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Bond prices and interest rates
Why a bond loses value when rates go up.
Price = Σ coupon / (1 + r)ⁿ + face / (1 + r)ᴺProblem / motivation
You hold a bond paying 3% a year. The central bank raises its rates and new bonds now pay 5%. What is yours worth?
Let's start with the object. A bond is a LOAN cut into resellable pieces: a state or a company borrows by issuing securities — whoever buys a €1,000 security lends €1,000, collects each year an interest fixed in advance (the COUPON: here 3%, that is €30), and gets their €1,000 back on an agreed date (MATURITY). That amount repaid at the end is called the FACE value.
If you hold the security to the end, the story stops there: €30 a year, then €1,000. But there is a second-hand market — you can sell your bond on before maturity. And there a question arises: how much is this piece of paper promising money TOMORROW worth TODAY? The opening answer is already clear: nobody will pay you the full price for your 3% when new issues pay 5% — your price must fall until it becomes competitive again. The question is BY HOW MUCH: that is the whole point of the course, and the key to the most counter-intuitive relation in markets — prices and rates move in opposite directions.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
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0 idea(s) proposedFormalization
Let's take our bond again: face value €1,000, coupon €30, maturity 3 years. Owning it means holding three appointments with money: €30 in one year, €30 in two years, €1,030 in three years (last coupon + repayment). Today's price must sum up those three promises in a single number — and for that, you need to know how to convert future money into today's money.
The conversion comes from the alternative investment (assumption 3). At the market rate r = 5%, €100 today becomes €105 in a year: receiving €X in a year is therefore worth X ÷ 1.05 today. In two years, the money would have compounded TWICE (× 1.05 × 1.05 = × 1.05²): €X in two years is worth X ÷ 1.05². Remember the bridge: “investing at 5%” = “multiplying by 1.05”; discounting means making the journey in reverse — dividing. That is why we divide, and never subtract.
The price adds up each converted appointment: Price = coupon ÷ (1+r)¹ + coupon ÷ (1+r)² + (coupon + face) ÷ (1+r)³. For any maturity, we write n for the year number (1, 2, 3…) and N for maturity: the symbol Σ (“sigma”) is merely shorthand saying “add up the N pieces”. That is the formula in the header — and exactly the sum the simulator unrolls for whatever maturity N you choose.
One marker before computing. If the market demands exactly what the coupon gives (r = 3%), discounting gives back the face value to the cent: 30 ÷ 1.03 + 30 ÷ 1.03² + 1,030 ÷ 1.03³ = a round €1,000. The security is said to trade AT PAR. Everything else follows: a market more demanding than the coupon (r > 3%) → price BELOW par — the security is said to trade at a DISCOUNT; a less demanding market → ABOVE par. Click each term:
Solving / calculation
Our bond (face value €1,000, coupon €30, 3 years), but the market has moved to 5%. Let's unroll the formula flow by flow, check the pivot, then try it yourself.
- Year 1 coupon, discounted
30 / 1.05≈ €28.57 - Year 2 coupon (two years of waiting: we divide twice)
30 / 1.05²≈ €27.21 - Year 3: last coupon + repayment of the face value
(30 + 1,000) / 1.05³≈ €889.75 - The price: the sum of the three converted appointments
28.57 + 27.21 + 889.75≈ €945.54 (exact sum — on the rounded figures displayed, your calculator will say 945.53: quite normal) - Cross-check at the pivot: if the market demanded 3% (= the coupon)
30/1.03 + 30/1.03² + 1,030/1.03³= exactly €1,000 — at par ✓
At 5%, the security is worth only €945.54: about €54.5 less than its face value, that is −5.4%. Nothing has changed in its promises — €30, €30, €1,030 — but each promise is now divided by a more demanding rate. And the tipping point reads at par: a 3% coupon against a market at 5% → below par; against a market at 2% → above. The simulator lets you set everything — including maturity, to see that a LONG security takes a far heavier blow.
Price = Σ coupon/(1+r)ⁿ + face/(1+r)ᴺSet the market rate, the coupon, the face value and MATURITY: the price recomputes itself by discounting each flow. Look for par (r = the coupon rate), then push maturity to feel the sensitivity.
Economic interpretation
Three flows and a division: that is all it takes to read whole swathes of finance — and your own investments.
Holding a bond means being exposed to rates: if they rise, your security loses capital even though the issuer honours every payment — that is exercise 3. And the more DISTANT the maturity, the heavier the loss: far-off flows undergo the division year after year (push the maturity slider in the simulator: same conditions, N = 2 then N = 10). Professionals sum up this sensitivity in a single figure, DURATION — to a first approximation, the average waiting time of the flows, where each flow counts in proportion to its share of the price: our small coupons weigh little, the big final repayment almost everything, hence a duration close to 3 years for our security. Above all remember the idea: the longer it is, the more it moves.
So far: rate → price. Investors also make the REVERSE journey: start from the observed price and look for the rate that equalises price and discounted flows — that rate is called the yield to maturity, and it is what is compared from one security to another, never the advertised coupon. In a single-flow case, the journey can be done in your head: paying €970 for a security that will pay €1,030 in a year means earning 1,030 ÷ 970 − 1 ≈ 6.2% — whatever its original coupon. Hence the rule: below par, the yield EXCEEDS the coupon rate (you collect the coupons AND the climb back to face value).
The course “Return and risk” set the riddle: why a bond at 12% when the state pays 3%? Our formula answers: if lenders doubt they will be repaid, they demand a far higher r — and a higher r crushes the price of promises. For equal promised flows, the doubtful security is worth less; the gap in required rates is the default premium — the “spread”, in market jargon. The advertised return pays for the doubt, not for generosity.
Reread the opening: “the central bank raises its rates”. The policy rate transmits to market rates (course “The policy rate and transmission”), and each rise mechanically lowers the price of bonds already issued — those of savers, of life insurers, of banks. And this discounting logic does not stop at bonds: valuing a SHARE means discounting its future dividends — the course “Valuing a share: the Gordon-Shapiro model” applies exactly the same move.
Limits / critiques
Exercises
Market rates FALL from 4% to 2%. The price of a fixed-coupon bond already issued will…
Your bond was quoted at €1,040 yesterday; after a rise in rates, it is quoted at €988. What capital loss, in % (÷ starting price)?
A ZERO-COUPON bond: no coupon, face value €1,000 paid in a year. Market rate 4%. What is its price (in €)?
True or false: a bond whose coupon exactly equals the market rate trades at par.
Face value €1,000, coupon €50 (5%), maturity 2 years, market rate 10%. What is the price (in €, to the nearest €2)?
You pay €960 for a security that will pay €1,020 in a year (last coupon + face value). What is your yield to maturity (in %)?
Two identical bonds (3% coupon), maturities 2 years and 10 years. Rates rise by one point. Which loses more?
The state borrows at 3%; a fragile company has to promise 12%. Why?