How is the swap rate curve constructed, and why does bootstrapping from deposit rates to swap rates matter for valuation?

The swap rate curve (also called the par swap curve) is the foundation for pricing nearly all fixed-income derivatives. It is constructed by bootstrapping discount factors from progressively longer-maturity instruments, ensuring consistency across the entire term structure.\n\nInstrument Hierarchy:\n\n`mermaid\ngraph TD\n A[\"Overnight to 1W
Fed Funds / SOFR deposits\"] --> B[\"1M to 3M
Money market deposits\"]\n B --> C[\"3M to 2Y
Eurodollar or SOFR futures\"]\n C --> D[\"2Y to 30Y
Interest rate swaps\"]\n D --> E[\"30Y+
Long-dated swaps or bonds\"]\n A --> F[\"Short-end discount factors\"]\n C --> G[\"Mid-curve discount factors\"]\n D --> H[\"Long-end discount factors\"]\n F --> I[\"Complete Zero Curve\"]\n G --> I\n H --> I\n`\n\nBootstrapping Logic:\n\nFor the short end (0-3 months), discount factors come directly from deposit rates. A 90-day deposit at 5.20% annual gives:\n\nDF(0.25) = 1 / (1 + 0.052 x 90/360) = 1 / 1.013 = 0.98717\n\nFor the mid-curve (3 months to 2 years), Eurodollar or SOFR futures imply forward rates, which are chained onto existing discount factors:\n\nDF(0.50) = DF(0.25) / (1 + f(0.25, 0.50) x 0.25)\n\nFor the long end (2-30 years), par swap rates are used. A 3-year par swap at 4.85% means the fixed leg equals the floating leg. Given DF(1) and DF(2) from earlier steps:\n\n0.0485 x [DF(1) + DF(2) + DF(3)] + DF(3) = 1\n\nSolve for DF(3) algebraically.\n\nWorked Example:\nNorthgate Capital needs to value a 3-year receiver swap. The bootstrapped curve yields:\n\n| Tenor | Instrument | Rate | Discount Factor |\n|---|---|---|---|\n| 6M | Deposit | 5.15% | 0.97490 |\n| 1Y | Futures-implied | 5.05% | 0.95108 |\n| 2Y | 2Y Swap | 4.90% | 0.90812 |\n| 3Y | 3Y Swap | 4.85% | 0.86753 |\n\nThe fixed leg PV per $1 notional: 0.0485 x (0.95108 + 0.90812 + 0.86753) = 0.0485 x 2.72673 = $0.13225\n\nThe floating leg PV equals 1 - DF(3) = 1 - 0.86753 = $0.13247\n\nThe tiny difference ($0.00022) reflects rounding; at par the swap has zero NPV.\n\nWhy Ordering Matters:\nEach successive discount factor depends on all previous ones. An error in the 6-month deposit rate propagates into every longer-tenor discount factor. This is why the short end uses the most liquid instruments and why curve construction teams monitor input quality obsessively.\n\nInterpolation Methods:\nBetween node points, dealers use cubic spline, monotone convex, or piecewise linear interpolation on either zero rates or log discount factors. The choice affects forward rate smoothness and hedging stability.\n\nPractice curve construction problems in our FRM question bank.