# Why use swap-rates in a yield curve?

I have a question concerning interest yield curves. Many institutions use the Libor-swap rate curve as a yield curve. Let's be precise and say that we want the yield curve to be the curve that gives us the rate at which a well-rated bank can lend money for any tenor. When the tenor is less than 12 months then that's basically the Libor rate of this tenor. But when the tenor is greater than 1 year, then why should this be the swap-rate? Is there a theoretical justification that the swap-rate corresponds really to this rate?

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possible duplicate of Where do swap rates and/or long-term forward rates come from? –  Phil H Jun 10 '13 at 9:28

## 5 Answers

I agree with joelhoro's answer that swap rates are not used directly as funding rates, although I noticed that quite a few answers here do assume (erroneously) that the swap rate is taken verbatim to construct the yield curve. Instead, one performs bootstrapping. For example, the value of receiving fixed while paying floating in a swap = $V_\text{fixed} - V_\text{float} = R_\text{swap} \sum_{i=1}^{2T} \text{day}(t_{i-1}, t_i) d(0, t_i) - \sum_{i=1}^{4T} \left[ d(0,t_{i-1}) - d(0,t_i) \right]$, where $\text{day}(t_1, t_2)$ is the day-count fraction, $T$ is the maturity, and $d(t_1, t_2)$ is the discount factor.

We solve for the swap rate by setting the value to zero: $R_{\text{swap}} = \frac{1 - d(0,T)}{\sum_{i=1}^{2T} \text{day}(t_{i-1}, t_i) d(0, t_i)}$. Hence, given a set of market (breakeven) swap rates for various maturity dates, one can work out the various discount factors $d(0, t_i)$, where $i = 1, \ldots, 2T$.

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Liquidity, liquidity, liquidity!

we want the yield curve to be the curve that gives us the rate at which a well-rated bank can lend** money for any tenor

So, that starting premise may be causing confusion. What you describe is not what the swap curve was ever meant to show.

The widespread use of the swap (par) curve has to do with the liquidity of interest rate swaps, and the liquidity has to do simply with the fact that the most active participants are going from fixed-rate bonds (assets or liabilities) to synthetic floating rate, or the other way around.

So, for instance, in the US, fixed-rate corporate bonds typically pay coupons on a semiannual 30/360 basis, and floating-rate loans are typically linked to (quarterly ACT/360) LIBOR. So a very liquid market evolved with those two legs; there is far less use for an exchange for rolling short-term rates against a bullet long-term rate. (As to why those are the conventions in the US and why they vary by country is a different question.)

Liquidity -> frequent transactions -> reliable market levels, even if not in a particularly intuitive "format".

** Minor thing, but perhaps you meant borrow and not lend? Where the bank lends has to do with whom it is lending to. Where the bank borrows has to do with its own credit risk. Arguably, the CDS market does what you want the yield curve to do; it is a spread and the CDS coupon is paid periodically, yes, but that spread represents the "bullet" cost of the bank borrowing for that tenor.

It is actually impossible to derive bullet bank borrowing costs from the swap curve, not even when you bootstrap long-term zero rates. (Because a par swap rate is an "average" of rolling short-term borrowing costs - there is a significant credit premium involved in increasing that short tenor - and those costs come from a panel (BBA) in which weak banks are constantly being replaced by stronger banks.)

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Another point to mention is that using a LIBOR swap curve removes sovereign risks associated with using treasuries to construct yield curves.

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Replacing it with the bank risk... which is higher than the sovereign risk, at least for the USA or UK. –  quant_dev Jun 10 '13 at 7:36

Nobody is saying that the swap rate is used as such in calculations of funding rates. What happens is that the yield curve is constructed using a series of instruments, together with their current market prices.

These instruments are usually short term futures, deposits, ... and swaps. The yield curve is constructed by solving so that each of these instruments returns the current market price. For practical reasons this is usually done by bootstrapping, i.e. trying to find the front of the yield curve to satisfy the shortest maturity and then using that to determine the curve further back.

So swap and swap rates are used mostly because they are the most liquid long-term instrument, but people could choose to calibrate using other instruments if they were quoted on the market.

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Recall that an interest rate swap has two legs, one fixed and one floating, each paid by one party to the transaction.

Now, assume you go to a big bank like JPM, and want to borrow \$100MM at fixed rate. JPM will have to fund that position, which because it is a big bank it will do at floating interest rates.

But maybe JPM is worried about the effect such interest rate variation has on its Sharpe ratio, so they want to lock in the profits they made from arranging the loan with you. In this case, they will swap fixed for floating and be able to book the profits with certainty (modulo credit risk).

Therefore, the price of your loan was ultimately driven by JPMs cost of a fixed-float swap.

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