Price and yield

Present value from a cash-flow schedule

The primitive under every other calculation here. A bond is a known list of dated amounts; its full price is the sum of those amounts multiplied by discount factors. Whether the discount factors come from one yield or from a curve is the only structural choice.

Field	Value
Formula	P_full = sum over k of CF_k * DF(t_k). Discounting at a single yield: DF = (1 + y/f)^-n_k, with n_k = (k-1) + w
Worked, first flow	2026-11-15: 2.250 * (1 + 0.023420352)^-0.434783 = 2.250 * 0.989985137 = 2.227467
Worked, final flow	2034-11-15: 102.250 * 0.683539924 = 69.891957
Sum of all 17 flows	100.021739 = the full price
Quoted price	100.021739 - 1.271739 = 98.750000
Concentration	The redemption flow alone is 69.877 percent of the full price; all 16 coupons together are 30.123 percent

• The last field is why duration on a par-ish medium bond sits near two thirds of its maturity rather than at its midpoint: the principal dominates the present value, so the schedule's centre of mass is pulled toward the end.
• Discounting at a single yield and discounting on a curve give the same price only if the curve is flat. Everything called a spread in the curves section is a measure of the gap between those two calculations.
• The schedule must be generated from the maturity date backwards, not from the issue date forwards, or the intermediate dates will be wrong for any bond whose dates fall near a month end. That error changes both the flow dates and E, so it moves the price and the accrued together and can look self-consistent.
• For a bond in its final coupon period, US street convention switches to simple interest rather than compounding the single remaining flow. Systems that keep compounding produce a small yield error that grows as maturity approaches and is largest on the last day.

Also described at: Wikipedia · Wikidata

Street-convention yield to maturity

The single nominal rate, compounded at the coupon frequency, that discounts a bond's remaining cash flows back to its full price. There is no closed form; it is found by iteration. Street convention refers to the specific choice of compounding the fractional first period rather than treating it with simple interest.

Field	Value
Formula	Solve for y: P_full = sum over k of CF_k * (1 + y/f)^-((k-1)+w), where w = (days from settlement to the next coupon) / E
Worked, inputs	P_full = 100.021739, f = 2, w = 80/184 = 0.434783, 17 flows of 2.250 plus 100 at 2034-11-15
Solution	y = 4.684070 percent nominal semiannual
Check	Substituting back gives 100.021739 against the target 100.021739
Sensitivity of the solve	A 0.01 error in the full price moves the yield by roughly 0.15 basis points
Annual equivalent	(1 + 0.023420352)^2 - 1 = 4.738922 percent

• Yield is a summary of a price, not an input to it. Two bonds with the same yield and different coupons do not have the same exposure, the same reinvestment profile, or the same tax treatment, which is why yield comparisons across coupons need a spread measure rather than a yield difference.
• The convention is not universal. Street convention compounds the fractional period; true or ICMA yield does the same but some systems apply simple interest over the fractional period, and Japanese, Italian and certain other markets have their own quoted measures. On this bond the variants sit within a basis point or so, which is small enough to be missed and large enough to matter on a large position.
• Bisection is slow but cannot diverge, and a bond's price is monotonic in yield, so it always converges. Newton iteration on the analytic derivative is faster and can overshoot into a negative discount factor on a deeply distressed price; a bracketed method is the safer default for production.
• The reinvestment assumption embedded in yield is that every coupon is reinvested at y until maturity. Nothing enforces that, so realised return equals yield only under a condition nobody controls. Yield is a discount rate, not a forecast.

Source: SIFMA standard securities calculation methods; ICMA Rule 251 for the true-yield variant

Also described at: Wikipedia · Wikidata · 31 CFR Part 356, Appendix B (formulas and tables)

The fractional first period, w

Between coupon dates the first cash flow is less than a full period away, so every exponent in the discounting is shifted by a fraction. That fraction, w, is the single mechanism connecting a settlement date to a price, and it is the source of most off-by-one pricing errors.

