The Mathematics of Financial Models: Solving Real-World Problems with Quantitative Methods

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 rates² 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 ...