Quant Finance Foundations, Part 3. Previously: Present Value Explained.
A yield curve is the set of interest rates that apply across every maturity, and it resolves the problem the last post ended on. Discounting needs a rate, but there is no such thing as the interest rate. Instead, there is a different rate for every maturity.
Reading the curve tells you what the market currently believes about the economy. Moreover, building one is among the most common tasks a junior quant is handed. Both skills start with the present value arithmetic from the previous post.
What a yield curve is
Plot maturity on the horizontal axis and the annualised interest rate on the vertical axis. The result is a yield curve. A point at the three-year mark therefore answers one question: what rate applies to money borrowed or lent for three years, starting today?
Curves come in a few recognisable shapes, and each carries a message.
- Upward sloping (normal). Long rates exceed short rates. Lenders demand compensation for tying money up longer, and the market expects rates to stay stable or rise. This is the usual state of the world.
- Flat. Little difference across maturities. Typically a transition — the market is unsure which way policy goes next.
- Inverted. Short rates above long rates. This says the market expects the central bank to cut, which usually means it expects the economy to weaken. Inversions have preceded most modern recessions, which is why financial journalists write about them constantly.
- Humped. Rates rise then fall. Often reflects a specific expected policy path rather than a broad economic signal.
Three different things called “the curve”
This is where beginners get lost, because three related objects share the same picture.
| Curve type | What it is | Who uses it |
|---|---|---|
| Par yield | Coupon that prices a bond at exactly 100 | Quoted in the market |
| Zero (spot) rate | Rate for one cash flow at one date | Used for discounting |
| Forward rate | Rate for a future period, agreed today | Consumed by pricing models |
Par yields are what you observe directly in the market — the coupon that would make a bond of that maturity trade at exactly 100. They are quoted, but they are not what you discount with, because each par yield blends the rates of every period up to that maturity.
Zero rates (spot rates) apply to a single cash flow at a single date, with nothing in between. These are what you actually need for discounting. There is one zero rate per maturity, and the discount factor at time n is built from it directly.
Forward rates are rates for a future period agreed today — for example, the one-year rate starting two years from now. They are implied by the zero curve rather than observed, and they are what pricing models for swaps and futures are built on.
All three describe the same underlying object. Given any one, the other two follow by arithmetic. The market quotes par yields; the quant needs zero rates; the model consumes forwards.
Where the curve comes from
Nobody publishes a zero curve. It is constructed, and the process is called bootstrapping. The name is literal — you pull the curve up by its own bootstraps, solving each maturity using the ones you already solved.
The raw material is a set of liquid market instruments, each covering a different part of the maturity spectrum:
- Short end (overnight to ~3 months): deposit rates and overnight index swap rates.
- Middle (3 months to ~2 years): forward rate agreements or interest rate futures.
- Long end (2 years and beyond): interest rate swaps, which are the most liquid long-dated instruments in most markets.
Bootstrapping, one step at a time
The logic is recursive. Start with the shortest instrument, which has a single cash flow. Its rate is the zero rate for that maturity — nothing to strip out.
Move to the next maturity. That instrument has cash flows at earlier dates too, but you already know the zero rates for those. Discount them with what you have, and only one unknown remains: the discount factor for the final date. Solve for it.
Repeat. Each new instrument introduces exactly one new unknown, so each step is a one-variable solve. By the time you reach the thirty-year point you have a complete zero curve.
A concrete version. Suppose the one-year zero rate is 5.00%, and a two-year bond with a 5.50% annual coupon trades at par (100). Its cash flows are 5.50 at year one and 105.50 at year two:
100 = 5.50 / 1.05 + 105.50 / (1 + z₂)²
The first term is 5.238. So 94.762 = 105.50 / (1 + z₂)², giving (1 + z₂)² = 1.11330 and z₂ = 5.514%. Notice it sits slightly above the 5.50% par yield — the standard result when the curve slopes upward.
Interpolation: the quiet source of errors
Market instruments give you a handful of points. Real portfolios have cash flows on arbitrary dates, so you have to fill the gaps, and the choice of how is not cosmetic.
Linear interpolation on zero rates is simple and produces a jagged, economically implausible forward curve. Interpolating on the log of discount factors is equivalent to piecewise-constant forwards — better behaved, still not smooth. Monotone convex methods and splines produce smooth forwards but can introduce oscillation if applied carelessly.
Two desks using identical market data and different interpolation will produce different prices for the same instrument. On a large book the gap is material, which is why interpolation methodology is a documented, validated choice rather than a coding convenience.
What changed after 2008
The single-curve world described above is a simplification the industry abandoned. Before the crisis, one curve served both to project future floating rates and to discount cash flows. The crisis blew apart the assumption that interbank lending was risk-free, and basis spreads between tenors widened to levels nobody could ignore.
The modern setup is multi-curve: a collateral-based discount curve (OIS, or in newer conventions a risk-free rate such as SOFR or SONIA), plus separate forecast curves per tenor. The bootstrapping logic survives; there are simply several curves to build and they are solved jointly. The IBOR transition then rebuilt the underlying reference rates entirely, but the mathematical machinery stayed put.
Try it yourself
Bootstrap a curve by hand, in this order:
- Collect four or five quoted government bond or swap rates from a market you follow.
- Read the zero rate off the shortest instrument directly — it has a single cash flow.
- Move to the next maturity, discount its earlier cash flows with the rates you already have, and solve for the one remaining unknown.
- Repeat until you reach the longest maturity.
- Derive the implied one-year forward rates from the finished curve.
Then ask whether that path of future short rates matches what you would independently expect from the central bank. When the two disagree, you have found either an arithmetic mistake or a trade.
When the two disagree, you have found either a mistake in your arithmetic or a trade. Both are educational.
Frequently asked questions
What does an inverted yield curve mean?
It means short rates sit above long rates, which implies the market expects the central bank to cut. Historically, inversions have preceded most modern recessions.
Why can’t I just discount with the quoted par yield?
Because each par yield blends the rates of every period up to that maturity. Discounting requires zero rates, which is why bootstrapping exists.
What is bootstrapping a yield curve?
It is solving for zero rates one maturity at a time, using the rates you already solved to strip out the earlier cash flows. Each step therefore has exactly one unknown.
Why did banks move to multi-curve frameworks?
The 2008 crisis destroyed the assumption that interbank lending was risk-free. As a result, discounting moved to collateral-based curves while forecasting kept separate curves per tenor.
Next in this series: the derivatives you can price with nothing more than a curve and arithmetic — forwards, futures and swaps. Later posts build on this curve again when pricing options with a binomial tree.