Yield Curve and Forward Rate Explorer
The yield curve plots interest rates (spot rates) against their maturity. It is one of the most watched indicators in fixed income markets. Understanding spot rates, forward rates, and discount factors is fundamental to bond pricing, interest rate risk management, and understanding what markets are pricing in about future rates.
A spot rate \(s_t\) is the annualized yield on a zero-coupon bond maturing in \(t\) years — the rate you lock in today for borrowing or lending over the full horizon. The discount factor \(d(t)\) converts a cash flow at time \(t\) back to present value:
$$d(t) = \frac{1}{(1+s_t)^t}$$An implied forward rate \(f(t)\) is the break-even rate for a one-period loan starting at year \(t-1\) and ending at year \(t\), extracted directly from the spot curve by no-arbitrage:
$$f(t) = \frac{(1+s_t)^t}{(1+s_{t-1})^{t-1}} - 1$$When the spot curve is upward-sloping (normal), forward rates lie above the spot curve — the market implies short-term rates will be higher in the future. An inverted curve implies forward rates below current spot rates. A humped curve produces forward rates that eventually turn sharply downward. The yield to maturity (YTM) of a coupon bond is a single internal rate of return that sets the present value of all cash flows equal to the price, but it is not a spot rate — it is a complex average of the spot rates for each cash flow date.
The visualization
Choose a curve preset or adjust individual spot rates with the sliders. The chart updates the spot and forward curves instantly. Configure a coupon bond to see how it is priced using discount factors and what YTM results.
Curve Shape Presets
Spot Rates
Coupon Bond Settings
Bond Results
1-yr forward rate: f(t) = [(1 + st)t / (1 + st−1)t−1] − 1
Bond price (annual): P = Σ C · d(t) + F · d(T)
YTM: solve P = Σ CF / (1 + y)t (bisection)
| Year | Spot Rate | Discount Factor d(t) | Zero Price ($100 × d(t)) | Fwd Rate f(t) |
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Key takeaways
- Spot rates vs. forward rates. Spot rates discount individual cash flows from today to maturity. Forward rates are the break-even rates implied for a single period in the future — they reveal the rates the market is pricing in for future borrowing and lending, though they are not forecasts.
- Normal, inverted, and humped curves. A normal (upward-sloping) curve implies forward rates above spot rates and typically reflects expectations of rising rates or a term premium. An inverted curve — where short rates exceed long rates — has historically been a reliable recession indicator. A humped curve produces forward rates that first rise above and then fall sharply below the spot curve.
- Bond pricing with spot rates vs. YTM. The theoretically correct price discounts each cash flow at its own spot rate. YTM is a single internal rate of return that gives the same price — a useful summary, but it implicitly assumes all cash flows are reinvested at the same YTM, which is generally not realistic.
- What forward rates tell us. If markets are efficient, implied forward rates reflect a combination of rate expectations and risk (term) premia. A steeply rising forward curve means markets expect higher rates and/or demand a large premium for holding long maturity bonds.
- Premium, par, and discount bonds. A bond trades at a premium when its coupon rate exceeds the spot rates applicable to its cash flows (YTM < coupon rate), at a discount when the reverse holds, and at par when coupon equals YTM. Adjusting the spot curve and coupon in the tool illustrates these relationships directly.