What yield to maturity measures
Yield to maturity is the internal rate of return on a bond bought today and held to redemption. It is the one discount rate at which the present value of every remaining coupon plus the face value equals the price you actually pay. Quote a bond's price and its YTM and you have said the same thing twice — the market moves fluidly between the two because they are algebraically linked.
The reason YTM dominates fixed-income conversation is that it is comparable. A 4% coupon bond at 88 and a 7% coupon bond at 106 are not obviously rankable, but their yields are on the same scale immediately. That comparability is why bond traders talk in yields and settle in prices.
YTM captures both sources of return: the coupon income you collect along the way, and the capital gain or loss you book when a bond bought away from par is redeemed at par. The current yield sees only the first of those, which is why it is a poor guide to a deeply discounted bond. If you already know the yield and want the price, run the calculation the other way with the bond price calculator.
Why there is no formula, only a solver
Rearranging the pricing equation to isolate i is impossible for any realistic bond. Substituting x = 1 + i turns it into a polynomial of degree N, and the Abel–Ruffini theorem says polynomials of degree five or higher have no general solution in radicals. A ten-year semiannual bond has N = 20. So every yield you have ever seen on a screen was found by iteration.
The good news is that the iteration is exceptionally well behaved. Price is a strictly decreasing function of yield whenever all the cash flows are positive, so there is exactly one root and no ambiguity of the kind that plagues internal rate of return on projects with sign changes. This calculator brackets the root between a periodic yield of −90% and +500% and bisects until it converges, widening the upper end of the bracket if a price low enough to imply a yield above 500% per period needs it — slower than Newton–Raphson, but it cannot diverge or land on the wrong root. If you are computing an IRR on an irregular cash-flow stream instead, where multiple roots are a genuine hazard, use the money-weighted return calculator.
Two yields, one bond. The solver returns a periodic yield — the return per coupon period. Multiplying it by the number of periods per year gives the nominal bond-equivalent yield, which is the market convention and the number quoted everywhere. Compounding it instead gives the effective annual yield. A 5% semiannual periodic yield is a 10.00% BEY and a 10.25% effective annual yield. Both appear as outputs above; use BEY to compare bonds with each other and the effective rate to compare a bond against an annually compounded instrument such as a certificate of deposit.
Worked example: an 8% bond bought at 87.538
You buy a $1,000 face bond with an 8% semiannual coupon, ten years to run, at a clean price of $875.38. Each half-year pays $40, and there are N = 20 periods left.
- Start with the shortcut. The approximation is [C + (F − P)/n] ÷ [(F + P)/2] = [80 + (1000 − 875.38)/10] ÷ [(1000 + 875.38)/2] = 92.462 ÷ 937.69 = 9.86%. That is your opening bracket, not your answer.
- Try 9% (i = 4.5%). 1.04520 = 2.411714, so the discount factor is 0.414642. Coupons: 40 × (1 − 0.414642) ÷ 0.045 = 40 × 13.007956 = $520.32. Principal: $414.64. Price = $934.96 — too high, so the true yield is above 9%.
- Try 11% (i = 5.5%). 1.05520 = 2.917757, discount factor 0.342729. Coupons: 40 × 11.950382 = $478.015. Principal: $342.729. Price = $820.74 — too low, so the yield is below 11%.
- Interpolate. (934.96 − 875.38) ÷ (934.96 − 820.74) = 59.58 ÷ 114.22 = 0.5216, giving i ≈ 4.5% + 0.5216 × 1.0% = 5.02%, a BEY of about 10.04%. Linear interpolation across a convex curve overshoots slightly.
- Try 10% (i = 5%). 1.0520 = 2.653298, discount factor 0.376889. Coupons: 40 × 12.462210 = $498.49. Principal: $376.89. Price = $875.38 — an exact match, so the yield to maturity is 10.00%.
- Convert and cross-check. Effective annual yield = 1.05² − 1 = 10.25%. Current yield = $80 ÷ $875.38 = 9.14%. Capital gain at maturity = $1,000 − $875.38 = $124.62.
Notice the ordering: the 8% coupon rate is below the 9.14% current yield, which is below the 10.00% yield to maturity. That ladder is the signature of a discount bond, and it inverts exactly when the price crosses par.
How to read the yield you get
Compare the yield to the coupon rate first. Above the coupon rate means you bought at a discount and part of your return is the accretion to par; below it means you paid a premium that will be amortised away. The gap between the two also tells you roughly how far rates have moved since issue, since most bonds are priced near par when they are sold.
