FRM Exam Part I · Interest Rates

Bond Pricing and Yield to Maturity for FRM Part I

Q: What is the difference between YTM and the spot rate?

A spot (zero) rate discounts a single cash flow at one maturity. YTM is one rate that discounts all of a bond's cash flows to its price. For a coupon bond on a sloped curve they differ. For a zero-coupon bond they are the same.

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

List each coupon and the principal with its date. Discount each at the zero rate for that date, using the stated compounding. Add the present values. The total is the bond price.

Q: What is the bond-equivalent yield?

It is the yield on a bond paying semiannual coupons, found by doubling the semiannual rate. It ignores compounding within the year. To get an effective annual yield, use (1 + y/2)^2 − 1.

Q: Which day count conventions should I know for FRM Part I?

Know actual/actual for US Treasury bonds, 30/360 for many corporate and municipal bonds, and actual/360 for money market instruments. They set how accrued interest is calculated. Quoted bond prices are clean, so accrued interest is added to get the cash price.

Updated 11 October 2026 · Fact-checked

A bond's price is the present value of its coupons and principal. You can discount each cash flow at its own zero rate, or discount all of them at one yield to maturity (YTM). YTM is the single rate that makes the discounted cash flows equal the market price. You find it by trial or a calculator.

Understand Bond Pricing and Yield to Maturity

A bond is a set of promised cash flows: coupons on fixed dates and the principal at maturity. Its price today is what those cash flows are worth today. Money received later is worth less, so you discount each cash flow back to today.

The cleanest way is to use zero rates (spot rates). A zero rate is the yield on a zero-coupon bond for one specific maturity. Each cash flow is discounted at the zero rate for its own date. A 1-year coupon uses the 1-year rate. A 3-year coupon uses the 3-year rate. This is the no-arbitrage price, because the bond is just a bundle of zero-coupon bonds.

Yield to maturity takes a shortcut. It is the single discount rate that, applied to every cash flow, gives the bond's market price. You are given the price and solve for the rate. YTM is a blend of the zero rates along the curve, weighted by the size and timing of cash flows. It is not the same as a spot rate. They are equal only when the curve is flat, or for a zero-coupon bond.

A bond priced at par has YTM equal to the coupon rate. A bond priced below par (a discount) has YTM above the coupon rate. A bond priced above par (a premium) has YTM below the coupon rate.

Quoting matters. A semiannual bond quotes its yield as a bond-equivalent yield: the semiannual yield multiplied by 2, with no compounding effect. To compare it with an annually compounded yield, convert it. Day counts also matter, because accrued interest depends on them. Treasuries use actual/actual, and US corporates use 30/360. The quoted (clean) price excludes accrued interest. The cash (dirty) price includes it.

Key formulas to remember

Price using zero rates (annual compounding): P = Σ C_t ÷ (1 + z_t)^t; Each cash flow C_t uses the zero rate z_t for its own maturity t. The final cash flow includes the principal.
Price using YTM (semiannual): P = Σ (c/2) ÷ (1 + y/2)^k + F ÷ (1 + y/2)^n; c is the annual coupon, F the face value, n the number of half-years, y the bond-equivalent yield. k runs from 1 to n.
Price with continuous compounding: P = Σ C_t × e^(−z_t × t); Use when zero rates are quoted with continuous compounding.
Yield to maturity definition: P = Σ CF_t ÷ (1 + y)^t, solve for y; No closed form for coupon bonds. Use a financial calculator (N, I/Y, PV, PMT, FV).
Convert semiannual yield to effective annual: EAY = (1 + y/2)^2 − 1; The bond-equivalent yield y is twice the semiannual rate.
Dirty price: Dirty price = Clean price + Accrued interest; Accrued interest = coupon × (days since last coupon ÷ days in coupon period).
Par, premium, discount: Price < par ⇔ YTM > coupon rate; Price > par ⇔ YTM < coupon rate; Price = par when YTM equals the coupon rate on a coupon date.

How to solve Bond Pricing and Yield to Maturity questions

Use this order for any pricing or yield question. It keeps the compounding and the periods consistent.

1Identify what is given: face value, coupon rate, coupon frequency, maturity, and either the rates (zero or yield) or the price.
2Convert to periods. For semiannual bonds, use n = years × 2 and a periodic coupon = annual coupon ÷ 2.
3List every cash flow with its date. Add the face value to the last coupon.
4Check the compounding of the given rates (annual, semiannual or continuous). Use the matching discount factor.
5To price: discount each cash flow at its zero rate, or at the YTM if one yield is given, and add them up.
6To find YTM: enter N, PV (negative), PMT and FV in the calculator and solve for I/Y. Then multiply by the periods per year to get the bond-equivalent yield.
7Sanity check. A discount bond must have YTM above the coupon rate. A premium bond must have YTM below it.
8If a quoted price is involved, add accrued interest to get the cash price, using the stated day count.