Field	Value
Formula	w = (days from settlement to the next coupon date) / E, with both counts on the bond's own day-count basis. Then n_k = (k-1) + w, and the accrued fraction is A/E = 1 - w
Worked	80 days from 2026-08-27 to 2026-11-15, over E = 184, so w = 0.434782609
Complement	A/E = 104/184 = 0.565217391, and w + A/E = 1.000000000
Effect on the price	On a coupon date w = 1 and the full price at this yield would be 98.783577; at w = 0.434783 it is 100.021739
Growth through the period	The full price at a fixed yield rises by a factor of (1 + y/2)^(1-w) across the period, which is 1.013170937 at this settlement

• w and the accrued fraction are complements computed from the same two day counts. If a system computes them independently, on different bases, the price and the accrued disagree and the resulting quoted price is wrong by the difference. This is a real and common defect and it is invisible on a coupon date.
• The day-count basis used for w should be the bond's own. Using ACT/ACT for accrued and 30/360 for w, or the reverse, produces a yield error of a fraction of a basis point on this bond and considerably more on a short one.
• In the final coupon period there is only one flow and w is the whole exposure, so the choice between compounding and simple interest over w dominates the yield. That is precisely where street convention and true yield diverge most.
• The identity w + A/E = 1 is the cheapest available unit test for a pricing library, and it fails immediately if the schedule was generated in the wrong direction.

Also described at: 31 CFR Part 356, Appendix B (formulas and tables)

Current yield

Annual coupon over quoted price. It measures cash income against capital employed and nothing else: no capital gain, no timing, no compounding.

Field	Value
Formula	Current yield = C / P
Worked	4.500 / 98.750 = 4.556962 percent
Against yield to maturity	4.556962 percent against 4.684070 percent, a gap of 12.71 basis points
Where the gap comes from	The bond is priced at a discount, so the pull to par from 98.750 to 100 over 8.2247 years is a return that current yield ignores entirely
At a premium	At a quoted price of 106.000 the current yield would be 4.245283 percent while the yield to maturity is 3.647654 percent, so the sign of the gap flips

• Current yield is above yield to maturity for a premium bond and below it for a discount bond, always, and the two coincide only at par. That makes it a useful sanity check on which side of par a price sits and a poor basis for a purchase decision.
• It is still the right number for a specific question: how much cash the position generates against the money tied up in it. For an entity funding a portfolio out of coupon income, current yield is the operative constraint and yield to maturity is not.
• Because it ignores maturity it makes long premium bonds look attractive and short discount bonds look poor. Screens ranked on current yield systematically select instruments with the largest embedded capital loss.

Also described at: Wikipedia · Wikidata

Simple yield to maturity

Current yield plus the pull to par amortised in a straight line over the remaining life, with no compounding anywhere. Widely used in the Japanese government bond market and as a quick mental approximation elsewhere.

Field	Value
Formula	Simple yield = (C + (100 - P)/n) / P, where n is the years to maturity
Worked	(4.500 + (100 - 98.750)/8.224658) / 98.750 = (4.500 + 0.151982) / 98.750 = 4.710868 percent
Against yield to maturity	4.710868 percent against 4.684070 percent, out by +2.68 basis points
Direction of the error	It overstates the yield of a discount bond, because straight-line amortisation credits the capital gain earlier than compounding does
Where it is exact	Only at par, where the second term vanishes and it collapses to current yield

• On this bond the approximation is within 2.7 basis points, which is good enough for a conversation and not for a trade. The error grows with the distance of the price from par and with maturity.
• The n in the numerator is years to maturity on whatever basis the market uses, and the choice of ACT/365 against 30/360 against exact period count moves the answer. For a quoted convention the basis is specified; for a mental approximation it is not.
• It is the standard quoted measure for Japanese government bonds, so a JGB yield taken from one source and a Treasury yield from another may not be the same kind of number.

Running yield and the income component of return

The coupon income earned over a holding period, expressed as a rate on the capital employed. It is the part of return that arrives whether or not the price moves, and the part that funding cost is set against.

Field	Value
Formula	Running yield over h days = (C * h/B) / P_full, annualised. Net of funding at rate r on the full price, the carry is (C * h/B_bond - r * P_full * h/360) per 100 of par
Worked, income	Reference bond held 92 days from 2026-08-27: coupon earned = 2.250 for the period ending 2026-11-15 plus accrued of 0.149171 after it, against accrued given up of 1.271739
Net income	2.250 + 0.149171 - 1.271739 = 1.127432 per 100 of par
Financing at an illustrative 4.000 percent	100.021739 * 0.04 * 92/360 = 1.022444
Carry	1.127432 - 1.022444 = 0.104988 per 100 of par, before any price change
Annualised	0.104988 / 100.021739 * 360/92 = 0.410732 percent

• Carry is positive here because the coupon rate exceeds the assumed financing rate, which is the ordinary state of a positively sloped funding position. It reverses when short rates rise above the coupon, and the reversal is arithmetic rather than a market view.
• The income and the financing are on different day-count bases: the coupon on ACT/ACT, the repo on ACT/360. Netting them without converting overstates carry by roughly the 365/360 factor on the financing leg.
• Carry is not return. A position can have positive carry and negative return over the same window, and the decomposition in the curves section separates the two explicitly.
• Special-collateral repo changes the financing leg without touching the bond, so two holders of the same bond have different carry. That difference is the whole economics of the specials market.

