I am currently working under estimating a Yield Curve. From my understanding common procedures to construct a yield Curve like Nelson Siegel have the input of a series of different zero rates and maturity pairs and returns a well behaved curve behind it. The zero rates in the united states are obtained from t-bills (that are bullets) and longer t-bonds that pay coupons in the middle, this doesn't seem a problem as you have a stripping program that lets you value each coupon transforming it into a bullet and giving you a zero rate via bootstrapping. The problem is that in my country i only have a very short series of zero coupon bonds and have long term govt securities with coupons. A lack of a stripping program from the central bank makes that the only data i have is the Yield of the bond, but the yield (internal rate of return) has the underlying assumption that you can reinvest each coupon at the actual rate which is a big assumption. I can certainly build a yield curve with the Yields, but doing so grossly underestimates this reinvestment risk and may lead to problems when using the yield curve for interest risk. Is there a procedure for estimating the yield curve in countries without strips?
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$\begingroup$ You can link the credit rating of your country with a spread on the risk free rate. For instance floating rate+200bps. $\endgroup$– alexbougiasCommented Jun 14, 2019 at 15:45
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$\begingroup$ Even if there is no stripping program, you can still construct a zero coupon curve that reprices the coupon paying bonds. Yes, there may be many different curves that can fit the data, so you will have to decide on an interpolation method. Eg cubic spline. Etc. $\endgroup$– dm63Commented Jun 14, 2019 at 20:06
2 Answers
You do not need zero rates to estimate a parametric model of the yield curve, such as Nelson-Siegel. Suppose for instance that you have a cross-section of bond prices. Then:
- For given parameters for your yield-curve model, compute yield curve;
- with this yield curve, calculate theoretical bond prices;
- compute discrepancy between theoretical bond prices and observed bond prices.
Or suppose you have a cross-section of yields-to-maturity of bonds. Then:
- For given parameters for your yield-curve model, compute yield curve;
- with this yield curve, calculate theoretical bond prices;
- compute theoretical yields-to-maturity for theoretical bond prices;
- compute discrepancy between theoretical yields-to-maturity and observed yields-to-maturity.
You now have a link between your yield-curve paramaters and goodness-of-fit. You only have to find parameters for the model (step 1) that result in a small discrepancy between model quantities (prices or yields) and observed quantities. So, you put this computation into an objective function and feed it to a numerical optimization procedure.
As a further complication, in some countries the taxation and other treatment of zero-coupon bonds differs slightly from coupon bonds - for example, in Brazil LTNs vs. NTN-F's; in Colombia TES Corto Plazo v TES serie B; to a lesser extent in Mexico Cetes v MBONOs, etc. Practically this means that, unlike the U.S., a yield of an old coupon bond with year or less left to maturity will not be consistent with the yields of the zero-coupon bonds. You may prefer to buld two curves, one for zc and the other for coupon bonds, with some spread allowed between them.
