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CFA Level I Exam · Fixed-Income Bond Valuation: Prices and Yields

Bond Price Formula: Present Value at Yield to Maturity

Updated 7 October 2026 · Fact-checked

A bond's price is the present value of its remaining coupons and its principal, each discounted at the yield to maturity per period. Divide the annual yield and coupon by the payments per year, multiply the years by that number, then use N, I/Y, PMT and FV on your calculator to compute PV.

Understand Bond Prices and the Pricing Equation

A bond is a promise of fixed cash flows: coupons during its life and the principal (face value) at maturity. Like any asset, it is worth the present value of the cash flows you expect to receive. The discount rate is the yield to maturity (YTM), the single rate that makes the present value of the promised cash flows equal to the market price.

So the pricing equation has two parts. The coupons form an annuity. The principal is a single lump sum at the end. Add the present values of both. If the coupon rate equals the YTM, the bond prices at par. If the coupon rate is below the YTM, the price is below par (a discount). If it is above, the price is above par (a premium).

Match the periods. A bond paying semiannual coupons has a periodic coupon of half the annual coupon, a periodic yield of half the annual YTM (the usual bond-equivalent convention), and N equal to years times two. Mixing annual rates with semiannual periods is the most common source of wrong answers.

The formula above gives the price on a coupon date. Between coupon dates, the buyer also owes the seller the interest earned since the last coupon. The full price (dirty price) is the present value of all remaining cash flows at the settlement date. The accrued interest is the share of the next coupon earned so far. The flat price (clean price) is the full price minus accrued interest, and it is the price usually quoted.

Price and yield move in opposite directions. A higher required yield means heavier discounting, so a lower price. This is why the pricing equation sits under duration, convexity and spread work later in fixed income.

Key formulas to remember

Bond price (on a coupon date)
PV = PMT ÷ (1+r)¹ + PMT ÷ (1+r)² + … + (PMT + FV) ÷ (1+r)ᴺ
r is the yield per period, PMT the coupon per period, FV the principal, N the number of periods remaining.
Annuity form
PV = PMT × [1 − (1+r)⁻ᴺ] ÷ r + FV ÷ (1+r)ᴺ
Same value as the sum above. Use it for hand calculation or to check your calculator.
Periodic inputs
r = YTM ÷ m; N = years × m; PMT = annual coupon ÷ m
m is the number of coupon payments per year (1 for annual, 2 for semiannual).
Zero-coupon bond
PV = FV ÷ (1+r)ᴺ
No coupons, so only the principal is discounted.
Full price between coupon dates
Full price = PV at last coupon date × (1+r)^(t ÷ T)
t is days since the last coupon, T is days in the coupon period. Equivalent to discounting all remaining flows to the settlement date.
Accrued interest
AI = (t ÷ T) × PMT
Share of the current coupon earned by the seller.
Flat price
Flat price = Full price − AI
The quoted (clean) price.

How to solve Bond Prices and the Pricing Equation questions

Use this routine for any question that asks for a bond price from a yield.

  1. 1List the face value, coupon rate, coupon frequency, years to maturity and YTM. Note whether the date is a coupon date or between coupon dates.
  2. 2Convert to per-period inputs: PMT = annual coupon ÷ m, r = YTM ÷ m, N = years × m.
  3. 3Enter N, I/Y (as a percent per period), PMT and FV in the calculator, then compute PV. Ignore the sign; the price is the absolute value.
  4. 4Sense-check against par: coupon rate below YTM means a discount, above means a premium, equal means par.
  5. 5If the settlement date is between coupon dates, compute the price as of the last coupon date, then multiply by (1+r)^(t÷T) to get the full price.
  6. 6If the question asks for a quoted price, compute AI = (t÷T) × PMT and subtract it from the full price to get the flat price.
  7. 7Compare your result with the three options. Eliminate any option that breaks the par check.

Quickest way: Calculator-first with a par check

When to use it: Standard multiple-choice questions where you are given a yield and need the price on a coupon date.

  1. Clear the TVM memory (BA II Plus: 2nd, CLR TVM). Set P/Y = 1 and C/Y = 1 (2nd, P/Y) so that N and I/Y are per period.
  2. Key N, then I/Y, then PMT, then FV, each followed by the key. Example: 5 [N], 5 [I/Y], 60 [PMT], 1000 [FV], [CPT] [PV].
  3. Read the result as a positive number. On the HP 12C use n, i, PMT, FHV and PV in the same way.
  4. Check the answer is above 1,000 for a premium, below for a discount. This often removes two options before you finish keying.

Common mistakes in Bond Prices and the Pricing Equation

  • Using the annual yield with semiannual cash flows

    You enter N as the number of years and I/Y as the annual rate out of habit.

    Fix: Divide the yield and the coupon by m and multiply N by m before keying anything.

  • Using the coupon rate as the discount rate

    Both are quoted as percentages and the coupon is the first number in the question.