Also described at: Wikipedia · Wikidata

Yield to call and yield to worst

Yield to call is the yield computed to a call date and its call price rather than to maturity and par. Yield to worst is the minimum over maturity and every call date, and it exists to give a conservative floor on a callable bond's yield.

Field	Value
Formula	For each redemption date j: solve P_full = sum of CF_k*(1+y_j/f)^-n_k with the final flow equal to (C/f + call price_j). Yield to worst = min over all j including maturity
Worked, first call	To 2029-11-15 at 101.000: 5.214105 percent
Worked, second call	To 2031-11-15 at 100.500: 4.858008 percent
Worked, third call	To 2033-11-15 at 100.000: 4.705210 percent
To maturity	4.684070 percent
Yield to worst	4.684070 percent, at maturity
Crossover prices	The first call becomes the worst above a quoted price of 101.534366; the second above 101.224606; the third above 99.993846

• For a bond trading at a discount to every call price, the maturity yield is the worst and yield to worst carries no information. The measure only bites on premium bonds, which is exactly when a call is likely to be exercised, so the coincidence is not accidental.
• The crossover prices in the last field are exact and worth computing once per bond. They tell you at what price the reported yield to worst will silently switch to a different date, which is the moment a screen's duration figure jumps.
• Yield to worst is not a valuation. It prices the bond as if the worst outcome were certain, which understates value for the holder who is short the option only probabilistically. Option-adjusted spread in the curves section is the measure that prices the optionality rather than assuming it.
• Duration reported alongside yield to worst is normally the duration to the worst date, so a small price move across a crossover changes reported duration discontinuously. Effective duration does not have that discontinuity, which is one reason it is the risk number and yield to worst is the quoted one.
• The call schedule used here is illustrative and attached to the reference bond only to make the arithmetic concrete. A make-whole call, which redeems at a spread to Treasuries rather than a fixed price, is not a call for this purpose and should not be run through this formula.

Discount rate to price on a money-market instrument

A US Treasury bill, banker's acceptance or commercial paper quoted on a discount basis is quoted as a rate applied to face value, not to the amount invested. The price follows directly and is always above the price a same-numbered add-on yield would give.

Field	Value
Formula	P = 100 * (1 - d * t/360), where d is the quoted discount rate and t the actual days to maturity
Worked, 91 days	100 * (1 - 0.0400 * 91/360) = 100 * (1 - 0.010111111) = 98.988889 per 100 of face
Worked, 28 days	100 * (1 - 0.0400 * 28/360) = 99.688889
Worked, 182 days	100 * (1 - 0.0400 * 182/360) = 97.977778
On 10,000,000 face, 91 days	proceeds 9,898,888.89, discount 101,111.11
Inverting	d = (100 - P)/100 * 360/t, so at P = 98.988889, d = 4.000000 percent

• The discount rate understates the return because it divides the discount by face value rather than by the money actually invested. That is not a quirk to be corrected in the price formula; the price formula is correct and the rate is simply defined that way.
• Bills are quoted in discount terms on the secondary market and auctioned on a discount basis, while the Treasury separately publishes an investment rate on a bond-equivalent basis. Two published numbers on the same security differ by roughly 10 basis points here, and both are correct.
• The denominator is 360 and the numerator actual days, so a 360-day holding period accrues exactly the quoted rate and a 365-day one accrues more. On a discount instrument that surfaces as a price below the naive expectation.
• Never compare a discount rate with a bond yield directly. Convert to bond-equivalent yield first; the entry below gives the closed form.

Also described at: 31 CFR Part 356, Appendix B (formulas and tables) · TreasuryDirect: Treasury bills

Bond-equivalent yield from a discount rate

The conversion that makes a discount-quoted instrument comparable with a coupon bond: restate the return against the amount invested rather than face value, and against a 365-day year rather than 360.

