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FRM Exam Part I · Bond Yields and Return Calculations

Bond Pricing and Present Value for FRM Part I

Updated 11 October 2026 · Fact-checked

A bond's price is the present value of its coupons and principal. Discount each cash flow with its own spot rate, or all of them at one yield. The result is the dirty (full) price. Subtract accrued interest to get the clean (quoted) price.

Understand Bond Pricing and Present Value

A bond is a stream of fixed cash flows: coupons paid on set dates and principal paid at maturity. Money received later is worth less than money received today, so you discount each cash flow back to today. The bond price is the sum of those present values.

There are two ways to discount. With spot rates, each cash flow uses the zero rate for its own maturity. A 1-year payment uses the 1-year spot rate, a 2-year payment uses the 2-year spot rate. This is the no-arbitrage method, because it equals the price of a portfolio of zero-coupon bonds. With yield to maturity (YTM), one single rate discounts every cash flow. The yield is the single rate that makes the discounted cash flows equal the market price.

Price and yield move in opposite directions. If the coupon rate equals the yield, the bond prices at par. If the coupon is above the yield, it prices at a premium. If it is below, it prices at a discount.

Between coupon dates, the buyer owes the seller the coupon earned since the last payment. This is accrued interest. The dirty price (full price) is what the buyer actually pays. The clean price is the dirty price minus accrued interest, and it is the price quoted in the market. Quoting clean prices stops the quote from jumping down every time a coupon is paid.

Accrued interest depends on the day count convention. US Treasury bonds use actual/actual. US corporate and many agency bonds use 30/360. Money market instruments often use actual/360. Always use the convention the question gives you.

Key formulas to remember

Price using spot rates (annual compounding)
P = Σ C ÷ (1 + zₜ)ᵗ + F ÷ (1 + zₙ)ⁿ
zₜ is the spot rate for time t. C is the coupon, F the face value. Each cash flow gets its own rate.
Price using yield (semiannual)
P = Σ (c/2 × F) ÷ (1 + y/2)ᵏ + F ÷ (1 + y/2)²ⁿ, k = 1 to 2n
Divide the annual yield and coupon by 2. The number of periods is 2 × years.
Zero-coupon bond price
P = F ÷ (1 + z)ᵀ
A single cash flow. Discount factor d(T) = 1 ÷ (1 + z)ᵀ.
Accrued interest
AI = Coupon per period × (days since last coupon ÷ days in coupon period)
The day count convention sets how both day numbers are counted.
Clean and dirty price
Dirty price = Clean price + Accrued interest
Rearrange to get the clean price from a computed full price.
Par, premium, discount rule
Coupon = yield → par; coupon > yield → premium; coupon < yield → discount
Holds on a coupon date. A quick check on your answer.

How to solve Bond Pricing and Present Value questions

Use this routine for any bond pricing question. It keeps the compounding, the rates and the clean or dirty choice under control.

  1. 1Read what is asked: clean price, dirty price, yield or accrued interest. Note the face value, coupon rate, payment frequency and time to maturity.
  2. 2Write the cash flows on a timeline. Coupon per period = coupon rate × face ÷ frequency. Add the face value to the last coupon.
  3. 3Choose the discount method. If the question gives spot rates, use one rate per cash flow. If it gives a yield, use one rate for all, divided by the frequency.
  4. 4Match the compounding. Use the period rate (yield ÷ frequency) and the number of periods (years × frequency).
  5. 5Discount each cash flow and add them. On a financial calculator, enter N, I/Y, PMT, FV and compute PV when the yield is flat.
  6. 6If the valuation date is between coupons, treat the result as a dirty price and compute accrued interest with the stated day count.
  7. 7Clean price = dirty price − accrued interest. Report the one the question asks for.
  8. 8Sanity check: coupon above yield means price above par, and the reverse.

Quickest way: Calculator shortcut and par check

When to use it: Use when one yield is given and the valuation date falls on a coupon date.

