Where you are. You have priced bonds, inverted the price to a yield, and measured sensitivity. Every one of those used a single rate for every payment, as though money had one price no matter when it comes back. It does not. Lending for a year and lending for ten are different propositions and are priced differently, and this lesson replaces the single number with the line that finance actually quotes.
Two rates on the same screen
A trader’s screen shows the same issuer twice. Lend for one year: three percent. Lend for ten: five.
Same borrower. Same currency. Same certainty of repayment. Two prices, and they are not a mistake or an arbitrage; they are the market’s honest quotes for two genuinely different things.
Ask why the ten-year pays more and the first answer is usually risk, which cannot be right here because the issuer is identical. The real answers are about time: rates might rise over ten years and you would be stuck earning five while others earn more, and money committed for a decade cannot be used for anything else. Both are charges for time, not for the borrower.
The idea in one paragraph
The yield curve is the market’s rate for each length of loan, plotted from short maturities to long, and reading it is the closest thing finance has to a public forecast. Its shape says what the market expects: sloping up usually means rates are expected to rise, flat means no strong view, and sloping down - inverted - means the market expects rates to be lower in future than now, which historically has meant it expects trouble. Pricing off a curve is a small change to what you already do: discount each payment at the rate for its own maturity instead of one rate for all of them. And because the curve is drawn for one issuer, comparing two issuers’ curves gives you the extra yield demanded of the riskier one, which is a price on default itself.
The same bond, three curves
Take the five-year bond from lesson 5 and price it off three curves: one rising from three to five percent, one flat at four, one falling from five to three.
The flat curve prices it at exactly its face, which you can predict from lesson 7: a four percent bond discounted at four percent is worth par. The rising curve prices it below face; the falling curve above.
The reason is lesson 5’s observation about where the money is. Most of the bond’s value is the final payment, so the rate at the far end of the curve does most of the work. A rising curve punishes that payment with its highest rate; an inverted curve rewards it with its lowest.
The shape as a forecast
An upward-sloping curve is the ordinary condition. It says the market expects short rates to be higher later, and it pays lenders something for committing money for longer.
An inverted curve is the interesting one. It says the market expects rates to be lower in future than they are now, and rates usually fall because a central bank is cutting them, and central banks usually cut because the economy is weakening. That chain is why an inverted curve has a reputation as a recession signal.
One more curve, and the gap between them
Draw the curve for a government and the curve for a company in the same currency, and the company’s sits above it everywhere. The vertical gap at any maturity is what the market charges for the possibility that this borrower does not pay.
This is the missing number from lesson 5, where two bonds with identical schedules could be worth different amounts. The difference has a name and it is quoted in basis points, exactly as module 3 taught you to quote every other toll.
Check yourself
1. The same issuer quotes 3% for one year and 5% for ten. Why is “the ten-year is riskier” the wrong explanation?
Because the issuer is the same, so default risk is not what differs. What differs is time: the risk that rates rise and leave you locked into a now-poor return, and the cost of committing money for a decade rather than a year. Both are charges for duration of commitment, not for the borrower’s creditworthiness.
2. Why does the far end of the curve move a five-year bond’s price more than the near end?
Because most of the bond’s value is its final payment, which carries the face value as well as the last coupon. That payment is discounted at the far-end rate, so a change there moves far more present value than an equal change applied to a single small coupon near the start.
3. A curve inverts. What is the market saying, and why is that associated with bad news?
That it expects rates to be lower in future than they are now. Rates usually fall because a central bank cuts them, and central banks usually cut when the economy is weakening, so an expectation of falling rates is indirectly an expectation of trouble. The signal is a bet, not a certainty, and its timing has varied widely.
4. Two bonds have identical schedules and different prices. Where does the difference show up on a curve, and what is it called?
As the vertical gap between the two issuers’ curves at that maturity: the credit spread. It is the extra yield the riskier borrower must pay, quoted in basis points, and it is the market’s price for the possibility that the promises are not kept.
Do this
Price one bond off three curves and see the shape decide the answer.
python3 code/curve_shapes.pyThe starter leaves you price_off_curve, which discounts each payment at the rate for its own tenor rather than one rate for all. The assertions check that the flat curve prices the bond at exactly par and that the rising curve prices it below the flat one, which prices it below the inverted one. Success prints all three prices and the line:
cheapest off the upward curve: one bond, one set of promises, three prices - the curve is the priceThe completed version is in solutions/curve_shapes.py.
What you can now do. You can price off a curve rather than a rate, read a curve’s shape as the market’s expectation, and name the gap between two issuers’ curves. That completes the pricing machinery for instruments that already exist. The next lesson asks where they come from in the first place.