Field	Value
Formula	BEY = (100 - P)/P * 365/t, which simplifies to BEY = 365*d / (360 - d*t) with no reference to price
Worked, 91 days from price	(100 - 98.988889) / 98.988889 * 365/91 = 4.096981 percent
Worked, 91 days from the closed form	365 * 0.0400 / (360 - 0.0400 * 91) = 14.600000 / 356.360000 = 4.096981 percent
Uplift over the discount rate	4.096981 - 4.000000 = 9.70 basis points at 91 days
Uplift at 28 days	6.82 basis points
Uplift at 182 days	13.93 basis points, so the correction roughly scales with maturity

• Two separate corrections are folded into one formula: the denominator change from face to price, worth about 4 basis points at 91 days here, and the 365/360 year-length change, worth about 5.6. Both move the same way, so the total is always an uplift.
• The closed form is exact and does not need the price, which makes it the right thing to implement. Deriving it through the price and rounding the price first introduces an error at the fourth decimal of the yield.
• This formula holds for instruments of one semiannual period or less. Beyond about 182 days a bond-equivalent measure has to account for one intervening compounding, and the United States Treasury publishes its own investment-rate formula for that case. The exact published form is not reproduced here; take it from the source rather than extending this expression.
• A bill and a short coupon bond quoted on the same screen are on different conventions unless the screen has already converted. Whether it has is a data question with a 10-basis-point answer.

Source: US Treasury published bill auction results (discount rate and investment rate); SIFMA standard securities calculation methods

Also described at: Wikipedia · Wikidata · 31 CFR Part 356, Appendix B (formulas and tables)

Discount rate, add-on yield and money-market yield compared

Three rates that can be quoted on the same instrument and are never equal. The discount rate divides by face and 360; the add-on or money-market yield divides by price and 360; the bond-equivalent yield divides by price and 365.

Field	Value
Formula	d = (100-P)/100 * 360/t. Add-on = (100-P)/P * 360/t. BEY = (100-P)/P * 365/t. Add-on = 360/365 * BEY, always
Worked, 91 days at a 4.000 percent discount rate	price 98.988889; add-on 4.040858 percent; BEY 4.096981 percent
Ordering	d = 4.000000 < add-on 4.040858 < BEY 4.096981, and this ordering is structural
Add-on to BEY	4.040858 * 365/360 = 4.096981 percent, matching the BEY column
Reverse conversion	An add-on yield of 4.100 percent for 91 days implies a discount rate of 360*0.041/(360 + 0.041*91) = 4.057944 percent
On 10,000,000, 91 days	the interest is 101,111.11 on all three quotes; only the rate differs

• The interest amount is identical in every case. The three rates are three ways of dividing the same cash by different bases, which is why arguing about which is right is a category error and converting is not.
• An instrument quoted add-on and one quoted discount at the same numeric rate are not the same price. On a 91-day trade the difference is roughly 4 basis points of yield, and on a large short-dated book that is not a rounding item.
• Commercial paper is quoted discount in the United States and interest-bearing in some other markets, so a CP programme spanning both has two quoting conventions for one funding cost.
• Certificates of deposit and interbank deposits are add-on instruments: interest is added to the principal at maturity rather than deducted from the face at issue. That is a structural difference, not a quoting one, and it changes the amount that settles at both ends.

Converting between compounding frequencies

A nominal rate is meaningless without its compounding frequency. Converting is a two-step operation through the effective annual rate, and skipping it is the most common way a genuine yield difference is manufactured out of nothing.

Field	Value
Formula	Effective annual rate: EAR = (1 + y_m/m)^m - 1. Nominal at frequency k: y_k = k*[(1 + EAR)^(1/k) - 1]. Continuous: y_c = ln(1 + EAR)
Worked, semiannual to annual	(1 + 0.023420352)^2 - 1 = 4.738922 percent
Worked, semiannual to quarterly	4 * [(1 + 0.047389217)^(1/4) - 1] = 4.656961 percent
Worked, semiannual to continuous	ln(1 + 0.047389217) = 4.630061 percent, equivalently 2*ln(1 + y/2)
Worked, annual to semiannual	An annual 5.000000 percent is 2*[(1.05)^(1/2) - 1] = 4.939015 percent nominal semiannual
Spread of the basis	The same yield spans 4.630061 to 4.738922 percent across the bases, a range of 10.89 basis points

• The direction of the effect is fixed: for a given effective rate, more frequent compounding means a lower nominal rate. A semiannual US Treasury yield is therefore numerically below the annual-equivalent yield a European convention would report on the same bond.
• European government bonds are conventionally quoted on an annual basis and US Treasuries on a semiannual one. Comparing a bund yield and a Treasury yield without conversion introduces roughly 5 basis points of pure convention on a 4.7 percent level, and the size of the artefact grows with the level of rates.
• Curve construction and option pricing normally run on continuously compounded rates because the algebra is cleaner, and then quote back out. Discount factors are basis-free, which is why passing discount factors between systems is safer than passing rates.
• The conversion is exact, not an approximation, and the shortcut of adding a fixed number of basis points is only accurate near the level at which it was calibrated.

