Bond Yield to Maturity Calculator

Background.

A five-year bond with a 5% coupon is offered at 950 per 1,000 of face value. The coupon says 5%, the dealer's screen says a little over 6%, and both are right. The coupon rate was fixed on face value when the bond was issued. Yield to maturity is the return the remaining cash flows give whoever buys the bond today at 950 and holds it to the end: ten half-yearly coupons of 25, then 1,000 back at maturity. On top of the coupons that buyer gains 50 as the price is pulled up to par, and the single rate that accounts for all of it works out to a nominal 6.1776% a year.

Three yields describe the same bond at the same moment, and comparing bonds on mismatched yields is the usual way the comparison goes wrong. The coupon rate (5%) is cash interest as a share of face value. Current yield (5.2632%) is the annual coupon divided by the price you pay; it measures income but cannot see the discount. Yield to maturity (6.1776%) is the internal rate of return on price, coupons and redemption together, assuming every payment arrives on schedule and you keep the bond until it matures. Buy the same bond at a premium and the order reverses: the yield to maturity falls below the coupon rate, because the premium you paid is lost by the time the bond repays 1,000.

The unknown rate sits inside every discount factor, from the first coupon to the last, so for an ordinary coupon bond no rearrangement of the price equation isolates it. The solver here searches for it numerically, then reprices the bond at the rate it found. That repriced figure, shown as the price check, should equal the price you entered, which is your confirmation that the yield fits the cash flows.

Yield to maturity is a statement about arithmetic, not a forecast. What you actually earn will differ if you sell before maturity at a different market yield, if the issuer calls the bond or misses a payment, or if the coupons you reinvest earn less or more than the yield itself. FINRA's investor material lists yield to maturity among the common measures of bond yield, and comparison is what it is best at: putting bonds with different coupons, prices and maturities on one scale before you look at credit quality, call features and tax.

What is bond yield to maturity calculator?

Yield to maturity is the discount rate that makes the present value of a bond's remaining coupons and its redemption payment equal to the bond's price, assuming the bond is held to maturity and pays as promised. The solved rate is a per-period rate tied to the coupon frequency. For a semiannual bond the six-month rate is conventionally doubled and quoted as a nominal annual yield, sometimes called the bond-equivalent yield; compounding the six-month rate instead gives the effective annual yield, which is slightly higher.

A bond priced below its face value trades at a discount and its yield to maturity exceeds its current yield, which in turn exceeds the coupon rate. A bond priced above face value trades at a premium and the ranking runs the other way. At exactly par all three coincide on a nominal basis. Because yields move inversely to prices, a rise in market interest rates lowers the price of a bond already issued until its yield to maturity matches what new bonds of similar credit and maturity offer.

The method fits fixed-rate bullet bonds with a known coupon, frequency and redemption value. It leaves out default risk, taxes, accrued interest, call and put options, sinking funds, floating or inflation-linked coupons, amortising principal and irregular first or last coupon periods.

How to use this calculator.

1. 01Enter the price and the face value on the same basis: 950 and 1,000 in currency, or a quote as a percentage of par such as 95 against 100 — both give the same yield. Use the quoted clean price; there is no settlement date here, so accrued interest is not handled.
2. 02Enter the annual coupon rate as a percentage of face value (0 for a zero-coupon bond) and choose how often coupons are paid. US Treasury notes and bonds pay every six months, as most US corporate bonds do; the prospectus or your broker's bond detail screen gives the schedule for anything else.
3. 03Enter the years remaining to maturity. Years multiplied by payments per year must come to a whole number of coupon periods: 5.5 years is 11 semiannual periods, but 5.3 years would be 10.6 and is refused rather than rounded. For a bond part-way between coupon dates this calculator cannot give the yield, because it has no settlement date or accrued interest. The term must be at least one coupon period long (6 months for semiannual coupons); a shorter term is rejected, because there would be no yield to solve before the first payment.
4. 04Leave the redemption value equal to the face value for an ordinary bond, and remember to change it whenever you change the face value, since it does not follow automatically. To estimate a yield to call, enter the call price as the redemption value and the years to the call date as the maturity.
5. 05Read the nominal yield to maturity first and compare it only with yields quoted on the same convention. Check that the price check equals the price you entered, and use the premium or discount figure to see how much of the return comes from the price moving to par.

The formula.

A fixed-rate bond is worth the present value of its coupons plus the present value of its redemption payment: P = C × [1 − (1 + r)⁻ⁿ] ÷ r + R ÷ (1 + r)ⁿ. Here C is the coupon per period (face value × annual coupon rate ÷ payments per year), n is the number of coupon periods left (years × payments per year, which must be a whole number), R is the redemption value and r is the periodic yield. The first term is the standard annuity formula, which compresses the stream of equal coupons into one expression; the second discounts the single repayment at maturity.

Given P, the job is to find r, and because r appears in both a power and a denominator there is no closed-form answer for a coupon bond. The solver uses bisection: it starts with a range of yields wide enough to contain any real bond, from just above −100% to 10,000% per period, prices the bond at the midpoint, keeps whichever half still contains the target price and repeats until the repriced bond matches the entered price. Bisection is slower than Newton's method but it cannot overshoot or diverge, which matters for steep discounts and very long maturities. A zero-coupon bond needs no search, since P = R ÷ (1 + r)ⁿ rearranges to r = (R ÷ P)^(1/n) − 1.

Once r is known, the nominal yield to maturity is r × m and the effective annual yield is (1 + r)^m − 1, where m is the number of payments a year. Current yield is simpler: annual coupon ÷ price. A well-known shortcut, (annual coupon + (face − price) ÷ years) ÷ ((face + price) ÷ 2), gives 6.1538% for the example bond against the exact 6.1776%; it is fine for a quick sense-check and drifts further off as discounts and maturities grow.

The equation assumes equal coupon periods, cash flows paid in full and on time, and a whole number of periods from today. It does not model a purchase between coupon dates, where professional yield calculations use a settlement date, a day-count convention and fractional first periods.

A worked example.

Frequently asked questions.

References& sources.

1. [1]U.S. Securities and Exchange Commission (n.d.). Bonds. Investor.gov.
2. [2]Financial Industry Regulatory Authority (n.d.). Bond Yield and Return. FINRA.
3. [3]Financial Industry Regulatory Authority (n.d.). Understanding Bond Prices, Rates and Yields. FINRA.
4. [4]Municipal Securities Rulemaking Board (n.d.). Bond Basics. MSRB Education Center.
5. [5]Municipal Securities Rulemaking Board (n.d.). Yield. MSRB Education Center.
6. [6]U.S. Department of the Treasury (n.d.). Treasury Bonds. TreasuryDirect.