Bond Pricing & Yield

Last time we built the machinery for valuing any stream of cash flows: discount each payment back to today, then add them up. Bonds are where that machinery gets its first real workout, and they are a satisfying place to start because the cash flows are not a mystery.

When we buy a bond, we know what gets paid and when. The coupon amounts, the payment dates, the face value at maturity: all of it is spelled out in the contract, which leaves us nothing to forecast.

That makes a bond the closest thing to a pure time-value-of-money problem. Pricing one takes nothing beyond the present value formula we already have.

Zero Coupon Bonds

The simplest bond pays nothing at all until the end. A zero coupon bond (or “zero”) is a lump sum arriving at some future date: we buy it at a discount, we collect the full face value at maturity, and the profit is baked into that discount.

Pricing one is the present value formula from the last post, with the payment renamed to a face value:

$$P=\frac{F}{(1+r{)}^n}$$

where $F$ is the face value, $r$ is the discount rate per period, and $n$ is the number of periods until maturity.

Take a zero with a 1,000 face value maturing in 3 years, discounted at 4% annually:

$$P=\frac{1,000}{(1.04{)}^3}=\frac{1,000}{1.1249}=889.00$$

We pay 889 today, collect 1,000 in three years, and the 111 difference is our return. Because there are no coupons to reinvest before maturity, that return is unambiguous. Treasury bills work this way, sold at a discount and redeemed at face value.

Zeros make the relationship between price and return transparent. A lower price means a higher return, and a higher price means a lower return, with no coupon income in between to muddy the comparison.

Coupon Bonds

Most bonds are not zeros. They pay periodic interest (the coupon) on a regular schedule, usually semi-annually, and return the face value at maturity. That gives two streams of cash: an annuity of coupon payments, and a lump sum (the face value) at the end.

The price is the present value of both streams:

$$P=C\times \frac{1-(1+y{)}^{-n}}{y}+\frac{F}{(1+y{)}^n}$$

where $C$ is the coupon payment per period, $y$ is the yield per period, $n$ is the total number of periods, and $F$ is the face value. The first term is the present value of the coupon annuity, the annuity formula from the last post doing its work. The second term is the present value of the face value, priced like a zero.

Take a bond with a 1,000 face value, a 6% annual coupon paid semi-annually, and 5 years to maturity. That gives us 30 every six months for 10 periods, plus 1,000 at the end. At a market yield of 4%, or 2% per semi-annual period, we discount the two streams and add them:

$$P=30\times \frac{1-(1.02{)}^{-10}}{0.02}+\frac{1,000}{(1.02{)}^{10}}=269.48+820.35=1,089.83$$

At 1,089.83 the bond costs more than its face value, which is what we should expect: it pays a 6% coupon while the market demands only 4%. Investors bid the price up until the deal is fair.

Premium, Discount, and Par

The comparison between a bond’s coupon rate and the market’s yield sorts every bond into one of three cases.

Scenario	Coupon vs. Yield	Price vs. Face	Why
Premium	Coupon rate exceeds yield	Above face value	Coupons are more generous than the market requires, so investors bid the price up
Discount	Coupon rate falls below yield	Below face value	Coupons are not enough to satisfy the market, so the price drops to compensate
Par	Coupon rate equals yield	At face value	Everything lines up

Drag the coupon rate and yield sliders below and watch the price cross above and below face value.

Notice the breakdown bar beneath the price. For short-maturity bonds the face value dominates, since most of the money comes back soon. For long-maturity bonds the coupon payments take a much larger share of the price, because discounting over many periods cuts the distant face value down further than it cuts the nearer coupons.

Yield to Maturity

So far the yield has been the input and the price the output. In practice the question runs the other way: the price is already quoted on the screen, and what we want to know is the return that price implies.

Yield to maturity (YTM) is the discount rate that makes the present value of all the bond’s future cash flows equal to its current market price. It is the $y$ that solves:

$$P=C\times \frac{1-(1+y{)}^{-n}}{y}+\frac{F}{(1+y{)}^n}$$

For a zero coupon bond we can solve for $y$ algebraically. For a coupon bond we cannot, because $y$ appears in both the exponents and the denominators, so the solution needs a numerical method (in practice, a routine that tries progressively better guesses until it converges). Every financial calculator and spreadsheet does this automatically.

