Bond Price Calculator — YTM, Duration & Convexity
Calculate bond price from yield to maturity, coupon rate, and maturity. Find YTM from price, compute Macaulay and modified duration, convexity, and accrued interest for clean vs dirty price.
Quick Presets
Calculation Mode
What Is the Bond Price Calculator — YTM, Duration & Convexity?
This bond calculator handles two modes: compute the fair price of a bond given its yield to maturity, or compute the implied YTM given a quoted market price. Beyond price and yield, it calculates the full suite of fixed-income analytics that institutional bond traders use every day: Macaulay duration, modified duration, convexity, DV01, current yield, clean vs. dirty price, and a price sensitivity table across a range of yield shifts.
- ›Two calculation modes — Price from YTM uses the discounted cash flow formula directly. YTM from Price uses iterative Newton-Raphson search converging in milliseconds.
- ›Clean vs. dirty price — dirty price is what you actually pay; clean price removes accrued interest for quoting purposes. The difference matters when buying mid-coupon-period.
- ›Duration and convexity — duration measures linear price sensitivity to yield; convexity captures the curvature. Together they give a second-order approximation of price change for any yield shift.
- ›Sensitivity table — bond price at ±50, ±100, ±200 basis points so you can immediately see the asymmetric impact of rising vs. falling rates.
- ›Presets — 10-Year Treasury, Investment Grade Corporate, High Yield, and Zero Coupon bonds load real-world parameter sets.
Formula
Bond Price (from YTM)
P = Σ [C / (1 + y/m)^(t)] + F / (1 + y/m)^(n×m)
where t runs from 1 to n×m (total coupon periods)
Macaulay Duration
D = Σ [t × PV(CFt)] / P
Modified Duration
D_mod = D / (1 + y/m)
DV01 (Dollar Value of 1 Basis Point)
DV01 = D_mod × P × 0.0001
Convexity
Convexity = Σ [t(t+1) × PV(CFt)] / [P × (1+y/m)² × m²]
| Symbol | Name | Description |
|---|---|---|
| P | Bond price | Present value of all future cash flows |
| F | Face value | Par value — amount repaid at maturity (typically $1,000) |
| C | Coupon payment | Periodic interest payment = (coupon rate × F) / m |
| y | YTM | Yield to maturity — annual rate used to discount cash flows |
| m | Payment frequency | 1 = annual, 2 = semiannual (most US bonds) |
| n | Years to maturity | Time until the bond matures and face value is repaid |
| D | Macaulay duration | Weighted average time (in years) to receive cash flows |
| D_mod | Modified duration | % price change per 1% change in yield; D / (1 + y/m) |
| DV01 | Dollar value of 1bp | Dollar price change for a 1 basis point (0.01%) yield move |
How to Use
- 1Choose a mode: Select "Price from YTM" to compute fair value from a given yield, or "YTM from Price" to find the implied yield given a current market price.
- 2Enter face value: Typically $1,000. Enter the par value printed on the bond.
- 3Enter coupon rate: The annual coupon rate as a percentage (e.g. 5 for 5%). For zero-coupon bonds enter 0.
- 4Enter YTM or price: Depending on mode, enter the yield to maturity (%) or the current market price ($).
- 5Set maturity and frequency: Enter years to maturity and payment frequency. Most US Treasury and corporate bonds are semiannual (2).
- 6Press Calculate: View bond price, YTM, duration, modified duration, convexity, DV01, clean/dirty price, and the sensitivity table.
- 7Use presets: Click 10-Year Treasury, IG Corporate, High Yield, or Zero Coupon to explore realistic examples.
Example Calculation
10-year bond, 5% coupon, $1,000 face value, 4.5% YTM (semiannual)
Parameters: F=$1000, coupon=5%, YTM=4.5%, n=10yr, m=2
Semiannual coupon C = 1000 × 0.05 / 2 = $25.00
Semiannual yield y = 0.045 / 2 = 0.0225
Total periods N = 10 × 2 = 20
PV(coupons) = 25 × [1 − (1.0225)⁻²⁰] / 0.0225 = $398.64
PV(face) = 1000 / (1.0225)²⁰ = $641.86
Bond Price = $398.64 + $641.86 = $1,040.50
Current yield = 50 / 1040.50 = 4.806%
Macaulay duration ≈ 8.11 years
Modified duration ≈ 7.93
DV01 ≈ $0.825 per $1,000 face
Price vs. yield relationship
Because YTM (4.5%) is below the coupon rate (5%), this bond trades at a premium ($1,040.50 > $1,000). If YTM rose to 5%, the price would fall to exactly $1,000 (par). At 5.5% YTM, the price would fall to about $961 — a discount bond. Duration of ~7.93 means a 1% rate rise cuts the price by approximately 7.93%.
