Yield to Maturity

Put two bonds side by side. One pays a 3% coupon, matures in seven years and costs 91. The other pays 8%, matures in four years and costs 108. Which is the better deal? You cannot answer that by staring at coupons, or prices, or maturities, because each bond is a different-shaped pile of cash arriving on different dates.

The market's fix is to compress the whole schedule into one number: the interest rate a savings account would have to pay to turn today's price into exactly that stream of payments. That number is the yield to maturity. It is the miles-per-gallon of a bond — an imperfect summary that nonetheless lets you compare things that are otherwise incomparable.

The definition, backwards

Bond Pricing and Accrued Interest goes one direction: given a rate, compute the price. YTM goes the other: given the price, find the rate. Same equation, different unknown. With $P$ the price, $C$ the coupon per period, $F$ the face amount, $n$ periods remaining and $y$ the yield per period,

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

In plain English: find the one rate $y$ that, applied to every future payment, makes them all add up to what the bond costs today. Because $y$ appears inside several powers, there is no algebraic formula for it — you solve numerically, by trial or by Newton's method. Every calculator and every bond system does exactly that under the hood.

Yields move opposite to prices, and they do so along a curved line. That curvature is not a detail; it is Bond Duration and Convexity in one picture.

Worked example: solving for the yield

A bond pays a 6% annual coupon on 100 of face, has three years left, and trades at a clean price of 97. Since it costs less than par, the yield must be above 6%. Guess and check.

Try 7%. Discount the three payments of 6, 6 and 106:

$$\frac{6}{1.07} + \frac{6}{1.07^{2}} + \frac{106}{1.07^{3}} = 5.607 + 5.241 + 86.528 = 97.376 .$$

That is above 97, so 7% discounts too gently. Push the rate up.

Try 7.5%.

$$\frac{6}{1.075} + \frac{6}{1.075^{2}} + \frac{106}{1.075^{3}} = 5.581 + 5.192 + 85.326 = 96.099 .$$

Now we have overshot: 96.099 is below 97. The answer sits between 7% and 7.5%. Interpolate linearly between the two attempts:

$$y \approx 7\% + 0.5\% \times \frac{97.376 - 97.000}{97.376 - 96.099} = 7\% + 0.5\% \times 0.294 = 7.15\% .$$

Read that as: we needed to give up 0.376 of price out of the 1.277 that half a percent of yield buys, so we move about 29% of the way. Checking 7.15% gives 96.99, close enough. The YTM is about 7.15%.

Note where that beats the naive measures. The current yield is just coupon over price, $6 / 97 = 6.19\%$. YTM is higher because you also collect the 3 points of price appreciation as the bond pulls back to 100 at maturity — roughly 1 a year on a 97 investment, about another 1%. Current yield sees the income and misses the pull to par.

Worked example: the return you actually get

Buy that bond at 97 and hold it for three years. You receive 6, 6, 106 — but the first two coupons have to be reinvested somewhere, and the YTM calculation quietly assumes they earn 7.15%. Suppose rates fall and you can only reinvest at 3%:

• Coupon at year 1, compounded two years: $6 \times 1.03^{2} = 6.365$
• Coupon at year 2, compounded one year: $6 \times 1.03 = 6.180$
• Coupon at year 3: $6.000$
• Principal: $100$

Total at maturity: 118.545. Your realised annual return is

$$\left(\frac{118.545}{97}\right)^{1/3} - 1 = 1.2221^{1/3} - 1 = 6.92\% .$$

So the "7.15% yield" delivered 6.92%. The gap is small here because the coupons are a small slice of the total, but on a 30-year high-coupon bond, reinvestment can account for well over half the promised return, and the shortfall becomes enormous.

What YTM assumes

Four assumptions are baked in, and none is guaranteed:

1. You hold to maturity. Sell early and your return is whatever the price does.
2. Every coupon is reinvested at the yield itself.
3. The issuer pays in full and on time, so YTM on a risky bond is a promised yield, not an expected one.
4. One flat rate applies to every date, even though the market plainly charges different rates for different horizons — the reason Bootstrapping the Zero Curve exists.

Conventions matter too. US bonds quote a bond-equivalent yield: the semi-annual rate doubled, not compounded. A 3.5% semi-annual rate is quoted as 7.00%, though it truly compounds to 7.12%. Comparing a US corporate to an annual-pay European bond without converting is a classic unforced error.

Key terms

• Yield to maturity — the discount rate equating price to the present value of cash flows.
• Current yield — coupon divided by price; ignores the pull to par.
• Pull to par — the drift of a premium or discount bond toward 100 as maturity nears.
• Reinvestment assumption — the implicit claim that coupons earn the YTM.
• Bond-equivalent yield — the US convention of doubling the semi-annual rate.