Also described at: Wikipedia · Wikidata

Zero-coupon yield, discount factors and strips

A zero-coupon instrument has one flow, so its price and its yield determine each other exactly with no reinvestment assumption and no iteration. It is the cleanest object in the market and the building block a curve is expressed in.

Field	Value
Formula	P = 100 * (1 + z/f)^-n, so z = f * [(100/P)^(1/n) - 1]. DF = P/100. Macaulay duration = n/f exactly; convexity = n*(n+1)/(f^2 * (1+z/f)^2)
Worked, price from rate	The reference bond's final flow: n = 16.434783 periods at 4.684070 percent gives DF = 0.683539924, so a 100 strip to that date prices at 68.353992
Worked, rate from price	2 * [(1/0.683539924)^(1/16.434783) - 1] = 4.684070 percent
Duration	Macaulay = 16.434783/2 = 8.217391 years; modified = 8.029341
Convexity	16.434783 * 17.434783 / (4 * 1.047389217) = 68.393119
Against the coupon bond	Same maturity, but modified duration 8.029341 against 6.706168 and convexity 68.393119 against 53.827761

• A zero's Macaulay duration is its maturity, exactly, with no dependence on yield. That is the only instrument for which the statement is true, and it is why zeros are the natural basis for a key-rate framework.
• The strip has both more duration and more convexity than the coupon bond of the same maturity. A duration-matched trade between a coupon bond and a strip is therefore not convexity-matched, and the residual is a long-gamma position that shows up in large moves.
• Discount factors are the interchange format. A rate needs a compounding convention and a day count to be interpreted; a discount factor needs a date. Systems that exchange rates rather than factors accumulate convention errors at every boundary.
• The formula assumes the quoted zero rate is on the same compounding frequency f as the rest of the calculation. A continuously compounded zero substituted into this expression will be out by the conversion in the entry above.

Also described at: Wikipedia · Wikidata

What yield to maturity assumes, and when it is realised

Yield to maturity is the internal rate of return of a bond's cash flows. It is realised as a holding-period return only if every coupon is reinvested at that same rate until maturity, which is an assumption about the future, not a property of the bond.

Field	Value
Formula	Realised annualised return = f * [(terminal wealth / P_full)^(1/n) - 1], where terminal wealth = sum of CF_k * (1 + r/f)^(n - n_k) at the actual reinvestment rate r
Setup	Reference bond held to maturity: P_full = 100.021739, n = 16.434783 semiannual periods, 16 coupons of 2.250 plus 102.250 at maturity
Worked, reinvesting at the yield	Terminal wealth = 146.329037, giving a realised return of 4.684070 percent, equal to the yield by construction
Reinvest at 2.000 percent	Terminal wealth = 141.468497, realised return 4.263787 percent
Reinvest at 7.000 percent	Terminal wealth = 151.086285, realised return 5.082914 percent
Range	A reinvestment rate between 2 and 7 percent moves the realised return over 82 basis points on a bond whose yield never changed

• The reinvestment exposure is concentrated in the coupons, and the coupons are a minority of this bond's present value, which is why the realised-return range above is a few tens of basis points rather than hundreds. On a high-coupon long bond the range is far wider; on a strip it is zero.
• This is the precise sense in which duration is a hedge and yield is not a forecast. Immunisation works by setting duration equal to the horizon so that the reinvestment gain and the price loss offset, and it works because the two effects have opposite signs.
• A portfolio's yield is not the market-value-weighted average of its holdings' yields except by coincidence; the correct aggregate is the internal rate of return of the combined cash flows. The weighted average is a common shortcut and it drifts most when the holdings differ in maturity.
• Holding to maturity does not remove interest-rate risk, it converts price risk into reinvestment risk. Both are real; only one is marked.

Also described at: Wikipedia · Wikidata