How this calculator works
A bond's price is the present value of everything it will pay: a fixed coupon at each period and the face value at maturity. Each cash flow is discounted at the yield to maturity for the number of periods until it arrives. When the yield is above the coupon rate the bond trades below face value, at a discount; when the yield is below the coupon rate it trades at a premium. The convention used here divides the annual yield by the coupon frequency and compounds at that periodic rate. That is the usual street convention for a nominal yield, not an effective annual yield.
Formula
Price = sum from t = 1 to n of C / (1 + y)^t plus F / (1 + y)^n, where C = F x coupon rate / frequency, y = annual YTM / frequency, n = years x frequency, and F is the face value. With y = 0 the price is C x n + F.
Worked example
A $1,000 bond paying a 5% annual coupon semi-annually with 10 years left and a 4.5% yield has C = $25, n = 20 and y = 2.25% per period. The price is the present value of 20 payments of $25 plus $1,000 at period 20, which is $1,039.91, a premium of $39.91 above face value.
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Frequently asked questions
Why does the price move opposite to the yield?
The coupons are fixed, so a higher discount rate reduces the present value of every future payment. The bond must therefore be cheaper for its yield to rise.
What about accrued interest?
This is a clean price on a coupon date. Between coupon dates a buyer also pays accrued interest to the seller, and the dirty price is the clean price plus that accrual.
Can I price a zero-coupon bond?
Yes. Enter a coupon rate of 0%. The price is then the face value discounted for the whole term, with no coupon payments.
What if the yield is negative?
The calculator accepts yields above -50% and discounts accordingly. A negative yield means the investor pays more than the total cash received, which has happened in some government bond markets.