Understanding Bond Price — YTM, Duration & Convexity
Financial Disclaimer
This calculator is for educational and planning purposes only. It does not constitute financial advice. Consult a qualified financial advisor before making investment or retirement decisions. Tax rules and contribution limits change annually; verify current limits at irs.gov.
The Price-Yield Relationship
The most fundamental concept in fixed income is that bond prices and yields move in opposite directions. When market interest rates rise, existing bonds paying lower coupons become less valuable — their prices fall. When rates fall, existing bonds paying higher coupons become more valuable — their prices rise. This relationship is not linear; it is convex, meaning prices fall slower when yields rise than they rise when yields fall.
- ›Premium bond: coupon rate > YTM → price > face value. The bond pays more than the market requires, so investors pay a premium.
- ›Discount bond: coupon rate < YTM → price < face value. The bond pays less than the market requires, so investors pay less than par.
- ›Par bond: coupon rate = YTM → price = face value. The coupon exactly matches the required yield.
Duration — Measuring Interest Rate Risk
Duration is the bond market's measure of interest rate sensitivity. Macaulay duration tells you the weighted average time (in years) until you receive the bond's cash flows — intuition: a zero-coupon bond has duration exactly equal to its maturity because all cash flow comes at maturity. Modified duration converts this to a price sensitivity measure: a modified duration of 7 means the bond price changes approximately 7% for each 1% change in yield.
Convexity — The Second-Order Effect
Duration gives a linear approximation of price change. Convexity captures the curvature of the price-yield relationship. A bond with higher convexity loses less price when yields rise and gains more price when yields fall, compared to a bond with identical duration but lower convexity. In practice, investors pay a premium for convexity — especially relevant for mortgage-backed securities (negative convexity) vs. standard government bonds.
Clean vs. Dirty Price
When a bond is quoted in the market, the clean price (also called flat price) excludes accrued interest. The dirty price (full price) includes the interest that has accrued since the last coupon payment. When you actually settle a bond trade, you pay the dirty price — the clean price plus accrued interest. Coupon bonds are always quoted clean; accrued interest is added at settlement.
Frequently Asked Questions
Why does the bond price go down when interest rates rise?
Intuition: Imagine you own a bond paying 3% when new bonds pay 5%. To sell yours, you must lower the price until a buyer gets an effective 5% return on your 3% coupon bond. That lower price is the market price.
- ›Price ↑ when yields fall — existing high coupons become more valuable
- ›Price ↓ when yields rise — existing low coupons become less valuable
- ›This relationship holds for all bonds, regardless of credit quality
- ›Duration quantifies how much the price changes per 1% yield move
What is the difference between coupon rate and yield to maturity?
- ›Coupon rate: fixed % of face value, set at issuance, printed on the bond certificate
- ›Current yield: annual coupon / current price — ignores the time-value of the maturity payment
- ›YTM: the total annualised return if held to maturity at the current price — the most complete yield measure
- ›YTM accounts for coupon payments, price discount or premium, and time value of money
What is Macaulay duration and why does it matter?
- ›Zero-coupon bond: duration = maturity (all cash flow at end)
- ›Coupon bond: duration < maturity (earlier coupons shorten effective wait)
- ›Higher coupon → shorter duration (earlier cash flows dominate)
- ›Longer maturity → longer duration (more exposure to yield changes)
- ›Modified duration = Macaulay duration / (1 + y/m) = % price change per 1% yield
What is DV01 and how is it used?
- ›DV01 = Modified Duration × Price × 0.0001
- ›For $1M face value with DV01 of $700: a 10bp rate rise costs $7,000
- ›Used for rate hedging: match DV01 of your bond with an offsetting position
- ›Treasury futures, interest rate swaps, and options all express risk in DV01 terms
What is the difference between clean price and dirty price?
- ›Bonds are always quoted at the clean price in the market
- ›Accrued interest = (days since last coupon / days in coupon period) × coupon amount
- ›Dirty price = clean price + accrued interest
- ›At a coupon payment date, clean = dirty (no accrued interest)
- ›This avoids the appearance of the price dropping sharply on each coupon date
How does convexity help bond investors?
Price approximation including convexity:
ΔP/P ≈ −D_mod × Δy + ½ × Convexity × (Δy)²
- ›For small yield moves, duration alone is sufficient
- ›For large moves (>50bp), add convexity for meaningful accuracy improvement
- ›Higher convexity → bond performs better than duration predicts
- ›Callable bonds and mortgage-backed securities can have negative convexity
What preset bond types are available?
- ›10-Year Treasury: benchmark risk-free rate, semiannual, high duration
- ›IG Corporate: investment grade spread over Treasuries, slight premium/discount
- ›High Yield: trades at discount reflecting credit spread and default risk
- ›Zero Coupon: all return comes from price appreciation; longest effective duration for its maturity