Building Yield Curves from Cash, Futures, and Swap Quotes

# Why is "full" Yield Curve (term structure of interest rates) 3 component based?


# Why is "full" Yield Curve (term structure of interest rates) 3 component based?












I am trying to understand bond-valuation and construction of yield curve. I don't have any exposure to bootstrapping or what-so-ever as of now. So it's appreciated to have an example but not too technical (not too advanced mathematical terms - but straight forward).

In Wiki shows that yield curve has three types of rates involved.

I have seen online that some uses interpolation (non-/linear) to construct the full yield curve with tenors. And for lending and borrowing, most institutions take yield curve as the benchmark.

My questions:

- Why is the curve constructed using cash, future and swap rates?

- What are these future rates? Bond futures or interst rate futures?

- Cash rates plotted in the curve are directly used for lending/borrowing within institutions. What about the the futures rates and swap rates in the yield curve? Are these taken as it is or just as a reference to derive "in-house" rates?

- Are there risks lenders/borrowers facing by taking rates plotted in the yield curve as it is?

- Are there functions availabile in C# to interpolate and build full yield curve using USD-daily tresury yield curve?

## Answer by Helin (score 5)

https://quant.stackexchange.com/a/14745

@Arrigo's answers are quite good; I'll try to beef up his points a bit more.

- Yield curves should be constructed using instruments of similar credit risks. If you're building a US Treasury yield curve, then you should use Treasury bills, notes, and bonds (although lots of people actually exclude Treasury bills because of market segmentation concerns). On the wikipedia page, they're building a USD swap curve, which should reflect credit risks embedded in LIBOR. Of course, LIBOR cash rates, Eurodollar futures, and conventional interest rate swaps are all LIBOR-based and belong to the same term structure. The choice of what instruments to use is actually art. Some market participants include EVERYTHING that has active quotes. Others use only the most liquidly traded. There's really no consensus here. In the post financial crisis world, a swap curve is built with at least the following: 1) LIBOR cash rate, 2) Eurodollar futures, 3) par swap rates, 4) fed funds futures, 5) OIS rates, 6) Libor/FF basis swaps & 7) tenor basis swaps.

- These are eurodollar futures rates. They are essentially 3M Forward Rate Agreements, but not exactly the same. Eurodollar future contract is an exchange-traded product and require daily variation margins, while FRAs are OTC products. Because of this, eurodollar futures-implied rates are higher than FRA rates; this difference is known as the "convexity bias." In practice, eurodollar futures rates are used, instead of FRA quotes, since the former is much more actively traded and provide better price discovery.

- Eurodollar futures are quoted as "100 – interest rate." So the first step would be do "100 – price" to get the "futures" rate. Then we make a convexity-adjustment to obtain the "forward" rate. Different shops have different models to compute the necessary adjustment. Swap rates can be used directly for curve construction.

- Most of the standard swaps are actually centrally-cleared nowadays (as required by Dodd-Frank), so frankly there's not much default risk. Similarly, Eurodollar futures are traded on exchanges, and your counterparty is the exchange. Again, virtually no default risk here (unless the exchange goes under – very unlikely). Libor cash rates are not tradable.

- I don't know much about C#. I'd only mention that to build a "Treasury" yield curve, you should use Treasury bills, notes, and bonds as inputs, NOT swap rates or eurodollar futures rates. The US Treasury Department builds their curves using only on-the-run issues, while the vast majority of dealer Treasury curves are built with a large number of off-the-run issues.

I also recommend that you read this post What is the Swap Curve?, which provides some color on the transition to OIS discounting. The wikipedia page you referenced desperately needs updating. It's outright wrong to use 1M and 3M libor rates in the same curve, since 3M libor is clearly riskier than 1M libor (there's a "basis" between them). Likewise, the correct discount rate for building a swap curve is NOT libor, but OIS.

## Answer by roym00 (score 2)

https://quant.stackexchange.com/a/14744

I'll try to give you an answer.

- I think the term structure is built from those financial products because they are the most liquid for those maturities: theoretically, a liquid instrument has a price coming from a large consensus which you can think of the market. This is an "academical" reason, probably there are other reasons also (I'm still learning).

- Futures in the wiki page stands for Forward Rate Agreement (FRA), which is another derivative on interest rates (usually for short maturities). I recommend the Hull's book to find what this derivatives is; if you know what an interest rate swap (IRS) is, then a FRA is an IRS where only 1 cash-flow is exchanged (only one payment).

- The term structure does not contain FRA rates and interest rate swap (IRS) rates, but spot rates that are computed from those numbers (you find the numbers in the market). The rate of a FRA or of an IRS is a fixed rate that makes zero the fair value of the derivative at inception. To get the spot rate, for each maturity of the term structure you invert the formula used to compute the FRA or IRS rate. I recommend Hull's book again to know more about it.

- I don't have enough experience to answer it fully. As a hint, I'd say it depends on the term structure (there are many, not just one). I'll give you the example of the LIBOR rate: LIBOR is a trimmed average of reported funding rates for unsecured borrowing deals from a panel of large banks with the highest credit quality. From this definition, LIBOR contains some information about credit risk (this was almost completely forgotten before the crisis), meaning that banks with lower credit quality should be charged higher rates when they ask for borrowing. As far as I know, the Overnight Indexed Swap (OIS) rate is today's standard choice as risk-free rate to in any pricing formula.

- I would stick with the suggestion of @Matt Wolf. I don't know personally any library directly written in C#. I know that QuantLib is available in C# in some way, but I never used the library itself.

Hope this helps.