What is yield to maturity (YTM)?

Yield to maturity is the total return an investor earns by holding a bond until it matures, assuming all coupon payments are reinvested at the same rate. It accounts for the purchase price, face value, coupon payments, and time to maturity. YTM is essentially the bond's IRR. If a bond trades below par (discount), YTM exceeds the coupon rate. If it trades above par (premium), YTM is below the coupon rate.

What is the difference between clean price and dirty price?

The clean price is the quoted market price of a bond excluding accrued interest. The dirty price (invoice price) is what the buyer actually pays: clean price plus accrued interest since the last coupon date. Bonds are quoted at their clean price in markets, but settled at the dirty price. Accrued interest compensates the seller for interest earned since the last payment.

What is bond duration?

Duration measures a bond's sensitivity to interest rate changes. Expressed in years, it represents the weighted average time to receive the bond's cash flows. A bond with a duration of 7 years will see its price fall roughly 7% for each 1 percentage point rise in yields. Longer duration means more price sensitivity. Zero-coupon bonds have duration equal to their maturity; coupon bonds have shorter duration than their maturity.

What are day-count conventions in bond pricing?

Day-count conventions determine how accrued interest is calculated between coupon dates. The main ones are: 30/360 (assumes 30 days per month, 360-day year, common for US corporate and municipal bonds), Actual/360 (actual days elapsed divided by 360, common for money market instruments), Actual/365 (actual days divided by 365, common in UK markets), and Actual/Actual (actual days in period divided by actual days in coupon period, used for US Treasuries).

Bond Calculator 2026 - Price, YTM, Dirty Price, Clean Price & Accrued Interest

Bond Parameter Solver

Solve for Price, YTM, Coupon Rate, or Face Value. Enter any three known values and the calculator finds the fourth.

Face Value ?

Market Price ?

Annual Coupon Rate ?

Yield (YTM) ?

Years to Maturity

Coupon Frequency

Dirty Price / Clean Price Calculator

For bonds traded between coupon dates. Calculates dirty price, clean price, and accrued interest using any day-count convention.

Face Value

Annual Yield (YTM)

Annual Coupon Rate

Coupon Frequency

Settlement Date ?

Maturity Date

Day-Count Convention ?

30/360 Actual/360 Actual/365 Actual/Actual

Bond Parameter Results

Solving for: Price

Bond Price

$925.61

Price / Par

92.56%

Current Yield

5.40%

Macaulay Duration

7.80 yrs

Total Coupons

$500

Capital Gain/Loss

+$74.39

Total Return

$574.39

Modified Duration

7.57

Bond Cash Flows and NPV Profile

Coupon Payment Schedule

Period	Date	Coupon	PV Factor	PV of Coupon	Cum. PV

What Is a Bond?

A bond is a fixed-income debt instrument where an investor lends money to a borrower (a corporation, government, or municipality) for a defined period at a fixed interest rate. The borrower (issuer) commits to paying periodic interest (coupons) and returning the principal (face value) at maturity.

Bonds allow governments to fund public spending and companies to raise capital without diluting equity. From an investor's perspective, a bond represents a contractual claim: the issuer must pay coupons on schedule and return the face value at maturity, regardless of profits.

The Core Components

Face Value (Par): The principal amount the issuer repays at maturity. US bonds typically have a $1,000 face value. This is also the base for coupon calculations.
Coupon Rate: The annual interest rate, expressed as a percentage of face value. A 5% coupon on a $1,000 bond pays $50 per year ($25 semi-annually).
Maturity Date: When the issuer repays the face value. Short-term bonds mature in 1 to 3 years; long-term bonds may run 30 years or more.
Coupon Frequency: How often interest is paid. Most US bonds pay semi-annually; some corporate bonds pay quarterly or monthly.
Yield to Maturity (YTM): The total return earned by holding the bond to maturity, accounting for coupon payments and the difference between purchase price and face value.

Bond Price Formula

A bond's price is the present value of all future cash flows: periodic coupon payments plus the face value at maturity, discounted at the yield per period.

Price = C×[1-(1+r)^(-N)]/r + F/(1+r)^N C = coupon per period · r = yield per period · N = total periods · F = face value

For a $1,000 bond with a 5% coupon, 10-year maturity, semi-annual payments, and a 6% yield:

C = $1,000 × 5% / 2 = $25 per period
r = 6% / 2 = 3% per period
N = 10 × 2 = 20 periods
Price = 25 × [1 - (1.03)^(-20)] / 0.03 + 1000 / (1.03)^20 = $925.61

The bond trades at a discount because its coupon rate (5%) is below the market yield (6%). Investors require the lower price to earn an equivalent return to newly issued bonds at 6%.

