The Bond Yield Curve Explained
Every article so far discounted all of a bond's cash flows at one yield. Real markets don't work that way — and the difference is measurable.
So far in this series, pricing a bond has meant picking one yield and discounting every cash flow with it. That's a useful simplification for learning, but it isn't how the market actually prices anything precisely — because the 1-year interest rate, the 5-year rate, and the 10-year rate are rarely the same number. Those different rates across different maturities together form the yield curve, and this article shows exactly what gets lost by ignoring it.
A set of rates, not one rate
Imagine today's market shows approximately these yields:
| Maturity | Yield |
|---|---|
| 1 year | 5.0% |
| 2 years | 5.2% |
| 3 years | 5.4% |
| 5 years | 5.7% |
| 10 years | 6.0% |
| 30 years | 6.3% |
That curve carries real information: it tells you how the market currently prices time and risk across every horizon at once, not just at whatever single maturity you happen to be looking at.
Three common curve shapes
Why YTM is a summary, not the whole story
When you calculate a bond's YTM, you get one number describing that bond's yield. Convenient — but the bond's actual cash flows are each exposed to a slightly different point on the curve. A coupon arriving in Year 1 and the principal arriving in Year 10 shouldn't really be discounted at the same rate, because the market doesn't lend money at the same rate for one year as it does for ten.
YTM is a useful summary number. The term structure — the full curve of maturity-specific rates — carries more detailed information than any single yield can.
Spot rates: the building block
A spot rate is the rate for a single cash flow at a single maturity — most cleanly understood through a zero-coupon bond, since it has exactly one payment. If a 1-year zero-coupon bond's price implies a 1-year rate of 5%, that's the 1-year spot rate. If a 5-year zero implies 5.8%, that's the 5-year spot rate. Each maturity carries its own spot rate, and together they form the spot curve.
This is exactly why zero-coupon bonds keep reappearing throughout this series — as introduced early on — they're the cleanest possible building block for everything involving the shape of rates over time.
Pricing a bond properly, with a full curve
The pricing formula from earlier in this series used one yield for every cash flow:
A term-structure approach instead discounts each cash flow at the spot rate matching its own maturity:
A worked example
Take a bond paying ₹80, ₹80, and ₹1,080 in years 1, 2, and 3, with spot rates of 5%, 5.5%, and 6% for those respective maturities:
| Year | Cash flow | Spot rate | Present value |
|---|---|---|---|
| 1 | ₹80 | 5.0% | ₹76.19 |
| 2 | ₹80 | 5.5% | ₹71.88 |
| 3 | ₹1,080 | 6.0% | ₹906.79 |
| Total | ₹1,054.86 |
Compare that to pricing the same bond with one flat rate, using only the 3-year spot rate of 6% for every cash flow — the natural mistake if you had only that one number and not the full curve:
The difference is small here — about ₹1.40 — but it's real, and it's not noise. It's the direct, measurable cost of collapsing an entire curve into one number when the curve isn't actually flat. For longer-dated or more irregular cash-flow schedules, that gap widens considerably.
Price this bond at a flat 6% in the calculator →The calculator prices off a single yield you supply, exactly like the flat-rate comparison above — it doesn't yet construct a full spot curve from multiple market rates, which is a more advanced, separate calculation covered by bootstrapping below.
Where spot rates actually come from
The market doesn't hand you a neat list of spot rates directly. They're derived from the prices of traded securities with known cash flows, through a process called bootstrapping: start with the shortest-maturity instrument, solve for its spot rate, then use that to solve for the next maturity out, and so on, working progressively further along the curve. This is a genuinely advanced technique — worth knowing exists, not something you need to compute by hand to use the concepts above correctly.
Forward rates: what the curve implies about the future
A forward rate is a future-period rate implied by today's curve — for example, the one-year rate that will apply starting one year from now, as implied by where today's 1-year and 2-year spot rates sit relative to each other.
Suppose the 1-year spot rate is 5% and the 2-year spot rate is 6%. A two-year investment must be consistent with earning the 1-year rate for the first year, then some forward rate for the second:
So today's curve implies a one-year rate of roughly 7.0% beginning a year from now — noticeably higher than either the 1-year or 2-year spot rate on their own. That's the curve's embedded expectation about future short-term rates, not a guarantee of what will actually happen.
Why curve shape matters: non-parallel moves
Standard modified duration assumes the whole curve shifts by the same amount everywhere — a parallel shift. Real curves rarely move that cleanly. The 2-year point might jump sharply on a policy announcement while the 10-year point barely moves, or vice versa. Key-rate duration extends the idea from earlier in this series by measuring sensitivity to a move at one specific point on the curve, holding the rest fixed — answering "where along the curve is my risk actually concentrated?" rather than treating the curve as one number that moves uniformly.
Common mistakes
The hierarchy to keep in mind
With this article, the series has covered every tool needed to price and risk-manage a plain, option-free bond correctly. The next step is where fixed income gets genuinely more interesting: what happens when a bond's cash flows aren't fixed at all, because the issuer or the investor has a contractual choice built in. That's callable and putable bonds — the beginning of embedded options.