    Fix: The coupon only sets the cash flow. Discount at the YTM. The coupon rate is used for PMT only.

  • Forgetting the principal in the final period

    You focus on the annuity and leave out the lump sum.

    Fix: Always enter FV. The final cash flow is the last coupon plus the face value.

  • Treating the quoted price as the amount paid between coupon dates

    Quoted prices are flat, so the accrued interest is not visible.

    Fix: The buyer pays the full price: flat price plus accrued interest.

  • Computing accrued interest on the whole coupon

    You forget that only the elapsed fraction of the period has been earned.

    Fix: Use AI = (t ÷ T) × PMT, with t days since the last coupon and T days in the period.

  • Getting the sign of the premium or discount backward

    You remember that price and yield move in opposite directions but apply it to the wrong comparison.

    Fix: Compare the coupon rate with the YTM: coupon below YTM gives a discount, coupon above gives a premium.

Worked examples

Example 1

A 5-year bond with face value USD 1,000 pays an annual coupon of 6%. The YTM is 5%. What is the price, to the nearest cent?

Show the solution
  1. Annual coupon, so m = 1: PMT = 60, r = 5%, N = 5, FV = 1,000.
  2. Present value of coupons: 60 × [1 − 1.05⁻⁵] ÷ 0.05. Since 1.05⁻⁵ = 0.783526, the annuity factor is 0.216474 ÷ 0.05 = 4.329477, so coupons = 259.77.
  3. Present value of principal: 1,000 × 0.783526 = 783.53.
  4. Add: 259.77 + 783.53 = 1,043.29 (rounding differences of a cent are possible).
  5. Calculator: 5 [N], 5 [I/Y], 60 [PMT], 1000 [FV], [CPT] [PV] shows −1,043.29.
  6. Par check: coupon 6% is above YTM 5%, so a premium is expected. 1,043.29 is above 1,000.

Answer: USD 1,043.29

Example 2

A 4-year bond with face value USD 1,000 pays a 5% coupon semiannually. The YTM is 6%, quoted as an annual rate with semiannual compounding. The price is closest to: A) 782.66, B) 964.90, C) 965.35.

Show the solution
  1. Semiannual, so m = 2: PMT = 1,000 × 5% ÷ 2 = 25, r = 6% ÷ 2 = 3%, N = 4 × 2 = 8.
  2. Present value of coupons: 25 × [1 − 1.03⁻⁸] ÷ 0.03. Since 1.03⁻⁸ = 0.789409, the factor is 0.210591 ÷ 0.03 = 7.019692, so coupons = 175.49.
  3. Present value of principal: 1,000 × 0.789409 = 789.41.
  4. Add: 175.49 + 789.41 = 964.90.
  5. Calculator: 8 [N], 3 [I/Y], 25 [PMT], 1000 [FV], [CPT] [PV] shows −964.90.
  6. Trap check: option A comes from using 6% per period with N = 8 and PMT = 25 (the yield was not halved). Option C comes from treating the bond as annual, with 50, 6% and N = 4. Both ignore the semiannual convention.
  7. Par check: coupon 5% is below YTM 6%, so a discount is expected. Option B fits.

Answer: B) 964.90

Exam tips

  • Write the per-period inputs (PMT, r, N) on your scratch sheet before touching the calculator. This is where most marks are lost.
  • Use the par check to eliminate options. Numerical options are in ascending order, so a premium bond must be in the option above 1,000 and a discount bond below.
  • Watch for distractors built from the classic errors: annual inputs on a semiannual bond, or the coupon rate used as the discount rate.
  • If the question says 'quoted' or 'flat' price, subtract accrued interest from the full price. If it says 'amount paid' or 'full price', do not.
  • With roughly 90 seconds per question, trust the calculator and avoid the long-hand annuity formula unless asked for the structure of the equation.

Practice questions from Fixed-Income Bond Valuation: Prices and Yields

Bond Prices and the Pricing Equation in other exams

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

Bond Prices and the Pricing Equation: frequently asked questions

What is the bond price formula in CFA Level I?

The price is the present value of the coupons plus the present value of the principal, discounted at the yield to maturity per period. In calculator terms, enter N, I/Y, PMT and FV, then compute PV. The result is the price on a coupon date.

How do I price a bond with semiannual coupons?

Halve the annual coupon and the annual YTM, and double the number of years to get N. Then solve for PV in the usual way. The annual YTM is a bond-equivalent yield quoted with semiannual compounding unless the question says otherwise.

What is the difference between full price and flat price?

The full price includes accrued interest and is what the buyer actually pays. The flat price excludes accrued interest and is the quoted price. Flat price = full price − accrued interest.

Why does a bond trade at a discount when YTM is above the coupon rate?

Buyers require a return above what the coupon provides. The price must fall below par so that the gain at maturity makes up the difference. At that price, the present value of the promised cash flows at the YTM equals the market price.