Quickest way: Calculator TVM shortcut with par and direction checks

When to use it: Use it for YTM questions with a single yield, or when you need a fast price from a given YTM.

Set the calculator to one payment per period. Use periodic values: N = number of periods, PMT = periodic coupon, FV = face value.
Price: enter I/Y as the periodic yield (y ÷ 2 for semiannual) and compute PV. Ignore the sign.
YTM: enter PV as negative of the price and compute I/Y. Multiply by 2 for the bond-equivalent yield.
Before calculating, decide if the answer should be above or below the coupon rate. Eliminate options that break this.
For zero-rate pricing, there is no shortcut. Compute each discount factor and add the terms.

Common mistakes in Bond Pricing and Yield to Maturity

Using the annual coupon and annual yield with semiannual periods
You forget that n and the rate must be per period.
Fix: Halve the coupon and the yield, and double the years to get n.
Treating YTM as the spot rate for the bond's maturity
Both are called a yield for the same maturity.
Fix: A spot rate discounts one cash flow. YTM is one rate that fits all cash flows. They differ unless the curve is flat or the bond is a zero.
Leaving the principal out of the final cash flow
You list coupons and forget the face value repayment.
Fix: The last cash flow is the final coupon plus face value.
Calling the semiannual bond-equivalent yield the effective annual yield
The quoted yield is simply twice the semiannual rate.
Fix: Convert with (1 + y/2)^2 − 1 when annual compounding is asked.
Mixing clean and dirty prices
Quotes are clean, but the buyer pays the dirty price.
Fix: Add accrued interest to the quoted price to get the cash paid. Compute YTM from the dirty price.
Using a discount factor with the wrong compounding
The zero rates are continuous but you use (1 + z)^t.
Fix: Read the stated compounding first. Use e^(−zt) for continuous rates.

Worked examples

Example 1

Zero rates (annual compounding) are 1-year 3.0%, 2-year 3.5% and 3-year 4.0%. A 3-year bond pays an annual coupon of 5% on a face value of $100. What is its price?

Show the solution

Cash flows: $5 at year 1, $5 at year 2, $105 at year 3.
Year 1: 5 ÷ 1.03 = 4.8544.
Year 2: 5 ÷ 1.035² = 5 ÷ 1.071225 = 4.6677.
Year 3: 105 ÷ 1.04³ = 105 ÷ 1.124864 = 93.3434.
Sum: 4.8544 + 4.6677 + 93.3434 = 102.8655.

Answer: Price ≈ $102.87

Example 2

A 2-year bond with face value $100 pays a 6% coupon semiannually and is priced at $100.00 in the market with a bond-equivalent yield of 6.00%. Using the same bond, what is the price if the yield rises to 8.00% (bond-equivalent, semiannual)?

Show the solution

Periodic coupon = 6 ÷ 2 = $3. Periodic yield = 8% ÷ 2 = 4%. n = 4.
Annuity factor = (1 − 1.04^−4) ÷ 0.04.
1.04⁴ = 1.169859, so 1.04^−4 = 0.854804.
Annuity factor = (1 − 0.854804) ÷ 0.04 = 0.145196 ÷ 0.04 = 3.6299.
PV of coupons = 3 × 3.6299 = 10.8897.
PV of face = 100 × 0.854804 = 85.4804.
Price = 10.8897 + 85.4804 = 96.3701.

Answer: Price ≈ $96.37, a discount to par because the 8% YTM is above the 6% coupon.

Exam tips

Read the compounding and frequency in the first line. Most lost marks come from using the wrong periodic rate.
Use the par, premium and discount rule to remove wrong options before you calculate.
Practice the calculator keystrokes until YTM takes under a minute. Clear the TVM registers between questions.
When a question gives a zero curve, do not use a YTM formula. Discount each cash flow separately.
Watch for clean versus dirty price wording and for the stated day count (actual/actual or 30/360).

Practice questions from Interest Rates

Bond Pricing and Yield to Maturity 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 IYield Measures for Fixed-Rate Bonds

Bond Pricing and Yield to Maturity: frequently asked questions

What is the difference between YTM and the spot rate?

A spot (zero) rate discounts a single cash flow at one maturity. YTM is one rate that discounts all of a bond's cash flows to its price. For a coupon bond on a sloped curve they differ. For a zero-coupon bond they are the same.

How do I calculate a bond price using zero rates?

List each coupon and the principal with its date. Discount each at the zero rate for that date, using the stated compounding. Add the present values. The total is the bond price.

What is the bond-equivalent yield?

It is the yield on a bond paying semiannual coupons, found by doubling the semiannual rate. It ignores compounding within the year. To get an effective annual yield, use (1 + y/2)^2 − 1.

Which day count conventions should I know for FRM Part I?

Know actual/actual for US Treasury bonds, 30/360 for many corporate and municipal bonds, and actual/360 for money market instruments. They set how accrued interest is calculated. Quoted bond prices are clean, so accrued interest is added to get the cash price.