The 'zero coupon yield curve' (ZCYC for short) starts from the basic premise of 'time value of money' - that a given amount of money due today has a value different from the same amount due at a future point of time. An individual willing to part with his money today has to be compensated in terms of a higher amount due in future - in other words, he has to be paid a rate of interest on the principal amount. The rate of interest to be paid would vary with the time period that elapses between today (when the principal amount is being foregone) and the future point of time (at which the amount is repaid). At any point of time therefore, we would observe different spot rates of interest associated with different terms to maturity; longer maturity offering a 'term spread' relative to shorter maturity. The term structure of interest rates, or ZCYC, is the set of such spot interest rates. This is the principal factor underlying the valuation of most fixed income instruments.

Fixed income instruments can be categorized by type of payments. Most fixed income instruments pay to the holder a periodic interest payment, commonly known as the coupon, and an amount due at maturity, the redemption value. There exist some instruments that do not make periodic interest payments; the principal amount together with the entire outstanding amount of interest on the instrument is paid as a lump sum amount at maturity. These instruments are also known as 'zero coupon' instruments (Treasury Bills provide an example of such an instrument). These are sold at a discount to the redemption value, the discounted value being determined by the interest rate payable (yield) on the instrument.

Fixed income instruments can also be categorized by type of issuer. The rate of interest offered by the issuer depends on its credit-worthiness. Sovereign securities issued by the Government of any country, with minimal default risk, usually offer lower rates of interest than a non-sovereign entity with some default risk. The 'credit spread' that has to be added by a non-sovereign entity with non-zero probability of default risk, over and above the interest rates offered by a sovereign body, is directly related to the default risk of the issuer - higher the default risk, higher is the spread.

In empirical models of the ZCYC, the discounted stream of cashflows gives the underlying valuation of the bond. If the term structure is the only factor that influences the pricing of the bond, the present value relation, as we have mentioned earlier, should give us 'the' price of the bond. With the PV relation, and with information available on 'trade date', 'traded price', 'coupon rate'and 'date of maturity' of a bond, this essentially leaves as unknown only the set of interest rates. Trades on a given day provide us with such information for the sample of traded bonds. Estimation of the ZCYC now involves estimation of the appropriate set of interest rates that go into deriving the present value relation. This is done by specifying a functional form of the interest rate-maturity relation/discount function/forward rate function. The present exercise estimates the ZCYC using the 'Nelson-Siegel' (NS) functional form [Nelson & Siegel (1987)] using data on secondary market trades in Government securities reported on the Wholesale Debt Market segment of the National Stock Exchange (NSE-WDM).

The ZCYC depicts the relationship between interest rate and maturity for a set of 'similar' securities, as on a given date. To derive the 'true' term structure, we need to have a sample of bonds that are identical in every respect except in term to maturity. Government securities do, in practice, differ in coupon rates; nonetheless, these come closest to satisfying the requirement, and hence most empirical studies have concentrated on this segment of the securities market.

We have mentioned earlier that the underlying valuation of the bond is given by the discounted stream of cash flows. This relation should give us 'the' price of the bond if the term structure is the only factor that influences the pricing of the bond. In practice, however, observed prices differ from this 'average' price. Factors other than the term structure that affect the price of a bond include, for instance, tax regulations (differential tax rates for income and capital gains) that affect the relative valuations of bonds with different cash flows. Further, illiquid bonds trade at a premium relative to liquid bonds of the same residual maturity. Other bond characteristics also influence valuation. For trades in the same bond conducted on the same day, dispersion in prices could be attributed to transaction costs that vary with the size of the trade, an intra-day effect on account of new developments during the day, or other factors (expectations about the directionality of the term structure being an example) that have not been explicitly accounted for in the estimation.

The uses that an estimate of the term structure can be put to are immense. Once an estimate of the term structure based on default-free government securities is obtained, it can be used to price all non-sovereign fixed income instruments after adding an appropriate credit spread. It can be used to value government securities that do not trade on a given day, or to provide default-free valuations for corporate bonds. Estimates of the ZCYC at regular intervals over a period of time provides us with a time-series of the interest rate structure in the economy, which can be used to analyze the extent of impact of monetary policy. This also forms an input for VaR systems for fixed income systems and portfolios.

**Also see**:

Estimating the Zero Coupon Yield Curve - A technical paper (.pdf)

Role of Idiosyncratic Factors in Pricing (.pdf)