How Bonds Work: Coupons, Prices and Why Yields Move Opposite

When Alexander Hamilton sent his Report on the Public Credit to Congress in January 1790, the uncomfortable part of the proposal had nothing to do with arithmetic. The certificates the government had handed to soldiers and suppliers during the war had long since been sold on, frequently for a few cents on the dollar, to men who had never been near a battlefield. Madison wanted the original holders paid. Hamilton refused, on the ground that a debt which cannot change hands cleanly is worth very little to anybody, and that the person owed is whoever holds the paper on the day it comes due. Congress sided with Hamilton, and the principle has held for the two centuries since.

That principle is the whole of what a bond is: you hand over a fixed sum, the borrower pays you interest on a stated schedule, and on a stated date you get your money back. The loan itself trades. It changes hands at whatever price the market will pay that morning, and that price moves every day that interest rates move.

Almost everything people find confusing about bonds falls out of that arrangement. The payments are fixed. The price is not.

The four numbers that describe any bond

A two-year Treasury note and a thirty-year corporate issue are described by the same four numbers, and once you can read them off a term sheet you can read every bond ever written.

Term	What it means	Typical example
Face value (par)	The amount repaid on the maturity date	$1,000 per bond
Coupon rate	Annual interest as a percentage of face value	5%, paid as $25 twice a year
Maturity	The date the face value is repaid	10 years from issue
Issuer	Who owes you the money	US Treasury, Apple, the State of California

Face value never moves. Neither does the coupon rate, which is where the phrase fixed income comes from. A 5% coupon on a $1,000 bond pays $50 a year for as long as the bond exists, whatever happens to interest rates, to inflation, or to the issuer’s share price.

Yield describes something else entirely. It belongs to the purchase. Yield is the return earned given what was paid, so when the price changes the yield changes with it, because the numerator has stayed exactly where it was and the denominator has moved.

Why the price has to move against the yield

Suppose you bought a $1,000 bond with a 5% coupon and ten years left. It pays $50 a year, then returns your $1,000. Months later the Treasury is issuing ten-year paper at 6%. That pays $60 a year on the same $1,000 of face value.

Your bond still pays $50. It is a worse deal than what a buyer can get from the issuing desk that week. Nobody pays you $1,000 for it. The price falls until someone paying that lower price earns the same 6% in total, counting both the coupons received along the way and the climb from the discounted price back up to $1,000 at maturity.

The adjustment takes seconds. In a liquid market every headline about rising rates is also, without saying so, a headline about falling bond prices. They describe one event from two ends.

The arithmetic in full

The price of a bond is the present value of everything it will pay, discounted at the yield the market demands:

Price = C x [(1 - (1 + y)^-n) / y] + F / (1 + y)^n

Here C is the annual coupon in dollars, y is the market yield, n is the number of years to maturity and F is the face value. Treasuries pay twice a year in practice and the calculators work in half-year periods, but annual periods are used below so that every step can be checked on a pocket calculator.

Take the ten-year, 5% coupon, $1,000 bond, with the market yield now at 6%:

1. (1.06)^10 = 1.7908, so 1 / 1.7908 = 0.5584. That is what one dollar received in ten years is worth this morning.
2. The annuity factor is (1 - 0.5584) / 0.06 = 7.3601, which is what one dollar a year for ten years is worth this morning.
3. The coupons are worth $50 x 7.3601 = $368.00.
4. The face value is worth $1,000 x 0.5584 = $558.40.
5. Add them: $368.00 + $558.40 = $926.40.

The market yield rose by a single percentage point. Nothing about the bond changed: the coupon, the maturity date and the $1,000 repayment are all where they were, and the price still fell from $1,000 to $926.40, a loss of 7.36% of its value.

There is a shortcut. It makes the size of the move obvious without any of that. Your bond pays $10 a year less than the market now insists on. Ten dollars a year for ten years, discounted at 6%, is $10 x 7.3601 = $73.60, and subtracting that shortfall from par gives you $926.40 without ever discounting the face value separately.

The same one point rise, at four maturities

Maturity decides the damage. The same one point move in yields does very different work depending on how much of the bond’s life is still ahead of it, and each row below is that same 5% coupon bond, priced first at a 6% yield and then at a 4% yield.