Premium vs Discount vs Par

Discount bond (Price < Face): Coupon rate < YTM. Buyer earns a capital gain at maturity.
Par bond (Price = Face): Coupon rate = YTM. No capital gain or loss.
Premium bond (Price > Face): Coupon rate > YTM. Buyer accepts a capital loss at maturity, offset by higher coupons.

Yield to Maturity: What It Really Measures

YTM is the single most important metric in bond analysis. It is the internal rate of return of the bond's cash flows: the discount rate that makes the present value of all future coupon payments and the face value exactly equal to the current price. In effect, YTM is what you earn annually by holding the bond to maturity and reinvesting all coupons at that same rate.

YTM vs Current Yield vs Coupon Rate

Metric	Formula	What It Measures	Limitation
Coupon Rate	Annual Coupon / Face Value	Contractual interest rate on face value	Ignores market price
Current Yield	Annual Coupon / Current Price	Income return on the price paid	Ignores capital gain/loss at maturity
Yield to Maturity	Solved iteratively (IRR)	Total annualized return held to maturity	Assumes reinvestment at YTM
Yield to Call	IRR using call date and price	Return if bond is called early	Only applies to callable bonds

Calculating YTM

There is no closed-form formula for YTM. It requires iterative solving (bisection search or Newton-Raphson). Given price = $925.61, face = $1,000, coupon rate = 5% semi-annual, 10 years: YTM = 6.00% (the rate that satisfies the price equation above). This calculator solves YTM using binary search accurate to 0.001%.

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YTM assumes all coupons are reinvested at exactly the YTM rate. In practice, reinvestment rates vary. The "realized yield" or "total return" accounts for actual reinvestment rates and is a better measure for investors who plan to reinvest coupons at different rates.

Clean Price, Dirty Price, and Accrued Interest

Most bond trades happen between coupon payment dates. When a bond is sold between dates, the seller has earned interest since the last coupon but hasn't received it yet. To compensate, the buyer pays the seller for that earned but unpaid interest.

This leads to two prices:

Clean Price: The price quoted in markets, excluding accrued interest. Used for comparisons because it does not change with every passing day due to interest accumulation.
Dirty Price (Invoice Price): What the buyer actually pays: clean price plus accrued interest. This is the true economic transaction price.

Dirty Price = Clean Price + Accrued Interest Accrued Interest = Coupon × (Days Since Last Coupon / Days in Coupon Period) Example: $25 semi-annual coupon, 47 days since last coupon, 182-day period (30/360): AI = $25 × (47 / 180) = $6.53

After settlement, the clean price slowly rises as accrued interest builds, then drops sharply on the coupon date when the interest is paid out. The dirty price shows this sawtooth pattern; the clean price is smoother and more useful for market comparisons.

Day-Count Conventions

A day-count convention determines how many days are counted between two dates and what denominator is used to calculate fractions of a coupon period. Different bond markets use different conventions.

30/360 (Bond Basis)

Assumes each month has 30 days and the year has 360 days. The most common convention for US corporate bonds, agency bonds, and municipal bonds. Simple to calculate manually but introduces small distortions at month-end.

Actual/360

Uses the actual number of calendar days elapsed but divides by 360. Common for money market instruments such as US commercial paper, bankers' acceptances, and some bank loans. Slightly inflates effective interest costs compared to Actual/365.

Actual/365 (Fixed)

Uses actual days elapsed divided by a fixed 365-day year. Standard in UK gilts and some international markets. Does not adjust for leap years.

Actual/Actual (ICMA / US Treasury)

Uses actual days in the accrual period divided by the actual number of days in the full coupon period. US Treasury bonds and notes use this convention. It is the most economically accurate method because it respects the true length of each period.

Convention	Numerator	Denominator	Common Use
30/360	Adjusted calendar days (30/mo)	360	US corporate, municipal, agency bonds
Actual/360	Actual calendar days	360	Money market, commercial paper
Actual/365	Actual calendar days	365	UK gilts, some international
Actual/Actual	Actual days in period	Actual days in coupon period	US Treasuries, ICMA standard

Bond Types

Government Bonds

Issued by national governments. US Treasury securities are the global benchmark for risk-free rates:

Treasury Bills (T-bills): Maturities of 4 weeks to 52 weeks. Sold at a discount to face value; no coupon payments. The difference between purchase price and $1,000 face value is the return.
Treasury Notes (T-notes): Maturities of 2 to 10 years. Pay semi-annual coupons. The most traded fixed-income instrument in the world.
Treasury Bonds (T-bonds): 20 and 30-year maturities. Semi-annual coupons. Long duration makes them more sensitive to rate changes than shorter maturities.
TIPS (Treasury Inflation-Protected Securities): Principal adjusts with CPI inflation. The coupon rate is fixed but applies to the inflation-adjusted principal, providing a real return above inflation.

