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CHAPTER 2. Building Zero Curves

One important task that financial practitioners face daily is the need to present value (or discount) a cash flow that is going to be paid (or received) some time in the future. While it is straightforward to do this if a discount rate for the respective cash flow is known, in practice this is far from truth as:

■ Instruments have varying definitions and settlement criteria associated with the rates that are used for trading the respective instruments.[1]

■ Instruments that trade in the marketplace are usually not zero coupon bonds – hence making it difficult to extract the exact discount rates.

■ The maturity (or coupon) date of instruments typically does not mirror the dates of the cash flows that need to be discounted.

Given the above, it is important for any practitioner to have a consistent and objective process that can be used to discount any future cash flow. To help with this, I will focus my discussion in this chapter on the use of liquid market instruments to construct a curve of zero rates[2] so as to be able to discount cash flows from varying maturities. Starting with an overview on how various interest-bearing instruments are valued and traded, the chapter goes on to discuss the construction of the zero curve using a linear function assumption. The chapter concludes with the use of a cubic polynomial to construct the zero-rate curve following which a discussion on the difference in the types of zero curves produced using these methods is given.

  • [1] For example, swaps are traded using swap rates, bonds are traded using yields- to-maturity (or bond prices), and Treasury bills are traded using quoted prices (which are actually yields). This naturally leads one to the following two important questions: 1. Which rate is the right one to use? 2. How does one find the relative value of one instrument vis-a-vis another?
  • [2] A zero rate is an interest rate applied to a specific time to present (future) value cash flow from (to) that time, where all zero rates are expressed as continuously compounded rates. Thus, if the one-, two-, and three-year zero rates are 2, 3, and 5 percent respectively, then receiving a dollar one, two, and three years from now is equivalent to receiving e-1*0•02 = 0.9802, e-2*0.03 = 0.9418, and e-3*0.05 = 0.8607 respectively today. These numbers are also sometimes called zero-coupon discount factors or discount factors for short. Each zero rate is uniquely associated with a maturity date (or term), and the collection of all these zero rates across varying maturities is called a zero-rate curve or zero-rate term structure or zero curve. Since a zero rate is used to present a future dollar – the quality of the counterparty paying this dollar would affect the magnitude of this present value. Hence the present value of a dollar received from a AAA entity for a given maturity would never be greater than the present value of a dollar received from a BBB entity for the same maturity. This difference in present values is simply due to the value of the AAA/BBB credit spread as perceived by the market participants.
 
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