Years to maturity	Price at 6% yield	Price at 4% yield
2	$981.67	$1,018.86
5	$957.88	$1,044.52
10	$926.40	$1,081.11
30	$862.35	$1,172.92

A two-year bond gives up under 2% when yields rise a point. A thirty-year bond gives up nearly 14% on the same move. The reason is not mysterious: the holder of the two-year is stuck with a below-market coupon for two more years, while the holder of the thirty-year is stuck with it for thirty, and that sensitivity has a name, a formula and a long history. It is set out in duration and interest rate risk.

Notice as well that the gains at 4% exceed the losses at 6%. Thirty years out, the bond picks up $172.92 when yields fall a point. It surrenders $137.65 when they rise by the same amount. The asymmetry is called convexity, and it runs in the bondholder’s favour.

What actually pays you

Buy that ten-year bond at $926.40 and two things pay you. The first is $50 a year in cash. Against a price of $926.40 that is a current yield of $50 / $926.40 = 5.40%.

The second is quieter. The $73.60 of discount closes as the bond approaches maturity, because on the final day the issuer pays face value regardless of what any particular holder paid for the paper. Clerks were calling that drift the pull to par long before anyone had a computer to model it. The name stuck. Add the income and the capital gain together across the ten years and the total return comes to the 6% yield to maturity you bought at, which is what the yield was telling you in the first place.

Current yield and yield to maturity answer different questions. Quote one while meaning the other and you overpay. The differences between current yield, yield to maturity and yield to call are worked through in bond yields explained, and you can price any of them yourself with the bond yield calculator.

The issuer changes the promise, and nothing else

Every line of arithmetic above holds for every bond ever issued. The issuer changes only two things. One is the probability that you are paid at all, and the other is the way the income is taxed.

The US Treasury borrows through bills, notes, bonds and inflation-protected securities, sold at auctions that run on a published calendar, bills weekly and notes and bonds monthly or quarterly, and settled through the Federal Reserve, and the interest is exempt from state and local income tax. The instruments and the auction mechanics are in the Treasury securities guide.

Companies borrow through corporate bonds. They pay more than Treasuries because companies can stop paying, and the extra yield is the credit spread, which widens when lenders lose their nerve, usually well before anything visible has broken. See corporate bonds and credit spreads.

States, cities, school districts and public authorities issue municipal bonds, where the interest is generally free of federal income tax, and that exemption changes how the yield has to be compared against anything taxable. A full example is worked through in municipal bonds explained.

Three ways bonds fail

Three failures account for most of the money lost in what people describe as the safe part of a portfolio. The first one catches the most holders.

Rising rates cut market values, and the losses are not small. On the Bloomberg US Aggregate index, 2022 was the worst calendar year since the index began, a year in which the most conservative holding in most portfolios fell alongside the most aggressive one, which is precisely the correlation those holdings were bought to avoid.

Inflation hollows out what the fixed payments buy. A 4% coupon with inflation running at 5% loses purchasing power every year the bond exists. Every payment arrives in full and on time. The nominal yield on the quote screen says nothing at all about this, so subtract the inflation you expect before deciding that a yield is adequate.

Credit failure is rarer and more complete. Companies, and occasionally municipalities, miss payments. Bondholders then recover some fraction of face value through a restructuring or a bankruptcy court, and that is the risk the credit rating and the spread are attempting to price, with mixed success.

Owning them

You can bid at auction through TreasuryDirect, or buy individual bonds through a broker, where the dealer’s markup sits inside the price and never appears on the confirmation as a commission. Most investors hold bonds through funds. That buys instant diversification and monthly income, and it gives away the one feature that makes an individual bond predictable. A fund of ten-year bonds is permanently a fund of ten-year bonds. It sells what has aged and buys what is new, so it never pulls to par and there is no date on which you are made whole. That distinction is the subject of bond ETFs explained.

Underneath all of these prices sits the short-term interest rate set by the Federal Reserve and, more to the point, the market’s guess about where that rate is heading, which is why bond investors spend so much of their time reading FOMC statements line by line. Start with how the Federal Reserve works, then look at how yields line up across maturities in the yield curve explained, and test what has stuck with the bonds and yields quiz.