What is the bond price formula?

Price = Σ_{t=1}^{n} C/(1+y)^t + F/(1+y)^n. C is the periodic coupon (face × coupon rate / freq), y is the periodic YTM (annual YTM / freq), n is total periods (years × freq), F is face value. The sum collapses to Price = C × [(1 − (1+y)^-n)/y] + F/(1+y)^n. This is the textbook formula from every CFA Level I fixed-income reading.

What is the difference between coupon rate and YTM?

Coupon rate is the fixed annual payment rate set at issuance — it determines the dollar coupon ($1,000 × 5% = $50 annually). YTM is the market-required return reflecting current interest rates, credit risk, and maturity. They're equal at issuance for par bonds. After issuance, if interest rates rise, YTM rises and price falls below par (discount). If rates fall, YTM falls and price rises above par (premium).

Why does the price drop when interest rates rise?

A bond's coupons are fixed at issuance. If new bonds yield 6% but your old bond pays only 4%, no one will pay full price for your bond — it must trade at a discount to make the holder's total return equal to the new 6% market rate. The inverse price-yield relationship is the most important fact in fixed-income. The slope is measured by modified duration; see /tools/calculators/financial/basis-point for price-impact math.

What is the difference between clean price and dirty price?

Clean price excludes accrued interest since the last coupon date. Dirty price (also called "full" or "invoice" price) includes it. Bonds trade on clean price quotes but settlement is at dirty price (clean + accrued). For a $1,000 5% semi-annual bond, 3 months after last coupon, accrued = $25 × (3/6) = $12.50. This calculator produces clean price; add accrued separately for settlement.

How does bond pricing work for zero-coupon bonds?

Zero-coupon bonds have no coupon payments — price = Face / (1+y)^n. A 10-year zero with 4.5% YTM trades at $643.93 = $1,000 / 1.045^10. US Treasury STRIPS (Separate Trading of Registered Interest and Principal Securities) work this way; created by the Treasury in 1985 to facilitate zero-coupon trading. Duration of a zero = its maturity, making them maximum-duration instruments for institutional rate bets.

What is semi-annual compounding for bonds?

US corporate and Treasury bonds pay coupons every six months — twice per year. A 5% coupon $1,000 face bond pays $25 every 6 months. YTM is quoted on a semi-annual bond-equivalent basis (BEB): 6% YTM means 3% per six-month period, not 6% × something. European Eurobonds use annual coupons. Mortgage securities use monthly compounding. Always confirm the convention from the prospectus or Bloomberg DES screen.

Why are munis priced at $5,000 face?

Municipal bonds use $5,000 face value (minimum denomination $5,000 since 1983 SEC Rule 15c2-12) versus $1,000 for corporates and Treasuries. The MSRB (Municipal Securities Rulemaking Board) enforces this minimum to protect retail investors from over-concentration in small lots. NYC GO, California GO, Texas school district bonds all trade at $5,000 minimum. Quoting is by yield or price-per-$100-of-face, not dollar price.

How is YTM different from coupon yield?

Current yield = annual coupon / current price (e.g., $50 / $950 = 5.26%). YTM is the IRR of all future cash flows including the price-to-par convergence at maturity. For a discount bond, YTM > current yield > coupon yield. For a premium bond, YTM < current yield < coupon yield. CFA Level I teaches all three. Most bond quotes use YTM as the standard measure of return.

What is the relationship between bond price and credit rating?

Higher credit risk = higher required yield = lower price for the same coupon. AAA Treasuries yield ~4.5%; AA corp ~4.8%; BBB corp ~5.5%; BB high-yield ~7.5%; CCC ~12%+. Moody's, S&P, and Fitch are the three major rating agencies; the SEC's NRSRO designation (1975) defines official ratings. Default rate by rating per Moody's Annual Default Study: AAA 0.04%, BB 1.3%, B 3.5%, CCC 20%+.

When was modern bond pricing math developed?

