One bond, two questions
A coupon bond is a simple promise: pay a fixed amount every so often (the coupon) and return a lump sum (the face value, also called par) on a set date (maturity). Everything this calculator does follows from that one promise, asked in either direction. The first tab starts from a required yield, the return the market currently expects from a bond like this one, and works out the price that yield implies. The second tab starts from a price actually paid and works out the yield to maturity that price implies instead. Both tabs share the same bond description, face value, coupon rate, years to maturity and how often the coupon is paid, because they are two views of the same instrument, not two different tools.
How a bond's price is built
A bond's price is the present value of everything it will pay: every coupon between now and maturity, plus the face value repaid at the end. Discount each of those future amounts back to today at the required yield and add them up. Written as a formula, with C the coupon paid each period, F the face value, r the yield per period and n the number of periods to maturity:
The first term is the present value of the coupon annuity; the second is the present value of the single lump sum repaid at maturity. C is the face value times the annual coupon rate, split across however many payments happen each year, and r is the annual required yield divided the same way, since bonds compound at their own payment frequency. Most government and corporate bonds in the United States pay twice a year, so a coupon quoted as an annual rate is actually delivered in two equal instalments, each discounted at half the annual yield.
Worked example: a $1,000 face value bond with a 5% annual coupon, paid semi-annually, 10 years to maturity and a 6% required yield. That is a $25 coupon every half-year for 20 periods, discounted at 3% per period, plus $1,000 discounted 20 periods at the same 3%. Add the two present values together and the price comes to $925.61, a figure that also appears in finance textbooks covering this exact formula, and the cash-flow table above shows every one of those 20 discounted payments individually, summing to the same total.
The coupon-versus-yield seesaw
A bond's coupon rate is fixed the day it is issued and never changes. The required yield moves with the market every day. When those two numbers disagree, the price is what reconciles them, and it always moves the opposite way to the yield. If the required yield rises above the coupon rate, nobody will pay full face value for a bond that only pays the lower coupon, so the price falls below face value until the fixed coupon plus the built-in capital gain to maturity together match the higher yield; that is a discount bond. If the required yield falls below the coupon rate, the fixed coupon is now more generous than the market demands, so buyers bid the price above face value until the return, coupon minus the built-in capital loss to maturity, comes back down to the required yield; that is a premium bond. When the two rates are identical, the price sits exactly at face value, a par bond. This is the single fact behind the well-known rule that bond prices fall when interest rates rise: the coupon is locked in, so the price is the only part of the equation left to move.
Reading the cash-flow table and the price-yield curve
The cash-flow table lists every period this bond pays something, the coupon each time and the face value on top in the final period, then discounts each one back to a present value at the required yield; the column totals reproduce the headline price exactly, to the penny, because that price is nothing more than those numbers added up. The chart above it plots the same bond's price at every yield from zero upwards, holding the coupon, face value and term fixed, with the entered yield marked. That line curves, it does not run straight: price falls quickly as yields climb from near zero, then the fall flattens out at higher yields. That curvature, known as convexity, is why a bond's price reacts more to a fall in yields than it loses to an equal-sized rise, a small but genuine asymmetry that a straight-line approximation of "price versus yield" would miss entirely.
Working out the yield from a price already paid
Turn the question around: given a price actually paid rather than a required yield, what yield does that price imply? The formula above cannot simply be flipped, because the yield sits both outside the annuity term and buried inside a power on both sides of the equation; there is no algebraic rearrangement that isolates it, in exactly the same way an amortising loan's interest rate cannot be isolated from its payment formula. This calculator solves it by bisection instead: since a bond's price falls strictly as its yield rises, trying a yield, checking whether the price it produces is too high or too low against the price actually paid, and halving the gap repeatedly converges on the one yield that reproduces that price, typically within a hundred or so tries and accurate to a tiny fraction of a basis point. This yield to maturity is the return earned if the bond is held to maturity, every coupon is reinvested at that same rate, and the issuer pays everything on schedule.
