Bond Price and Yield: Method

Bond Price and Yield: the Method

What it does

A fixed-coupon bond is a loan: the holder receives the same interest payment (the coupon) on fixed dates and the face value at maturity. The tool turns a yield into a price, or a price into a yield, and then measures how far the price moves when the yield moves. Every step is printed with its numbers so that it can be repeated by hand.

The method

Symbols: F face value; c_r coupon rate, percent per year; m coupons per year; N coupons remaining; C = F (c_r/100)/m the coupon paid each period; y the yield to maturity, percent per year; y_p = y/100/m the yield per period; f the fraction of the current coupon period already elapsed at settlement (0 on a coupon date).

Price. A payment received k periods from now is worth its amount divided by (1 + y_p)^k today. The price is the sum over the N remaining payments, the last one carrying the face value:

P = C/(1 + y_p) + C/(1 + y_p)² + … + (C + F)/(1 + y_p)^N = C [1 − (1 + y_p)^−N] / y_p + F (1 + y_p)^−N.

The second form sums the equal coupons as an annuity. At a zero yield nothing is discounted and P = N C + F. A zero-coupon bond has C = 0 and P = F (1 + y_p)^−N.

Between coupon dates each payment is f of a period nearer, so every exponent becomes k − f, which multiplies the price above by (1 + y_p)^f. This is the dirty (full) price P_d, the amount that changes hands. The seller has earned the accrued interest AI = f C; the quoted, or clean, price is P_c = P_d − AI. The fraction f is entered, or counted from the settlement date and the coupon dates stepped back from maturity: actual days over the actual days in the period (actual/actual), or 30-day months over 360/m days (30/360, U.S. bond basis).

Current yield = 100 m C / P_c: a year of coupons over the clean price. It counts income only.

Yield to maturity is the y that makes the price formula return the price paid. There is no direct formula, so the tool starts from the estimate (annual coupon + (F − P)/years) / ((F + P)/2) and applies Newton's method, y_new = y − (price at y − target price)/slope, keeping each trial between a yield known to be too low and one known to be too high; a step that would leave that bracket is replaced by its midpoint.

Macaulay duration D_mac = Σ t_k PV_k / P_d, with t_k = (k − f)/m the time of payment k in years and PV_k its present value: the average time at which the bond's value is received. Modified duration D_mod = D_mac/(1 + y_p) equals −(1/P) dP/dy: the percentage price change per point of yield. Convexity Cx = Σ t_k (t_k + 1/m) PV_k / [P_d (1 + y_p)²] equals (1/P) d²P/dy²: how fast that sensitivity itself changes.

Price change for a yield change Δy (decimal): ΔP ≈ −D_mod P Δy + ½ Cx P Δy². The tool prints the first term alone, both terms, and the exact repricing at y ± 1 point.

Yield to call repeats the yield calculation with the payments ending on the call date and the call price in place of F. Yield to worst is the lower of the yield to call and the yield to maturity.

What it assumes

Every payment is made in full and on time (no default). The yield is one rate applied to all payments and compounded once per coupon period, so yields of bonds with different coupon frequencies are not directly comparable. All coupon periods are equal: no odd first or last coupon. The fractional period is discounted at compound interest (the market, or "street", convention); the U.S. Treasury's auction formula (31 CFR Part 356, Appendix B) uses simple interest over that fraction and gives a slightly different price between coupon dates (for its example D, 99.730918 against 99.738573 per 100 here). The call date must be a coupon date. Taxes, fees and bid-ask spreads are left out.

When it misleads

The yield to maturity is earned only if the bond is held to maturity and each coupon is reinvested at that same yield; sell earlier or reinvest at other rates and the realised return differs. Duration and convexity describe a small, equal move in the yield of this one bond; real moves differ by maturity, and for a large move even the convexity estimate drifts from the exact price. For a callable bond the duration printed here is to maturity and overstates the sensitivity when a call is likely, and the yield to worst considers only the one call date entered. A bond with a floating coupon, an inflation-indexed principal, a sinking fund or a real chance of default needs a different model. A higher yield is not a better bond: it is the market's price for more risk, a longer wait, or both.

Worked example

The default case is example A of 31 CFR Part 356, Appendix B: an 8¾ % 30-year Treasury bond, semi-annual coupons, priced on its issue date at a yield of 8.84 %, face value 100.

Coupon: C = 100 × 0.0875 / 2 = 4.375. Yield per period: y_p = 0.0884 / 2 = 0.0442. Coupons: N = 30 × 2 = 60.

Discount factor of the last payment: (1.0442)⁻⁶⁰ = 0.074641. Annuity factor: (1 − 0.074641)/0.0442 = 20.935719.

Price: P = 4.375 × 20.935719 + 100 × 0.074641 = 91.593772 + 7.464121 = 99.057893, the figure printed in the regulation.

Current yield: 100 × 8.75 / 99.057893 = 8.833 %. It lies between the coupon rate (8.75 %) and the yield to maturity (8.84 %) because the bond is bought slightly below face value.

Duration: the sum of t × PV over the 60 payments is 1,084.205, so D_mac = 1,084.205 / 99.0579 = 10.945 years and D_mod = 10.945 / 1.0442 = 10.482. Convexity: 20,393.2 / (99.0579 × 1.0442²) = 188.81.

One point up in yield (9.84 %): duration alone gives −10.482 × 99.058 × 0.01 = −10.38; with convexity −10.38 + ½ × 188.81 × 99.058 × 0.0001 = −9.45; exact repricing −9.51. One point down (7.84 %): +10.38, +11.32 and +11.39.

A calculator for education and analysis, not investment advice.