use it for any bootstrapping or forward question with annual compounding, especially when four answer options are close together.

Write the discount factors in a column on your scratch paper and store each in calculator memory.
Keep a running sum of DFs. You need it for every new par equation.
For one-year forwards, skip the spot rates and divide adjacent DFs: DF(t) ÷ DF(t+1) − 1.
Use the approximation f ≈ (T × s_T − t × s_t) ÷ (T − t) only to sense-check. It is close but not exact, so do not choose an answer from it when options are within 0.1%.
Eliminate options that break the ordering rule: on an upward curve, forward above the longer spot, and spot above par.

Common mistakes in Spot, Forward and Par Rates

Treating the par yield as the spot rate for maturities beyond one year.
Par yields are quoted like yields to maturity, so they look like the rate for that maturity.
Fix: Only the shortest maturity has par yield equal to spot rate. Bootstrap for every longer maturity.
Forgetting to include the final coupon plus principal as 1 + c in the last term.
Students discount the coupon but leave the principal out, or add it to the wrong date.
Fix: Write the par equation fully: 1 = c × (sum of DFs) + 1 × DF(n). Then divide by (1 + c) after moving the earlier terms across.
Computing a forward rate by subtracting spot rates, such as 4.5% − 4.0%.
The difference seems logical, but forwards depend on compounded growth over different horizons.
Fix: Use (1 + s_T)^T ÷ (1 + s_t)^t, then take the root of the length of the forward period.
Taking the wrong root when converting a discount factor to a spot rate.
Students use the square root for every maturity.
Fix: Use the exponent 1/n, where n is the maturity in years. For a 3-year DF, take the cube root.
Rounding discount factors to 2 or 3 decimals too early.
It feels easier, but errors compound across bootstrapping steps.
Fix: Keep six decimals through the calculation and round only the final rate.
Mixing compounding conventions, using semiannual coupons with annual-compounded discounting.
Market quotes for bonds are often semiannual, and the exam may switch between them.
Fix: Match periods to coupons. For semiannual, use half-year periods, coupon c ÷ 2 and rates divided by 2.

Worked examples

Example 1

Annual-pay par yields are 1-year 3.00%, 2-year 3.50% and 3-year 4.00%. Find the 2-year and 3-year spot rates, and the 2y1y forward rate (the one-year rate starting in 2 years). Use annual compounding.

Show the solution

1-year: s1 = 3.00%, so DF(1) = 1 ÷ 1.03 = 0.970874.
2-year par bond: 1 = 0.035 × DF(1) + 1.035 × DF(2).
DF(2) = (1 − 0.035 × 0.970874) ÷ 1.035 = (1 − 0.033981) ÷ 1.035 = 0.966019 ÷ 1.035 = 0.933352.
s2 = (1 ÷ 0.933352)^(1/2) − 1 = 1.071407^0.5 − 1 ≈ 3.51%.
3-year par bond: 1 = 0.04 × (DF(1) + DF(2)) + 1.04 × DF(3). The sum of DFs is 0.970874 + 0.933352 = 1.904226.
DF(3) = (1 − 0.04 × 1.904226) ÷ 1.04 = (1 − 0.076169) ÷ 1.04 = 0.923831 ÷ 1.04 = 0.888299.
s3 = (1 ÷ 0.888299)^(1/3) − 1 = 1.125747^(1/3) − 1 ≈ 4.03%.
Forward 2y1y = DF(2) ÷ DF(3) − 1 = 0.933352 ÷ 0.888299 − 1 ≈ 5.07%.

Answer: 2-year spot ≈ 3.51%, 3-year spot ≈ 4.03%, 2y1y forward ≈ 5.07%.

Example 2

The 2-year spot rate is 4.00% and the 3-year spot rate is 4.50%, both annually compounded. What is the one-year forward rate starting in 2 years? A) 4.50% B) 4.75% C) 5.51% D) 6.00%

Show the solution

Set up no-arbitrage: (1.045)^3 = (1.04)^2 × (1 + f).
Compute (1.045)^3 = 1.092025 × 1.045 = 1.141166.
Compute (1.04)^2 = 1.0816.
1 + f = 1.141166 ÷ 1.0816 = 1.05507.
f = 5.51%.
Sense check: the approximation 3 × 4.5% − 2 × 4.0% = 5.5% is close, and f is above the 3-year spot as it should be on a rising curve.

Answer: C) 5.51%

Exam tips

Questions often give par yields and ask for a forward rate. Plan for two stages: bootstrap to discount factors, then divide adjacent DFs.
Check the ordering on an upward curve: forward above spot, spot above par. If your answer violates this, recheck the arithmetic.
The simple approximation for forwards is usually close to the correct answer. Options may be built to catch you out, so compute exactly when options differ by small amounts.
Read the compounding and coupon frequency in the question. Semiannual par yields need half-year periods.
Practise with your financial calculator memory so each discount factor is stored and reused. This saves time over a 4-hour, 100-question exam.

Practice questions from Bond Yields and Return Calculations

Spot, Forward and Par Rates in other exams

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

Spot, Forward and Par Rates: frequently asked questions

What is the difference between spot rate, forward rate and par rate?

A spot rate is the yield on a zero-coupon bond to a given maturity. A forward rate is the rate for a future period implied by today's spot curve. A par rate is the coupon that prices a coupon bond at par.

How do I bootstrap spot rates from par yields?

Start with the 1-year par yield, which equals the 1-year spot rate. For each longer maturity, write the par bond equation using the known discount factors and solve for the one new discount factor. Then convert it into a spot rate.

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

A par bond pays coupons early, and those are discounted at lower short-term spot rates. This pulls the average yield below the spot rate at the final maturity. The effect reverses on a downward-sloping curve.

Do I need to memorise the forward rate formula for FRM Part I?

Yes. You must be able to derive a forward from spot rates quickly, either with the compounding equation or by dividing adjacent discount factors. Know both forms and use the one that needs fewer calculator steps.