Current yield is a simpler, cruder figure printed alongside it: the annual coupon divided by the price paid, nothing more. It ignores the time value of every payment and, crucially, ignores whether the price paid will converge up or down to face value by maturity. For a discount bond, current yield sits between the coupon rate and the (higher) yield to maturity, since it catches the coupon but misses the capital gain still to come; for a premium bond the ordering flips, current yield sits between the (lower) yield to maturity and the coupon rate, since it misses the capital loss still to come. The gap between the two is the honest cost of using the quick shortcut instead of the full calculation, and this calculator always shows both side by side rather than picking one.
What yield to maturity assumes, and what it leaves out
Yield to maturity is only the return actually earned under two conditions: the bond is held all the way to maturity rather than sold early, and every coupon received along the way is reinvested at that same yield rather than spent or reinvested at a different rate. Real interest rates move around over a bond's life, so the reinvestment assumption in particular rarely holds exactly; it is a standard, published simplification, not a guarantee. Credit risk is a separate matter this calculator does not model at all: a required yield already reflects the market's view of how likely an issuer is to pay on time and in full, but a higher yield on a shakier issuer is compensation for that risk, not a discount on it, and no formula can substitute for checking who is actually making the promise. One more simplification worth naming plainly: bond prices quoted day to day are usually "clean" prices, excluding interest accrued since the last coupon, while the amount that actually changes hands is the "dirty" price, clean price plus accrued interest. This calculator prices and yields a bond exactly on a coupon date and does not model accrued interest between coupon dates.
Questions people ask
Why does a bond's price fall when interest rates rise?
Because the coupon is fixed for the life of the bond while the required yield moves with the market. When the market starts demanding a higher yield than the fixed coupon pays, nobody will pay full face value for it any more, so the price falls until the fixed coupon plus the capital gain built in by buying below face value together add up to the new, higher required yield.
What is the difference between yield to maturity and current yield?
Current yield is just the annual coupon divided by the price paid, a quick snapshot that ignores the time value of money and any gain or loss the price still has to make up to reach face value by maturity. Yield to maturity is the full internal rate of return, accounting for every payment's timing and the eventual convergence to face value, which is why the two rarely match exactly.
Why can't the yield be solved directly from a price, the way price is solved from a yield?
Because the yield appears twice in the pricing formula, once on its own and once raised to a power inside the discount factor, on both sides of the equation once it is rearranged. There is no algebraic way to isolate it, so this calculator uses bisection: it tries a yield, checks the price that yield produces against the price actually paid, and narrows the gap by half repeatedly until the two prices agree.
What does it mean for a bond to trade at a premium or a discount?
It describes where the price sits relative to face value. A premium bond, coupon rate above the required yield, prices above face value; a discount bond, coupon rate below the required yield, prices below it. A bond priced exactly at face value, coupon rate equal to the required yield, is trading at par.
What is a zero-coupon bond, and how is it priced?
A bond with a 0% coupon rate: it pays nothing until maturity, when it repays the full face value in one instalment. Its entire price is the present value of that single future sum, face value divided by (1 + yield per period) raised to the number of periods, with no coupon annuity term to add.
Does this calculator account for credit risk or the chance of default?
No. The required yield entered here already reflects whatever the market believes about an issuer's ability to pay, riskier issuers trade at higher yields, but this tool takes that yield as a given input rather than assessing the risk itself. It has no view on any specific issuer's creditworthiness.
Does this calculator handle accrued interest between coupon dates?
No. It prices and yields a bond as of a coupon date, the "clean" price. In practice a bond traded between coupon dates changes hands at the clean price plus interest accrued since the last coupon, the "dirty" price actually paid, which this tool does not model.
These figures are estimates of the underlying bond mathematics, not a live market quote, and do not account for credit risk, accrued interest, taxes or trading costs. Not financial advice; check a current quote from a broker or the issuer before relying on any figure here.