Bond Price Calculator

A bond's price is the present value of two cash-flow streams discounted at the same rate: the coupon payments, which form an annuity, and the single repayment of face value at maturity. The rate used is the market yield entered, converted to a per-period figure. That makes the output a price at a given yield rather than an independent valuation: a fair value would require the calculator to supply the discount rate rather than take it as an input.

When the market yield exceeds the coupon rate the price sits below face value, because the coupon stream alone does not deliver the return the market is asking for and the shortfall has to come from buying below par. When the yield is below the coupon rate the price sits above face. When the two are equal the price is exactly face value, whatever the term or the payment frequency. That is the par identity, which two of the What-If cards on this page demonstrate.

A worked example

Face value 1,000, a 5% annual coupon paid twice a year, a market yield of 6% and ten years to maturity. Each period pays 25 (1,000 x 5% divided by 2) and the periodic yield is 3% (6% divided by 2), across 20 periods. The twenty coupons are worth 371.93 today and the 1,000 repayment is worth 553.68, giving 925.61, a discount to par of 74.39.

What moves the number most

Face value scales the price exactly: every cash flow is proportional to it, so a 10% larger bond costs 10% more and the price per unit of face does not change at all. The coupon rate moves the price symmetrically: 1 percentage point is 74.39 at the sample figures, the same in both directions, because the coupon leg is linear in the rate. The market yield is the asymmetric one: 1 percentage point lower adds 74.39 but 1 percentage point higher takes away only 67.74, and that gap is the convexity of the price-yield curve. Payment frequency barely registers, worth around 0.09 either way at these figures.

The formula behind this

Both legs are discounted at the same periodic rate because they are the same bond's cash flows, since the market is pricing one instrument, not two. The coupon leg is an ordinary annuity of n payments and reduces to C multiplied by (1 minus (1+y) to the power of minus n), divided by y. The face-value leg is a single discounting of F across n periods. Adding them gives the price. Setting the coupon equal to the yield collapses the two terms to exactly F, which is why the par identity holds at any term and any frequency.

Duration measures how far the price moves when the yield changes. Macaulay duration is the average time to receive the cash flows, weighted by their present values; for the bond above it is 7.89 years, longer than the maturity alone suggests because the 1,000 repayment at year ten dominates the weighting. Modified duration divides that by one plus the periodic yield, giving 7.66 years. The first-order reading that follows, a 1 percentage point yield rise costing 7.66% of the price, overstates the actual fall of 7.32% and understates the corresponding rise, and that error is what convexity describes. Duration depends on the coupon and the yield as much as on the maturity: two ten-year bonds with different coupons have different durations.

What the calculator leaves out

Credit risk, call and put features, accrued interest between coupon dates, liquidity, tax treatment, and any movement in the market yield across the bond's life. Prices quoted in the market are usually clean prices, excluding accrued interest; this calculation prices the bond as though it sits on a coupon date. The result illustrates the discounting arithmetic rather than reproducing a market quote.