A Rupee on hand today is worth more than a Rupee to be received in the future because the money on hand today can be invested to earn interest to yield more than a Rupee in the future. This concept is known as Time Value of Money. Hence all future cash flows that an investor expects to receive need to be converted to their present values for comparison of investment opportunities. Present value describes the process of determining what a cash flow to be received in the future is worth in today's rupees. Therefore, the present value of a future cash flow represents the amount of money today which, if invested at a particular interest rate, will grow to the amount of the future cash flow at that time in the future. The process of finding present values is called Discounting and the interest rate used to calculate present values is called the discount rate. For example, the Present Value of Rs.100 to be received one year from now is Rs90.91 if the discount rate is 10% compounded annually (Rs.90.91 = Rs.100/(1 + 0.10)). Compounding can also be done using market conventions of semi-annual, continuous, etc.

The following equation can be used to calculate the Present Value of future cash flows given the discount rate and number of years in the future that the cash flow occurs.

Where PV = Present Value, C = Coupon Cash flows of a bond, R = Final Redemption Value of the bond and i is the rate of return for investment and m is the time period of various cash flows.

The above method is also commonly known as YTM (Yield to Maturity) method of valuation. The convention method used above assumes a single rate of discount i for all cash flows to be received in various time periods. But we know that cash flows are received in different time periods and hence it may not be appropriate to discount all future cash flows with a single discount rate. For realistic valuation of future cash flows, it would be appropriate to find different discount rates applicable to respective periods. That means we require a rate i1 for the first cash flow while i2 should be used for the second cash flow and so on and so forth.

The various rates (i1, i2, etc.) applicable for various time periods of cash flows are commonly know as spot rates. These spot rates can be obtained through various methods. If the zero coupon papers various maturities are frequently traded in the market, then these rates can be used to interpolate to construct a spot rate yield curve to derive spot rates for time periods in which exact reference is not available in the market. But in Indian market very few zero paper instruments are traded and hence it may not be effectively used to construct a spot rate yield curve.

The objective behind the Zero Coupon Yield Curve is to use the observed bond prices in the market and convert them to spot rates application to various terms and construct a yield curve of spot rates. This may be feasible as a coupon paying bond is a combination of many zero coupon bonds maturing at various time periods.

The valuation of a bond by an investor can be expressed in terms present value formulations as given in the following equation. Suppose that the spot rates of interest (rt) for every future period are known, then the present value of an m-period bond making a series of coupon payments C every period and with redemption value R is:

The spot interest rate rt (t=1,2,3,….m) is the interest rate applicable on a cash payment due in t periods. The set of spot rates is commonly known as term structure of interest rates. This valuation is more realistic than the YTM method of valuation.

The present value of a bond that an investor pays is the possible final payment for acquiring the bond. Hence the convention requires that the present value of a bond should be equal to the clean price of a bond (normally Gilts are traded on clean price basis) plus the accrued interest from the last coupon payment date. That is the total or dirty price paid by the buyer can thus be decomposed into two components: accrued interest and the clean or quoted price. If market conditions are stable, such that factors underlying the valuation of the bond do not change, then the dirty price of the bond will still increase daily by the amount of accrued interest.

where:

p = clean price of the bond
ai = accrued interest
C = annual coupon payment
R = redemption payment
tm = maturity of bond (in years)
r(tj) = spot rate applicable to payment j due at time tj
v = frequency of coupon payments

There are many methods through which estimation of spot rates can be done. Nelson-Siegel’s functional form is one such method through which the spot rates can be estimated (Parsimonious Modeling of Yield Curves Journal of Business, Volume 60, October 1987). Hence, empirical estimation of equation (3) will essentially require specifying a parametric relation between maturity and spot interest rates. The Nelson-Siegel formulation that we adopt in the present exercise of estimating Zero Coupon Yield Curve provides a framework for the derivation of such a relation. Functional form provided by Nelson-Siegel is given as:

where ‘m’ denotes related maturity for the cash flows in a bond and b=[ß0, ß1, ß2 and tau] are parameters to be estimated. Here ß0 is the level parameter and commonly interpreted as long term (long term in mathematical sense – approaching infinity) rate, ß1 is slope parameter, ß2 is curvature parameter and tau is scale parameter while (ß0 + ß1) gives the short term rate. Alternatively it can also be said that ß0 is the contribution of long term component, ß1 is the contribution of short term component, ß2 indicates the contribution of medium term component, tau is the decay factor and ß2 & tau determine the shape of the curve .

The appeal of the NS functional form lies in its flexibility to cover the entire range of possible shapes that the ZCYC can take, depending on the value of the estimated parameters. In the formulation of the forward rate function f(m,b), the first term represents the long-term component and is a non-zero constant. The second term, which monotonically declines to zero, is the short-term component, and the third term represents the medium-term component. In the spot rate function, the limiting value of r(m,b) as maturity gets large is ß0 which therefore depicts the long term component (which is a non-zero constant). The limiting value as maturity tends to zero is ß0 + ß1, which therefore gives the implied short-term rate of interest.

With the above specification of the spot rate function, the PV relation can now be specified using the discount function given by

for continuous compounding or we may use the following {1+r(m ,b%)/2)^(2*m)} for semi-annual compounding.

Once the functional form is specified and the parameter values are generated [ß0, ß1, ß0 and tau], these values are used to calculate the spot rates for any term greater than 0 using the equation 4 above. And these spot rates are used to calculate the present value (commonly known as the estimated price or model price) of the cash flows and combine them to get the value of the bond. The present value arrived at is the estimated price (p_est) for each bond. These estimated values now can be compared with the observed market prices. It is common to observe market prices (pmkt) that deviate from this value. But the objective of a good estimation is to reduce the difference between the observed market prices and the estimated prices. For the purpose of the estimation exercise, we postulate that the observed market price of a bond deviates from its underlying valuation by an error term ei, which gives us the estimable relation

for continuous compounding or we may use the following {1+r(m ,b%)/2)^(2*m)} for semi-annual compounding.

This equation can be estimated by minimising the sum of squared price errors under maximum likelihood estimation procedure. Constraints imposed relate to the non-negativity of the long-run rate (ß0 – long term in mathematical sense i.e. maturity approaching infinity), the short-rate (ß0 + ß1), and the parameter tau. The steps followed in the estimation procedure are as follows:

• A vector of starting parameters (ß0, ß1, ß2 and tau) is selected, it can be anything. We use previous day’s parameter values as the starting parameters for the day.

• The discount factor function is determined using these starting parameters,

• Numerical optimisation procedures are used to estimate a set of parameters (under a given set of constraints viz. non-negativity of long run and short run interest rates) that minimise the specified loss function,

• The estimated set of parameters are used to determine the spot rate function and therefrom the ‘model’ prices,

• The model prices are compared with the volume weighted average price to compute the error for each traded security.