Estimating the NSE
Zero Curve :
Frequently Asked Questions by
Susan Thomas
Indira Gandhi Institute of Development Research,
Goregaon, Mumbai - 400 065
Gangadhar Darbha
National Stock Exchange; Mahindra Towers,
Worli, Mumbai - 400 018
Sudipta Dutta Roy
National Stock Exchange; Mahindra Towers,
Worli, Mumbai - 400 018
Vadhana Pawaskar
National Stock Exchange; Mahindra Towers,
Worli, Mumbai - 400 018
What
is the ‘term structure of interest rates’?
The ‘term structure of ‘NSE Zero
Curve’ (NSE Zero Curve for short) is a relationship between
maturity and interest rates. The NSE Zero Curve 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 NSE Zero Curve,
is the set
of such spot interest rates. This is the principal factor underlying
the valuation of most fixed income instruments.
What are the different types of fixed income
instruments available to an investor?
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 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 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.
What is the motivation behind the use of NSE
Zero Curve for valuation of fixed income instruments?
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 NSE Zero
Curve.
Note: In the formulation above, C denotes coupon and R the
redemption amount, m is the time to maturity.
How is this different from valuation using
Yield to Maturity (YTM)?
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 NSE Zero Curve – 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.
Is the ‘zero coupon’ yield curve
only useful for talking about zero coupon bonds?
No. Besides zero coupon instruments, the NSE Zero Curve can be used
to price a wide range of securities including coupon paying bonds,
derivatives, interest
rate forwards and swaps. In arriving at the NSE Zero Curve 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.
How does one estimate the ‘term structure’ /
NSE Zero Curve?
In empirical models of the NSE Zero Curve, 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 (1) 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 NSE Zero Curve 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 NSE Zero Curve.
What is the functional form of the relationship
used for the NSE NSE Zero Curve?
The NSE NSE Zero Curve is estimated using the ‘Nelson-Siegel’ functional
form [Nelson & Siegel (1987)]. We specify the spot rate function as
follows:
The NSE Zero Curve 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)
Why
are only Government bonds used for the estimation of the NSE Zero
Curve?
The NSE Zero Curve 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.
Is
the ‘term structure’ the only factor influencing the
price of a bond?
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 intra-day effect on account of new developments during
the day and expectations about the directionality of the term structure
being an example.
How
can these be accounted for in the empirical estimation?
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.
What are the other uses that the NSE Zero
Curve can be put to, besides valuation of fixed income, default-free
instruments?
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 NSE Zero Curve
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.