Bond Price & Yield to Maturity Calculator

A bond's price is the present value of every cash flow it pays: each coupon payment plus the face value at maturity, each discounted back to today using the market yield to maturity. The formula is price = C x [1 - (1 + i)^-n] / i + F x (1 + i)^-n, where C is the periodic coupon payment, i is the periodic yield (annual yield divided by payments per year), n is the total number of coupon periods, and F is the face value. When the yield equals the coupon rate, the bond prices exactly at par (100% of face value).

Unlike price from yield, yield from price has no closed-form algebraic solution, because yield appears inside the discounting exponent for every cash flow at once. It is solved numerically: a solver repeatedly guesses a yield, computes the resulting price, and narrows the guess until the computed price matches the actual price within a small tolerance. This calculator uses bisection, which repeatedly halves a bracketed search range because bond price is a strictly decreasing function of yield, guaranteeing the search converges to the one true answer.

A bond's coupon rate is fixed at issuance, but market yields move with interest rates. When the market yield falls below the bond's coupon rate, investors will pay more than face value for that above-market income stream, so the bond trades at a premium. When the market yield rises above the coupon rate, the bond's fixed payments look less attractive than newly issued bonds, so it trades at a discount to compensate a buyer with a lower purchase price. A bond priced exactly at par has a yield to maturity equal to its coupon rate.

The coupon rate is fixed at issuance: the annual coupon payment as a percentage of face value, and it never changes. Current yield is the annual coupon payment divided by the bond's current market price, a simple income-only snapshot that ignores any capital gain or loss at maturity. Yield to maturity is the most complete measure: the single discount rate that makes the present value of every remaining coupon plus the face value equal to the current price, capturing both the coupon income and the price gain or loss if held to maturity.

Yes. Most U.S. corporate and Treasury bonds pay semiannual coupons, splitting the annual coupon rate into two payments per year and dividing the annual yield by two for discounting each period. More frequent compounding at the same nominal annual yield slightly increases a bond's price, because coupon cash is received and (implicitly) reinvested sooner. Comparing two bonds' yields only makes sense when both use the same compounding frequency, mixing an annual-pay yield against a semiannual-pay yield understates the difference between them.

This calculator prices a standard, option-free, fixed-rate coupon bond on a coupon payment date. It does not account for accrued interest between coupon dates (the difference between a bond's clean price and its dirty, or full, price), odd first or last coupon periods, embedded call or put options, credit risk or default probability, or tax treatment. Real bond quotes in the secondary market typically add accrued interest to the clean price shown here to arrive at the amount actually settled.

Price is a sum of cash flows each discounted by the same unknown yield raised to a different power. Going from yield to price is a direct calculation. Going the other way means solving a polynomial for its root, and for a bond with many coupon periods there is no closed-form algebraic solution. Calculators handle it by guessing a yield, pricing the bond, comparing the result to the actual price, and narrowing the guess until the two agree to within a tolerance. That is why a yield figure can differ slightly between tools with different convergence settings.

The clean price excludes accrued interest and is what dealers quote. The dirty price, also called the full or invoice price, is the clean price plus accrued interest and is what actually settles. A clean price stays smooth through a coupon period, which is why it is the quoting convention: the dirty price sawtooths upward as interest accrues and drops on each payment date, which would make price comparisons across dates misleading. Calculators generally work in clean price unless they state otherwise.

Yes, arithmetically. If a bond's price is high enough that the total of its remaining coupons and its par repayment is less than what a buyer pays today, the discount rate that equates the two is below zero. This has occurred in real government bond markets where investors accepted a negative yield in exchange for a safe place to hold principal. A calculator that constrains its search to positive yields will fail to converge on such a bond rather than returning the correct negative figure.