Municipal Bonds

Issued by state and local governments. Interest is exempt from federal income tax and often from state and local taxes for residents of the issuing state. The tax-equivalent yield formula converts a muni yield to a comparable taxable yield: Tax-Equivalent Yield = Muni Yield / (1 minus Tax Rate). A 4% muni for someone in the 32% bracket equals a 5.88% taxable equivalent.

Corporate Bonds

Issued by companies. Rated by agencies (Moody's, S&P, Fitch). Investment-grade bonds carry ratings of BBB/Baa or higher; high-yield (junk) bonds are rated BB/Ba or lower. The credit spread over Treasuries compensates investors for default risk.

Zero-Coupon Bonds

Pay no periodic interest. Sold at a deep discount to face value and redeemed at par at maturity. The entire return is the capital appreciation. US Treasury STRIPS are the most common. Because all cash flow occurs at maturity, zero-coupon bonds have the longest duration relative to their maturity and the most price sensitivity to rate changes.

Duration and Interest Rate Risk

Duration measures how sensitive a bond's price is to changes in interest rates. It has two main definitions:

Macaulay Duration

The weighted average time (in years) to receive the bond's cash flows, with each cash flow weighted by its present value as a fraction of the bond's total present value.

D_mac = [sum of t × PV(CFt)] / Bond Price t = time in years to each cash flow · PV(CFt) = present value of each cash flow

Modified Duration

Converts Macaulay Duration into a direct price sensitivity measure: the approximate percentage change in price for a 1% (100 basis point) change in yield.

D_mod = D_mac / (1 + YTM/k) k = coupon payments per year Price change % ≈ -D_mod × change in yield (decimal) Example: D_mod = 7.57, yield rises 1% → price falls ~7.57%

Duration Rules

Zero-coupon bond duration equals its maturity exactly.
Higher coupon rate → shorter duration (more cash comes early).
Longer maturity → longer duration and higher rate sensitivity.
Higher yield → shorter duration (future cash flows worth less).

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Convexity refines the duration approximation. Duration assumes a linear price-yield relationship, but the actual relationship is curved (convex). A bond with higher convexity gains more when yields fall and loses less when yields rise than the duration estimate predicts.

Frequently Asked Questions

Why do bond prices fall when interest rates rise?

A bond's coupon payments are fixed. When new bonds in the market offer higher coupons due to rising rates, existing lower-coupon bonds become less attractive. Their prices fall until the yield matches the new market rate. A bond promising $50 per year on a $1,000 face value looks unappealing when new bonds offer $60 per year, so buyers pay less than $1,000 for it, bringing the yield into line.

What is the difference between YTM and current yield?

Current yield is simply annual coupon divided by current price. For a $50 coupon on a $925 bond, current yield = 5.41%. YTM is the total annualized return including the capital gain (or loss) if you hold to maturity, where the $925 bond returns $1,000 at maturity. For a discount bond, YTM exceeds current yield; for a premium bond, YTM is below current yield. Current yield is easy to calculate but incomplete; YTM is the standard comparison metric.

What is the dirty price of a bond?

The dirty price is what you actually pay when buying a bond between coupon dates: the quoted clean price plus accrued interest. If a bond's clean price is $944 and 47 days of accrued interest equals $6.47, you pay $950.47 (dirty price). The dirty price reflects the bond's true economic value at that point in time. After settlement, the next coupon payment you receive compensates you for the accrued interest you paid at purchase.

What day-count convention is used for US Treasury bonds?

US Treasury notes and bonds use the Actual/Actual (ICMA) convention. The numerator is the actual number of days from the last coupon to settlement. The denominator is the actual number of days in the full coupon period (typically 181 to 184 days for semi-annual payments). US corporate bonds and municipal bonds use 30/360. Money market instruments such as T-bills and commercial paper use Actual/360.

What does it mean when a bond trades at a spread to Treasuries?

The spread is the difference in yield between the bond and a Treasury of the same maturity. It represents the extra compensation investors demand for taking on credit risk (the possibility the issuer defaults), liquidity risk (harder to sell quickly), and tax differences. A corporate bond yielding 5.50% when the 10-year Treasury yields 4.30% has a spread of 120 basis points (1.20%). Investment-grade bonds typically trade at 50 to 200 basis points over Treasuries; high-yield bonds at 300 to 800 basis points or more.