How bonds work: coupon, price and yield

Buy a share and you own a slice of a company: you receive whatever is left after everyone else has been paid, for as long as the company lasts. Buy a bond and you are the lender. The borrower owes you a schedule of payments, and when the last one arrives the arrangement ends.

Four terms define almost every bond.

A $1,000 bond with a 5 percent coupon paid twice a year pays $25 every six months until it matures, then hands back the $1,000. Those payments are contractual. A company can cut its dividend at a board meeting; it cannot skip a coupon without defaulting.

That is the whole exchange. A bondholder gives up the upside, because the payments do not improve when the borrower does well, and gets two things instead: amounts known in advance, and a place ahead of the shareholders if things go wrong. Owning is a claim on what is left over; lending is a claim on a stated number by a stated date.

In the United States, interest on many municipal bonds is exempt from federal income tax, which lets those issuers pay a lower coupon than a company of similar standing. That is a feature of one country's tax code and of the buyer's own position rather than of bonds themselves.

A bond's future payments are written down in advance, so pricing one is a discounted cash flow calculation and nothing more. Add up every remaining coupon and the face value, each discounted back to today at the return the market currently wants from that borrower over that term:

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

$P$ is the price, $C$ the coupon per period, $F$ the face value, $n$ the number of periods left, and $y$ the return the market wants, expressed per period. On a bond paying twice a year, $y$ is the annual yield halved and $n$ is the years remaining doubled, which is how yields are quoted in the markets where bonds pay twice a year. Markets whose bonds pay once a year compound annually instead, and the arithmetic follows the payment schedule.

A small one works by hand. Take a three-year bond, $1,000 of face value, a 5 percent coupon paid once a year, priced on a day when the market wants 7 percent from that borrower:

Nothing in that table is specific to bonds. It is the arithmetic the net present value calculator above runs on any dated set of cash flows: enter the coupons and the final repayment as inflows, discount at the market's rate, and the answer is a price.

The one judgement is $y$. It comes from what the market charges this borrower for this maturity: the yield on government debt of the same term, which is what the yield curve plots, plus a spread for the chance that this particular borrower does not pay.

Issuers usually set the coupon at whatever the market charges on the day, so a new bond prices at face value. That is par, and it holds exactly as long as market rates hold.

Take a $1,000 bond with a 5 percent coupon paid twice a year and ten years left to run. Every row below is the same bond, the same payments, the same borrower. All that moves is what the market wants from a bond like it:

From the buyer's side the mechanism is obvious. The coupon is stuck at $25 twice a year, so if new bonds pay 7 percent nobody hands over $1,000 to receive 5 percent. The price drops until the fixed coupons plus the $1,000 returned at maturity add up to 7 percent a year for whoever buys at that price. A payment that cannot rise can only offer a higher return by costing less.

So a bond bought at par can be worth less next week with nothing about the bond having changed. What changed was the alternative.

It is also why three numbers get called the yield:

• Coupon rate, 5 percent here: fixed against face value, never moves.
• Current yield, 5.83 percent at a price of $857.88: the annual coupon divided by the price. It counts the cash and nothing else.
• Yield to maturity, 7 percent: the discount rate that makes all remaining payments add up to today's price, so it counts the coupons and the climb from $857.88 back to $1,000 at maturity. It is the one that compares two bonds.

Duration answers one question: when yields move, how far does the price move? In its most useful form, modified duration, the answer reads as a percentage. A bond with a modified duration of 8 loses about 8 percent of its price when its yield rises by one percentage point, and gains about as much when the yield falls by one.

$$\frac{\Delta P}{P} \approx -D \times \Delta y$$

Here $D$ is modified duration and $\Delta y$ is the change in yield. Underneath it sits a plainer idea: the average time the money arrives, each payment weighted by its present value. That average is Macaulay duration and it is measured in years. Modified duration is the same average divided by one plus the yield per period, which is the small adjustment that turns a number of years into a percentage move, so the two are close but not interchangeable. Two things push both up:

• Longer maturity, because more of the money sits further away. The exception is a low coupon priced well below par: stretch its maturity far enough and the sensitivity peaks and then eases, because nearly all of its value is already the single repayment at the end.
• Lower coupon, because less cash comes back early. A zero-coupon bond's average is its maturity exactly, the longest that average can be, so for a given maturity nothing moves more.

