How yield to maturity works

A bond is a stack of dated cash: coupons along the way, face at the end. Yield to maturity is the one discount rate that makes the present value of that stack equal the price you pay:

$$P = \sum_{t=1}^{n} \frac{C}{(1+i)^t} + \frac{F}{(1+i)^n}$$

$C$ is the coupon each period, $F$ the face, $n$ the number of periods left. $i$ is YTM for one period. With two coupons a year, $i$ is the annual YTM divided by 2 and $n$ is years times 2.

On a $1,000.00 bond paying 5 percent in two payments a year for 10 years, each coupon is $25.00. At a 5 percent YTM, $i = 0.025$ and $n = 20$. Discount the twenty coupons and the face at that rate and they add to $1,000.00. When the coupon rate equals YTM, the bond prices at par.

The bond price calculator on this page is that sum. The market-rate box is YTM. Type a rate, get a price. The inverse, a price in and a rate out, is the same identity solved for $i$. How bond pricing works owns the price. This page owns the rate.

How present value works is the discounting. How bond duration works is how far the price moves when this $i$ moves.

Current yield is the annual coupon divided by the price. It is a cash yield. It does not know that a discount bond repays more than you paid, or that a premium bond repays less.

When the market wants 7 percent, the same $25.00 coupons and $1,000.00 face price at $857.88. Current yield is $50 / 857.88 = 0.0583$, 5.83 percent. YTM is 7 percent. The extra 1.17 points is the $142.12 of face value the buyer collects at maturity, averaged into the rate.

When the market will accept 3 percent, the price is $1,171.69. Current yield falls to 4.27 percent. YTM is 3 percent. Current yield now sits above YTM, because the buyer of a premium takes a known loss of $171.69 at maturity and current yield pretends that loss is not there.

At par the two agree, and they agree with the coupon rate. Par is the only price where those three are the same number. Current yield against YTM is that split on one sheet.

Write the price as a cash outflow at time zero and the coupons and face as inflows after that, and YTM is the internal rate of return of that series. NPV and IRR is the same solve on any dated list. A bond just happens to have level coupons and a single face repayment.

Two assumptions ride along. Coupons are reinvested at the YTM itself, which is the usual IRR reinvestment convention, not a promise about what rates will do. And the bond is held to maturity, so the gain or loss against face is actually collected. Sell earlier and the realised rate is a different number, because the sale price is a new $P$ and the remaining $n$ has changed.

In the United States a bond yield is quoted as a nominal annual rate in step with how the coupons arrive, so a 7 percent YTM on a semiannual bond is 3.5 percent a half-year, which compounds to a little more than 7 percent across a full year. Other markets quote some bonds annually. How APR and APY work is that convention. Do not compare a semiannual YTM with an annually compounded one and call the gap a finding.

The coupons are fixed. When $i$ rises, every remaining payment is discounted harder and $P$ falls. When $i$ falls, $P$ rises. That is the whole of the inverse relationship, and it is why a bond whose price has dropped is the same bond at a higher YTM for the next buyer.

The three sheets on this page are three points on that curve: 5 percent YTM at $1,000.00, 7 percent at $857.88, 3 percent at $1,171.69. Nothing about the coupon or the face moved. Only $i$ moved.

How far the price moves for a given move in YTM is duration, which is a different page. This page stops at the identity: given a price, here is the rate; given a rate, here is the price. The bond price and yield explorer is that curve as a picture you can drag.

YTM is the rate if every coupon arrives, is reinvested at that same rate, and the face is repaid. A bond trading at a discount because the issuer looks shaky can print a high YTM for the mechanical reason that $P$ is low. If the coupons stop, the YTM was a schedule, not a cash flow.

Credit risk sits outside this formula. So does inflation. A 7 percent YTM against 3.2 percent inflation is a real rate of 3.68 percent only after how real returns work does the division, and only if that 7 percent is actually collected.

Total return over a holding period that is shorter than maturity includes the mark-to-market move as well as the coupons. YTM does not. It is a hold-to-maturity rate on a stated schedule.

It is not a credit screen, not a duration, and not accrued interest between coupon dates. The three sheets are a $1,000.00 par bond at a 5 percent YTM, the same coupons at 7 percent ($857.88, current yield 5.83 percent), and at 3 percent ($1,171.69, current yield 4.27 percent). This is educational material, not financial advice.