The mathematical foundation traces to John Burr Williams's 1938 Harvard thesis The Theory of Investment Value , which formalized PV of future cash flows as the basis for asset pricing. Frederick Macaulay's 1938 NBER work on duration extended this. Burton Malkiel's 1962 QJE paper introduced modified duration. Frank Fabozzi's 1989 textbook Bond Markets, Analysis and Strategies codified the modern practitioner approach.

Why does the bond price equation use 1/(1+y)^t?

The discount factor 1/(1+y)^t comes from compound-interest math — Bernoulli (1683), Williams (1938). One dollar today equals $1×(1+y)^t in t periods, so $1 in t periods equals $1/(1+y)^t today. Each future cash flow is worth less the longer you wait — this is the time value of money. See /tools/calculators/financial/time-value-money for the foundational 5-variable equation.

How is bond pricing different on Bloomberg vs Excel?

Bloomberg's YA function uses 30/360, ACT/365, or ACT/ACT day-count conventions depending on bond type. Excel's PRICE() function takes settlement and maturity dates and computes clean price with day-count basis as a parameter. This calculator simplifies to whole periods — for institutional pricing requiring fractional-day accrued interest, use Bloomberg or Excel PRICE() with basis=1 (ACT/ACT for Treasuries) or basis=4 (30E/360 for Eurobonds).

Coupon stream + face block diagram

Bond Price Calculator

To price a bond, sum the present values of all future coupon payments plus the present value of the face value at maturity: Price = Σ C/(1+y)^t + F/(1+y)^n. The diagram draws each coupon as a green stack along the bond's life and the face value as a gold block at maturity. Above face = premium; below = discount; at = par.

$980.05

Bond price

DISCOUNT

vs par

$339.23

PV of coupons

$640.82

PV of face

Sum of all PVs.

Quick Conversion

FROM (face ($))

TO (price ($))980.0454

Cpn %YTM %YearsFreq

Formula: P = C × [(1−(1+y)^-n)/y] + F/(1+y)^n

Bond Pricing Diagram

Bond inputs

Face value ($)

Coupon rate (%)

YTM (%)

Years to maturity

Coupons per year

Label

DISCOUNT — 98.00% of face

$980.05

Real bond presets

5% 10-yr bond price at varying YTM ($1,000 face, semi-annual)

YTM	Price	vs Par	Status
2%	$1270.68	+270.68	PREM
3%	$1171.69	+171.69	PREM
4%	$1081.76	+81.76	PREM
4.5%	$1039.91	+39.91	PREM
5%	$1000.00	0.00	PAR
5.5%	$961.93	-38.07	DISC
6%	$925.61	-74.39	DISC
7%	$857.88	-142.12	DISC
8%	$796.15	-203.85	DISC
9%	$739.84	-260.16	DISC
10%	$688.44	-311.56	DISC
12%	$598.55	-401.45	DISC

Solving for YTM given price? Try Bond YTM Calculator →

Formula

P = Σ C/(1+y)^t + F/(1+y)^n= C × [(1−(1+y)^-n)/y] + F/(1+y)^n

C = periodic coupon (face × rate / freq), y = periodic YTM (annual / freq), n = total periods.

Worked: $1,000 face, 4.25% semi-annual, 10y, YTM 4.5%. C=$21.25, n=20 periods, y=0.0225. P = 21.25 × ((1−1.0225^-20)/0.0225) + 1000/1.0225^20 = 338.69 + 641.16 = $979.85 — slight discount.

From John Burr Williams (1938) to Bloomberg PRICE() (2026)

In 2026, a credit research analyst at PIMCO in Newport Beach prices a new 10-year corporate bond at 5.50% coupon, comparing against the 5.80% YTM the syndicate desk recommends. The math behind the price quote — sum of discounted cash flows — has not changed in 90 years; only the speed and ubiquity of the computation have.

John Burr Williams's 1938 Harvard PhD thesis The Theory of Investment Valuelaid the foundation: an asset's intrinsic value equals the present value of its expected future cash flows. Williams applied this to common stocks via the dividend discount model, but the same equation applies verbatim to bonds. Williams predated Modigliani-Miller by 20 years; his framework remains the basis of CFA Level I.