The routine returns a $y$, but a $y$ per what? The rate we get back is a per-period rate, and for a semi-annual bond that is the six-month rate. Convention quotes YTM as an APR, the per-period rate multiplied by the number of periods per year, so a $y$ of $2\%$ per semi-annual period is quoted as a YTM of $4\%$. For the true annualized return we go back to the EAR from the last post: $(1.02{)}^2-1=4.04\%$.

YTM also assumes every coupon is reinvested at the same rate, which is slightly unrealistic. Still, it is the standard benchmark for comparing bonds.

YTM is not a guaranteed return. If we sell before maturity, if rates move, or if the issuer defaults, our realized return will differ. As a measure of what is priced into a bond right now, though, it is hard to beat.

The Price-Yield Seesaw

This is the most important relationship in fixed income: bond prices and yields move in opposite directions. When interest rates go up, bond prices go down. When rates fall, prices rise.

This is not a market opinion or a behavioral quirk. It is arithmetic. A bond’s cash flows are fixed, and its coupons do not change when the market moves.

If new bonds are issued at 6%, the 4% bond we already hold looks less attractive, so its price drops until the yield it offers matches the market. If new bonds pay only 2%, our 4% bond looks generous instead, and its price gets bid up.

The relationship is also convex, meaning the curve bends. A 1% drop in yield produces a bigger price increase than a 1% rise in yield produces a price decrease. The asymmetry favors bondholders, who gain more from falling rates than they lose from rising ones, all else equal.

Set the coupon and maturity with the sliders below, then sweep the highlight yield and watch the price trace out that convex curve.

Toggle the comparison on to overlay a second maturity. Longer-maturity bonds trace a steeper curve, since their prices are more sensitive to yield changes. More of a long bond’s value comes from cash flows far in the future, where discounting has a bigger impact: a 30-year bond might swing 15% on a 1% rate move, while a 2-year bond barely flinches.

Duration

That sensitivity to interest rates has a name: duration. The next post derives it in full. For now, take it as a rule of thumb for the percentage price change per 1% move in yield. A duration of 7 means roughly a 7% price change for every 1% change in yield.

Two rules of thumb follow from when a bond’s cash flows arrive.

• Higher coupon → lower duration. More income comes in early, so a smaller share of the value depends on the distant face value.
• Longer maturity → higher duration. More of the cash flows come due far ahead, exposed to discounting.

A zero coupon bond puts all of its value at the end, which gives it the highest duration, and the highest rate sensitivity, of any bond of the same maturity.

The Yield Curve

Duration describes how one bond responds to its own yield. Yields across different maturities also form a pattern, and that pattern tells us how the market sees the future.

The yield curve plots yields on the vertical axis against maturity on the horizontal. The yields are usually Treasury yields, since Treasuries are the closest thing to credit-risk-free, but the construction applies to any set of similar bonds.

Click through the presets below and watch the shape of the curve change.

The normal shape is upward-sloping, with longer maturities demanding higher yields. Locking money up for longer means more exposure to inflation and more time for something to go wrong, and investors want compensation for both.

An inverted curve (short rates above long rates) is the unsettling one. Inversion means the market expects rates to fall, which usually happens when the economy is slowing down and the central bank is expected to cut. Every U.S. recession since the 1960s was preceded by a yield curve inversion, though the timing between inversion and recession varies from months to over a year.

A flat curve is the ambiguous middle ground. Either the market cannot decide whether things are getting better or worse, or rates at all maturities have converged for some other reason. The explorer includes a fourth preset, a humped curve, where medium maturities yield more than both the short and the long end. It usually shows up in transition, on the way into or out of an inversion.

The shape matters beyond predicting recessions, because everything priced off interest rates moves with it: mortgage rates, corporate borrowing costs, and bank profitability (banks borrow short and lend long, so a flat or inverted curve squeezes their margins). The yield curve may be the most-watched chart in finance.

What’s Next

We gave duration as a rule of thumb: a duration of 7 means roughly a 7% price change per 1% yield move. Next we’ll do the math, defining Macaulay duration, modified duration, and convexity, and showing why the linear approximation is not quite enough.