Level-coupon bond price from yield to maturity

The price assembles from two present values. The coupon stream is an annuity — identical payments at identical spacing — and its worth today shrinks as the yield used to discount it grows. The face value is a single distant payment, discounted across every period at once, and therefore the more yield-sensitive piece the longer the bond runs. Their sum is the only fair price consistent with the quoted yield.

Where that sum lands relative to the face value is the market’s verdict on the coupon. A coupon richer than the prevailing yield makes the bond worth more than its face — a premium; a leaner coupon prices it below face — a discount; and when coupon and yield agree the bond stands exactly at par. The verdict is arithmetic, not sentiment: the price is wherever the promised payments, at today’s discounting, actually add up.

Price and yield move opposite one another, and the workspace draws the relationship: sweeping the yield traces the price falling along a curve that is steep when yields are low and flattens as they climb. That bow is the geometry behind the cluster’s other two pages — the duration pages measure the curve’s slope; this page is the curve itself.

The relation also runs backwards, and the workspace offers that direction: given a market price, it solves for the yield to maturity the price implies. That reverse reading is how a quoted bond is actually judged — the price is observable on the screen, and the yield it implies is what gets compared against everything else money could do.

Pricing a bond starts by refusing to add its payments up. A coupon arriving many years from now and a coupon arriving next period are not the same money, because the near one can be put to work in the interval and the far one cannot. Discounting is the correction that makes them commensurable: each payment is converted into what it is worth today at the yield the market demands, and only then may the pieces be added.

Once that discipline is accepted, the price assembles from two parts. The coupons form an annuity — identical payments at identical spacing — whose worth today shrinks as the discounting rate grows. The face value is a single payment at the far end, discounted across every remaining period at once, which makes it the more yield-sensitive piece the longer the bond has to run. The price is their sum, and the split between them is worth inspecting: it tells you whether you are buying a stream of income or a distant repayment wearing a coupon.

Nothing in this is a forecast. The calculation says what the promised payments are worth at a stated yield, no more — it does not judge whether the issuer will pay, whether the yield is fair, or whether the bond is a good idea. Those are separate questions with separate evidence, and this page deliberately answers none of them.

The market cannot rewrite a coupon, so it rewrites the entry price instead. If the coupon a bond pays is leaner than the yield currently on offer elsewhere, nobody buys it at face — the price falls until the shortfall in income is made up by the gain from paying less than the redemption amount. That is a discount. If the coupon is richer than prevailing yields, buyers compete the price above face, giving back the excess income as a premium that erodes to nothing by maturity.

Hold the yield still and slide the coupon rate upward, and the price rises through the face value at exactly the point where the coupon matches the yield. That is par, and it is not a special state of a bond so much as a crossing point of a comparison. A bond issued at par drifts to a premium or a discount the first time yields move, with no action from anybody.

This is why the words are so often misread as quality judgements. A bond trading at a discount is not damaged and one trading at a premium is not superior; both are quoted so that a buyer at today’s price earns today’s yield. The verdict is arithmetic, and the arithmetic is symmetric.

Because the yield sits in the denominator of every discount, price and yield move in opposite directions by construction. A rise in the demanded return makes every promised payment worth less today, so the price falls; a fall in yields lifts it. This is the single most quoted fact about bonds, and it is not a market behaviour that could one day stop — it is a property of the arithmetic.

The relationship is not a straight line, though. Sweep the yield in the workspace and the price traces a bowed curve: steep where yields are low, flattening as they climb, so that equal increments of yield do progressively less damage. The bow is why a single sensitivity number can only ever be an approximation, and why the two lessons after this one exist — one to measure where the bond’s weight sits in time, the other to convert that into how hard the price moves.

Time is the amplifier. The longer the bond has to run, the further the redemption payment must be discounted and the more coupons are exposed to the new rate, so the same shift in yield moves a long bond’s price far more than a short one’s. Sweeping the maturity input with everything else held still is the quickest way to feel that, and it is the observation the Macaulay lesson turns into a number.

In practice the price is the observable thing and the yield is what has to be inferred. The workspace runs that direction too: give it a market price and it solves for the yield to maturity that price implies. That inferred yield is the number actually used for comparison, because it is stated on the same scale as every other opportunity for the money — which a price, expressed in currency per unit of face, is not.

Two conventions have to be respected for any of it to mean anything. The first is schedule: the yield must be quoted on the same compounding footing the coupons follow, and mixing an annual yield into a semiannual bond misprices the stream quietly rather than loudly. The workspace assumes a single annual coupon when the frequency is left unstated, declared as an illustration — semiannual payment is the widespread market convention, and setting the frequency explicitly is the safer habit.

The second is what the price excludes. The figure computed here is a clean, on-coupon-date price: it carries no accrued interest for the days since the last coupon, no dealer spread, no fees and no tax. A settlement amount adds those on top. It also assumes the promise is kept — no call, no sinking fund, no default — so it is the value of the contract as written, which is the right baseline for study and never a substitute for a dealer quote.

The engine below is the same certified one the calculator page runs, verified and mounted mid-lesson. It arrives pre-filled with the pack’s declared reference bond — an annual-coupon issue with a handful of years to run, its coupon a notch below the yield the market demands — which is the instrument the whole arc follows. Lift the coupon until it matches the yield and watch the price arrive exactly at face; push it further and watch a premium appear. Then leave the coupon alone and sweep the yield instead, and the bowed curve of the previous section draws itself.

Read the output as the clean price consistent with the yield you typed — what the promised payments are worth today, not a dealer quote and not a recommendation. Compare it against the face value to name the position: below is a discount, above a premium, level is par. Every figure is computed live by the verified engine; this page stores none.