FRM Exam Part I · Properties of Interest Rates

Zero Rates, Bond Pricing and Bootstrapping for FRM Part I

Q: What is the difference between a par yield and a zero rate?

A zero rate applies to a single payment at maturity. The par yield is the coupon rate that makes a coupon bond price equal its face value. It depends on all the zero rates up to maturity, so it blends them.

Q: How do I calculate a bond price using zero rates?

Discount each cash flow at the zero rate for its own date, then add the present values. The final cash flow is the last coupon plus the face value.

Q: How does bootstrapping a zero curve work?

Start with the shortest maturity and read its zero rate directly. For each longer bond, discount the known earlier coupons with the zero rates you have, subtract from the price, and solve for the last discount factor. Convert it to a zero rate and repeat.

Q: Is bond yield the same as the zero rate?

No. Bond yield is one rate that equates the price to all cash flows. Zero rates differ by maturity. They match only when the curve is flat, or for a zero-coupon bond.

Updated 11 October 2026 · Fact-checked

A zero rate is the yield on a zero-coupon bond paying one cash flow at maturity. You price a coupon bond by discounting each cash flow at the zero rate for its own date. Bootstrapping builds the zero curve step by step, solving for each new zero rate from bond prices and the shorter zero rates already found.

Understand Zero Rates, Bond Pricing and Bootstrapping

A zero rate (spot rate) for maturity T is the return you earn on a investment that pays nothing until T, then pays everything at once. A 2-year zero rate of 4% means ₹100 today grows to 100 × (1.04)² if annual compounding is used. It has no reinvestment question, so it is the cleanest measure of the time value of money for a given maturity.

A coupon bond is a bundle of zero-coupon bonds. Each coupon and the principal is a separate cash flow with its own date. To price the bond, discount each cash flow at the zero rate for that date, then add them up. Using one rate for all cash flows is not how a no-arbitrage price is built.

The bond yield (yield to maturity) is different. It is the single discount rate that makes the present value of all cash flows equal the market price. It is a kind of average of the zero rates, weighted by the cash flows. Two bonds with the same maturity but different coupons can have different yields, because they weight the zero rates differently. The par yield is the coupon rate that makes a bond price equal its face value.

Bootstrapping builds the zero curve from market instruments. Start with the shortest maturity, where the zero rate is read directly (a zero-coupon bill or a short rate). For the next maturity, use a coupon bond price. All cash flows except the last are discounted with zero rates you already know. The final cash flow (coupon plus principal) is the only unknown, so you solve for its discount factor. Repeat for each longer maturity.

Key formulas to remember

Discount factor from a zero rate (annual compounding): d(T) = 1 ÷ (1 + z)^T; z is the T-year zero rate. With continuous compounding, d(T) = e^(−zT).
Bond price using zero rates: P = Σ C_t × d(t) + F × d(T); Each cash flow is discounted at its own zero rate. C is the coupon, F the face value.
Bond yield (YTM): P = Σ C_t ÷ (1 + y)^t + F ÷ (1 + y)^T; One rate y for all cash flows. Solve with a financial calculator (N, PV, PMT, FV, then CPT I/Y).
Par yield (annual coupons): c = (1 − d(T)) ÷ Σ d(t); c is the annual coupon rate as a fraction of face value, with the sum over all coupon dates t = 1 to T. Price then equals face value.
Bootstrapping the final zero rate: d(T) = (P − Σ C × d(t) for t < T) ÷ (F + C); Use for a bond whose last cash flow is F + C. Then z = (1 ÷ d(T))^(1/T) − 1.
Convert between compounding conventions: (1 + z_annual) = e^(z_cc); z_cc = ln(1 + z_annual). For m compounding periods per year, (1 + z_m ÷ m)^m = e^(z_cc).

How to solve Zero Rates, Bond Pricing and Bootstrapping questions

Use this method for any question on zero rates, bond prices or bootstrapping.

1Read the compounding convention and coupon frequency. Note whether rates are annual, semiannual or continuous.
2List every cash flow with its date. The last one is coupon plus face value.
3Convert each zero rate to a discount factor for its own date.
4For pricing, multiply each cash flow by its discount factor and add them up.
5For bootstrapping, start at the shortest maturity. Subtract the present value of known earlier coupons from the bond price, then divide by the final cash flow to get the new discount factor.
6Convert the discount factor back to a zero rate: z = (1 ÷ d)^(1/T) − 1, or z = −ln(d) ÷ T if continuous.
7For yield or par yield, use the right tool: a calculator for YTM, the par yield formula with discount factors for par rates.
8Check the answer. A bond priced at par should have yield equal to coupon. On an upward curve, the zero rate for a maturity should sit above the yield of a coupon bond of that maturity (the bond yield is below the zero rate).

Quickest way: Discount factor table method

When to use it: Use when the question gives several zero rates or bond prices and asks for a price, a par yield or a new zero rate.