Next, compare the yield to a Treasury of the same maturity. The difference is the credit spread, and it is the only part of the yield that pays you for taking risk. A 9% yield on a single-B corporate when Treasuries yield 4% is a 500 basis point spread; the same 9% when Treasuries yield 8% is a much thinner cushion. On municipal bonds, do this comparison only after grossing the yield up with the tax-equivalent yield calculator, because a tax-exempt 3.5% is not a taxable 3.5%.
Then check whether the yield is achievable. YTM assumes you hold to maturity, that the issuer pays in full, and that every coupon is reinvested at the yield itself until redemption. That last assumption is the one that most often fails. If you receive $40 twice a year for ten years and can only reinvest at 4% while the bond's YTM is 10%, your realised compound return over the holding period will be below 10%. The assumption bites hardest on high-coupon, long-maturity bonds, where reinvested coupons are a large share of the terminal value, and not at all on a zero-coupon bond, which has no coupons to reinvest and therefore delivers its quoted yield with certainty if held to maturity.
Finally, if the bond is callable, treat the YTM as a ceiling rather than an expectation. Issuers call bonds when refinancing is cheap, which is precisely when you would rather they did not. Run the yield to call calculator for every call date and quote the lowest of those and the YTM — the yield to worst.
Yield, current yield and the shortcut across prices
| Clean price | Yield to maturity | Current yield | Approximate YTM |
|---|---|---|---|
| 117.17 | 3.00% | 4.27% | 3.02% |
| 108.18 | 4.00% | 4.62% | 4.02% |
| 100.00 | 5.00% | 5.00% | 5.00% |
| 92.56 | 6.00% | 5.40% | 5.97% |
| 85.79 | 7.00% | 5.83% | 6.91% |
| 79.61 | 8.00% | 6.28% | 7.84% |
Away from par, current yield always sits between the 5% coupon rate and the yield to maturity; at par all three collapse to the same number. In this table the shortcut overstates the yield on the premium prices and understates it on the discounts, and its error widens the further the price is from par — 3 basis points at 92.56, 9 at 85.79 and 16 at 79.61.
Bond-equivalent yield is a convention, not a compounding rule
A quoted 6% semiannual yield does not mean 6% a year compounded. It means 3% every six months, which compounds to 6.09%. The convention exists so that Treasuries, corporates and municipals — nearly all of which pay semiannually — can be compared without conversion. It becomes a trap the moment you compare a bond against something quoted differently: an annually compounded eurobond yield, a certificate of deposit's APY, or a money-market instrument on a discount basis. Convert everything to the effective annual yield before you compare across instrument types.
Assumptions and limits you should know about
- Reinvestment at the yield. YTM is an internal rate of return, and every IRR implicitly assumes intermediate cash flows earn the IRR. Realised return differs whenever reinvestment rates differ from the yield.
- No default and no restructuring. Every promised payment is treated as certain. On distressed credit, quoted yields describe the promise, not the expectation.
- No call, put or sinking fund. This calculator prices to the stated maturity. Bonds with embedded options need yield to worst, and proper valuation needs an option model.
- A coupon-date price. The calculation assumes a whole number of coupon periods remain. Between coupon dates the market solves the same equation with a fractional first period against the dirty price, which is a different equation and returns a slightly different yield. Price a bond on its coupon date and the two agree exactly; the gap widens with the fraction of the period elapsed.
- Pre-tax. Coupon income and accreted discount are generally taxed differently, and US market discount rules can convert part of a discount bond's gain into ordinary income. After-tax yield can rank two bonds differently from pre-tax yield.
- Nothing about interest-rate risk. Yield tells you what you earn if nothing changes. It says nothing about what happens to the price if rates move — for that you need the duration of the bond.
Where YTM sits among the other yield measures
Each yield measure answers a narrower or wider question, and they are worth keeping straight. Coupon rate answers only what the issuer pays relative to face value; it is fixed at issue and tells you nothing about your return. Current yield answers what income you collect per dollar invested this year, which matters if you are living off the coupons but ignores the pull to par entirely. Yield to maturity answers what you earn per year if you hold to redemption and reinvest at the same rate. Yield to worst answers the same question under the least favourable redemption the issuer can force on you. Realised compound yield answers what you actually earned, computed after the fact with the reinvestment rates that really occurred.
Professional analysis goes one step further and abandons a single yield altogether. A bond's arbitrage-free value discounts each cash flow at the spot rate for its own date, and the difference between the market price and that value is expressed as a zero-volatility spread — the parallel shift you must add to the whole spot curve to reproduce the price. For bonds with embedded options, the option-adjusted spread removes the value of the option before comparing. Both are refinements of the same idea this calculator implements: find the rate that makes the arithmetic balance, then interpret the rate.
For most decisions, YTM plus a duration figure is enough. YTM tells you what the bond pays if you are right about holding it; duration tells you what it costs if you are wrong.