Yield to Maturity Formula: The Exact Equation and the Approximation

Yield to maturity (YTM) is the single discount rate that makes a bond's market price equal the present value of everything the bond still pays — every remaining coupon plus the face value at maturity. It is the bond's implied annual return if you buy at today's price, hold it to the end, and collect every payment. No algebra isolates that rate, so in practice you estimate it with an approximation formula or let a calculator iterate to the exact answer; this page shows both.

You need YTM whenever a bond trades away from face value. A quoted price of $946.30 tells you what a bond costs, but not what it earns — YTM converts that price into a rate you can compare against other bonds, savings rates, or a required return.

This is the same equation used in bond valuation, run in reverse. There you know the market rate and solve for the price. Here the market hands you the price, and you back out the rate hiding inside it. Every term is a cash flow divided by a discount factor: each coupon C discounted back from its payment date, and the face value F discounted back from maturity. YTM is whatever value of y makes all those pieces add up to the actual price.

Why you cannot solve it directly

The unknown y appears in the denominator of every term, raised to a different power in each one. Multiply the equation out and you get a polynomial of degree n in (1 + y) — and for a bond with more than a few periods left, no algebraic rearrangement isolates y on one side. This is not a gap in your algebra; a general closed-form solution does not exist.

So every YTM you have ever seen was found by iteration: guess a rate, compute the price it implies, compare against the actual price, adjust, repeat. In practice nobody does this by hand. You have three honest options.

• Trial and error — useful once, to see how it works. If your guessed rate prices the bond above its market price, guess higher; if below, guess lower.
• A financial calculator — enter N, PMT, PV (as a negative), and FV, then solve for I/Y.
• A spreadsheet — =RATE(n, coupon, -price, face) runs the iteration for you, or =IRR on the full cash flow column. For real-world bonds with settlement dates, =YIELD handles the day counts.

There is also a shortcut fast enough for an exam — and good enough to sanity-check a calculator answer.

The approximation formula

The approximation replaces the discounting machinery with a simple ratio: average annual earnings divided by average money invested. The numerator adds the coupon to the gain (or loss) from the price pulling toward face value, spread evenly over the remaining years. The denominator averages what you paid and what you get back.

Take a bond with a $1,000 face value and a 4.5% annual coupon, six years from maturity, trading at $946.30. It trades at a discount, so its YTM should come out above its coupon rate — keep that expectation in mind as a check.

Now the exact answer. In a spreadsheet, =RATE(6, 45, -946.30, 1000) iterates on the pricing equation and returns y = 5.578%, about 5.58%. You can verify it: discounting six $45 coupons and the $1,000 face value at 5.578% gives a present value of $946.28 — the market price to within two cents of rounding.

So the approximation lands at 5.54% against an exact 5.58% — off by roughly 0.03 percentage points. That is typical: the shortcut is reliable near par with moderate maturities, and drifts for long maturities and deep discounts or premiums, because averaging ignores compounding. Estimate and sanity-check with it; report the RATE answer when precision matters.

YTM vs coupon rate vs current yield

Students lose points by mixing up three rates that all describe the same bond. Using the bond above:

• Coupon rate: 4.50%. Fixed by the bond's contract — the $45 coupon divided by the $1,000 face value. It never changes, no matter what the price does.
• Current yield: 4.76%. The $45 coupon divided by the $946.30 price. It reflects what you paid, but counts only coupon income and ignores the $53.70 you gain as the price pulls to par.
• Yield to maturity: 5.58%. The full return — coupons plus the pull to par, with proper discounting. This is the only one of the three that answers "what does this bond earn me?"

The ordering is a useful self-check. For a discount bond, coupon rate < current yield < YTM, exactly as here (4.50% < 4.76% < 5.58%). For a premium bond the order flips, and at par all three rates are equal. If your computed YTM breaks this ordering, one of your inputs is wrong.

Common mistakes

Treating the coupon rate as the yield. The coupon rate is set when the bond is issued and printed on the contract; YTM is set by today's price. They agree only when the bond trades exactly at par. An exam question that gives you a price other than face value is testing whether you know the difference.

Ignoring the semiannual convention. Most US bonds pay coupons twice a year. The correct treatment is to work per period — halve the coupon, double the period count, solve the pricing equation for the semiannual rate, then double that rate to quote the annual YTM (the bond-equivalent yield convention). The two classic errors are reporting the semiannual rate as if it were annual, and discounting semiannual cash flows at a full annual rate. The example above uses annual coupons to keep the steps visible; a semiannual version of the same bond would use $22.50 per period over 12 periods.

Assuming YTM is a guaranteed return. The calculation quietly assumes every coupon is reinvested at the YTM itself, the issuer never defaults, and you hold to the final payment. If reinvestment rates fall or you sell early, your realized return will differ from the YTM you computed. YTM is the bond's implied rate, not a promise.

By the FinanceBrain Team · Last verified July 11, 2026 · How we produce and verify articles