Par, Zero, and Forward Curves in Yield-Curve Construction

# Bootstrapping - why PAR instruments


# Bootstrapping - why PAR instruments












I understand the mechanism of use bootstrapping to create a yield curve.

However, why do we choose a PAR (bond/swap) in creating the curve?

Is it for simplicity? Is it possible to use non-PAR instruments? Do we lose anything by using PAR instruments?

Thanks.

## Answer by nbbo2 (score 2, accepted)

https://quant.stackexchange.com/a/45453

There are several representations of "the yield curve". You can choose which one you want to use.

For the "raw yield curve" (or what I call the Wall Street Journal yield curve, because it is used in newspapers) you simply get a list of bonds of various maturities and coupons and you plot points using the maturity on the x-axis and the YTM on the y-axis. Because the bonds have different coupons (i.e. because of the coupon effect) the curve will not be smooth and will be theoretically flawed (the YTM of an N-year 3% bond is not equal to the YTM of an N-year 5% bond unless the term structure is flat). Therefore no serious quant should use this representation of the yield curve, but you still see it sometimes. (By selecting only bonds which are reasonably close to par, this curve can be made to look a little nicer, but is still flawed. It can be seen as an approximation of the Par Curve described below).

A major academic advance of the 1980's was the use of the Zero Coupon yield curve. In this representation a so-called bootstrapping procedure is used to identify the discount rates which the market applies to a single future cash flow. These discount rates are annualized and correspond to the yields on hypothetical Zero Coupon bonds of various maturities. This is a very simple and clean representation of how the market prices all bonds.

Because most bonds are coupon bonds, it is convenient for traders to think in terms of coupon bonds. For this representation, the so-called Par Yield curve, we convert the Zero curve by a simple calculation into the coupon rates which would cause an N-year bond to be priced at Par (100). This representation is widely used in practice. It amounts to saying: if a new N-year bond were issued today (priced at par) what YTM would it have. Bonds at par are easy and convenient to think about. It also ties in nicely with swaps because swap rates are based on entering the swap with 0 NPV, i.e at par.

Also there is the Forward Rate representation of the yield curve, which is nice because it shows the interest rates you can earn by leaving your money invested fr an additional period of time in the future. It is perhaps less used than the other two.

You don't have to use the Par Curve, but you should know how it is related to the (equivalent) Zero Curve and the Forward curve.

## Answer by Dimitri Vulis (score 2)

https://quant.stackexchange.com/a/45452

You use the most liquid instruments whose observable quotes you believe. For the swap curve, you use swap rates from the most common market convention. E.g., in case of USD - the fixed coupon one would pay for a floating leg reset from 3 month Libor. Not just any par rate. In case of some other currency, you'd use annual frequency because that's the most common swap. For USD, you might decide that for tenors before 5Y you might prefer to use ED futures instead of swap rates because they are more liquid. Likewise if you're building a bond yield curve from treasury bonds, you use the most liquid instruments. In the U.S., treasury constantly sells at par new bonds with different coupon and maturity as old bonds. In Brazil, Colombia, and many other countries treasury sells more bonds with the same coupon and maturity as existing ones, and does not price them at par. A new instrument appears maybe once a year. You'd use these bonds and heir market prices.