What is the bond pricing formula?

A coupon bond is priced as the present value of all its future cash flows: P = C x [1 - (1+r)^(-n)] / r + F x (1+r)^(-n), where C is the periodic coupon payment (annual coupon / frequency), r is the periodic yield (YTM / frequency), n is the total number of periods, and F is the face value. When the coupon rate equals the YTM the bond trades at par; when the YTM is above the coupon rate the bond trades at a discount; when below, at a premium.

How is yield-to-maturity calculated from a market price?

There is no closed-form solution for YTM given price — it must be solved numerically. This calculator uses Newton-Raphson iteration (up to 200 steps) with a bisection fallback to find the periodic yield r such that the present-value formula reproduces the observed market price to within a tolerance of 0.000001.

What is Macaulay duration and why does it matter?

Macaulay duration is the weighted-average time (in years) until you receive each cash flow, where each weight is that payment's present value as a fraction of the total bond price. A bond with a 7-year Macaulay duration will reprice as if all its cash flow arrives in year 7. It is the cornerstone metric for matching asset and liability cash flows in immunisation strategies.

What is modified duration?

Modified duration = Macaulay duration / (1 + r_periodic). It measures approximate percentage price sensitivity: if modified duration is 6 and the yield rises by 1%, the bond price falls by about 6%. The approximation is improved by adding the convexity correction, which the calculator does automatically in the +/- 1% scenario rows.

What is convexity and why is it important?

Convexity captures the curvature in the price-yield relationship. Duration alone gives a linear approximation; for large yield moves the error is significant. Adding the convexity term (0.5 x Convexity x (delta_y)^2 x Price) improves the estimate. Positive convexity (all plain-vanilla coupon bonds have it) means prices rise more than duration predicts when yields fall, and fall less than duration predicts when yields rise — a favourable asymmetry.

What is the difference between clean and dirty price?

The clean price is what is quoted in the market and what this calculator computes. The dirty price (settlement price) adds accrued interest: the fraction of the next coupon that has built up since the last payment date. Between coupon dates the buyer pays the seller the accrued interest as compensation for the coupon income they will not receive. At a coupon payment date the two prices are equal.

Bond Price Calculator

A bond price calculator that works in both directions: give it the yield-to-maturity (YTM) and it prices the bond, or give it the market price and it back-solves the YTM using Newton-Raphson iteration. Beyond the headline number it also computes Macaulay duration, modified duration, convexity and the expected price change for a one-percentage-point move in yields — the three metrics every fixed-income analyst reaches for first.

Fixed-income maths is deceptively straightforward in concept but tedious by hand, especially for semi-annual or quarterly bonds where you can have 80 or 120 periods to discount. This tool automates all of that and shows its working so you can see exactly which formula produced each number.

How it works

A coupon bond is simply a stream of known cash flows: periodic interest payments plus the face value returned at maturity. Each cash flow is discounted at the bond’s yield-to-maturity — the single constant rate that, when applied to all periods, makes the present value equal the observed price.

Pricing formula

The closed-form present-value formula is:

P = C * [1 - (1 + r)^(-n)] / r   +   F * (1 + r)^(-n)

where:

C = periodic coupon = (face * annual_rate) / frequency
r = periodic yield = YTM / frequency
n = total periods = years * frequency
F = face (par) value

When r = 0 (zero yield), this simplifies to P = C * n + F.

Solving for YTM

There is no algebraic inverse of the pricing formula — the yield is found numerically. This calculator applies Newton-Raphson iteration using the closed-form price derivative (dP/dr) as the gradient, converging to within 1e-8 in fewer than 30 steps for any realistic bond. If Newton-Raphson diverges (e.g. very deep discount bonds), a bisection fallback guarantees convergence.

Duration and convexity

Macaulay duration is the present-value-weighted average time to receive each cash flow:

D_mac = sum(t * PV(CF_t)) / P   (expressed in periods, then divided by frequency to get years)

Modified duration converts this to a direct price-sensitivity metric:

D_mod = D_mac / (1 + r_periodic)

This says: if the yield moves by 1%, the price changes by approximately D_mod * 1%.

Convexity captures the curvature in the price-yield curve:

C_x = sum(t(t+1) * PV(CF_t)) / [P * (1 + r)^2 * freq^2]

The full duration + convexity approximation for a yield change of dy is:

dP ≈ -D_mod * dy * P   +   0.5 * C_x * dy^2 * P

The calculator applies this formula to show estimated prices if the YTM rises or falls by exactly one percentage point.

Worked example

A £1,000 face value bond paying a 5% annual coupon semi-annually with 10 years to maturity and a required YTM of 6%:

Periodic coupon: £1,000 x 5% / 2 = £25.00
Periodic yield: 6% / 2 = 3.00% per period
Number of periods: 10 x 2 = 20

Price calculation:

P = 25 * [1 - (1.03)^(-20)] / 0.03   +   1000 * (1.03)^(-20)
P = 25 * 14.877  +  1000 * 0.5537
P = 371.93  +  553.68  =  925.61

The bond trades at a discount of £74.39 because the coupon rate (5%) is below the required yield (6%). The YTM back-calculation confirms: feeding 925.61 as the market price recovers exactly 6.00%.

Metric	Value
Price	925.61
YTM	6.00%
Current Yield	5.40%
Macaulay Duration	7.99 years
Modified Duration	7.76 years
Price if YTM rises 1%	~854.62
Price if YTM falls 1%	~1,002.68

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