Bootstrapping Zero-Coupon Rates from Interest Rate Swaps
Summary
The document explains how swap rates can be used to infer zero-coupon rates under a single-curve framework. Each swap rate gives a valuation equation involving the floating rates paid over its schedule. With swaps of progressively longer tenors, these equations form a triangular system that can support bootstrapping, much like deriving rates from Treasury bonds.
The method is underdetermined when the available swap maturities do not provide enough equations for all the unknown floating rates. The example uses semiannual payments and several swap tenors to illustrate that gap, so short-maturity rates may need to come from other instruments and longer maturities require simplifying assumptions. The answer cautions that single-curve methods are not suited to modern markets and that swap rates alone do not provide enough information for a useful curve. It does not specify a complete multi-curve procedure or the assumptions needed to fill missing maturities.
Key ideas
- Each swap rate implies an equation linking floating rates across its payment schedule.
- Swaps with progressively longer maturities can form a triangular system for bootstrapping rates.
- Sparse swap maturities leave more unknown rates than equations.
- Short maturities may require other instruments, while longer ones require assumptions.
- Single-curve bootstrapping is limited in modern markets, and swap rates alone are insufficient.
Tags
Full text
# How to derive zero-coupon rates from IRS? # How to derive zero-coupon rates from IRS? How can I calculate zero-coupon rates from historical IR swap rates? I have a record of IRS for the past 4000 days and I am want to compute the zero coupon rates based on them. ## Answer by Michaël Le Barbier (score 8) https://quant.stackexchange.com/a/9052 I assume that you are working in a single curve theory. While this theory used to do well, it is not adapted to today's market and — as Brian B pointed it out — you cannot get a useful information from swap rates alone. The swap rate $S(t)$ at $t$ for a given tenor $T$ and period $P$ is the fixed rate such that a swap starting at $t$ and ending at $t+T$ which exchanges $S(t)$ against the LIBOR-$P$ rate $R$ has value $0$ at $t$. (I write LIBOR for the simply compounded spot rate, which is what you are looking for.) Writing this statement in formulas gives you a linear equation whose unknowns are the rates $R(t)$, $R(t+P)$ and so on until $R(t+T)$. Considering swaps of increasingly large tenors leads to a triangular linear system that you almost can solve: but since you are likely to have much less data as what is really needed you end up using LIBOR rates to determine the zero-rate for short maturities and make simplifications to determine the zero-rate for longer maturities. For instance if you know the swap rates for $P = 6M$ and tenors $T$ in $1Y$, $2Y$ and $5Y$, writing the equations I described leaves you with 3 equations for 10 unknowns, and you have to make some simplifying assumptions to reduce this number of unkowns. The procedure is similar to the usual bootstrapping based on treasury bonds: the strategy is the same but the instruments used differ.
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