  1. Set the calculator to one payment per year or follow the period count you enter. Use N = years × frequency.
  2. Enter I/Y as yield ÷ frequency (as a percent), PMT as the period coupon, FV as the face value.
  3. Compute PV and flip the sign. That is the price.
  4. Check against par: coupon above yield gives a price above face value. If not, recheck N and I/Y.
  5. For spot rate questions with two or three cash flows, compute discount factors once and reuse them.

Common mistakes in Bond Pricing and Present Value

  • Using the annual yield and annual coupon with semiannual periods

    Candidates forget to divide by 2 and double the periods.

    Fix: For semiannual bonds always use y/2, c/2 and N = 2 × years.

  • Discounting every cash flow at one rate when spot rates are given

    Yield-based pricing is habit.

    Fix: With a spot curve, use zₜ for each cash flow t. Only a yield uses one rate.

  • Reporting the dirty price when the quote is clean

    Candidates ignore accrued interest.

    Fix: Read the question wording. Quoted price is clean. Cash paid is dirty.

  • Forgetting to add the face value to the final coupon

    Principal is treated as a separate item and dropped.

    Fix: Write the last cash flow as coupon + face value on the timeline.

  • Using the wrong day count for accrued interest

    Candidates apply actual/actual to everything.

    Fix: Use 30/360 for corporates unless told otherwise, and actual/actual for Treasuries.

  • Subtracting accrued interest twice or in the wrong direction

    Confusion between clean and dirty.

    Fix: Remember: dirty = clean + accrued. Clean is always the lower figure when accrued interest is positive.

Worked examples

Example 1

A 2-year annual-pay bond has face value $100 and a 5% coupon. The 1-year spot rate is 3% and the 2-year spot rate is 4%, both annually compounded. Find the price.

Show the solution
  1. Cash flows: year 1 = $5, year 2 = $5 + $100 = $105.
  2. PV of year 1 = 5 ÷ 1.03 = 4.8544.
  3. PV of year 2 = 105 ÷ 1.04² = 105 ÷ 1.0816 = 97.0782.
  4. Price = 4.8544 + 97.0782 = 101.9326.

Answer: Price ≈ $101.93. It is above par because the spot rates are below the 5% coupon.

Example 2

A US corporate bond pays a 6% semiannual coupon on face value $1,000 and uses 30/360. Coupons were last paid 2 months (60 days on 30/360) ago. The full price is $1,032.00. Find the accrued interest and the clean price.

Show the solution
  1. Semiannual coupon = 6% × 1,000 ÷ 2 = $30.
  2. Days in the coupon period on 30/360 = 180. Days since last coupon = 60.
  3. Accrued interest = 30 × 60 ÷ 180 = $10.00.
  4. Clean price = dirty price − accrued interest = 1,032.00 − 10.00 = $1,022.00.

Answer: Accrued interest = $10.00. Clean price = $1,022.00.

Exam tips

  • Check first whether the question asks for clean or dirty price. Wrong choice is a common trap answer option.
  • If options differ by about the accrued interest amount, the question is testing clean versus dirty.
  • Always match compounding to the rate given. Semiannual yields are often quoted as bond-equivalent yields.
  • Use the par rule to eliminate options. A 5% coupon bond priced at a 4% yield must be above par.
  • Keep four decimals in discount factors so rounding does not move you to a neighbouring option.

Practice questions from Bond Yields and Return Calculations

Bond Pricing and Present Value in other exams

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

Bond Pricing and Present Value: frequently asked questions

What is the difference between clean price and dirty price?

The dirty price is the full amount the buyer pays, including interest earned since the last coupon. The clean price is the dirty price minus accrued interest. Markets quote clean prices.

How do I calculate bond price using spot rates?

Discount each cash flow at the spot rate for its own maturity and add the present values. Include the face value in the final cash flow.

Which day count conventions do I need for accrued interest?

Know actual/actual for US Treasury bonds, 30/360 for most corporate bonds and actual/360 for money market instruments. The question normally states which one to use.

Does the FRM exam let me use a financial calculator?

GARP permits approved financial calculators. The time value of money keys are useful for yield-based pricing but not for pricing with a spot curve, where you compute each discount factor.