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The 'term structure of 'zero coupon yield curve' (ZCYC for short) is a relationship between maturity and interest rates. The ZCYC starts from the basic premise of 'time value of money' - that a given amount of money 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 offered a positive rate of return 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 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.
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Fixed income instruments can be categorised 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 exists 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 lumpsum 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 categorised 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.
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Modeled as a series of cashflows due at different points of time in the future, the underlying price of a fixed income instrument can be calculated as the net present value of the stream of cashflows. Each cashflow, in such a formulation (presented below), has to be discounted using the interest rate for the associated term to maturity [the appropriate discount factor being given by 1/(1+r(i))m(i), where m(i) is the time to maturity for the ith cashflow and r(i) the associated spot interest rate]. The appropriate spot rates to be used for this purpose are provided by the ZCYC.
Note: In the formulation above, C denotes coupon and R the redemption amount, m is the time to maturity.
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The yield to maturity (YTM) is the bond's internal rate of return - it is the single rate of interest that equates the discounted stream of cashflows to the price of a security.
In this formulation, all cashflows due in the future are all discounted using the same rate irrespective of when they fall due, unlike in term structure where each cashflow is discounted using the rate of interest associated with the time to that cashflow. At any point of time therefore, for a given set of bond prices, the YTM is instrument-specific. Being instrument-specific, YTM does not provide a unique mapping from maturity to interest rate space. The YTM cannot therefore be used to price any set of bonds apart from the specific bond to which it refers.
Note that use of YTM to discount the entire stream of cashflows due from a security carries with it the implicit assumption that, at any time before the terminal year, money can be re-invested at the same rate of interest, irrespective of term to maturity. Yet, we have argued that rates of interest would rise with maturity, so that the discounted value of a given cashflow would be less the further in the future it falls due. Herein lies the appeal of the ZCYC - it provides the entire set of interest rate-maturity pairs that can be used to discount a future payment. Further, these interest rate-maturity pairs, once derived, provide a unique mapping from maturity space to interest rates and hence can be used to discount any stream of cashflows, irrespective of their source.
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No. Besides zero coupon instruments, the ZCYC can be used to price a wide range of securities including coupon paying bonds, derivatives, interest rate forwards and swaps. In arriving at the ZCYC for a coupon bearing instrument (as shown above), what we have simply done is stripped the 'n' cashflows into 'n' zero coupon instruments, the first 'n-1' being coupon payments and the 'n'th being the terminal coupon plus redemption amount.
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In empirical models of the ZCYC, the underlying valuation of the bond is given by the discounted stream of cashflows. 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 defined as in (3) above, 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. Specifying an estimable relationship of the form , where ei denotes the residual, estimation of the ZCYC now involves estimation of the appropriate set of interest rates. Specification of a relation between maturity and interest rates therefore forms the first step in the estimation of the ZCYC.
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The NSE ZCYC is estimated using the 'Nelson-Siegel' functional form [Nelson & Siegel (1987)]. We specify the spot rate function as follows:
The ZCYC is estimated using data on secondary market trades in Government securities reported on the Wholesale Debt Market segment of the National Stock Exchange (NSE-WDM)
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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, different by coupon rates; nonetheless, these come closest to satisfying the requirement, hence most empirical studies have concentrated on this segment of the securities market.
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No. There are other factors, besides the term structure, that influence the pricing of a bond. These 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 example of which could be an intra-day effect on account of new developments during the day and expectations about the directionality of the term structure.
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To incorporate into the empirical estimation the impact of these other factors that influence the valuation of a bond, we need to have measures/proxies for these variables. There is substantial literature on incorporating the tax effect in econometric estimation. This essentially relates to discounting net (of taxes) cashflows and the estimation of the appropriate after-tax rates of discount. Proxies for liquidity include 'volume transacted on the given date', 'number of trades on the given date and 'amount outstanding'. Other bond characteristics such as 'age of the bond' and 'term to maturity' can be incorporated in the empirical specification. 'On-the-run issues' and 'benchmark securities' may trade at rates different from other securities with the same term to maturity. Since the PV relation is by itself highly non-linear in form, empirical studies usually include these variables as additional factors in the estimated equation and test for the extent of reduction in 'errors' (deviation between observed and model prices). The next phase of our exercise is aimed at capturing the impact of these variables. This should significantly improve the performance of the estimated model.
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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 analyse the extent of impact of monetary policy. This also forms an input for VaR systems for fixed income systems and portfolios.
Methodology
Research Papers: Estimating the NSE ZCYC | Role of Idiosyncratic Factors in Pricing
ZCYC for the Day and Time Series
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Last updated on March 12, 2008.
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