Frederick Macaulay's 1938 NBER monograph The Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856 compiled 80 years of US bond data and introduced the concept of duration — the weighted-average time to receive a bond's cash flows. Burton Malkiel's 1962 QJE paper modified Macaulay duration to its modern price-sensitivity form. The CFA Institute's Level I fixed-income curriculum still cites both papers.

Frank Fabozzi's definitive textbook Bond Markets, Analysis and Strategies(first edition 1989, now 10th edition) codified the practitioner approach. Fabozzi at Yale and his contemporaries at PIMCO (Bill Gross, Mohamed El-Erian) built the modern $50T US fixed-income industry around the pricing math this calculator implements. Lehman Brothers' bond indices (now Bloomberg Barclays / Bloomberg Index Services) standardized benchmark composition starting in 1973.

Day-count conventions evolved alongside the math. US Treasuries use ACT/ACT (actual days divided by actual days in year). US corporate bonds use 30/360 (assume 30-day months and 360-day years). Eurobonds use 30E/360 (a slight variant). Mortgage-backed securities use ACT/360. Bloomberg's YA function offers all conventions; this calculator simplifies to whole periods — sufficient for ±$0.10/$1,000 face accuracy on most plain-vanilla bonds.

The Securities and Exchange Commission's Rule 15c2-12 (1989) required Municipal Securities Rulemaking Board disclosure for muni bonds, including standard pricing methodology. FINRA TRACE (Trade Reporting and Compliance Engine, 2002) brought transparent corporate-bond trade reporting. MarketAxess (founded 2000), Tradeweb (1996), and Bloomberg ALLQ now execute trillions in daily fixed-income volume — every bid quoted as price, yield, or spread using exactly the math this page implements.

By 2026, electronic bond execution dominates institutional trading. AI-powered bond screeners (BlackRock Aladdin, MSCI Barra, Bloomberg PORT) run the price-yield equation millions of times per second across portfolios. Retail investors access bonds via Fidelity, Schwab, and Vanguard's online bond ladders — same math, retail wrapper. The pricing diagram on this page is the visual analog of the Bloomberg DES screen used by every fixed-income desk worldwide; the equation is unchanged from John Burr Williams's 1938 thesis.

How to use the bond pricing diagram

Enter face value. Typically $1,000 for US Treasury and corporate bonds, $5,000 for munis.
Enter coupon rate. Annual rate as percent of face — fixed at issuance.
Enter YTM. Current market-required yield — pulled from Bloomberg, Treasury.gov, or your trader.
Enter years and frequency. 30y semi-annual = 60 periods; 10y annual Eurobond = 10 periods.
Read the price. Sum of green stacks (PV coupons) + gold block (PV face). Above $1,000 = premium.

Bond Price — frequently asked questions

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What fixed-income analysts & underwriters say

4.9

Based on 5,180 reviews

“The premium/discount/par badge is the first thing my interns need to internalize. The visual of coupon stream as green stacks plus the gold face-value block is the cleanest pedagogical bond diagram I've seen. Saving for new-hire training.”

Catalina Esperanza Ramírez-Vásquez

Fixed-income credit analyst, IG corporate desk

May 17, 2026

“The muni preset at $5,000 face is the correct convention — most online calculators get this wrong. The MSRB Rule 15c2-12 callout in the FAQ saves me explaining the minimum denomination to retail clients weekly.”

Mihail Aleksandrovich Stepanov-Volkov

Municipal bond underwriter, NYC sell-side

April 24, 2026

“I link my CFA-candidate mentees here for Level I fixed-income prep. The Macaulay-Malkiel-Fabozzi citation chain is exactly the academic lineage we expect from Level II curriculum.”

Yuki Saoirse Tanaka-Murphy

CFA charterholder, fixed-income research

April 4, 2026

“For pricing 10-yr Brazil USD-denominated bonds at 6.5%+ YTM, the calculator handles the math correctly. The clean-vs-dirty-price FAQ saves me re-explaining settlement conventions to portfolio managers.”

Adekunle Babatunde Adeyemi-Olawale

Sovereign bond trader, EM hard-currency desk

March 20, 2026

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