Turn every given rate into a discount factor first and write them in a short list.
Compute the sum of discount factors once. It is reused for par yield and annuity values.
For par yield, compute (1 − d(T)) ÷ sum of d. You do not need to solve for price.
For bootstrapping, only the final discount factor is unknown. One division gives it.
Eliminate answer options that break basic logic, such as a zero rate below the yield on an upward curve.

Common mistakes in Zero Rates, Bond Pricing and Bootstrapping

Discounting every cash flow at the bond yield when the question gives zero rates.
Yield is the familiar single-rate method from basic bond pricing.
Fix: If zero rates are given, use a separate rate for each cash flow date.
Forgetting that the last cash flow includes the principal when bootstrapping.
Students divide by the coupon alone.
Fix: Divide by face value plus the final coupon.
Treating the par yield and the zero rate as the same thing.
Both are quoted per year for a given maturity.
Fix: Zero rate is for a single payment at T. Par yield is the coupon rate of a bond priced at par, so it averages several zero rates.
Mixing compounding conventions, such as using a semiannual rate as annual.
The rate is quoted without noticing the convention.
Fix: Use semiannual periods (rate ÷ 2, number of periods × 2) or convert first.
Using the wrong number of periods for the discount factor on a semiannual bond.
Students keep T in years but use a per-period rate.
Fix: Match the rate and the exponent: per-period rate with period count, annual rate with years.
Assuming bond yield is always above or below the zero rate of the same maturity.
Students memorize one case.
Fix: On an upward-sloping curve, a coupon bond yield is usually below the zero rate of the same maturity; on a downward curve it is usually above. Reason from the curve shape rather than memorizing.

Worked examples

Example 1

The 1-year, 2-year and 3-year annually compounded zero rates are 3%, 4% and 5%. Price a 3-year bond with a 6% annual coupon and face value 100.

Show the solution

Discount factors: d(1) = 1 ÷ 1.03 = 0.970874; d(2) = 1 ÷ 1.04² = 1 ÷ 1.0816 = 0.924556; d(3) = 1 ÷ 1.05³ = 1 ÷ 1.157625 = 0.863838.
PV of year 1 coupon: 6 × 0.970874 = 5.8252.
PV of year 2 coupon: 6 × 0.924556 = 5.5473.
PV of year 3 cash flow: 106 × 0.863838 = 91.5668.
Sum: 5.8252 + 5.5473 + 91.5668 = 102.9393.

Answer: Price ≈ 102.94

Example 2

A 1-year zero-coupon bond with face value 100 trades at 96. A 2-year bond with a 5% annual coupon and face value 100 trades at 99. Find the 2-year annually compounded zero rate.

Show the solution

1-year discount factor: d(1) = 96 ÷ 100 = 0.96.
PV of the first coupon: 5 × 0.96 = 4.80.
Remaining price for the final cash flow: 99 − 4.80 = 94.20.
Final cash flow is 105, so d(2) = 94.20 ÷ 105 = 0.897143.
z(2) = (1 ÷ 0.897143)^(1/2) − 1 = (1.114650)^(0.5) − 1 = 1.055770 − 1 = 0.05577.

Answer: 2-year zero rate ≈ 5.58%

Exam tips

Most questions give a price table and ask for one zero rate or a price. Practise the one-division bootstrapping step until it takes under a minute.
Check the compounding wording first. Annual, semiannual and continuous give different answers from the same numbers.
Know the logic of yield versus zero rate: on an upward-sloping curve, a coupon bond yield is below the zero rate for the same maturity.
Use the financial calculator only for YTM. For zero-rate pricing, a plain calculator with discount factors is faster and safer.
Par yield questions are shortcuts: (1 − d(T)) ÷ sum of d, and the result must be in annual terms unless you are told otherwise.

Practice questions from Properties of Interest Rates

Zero Rates, Bond Pricing and Bootstrapping in other exams

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

CFA Level IPar Curve and Bootstrapping Spot Rates

Zero Rates, Bond Pricing and Bootstrapping: frequently asked questions

What is the difference between a par yield and a zero rate?

A zero rate applies to a single payment at maturity. The par yield is the coupon rate that makes a coupon bond price equal its face value. It depends on all the zero rates up to maturity, so it blends them.

How do I calculate a bond price using zero rates?

Discount each cash flow at the zero rate for its own date, then add the present values. The final cash flow is the last coupon plus the face value.

How does bootstrapping a zero curve work?

Start with the shortest maturity and read its zero rate directly. For each longer bond, discount the known earlier coupons with the zero rates you have, subtract from the price, and solve for the last discount factor. Convert it to a zero rate and repeat.

Is bond yield the same as the zero rate?

No. Bond yield is one rate that equates the price to all cash flows. Zero rates differ by maturity. They match only when the curve is flat, or for a zero-coupon bond.