Same 5 percent coupon, same issuer, three maturities, market yields going from 5 to 7 percent:

The drift is visible in the middle row already, where 7.8 times two predicts 15.6 against a true 14.2, and the bottom row is where the approximation gives out: 15.5 times two percentage points predicts a fall of about 31 percent, and the true figure is 24.9. The gap is convexity: the line between price and yield bends, so duration overstates the loss on a large rise and understates the gain on a large fall. Over small moves it is close enough to do in your head.

This is the honest answer to whether bonds are safe. Nobody defaulted anywhere in that table, and the thirty-year bond still lost a quarter of its value.

Interest on a bond is owed rather than declared. Missing a payment, once any grace period written into the contract has run out, is a default: it hands the lenders enforceable rights, including in most cases the right to demand the whole principal back at once, rather than leaving them with a bad quarter. That is the main protection lending buys over owning.

If a company is wound up, claims are paid in rank order, and in a straightforward liquidation each rank is settled in full before the next receives anything:

1. The costs of the insolvency itself, and in most jurisdictions certain employee and tax claims, which come off the top before any bondholder sees a penny. 2. Secured bonds and loans, backed by named collateral the lender can seize and sell. 3. Senior unsecured bonds, a general claim on the company. 4. Subordinated bonds, which agreed by contract to wait behind the seniors. 5. Preferred shares, then common shares, which take whatever is left, often nothing.

Three points matter more than the order itself. Ranking high decides who is paid out of the money that exists, not how much exists, so recovery turns on what the business is worth broken up: senior secured claims have historically recovered a good deal more than subordinated ones, with a wide spread around both. The ranking comes from the bond's own contract together with the insolvency law of wherever the case is heard, so the details are jurisdictional rather than universal. And the strict order above describes a liquidation. Where a company is reorganised instead of broken up the outcome is a negotiated plan, and junior classes are routinely handed something while a senior class is still short, because the seniors trade a slice of value for a deal that closes.

Credit ratings sort issuers by the risk of default, from investment grade down to high yield. The market prices the same judgement continuously as a credit spread: the extra yield over government debt of the same maturity. That spread is not all reward. Part of it pays for the losses defaults are expected to cause, part for how much harder the bond is to sell, and only what is left after those is a risk premium in the strict sense, which is why a promised spread and an expected return are not the same number. A rating is an agency's opinion rather than a measurement, and spreads usually move well before ratings do.

A government borrowing in a currency it issues is a different case. It is never forced into default by an inability to find the money, because it can create the currency the debt is written in, which shifts the risk from default towards inflation and the exchange rate. That makes default a choice rather than an impossibility, and a few governments have chosen it on their own-currency debt anyway. The same government borrowing in someone else's currency has no such option, and defaults on foreign-currency sovereign debt have happened repeatedly.

Credit risk gets the attention. On high-quality bonds most of the price movement comes from somewhere else, and these are the usual sources. At the weaker end of credit, default climbs back up the list rather than replacing them.

• Interest rate risk. The one duration measures. Yields rise, prices fall, and any holder valuing the position at market prices shows the loss straight away.
• Inflation risk. A fixed coupon is fixed in nominal terms only. If prices climb faster than the market expected when the bond was issued, every remaining payment buys less, which is what inflation and purchasing power works through and what the real return calculator puts a number on. Several governments issue bonds whose principal tracks a price index, which moves that risk rather than deleting it.
• Reinvestment risk. A yield to maturity is only the one discount rate that makes the remaining payments add up to today's price. Turning it into the annual compound return actually collected takes a further assumption, that every coupon goes back to work at that same yield. So the 7 percent in the tables above is a quoted rate rather than a promise: if rates fall the coupons are reinvested at less and the realised return lands under it, and if rates rise it lands over. A zero-coupon bond is the one case with nothing to reinvest.
• Call risk. Many corporate and municipal bonds can be repaid early at the issuer's choice. Issuers call when rates have fallen, which returns your money exactly when there is nothing good to do with it, and caps how far the price can rise in the meantime.
• Liquidity risk. The largest government bond markets trade in size all day. A smaller sovereign's debt, or a single corporate issue, may not trade for days, so selling quickly means taking whatever bid is showing, and what liquidity means covers what that costs.
• Currency risk. A foreign bond pays in a foreign currency, and an exchange rate can move further in a month than a coupon pays in a year.

Holding to maturity settles the first of these, in the sense that the sum finally paid is the face value rather than whatever the market would have given for it, and it settles none of the rest. Everything here is educational material rather than financial advice.