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What is the so-called Swap Curve, and how does it relate to the Zero Curve (or spot yield curve)?

Does it only refer to a curve of swap rates versus maturities found in the market? Or is it a swap equivalent of a spot-yield curve constructed from bootstrapping a bond yield curve?

The context of this question is set against a backdrop of a plethora of terminology (that seems to be used interchangeably). I am looking into how the so-called Zero Curve (or spot yield curve) is constructed in order to discount various IR derivatives (including swaps) when pricing them.

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3 Answers 3

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Garabedian,

Typically, the "swap curve" refers to an x-y chart of par swap rates plotted against their time to maturity. This is typically called the "par swap curve."

Your second question, "how it relates to the zero curve," is very complex in the post-crisis world.

I think it's helpful to start the discussion with a government bond yield curve to clarify some concepts and terminologies. Consider the US Treasury market, using the outstanding Treasury notes and bonds (nearly 300 of them...), we can either use bootstrapping or more sophisticated spline models to construct a "fitted curve." Since this yield curve represents bonds of identical credit risks (basically risk-free), the zero coupon curve, the discount curve, the forward curve, and the par yield curve are just different representations of the same thing and can be translated very easily from each other. For simplicity, I'll assume annual compounding:

  • If you know the zero coupon rate $r_t$ for time $t$, then the discount factor is $1 / (1 + r_t)^t$.
  • If you know the 1-year zero coupon rate $r_1$ and 2-year zero coupon rate $r_2$, then you can compute the 1-year forward 1-year rate from $(1 + r_1)(1+f_{1,1})=(1+r_2)^2$.
  • You can also compute the 2-year par rate, just solve for $c$ from $$ \frac{c}{(1 + r_1)} + \frac{100 + c}{(1+r_2)^2} = 100. $$

Now let's return to the swap market. To be concrete, let's consider a 2-year USD par swap. This instrument has four fixed leg payment, and eight floating payment. The par swap rate is the fixed-leg interest rate that sets the present value of all the cash flows to 0. In other words, we'd solve for the $c$ in: $$ \sum_{i=1}^4 c \Delta_i d(T_i) = \sum_{j=1}^8 l_j \delta_j d(t_j), $$ where $d(t)$ is the discount factor for time $t$, $\Delta_i$ and $\delta_i$ are year fractions, and $l_j$'s are the 3M Libor forward rates.

Before the financial crisis, it is assumed that the discount curve and the forward curve are both based on Libor. This simplifies things a lot – just build a Libor forward curve so that it reproduces libors, futures rates, and par swap rates, and you're done. In this framework, all the translations (from zero curve to par curve to forward curve, etc.) above are still valid.

Unfortunately, the idea that Libor was the appropriate funding rate was completely invalidated during the crisis. In recent years, a common practice is to use the "OIS discounting"-based "multi-curve" approach. In the equation above, the $l_i$'s are still based on the 3M Libor forward curve, but the $d(t)$'s should be discount factors fitted to overnight indexed swaps.

Simply put, when you are building a swap curve, you now need to simultaneously calibrate both the OIS discount curve AND and Libor discount curve... Under this new paradigm, the simple translation that we used for government bonds above no longer works, since multiple curves are involved.

But it gets worse, since 1M Libor and 3M Libor have different credit risks, you can't even do something like $(1 + \text{Libor}_{\rm 1M}/12)(1 + \text{Libor}_\text{1 month forward 2 month} / 6) = 1 + \text{Libor}_{\rm 3M} / 4$! Instead, you need to build separate 1M and 3M Libor forward curves to account for the tenor basis.

As you can see, building a swap curve nowadays is a pretty involved task. What we now refer to as "a" swap curve is actually a collection of curves (OIS curve, 1M Libor, 3M Libor, 6M Libor, etc.) bundled together...

There are numerous literature you can find on this topic just by googling "multi-curve". For example, http://developers.opengamma.com/quantitative-research/Multiple-Curve-Construction-OpenGamma.pdf

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  • $\begingroup$ Thank you @haginile, your answer is very informative and clear. $\endgroup$
    – rex
    Commented Sep 1, 2014 at 6:24
  • $\begingroup$ Thank you so so much ! $\endgroup$
    – Ryan
    Commented Jun 18, 2020 at 15:04
  • $\begingroup$ This is a really good answer, thanks! $\endgroup$
    – KaiSqDist
    Commented Dec 14, 2023 at 0:13
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wiki about yield curve:

... the yield curve is a curve showing several yields or interest rates across different contract lengths (2 month, 2 year, 20 year, etc...). The curve shows the relation between the interest rate (or cost of borrowing) and the time to maturity. For example, the U.S. dollar interest rates paid on U.S. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve". More formal mathematical descriptions of this relation are often called the term structure of interest rates.

Basically, you have a graph tracing a rate $r(t)$ against time $t$.

wiki about swap:

An interest rate swap (IRS) is a financial derivative instrument in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate to a floating rate (or vice versa) or from one floating rate to another.

For example, party A pays fixed rate $r_A$ on specified notional, while counterparty B pays a floating rate (let's say LIBOR) $L_{3m}$ on the same notional. Duration of such contract is sometimes called Tenor.

