Yield to maturity (YTM) is the single discount rate that equates a bond's market price to the present value of all its future cash flows, both coupons and principal. It is the most commonly quoted measure of a bond's return and the number reported in Treasury auction results, on trading screens, and in yield curve charts.
YTM is calculated by solving for y in:
Price = C/(1+y) + C/(1+y)² + ... + (C+Face)/(1+y)ⁿ
where C is the coupon payment, Face is par value, and n is the number of periods to maturity.
YTM assumes two things that rarely hold exactly:
Despite these assumptions, YTM remains the standard because it compresses a bond's cash flow profile into a single comparable number. When the Treasury publishes daily yield curve rates, these are par yields, meaning the YTM of hypothetical bonds priced at par for each maturity.
YTM differs from related yield measures:
For zero-coupon bonds (like Treasury STRIPS), YTM and the spot rate are identical because there are no intermediate cash flows to reinvest.
The equation above is a discounted cash flow calculation read backwards.
Given a set of cash flows and a discount rate, the equation returns a price. Given the same cash flows and a market price, it returns a rate. That rate is the internal rate of return of the bond, and yield to maturity is simply the name the bond market gives it.
Seeing YTM this way explains its two most confusing properties.
Price and yield move in opposite directions. The cash flows of a Treasury are fixed. The only way for the market to pay less for a fixed stream is to discount it more heavily, so a falling price and a rising yield are the same event described twice.
A bond's coupon and its yield are different things. The coupon is fixed at issue and never changes. The yield changes every time the price changes. When the coupon is above the yield, the bond trades above 100 and is at a premium. When the coupon is below the yield, the bond trades below 100 and is at a discount. When the two are equal, the bond trades at 100 and is at par. This last case is what defines a par yield.
There is no closed form solution for the yield. Solving the equation requires iteration, which is why yield was a calculated field on a trading screen long before it was one in a spreadsheet.
A quoted bond price is not the amount a buyer pays.
Coupons accrue daily but are paid twice a year. A buyer who purchases between coupon dates receives the next full coupon even though the seller held the bond for part of that period. The buyer therefore compensates the seller for the interest earned so far. That amount is accrued interest.
Work an example. Take a note with a 4.00% coupon paying twice a year, so each period pays 2.00 per 100 of face. Treasuries accrue on an actual over actual basis, so the fraction elapsed is real days over the real length of the period. Sixty days into a 181 day period, that fraction is 60 over 181. Accrued interest is 2.00 multiplied by that fraction, or 0.663 per 100.
If the note is quoted at 99.00, the dirty price is 99.663. On 1 million of face value, the buyer pays 996,630 rather than 990,000.
Yield to maturity is always calculated from the dirty price, because that is the actual amount invested. Quoting the clean price keeps the number stable across a coupon period, which is why the market shows it. Both numbers are needed, and confusing them produces an error roughly the size of half a coupon.
YTM is a summary, and every summary discards something.
It assumes reinvestment at the same rate, which the page notes above. The consequence is worth stating directly. Two bonds with the same YTM and different coupons will not deliver the same realized return, because the higher coupon bond has more cash to reinvest and therefore more exposure to future reinvestment rates.
It also assumes a flat curve. The equation applies one rate to every cash flow, so a steeply sloped curve is compressed into a single average. This is harmless for quoting and unsuitable for valuation, which is the reason spot rates exist.
For a Treasury these limitations are manageable, because the cash flows are certain. They stop being manageable as soon as the cash flows are not. A bond that can be redeemed early has a schedule that may never occur, so a single yield to its stated maturity describes a future that the issuer controls.
Because the cash flows do not change. A Treasury pays the same coupons and the same principal regardless of what the bond costs. Paying less for that fixed stream means earning a higher return on the money invested, and that higher return is the yield.
Yes. The published rates are par yields, which are the yields to maturity of hypothetical bonds priced exactly at 100 for each tenor. They are fitted from traded securities rather than taken from any single bond, so no individual note will quote exactly at the published rate.
Current yield divides the annual coupon by the market price and stops there. It ignores the gain or loss from the price moving to 100 at maturity, and it ignores the timing of the cash flows. It is quick to calculate and understates the return on a discount bond, so it is rarely used for Treasuries.