Workspace
The calculator's own fields, action and results arrive with the verified pack when you load it. Nothing is computed in this page.
Bond pricing · yield to price
Level-coupon bond price from yield to maturity
Price a level-coupon bond from its yield to maturity — present value of the coupon stream plus discounted face value, or solve yield from price instead.
Verified engine
What the engine returns
The output is the price, and it settles below the face value: a discount, because the frozen coupon pays less than today’s yield, so the market only holds the bond if the entry price concedes the difference. The two present values that compose it — coupon stream and discounted face — are the split worth inspecting.
Inputs the result is built from
Face value
Annual coupon rate
Yield to maturity (annual)
Coupon frequency (times per year)
Years to maturity
How the result is derived
MethodThe coupon annuity’s present value plus the face value discounted over the remaining periods, both at the per-period yield; the reverse direction solves the yield to maturity that reproduces a given price.
StandardLevel-coupon bond pricing closed form
GuardA zero or negative face value is refused — the coupon is defined as a fraction of face, so without a positive face there is no instrument to price.
How the price moves with the yield
What a stream of coupons and a face value are worth today
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.
The coupon annuity’s present value plus the face value discounted over the remaining periods, both at the per-period yield; the reverse direction solves the yield to maturity that reproduces a given price.
When this calculation is used
- Pricing a bond from its stated coupon, remaining term and the yield the market currently demands.
- Reading a quoted price backwards into its implied yield to maturity, using the reverse direction.
- Checking whether a bond should stand at a premium or a discount by comparing its coupon against prevailing yields.
- Watching how sensitive the price is to the yield by sweeping the yield and reading the traced curve.
- Supplying the priced instrument whose timing and sensitivity the cluster’s duration pages then measure.
Worked example
The pack’s declared reference bond: an annual-coupon issue with a handful of years to run, its coupon rate a notch below the yield the market demands — the cluster’s shared instrument, met here first at its price.
The output is the price, and it settles below the face value: a discount, because the frozen coupon pays less than today’s yield, so the market only holds the bond if the entry price concedes the difference. The two present values that compose it — coupon stream and discounted face — are the split worth inspecting.
Lift the coupon until it matches the yield and the price climbs to par exactly; push it past and a premium appears. Then sweep the yield and watch the price trace its bowed curve — steep near low yields, flatter far out. Every figure on screen is computed by the certified engine after the page mounts; the page itself carries no prices.
What each input represents
Face value
The amount the issuer repays at maturity and the base the coupon rate is quoted against. Par, premium and discount are all positions relative to this figure, which is why prices are often quoted per unit of face rather than in raw currency.
Annual coupon rate
The stated annual coupon as a percentage of face value, fixed when the bond was issued. Divided across the payment schedule it sets the size of each periodic payment — the half of the bargain the issuer froze in the past, which the market’s moving yield is forever re-judging.
Yield to maturity (annual)
The single annual rate that discounts every remaining payment — the return the market currently demands for holding this promise to its end. It is the input the sweep varies, the quantity the reverse direction solves for, and the other half of the bargain: the present’s answer to the coupon the past chose.
Coupon frequency (times per year)
How many coupons arrive each year — the schedule that divides the annual coupon and yield into their per-period sizes and sets how many discounting steps remain. The workspace assumes one annual coupon when unstated, declared as an illustration; semiannual payment is the widespread market convention.
Years to maturity
The time remaining until the face value is repaid, part-years welcome. It sets how many coupons are still owed and how far the redemption payment is discounted — the lever that most amplifies the price’s response to the yield.
Assumptions and limits
- The bond pays level coupons on schedule and repays face at maturity — no calls, sinking funds or default scenarios.
- One flat yield discounts every payment; a sloped curve would price each payment off its own rate.
- The price is a clean, on-coupon-date figure: accrued interest between coupon dates, and the dirty price that includes it, are a separate calculation.
- The yield is quoted on the same compounding schedule the coupons follow — mixing conventions misprices the stream.
What the guards protect against
- A zero or negative face value is refused — the coupon is defined as a fraction of face, so without a positive face there is no instrument to price.
- A zero or negative yield is refused: the annuity term of the closed form divides by the per-period yield, so the formula cannot state a price there and declines rather than fabricates.
Provenance
Level-coupon bond pricing closed form
The coupon annuity’s present value plus the face value discounted over the remaining periods, both at the per-period yield; the reverse direction solves the yield to maturity that reproduces a given price.
Educational reference, not investment advice. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.