At the point of initiation of the swap, the fixed swap rate is chosen so that it has a net present value of zero. Such swap rate is called par swap rate and denoted $S(t,T)$, where $t$ is a starting date of the swap and $T$ is the end date. $(T-t)$ is tenor.

Finally, the swap rate curve is a yield curve where reference rate is par swap rate, e.g. $r(t)=S(t,T)$. For example, if you are looking at 2Year tenor swap curve it would be a graph tracing $S(t,t+2Y)$ against $t$.

Hope it's more clear now.

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    $\begingroup$ Can you say what swap exactly means, and what difference to yield curve? $\endgroup$
    – emcor
    Commented Aug 29, 2014 at 15:40
  • $\begingroup$ Use wiki for swap definition and another wiki for yield curve definition. Swap curve is simply a yield curve using par swap rate as yield rate. $\endgroup$
    – Bruno
    Commented Aug 29, 2014 at 15:53
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    $\begingroup$ This is a misleading answer... CMS is not the same thing as a vanilla swap... $\endgroup$
    – Helin
    Commented Aug 29, 2014 at 16:51
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    $\begingroup$ @haginile CMS is a swap stuck on a swap rate. PS: I tried to improve my answer. $\endgroup$
    – Bruno
    Commented Aug 29, 2014 at 17:14
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    $\begingroup$ Yes, just to clarify, vanilla USD 10y swaps pay semi-annual fixed coupons against 3M Libor for 10 years; i.e., the floating leg depends on future realizations of 3M libors. CMS, by contrast, pays semi-annual fixed coupons against a swap rate; e.g., you may have a CMS that depends on future realizations of 10y swap rates. Therefore, CMS is much much more complex instruments then swaps. Knowing only the swap curve alone won't help you price these instruments... $\endgroup$
    – Helin
    Commented Aug 30, 2014 at 1:15
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A "swap curve" is a bit of a misnomer this days, so let me give some historical context to show where it all comes from. There is a long build up, but I think it is useful to fully appreciate the concept once and for all.

A "curve" in finance is merely a function of time, $f(t)$.

The two most frequently used curves are discount curve $D(t)$, and volatility curve. The latter is not of interest for this subject, but discount curve gives the present value of a unit of currency at a future time $t$.

As the discount curve is supposed to be non-negative, it is often, in practice, implemented as an exponential of a "(term) yield curve" $y(t)$

$D(t) = e^{-y(t)t}$

Now there are many discount and discount-like curves, which have the same shape, but do not exactly represent the present value of a unit of currency. For example a "survival curve" used in credit has the same form, but has a slightly different meaning.

True discount curves are build differently in different contexts. The two most important discount curves in the past were "government curve" and "(libor) swap curve". The former was stripped from the government debt instruments, and its corresponding yield curve represented the cost of the borrowing by the government. It was supposed to be closest to the concept of the risk-free curve. Still is.

However in every currency, or rather, financial center, there existed an "IBOR" market, the market of unsecured interbank lending. There were multiple terms of such lending, from overnight to 1, 3, 6 and 12 months. The indicative cost of such lending was represented by IBORs.

IBORs became an important benchmark and various longer term debt products were invented, referencing the future IBOR rates of different tenors. Also to hedge longer term borrowings via fixed bonds, IBOR swaps were invented, which would swap fixed rates of given tenor for the future IBOR rate of some tenor, the tenors not necessarily being same. But in each currency there was a "standard" specification. For example, USD swap was 3 month floating vs 6 month fixed, EUR was 6 month floating vs 1 year fixed, GBP was 6 month floating vs 6 month fixed.

So now we can define was the swap curve meant. The swap curve was a "self discounting" curve derived from the products referencing IBOR with a given term. To price a such a swap on needs to know the discount factors to discount both fixed and floating future payments. There is a famous result that if you use same curve for computing forward floating rates as you use for discounting (and such curves are called "self discounting", then the value of the floating leg is $1-D(T)$, where $T$ is maturity of the swap.

Thus, at the time, swap curves meant self discounting curves referencing a concrete tenor, hence not only some abstract discount factor could be computed using this curve, but also the forward. That was the point. E.g. the standard USD swap curve, stripped from USD Libor deposits, 3m fras and standard 3m floating vs 6m fixed swaps could also be used to compute 3m Libor forwards till the end of the curve. It could also be used to come the swap forward rates, which was relevant for swaption pricing.

All these curves were independent and could only be used to price swaps of the given tenor, or compute forward of the given tenor. Legs of the tenor basis swaps were priced individually using the corresponding tenor swap curves.

This all changed with introduction of the OIS and now RFR discounting. Now the only self-discounting "swap" curves in the former sense are RFR swap curves. They are used both to discount cashflows of the RFR swaps and to compute the forwards.

The remaining IBOR swaps now need two curves to be priced. RFR curves are used for discounting, but "projection" curves are used to compute forwards.

As such, except for the RFR swap curves, there are no more swap curves in the traditional sense, and the curves used to compute forwards except for the RFR rates are better called "projection curves" to avoid confusion.

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