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9) yields the following equation for the discount function: k dfm = α0 + ∑αjm j j=1 by taking α0 as one. The discount function dfm is a kth-degree polynomial. Some drawbacks to using a polynomial are: ᔢ ᔢ ᔢ ᔢ This formulation does not have the ability to give more weight to values of m which are more likely to occur. It would take an extremely high-order polynomial to fit both the long and short end of the curve, and this would result in the polynomial taking on extreme values between observations at the long end.

24) to derive the discount factors. The second approach involves interpolating the nacq swap rates to get an interpolated swap rate value at each quarter. 24) as before, except that the bootstrap formula 23 CONCEPTS AND TERMINOLOGY now has to take into account the fact that the rates are compounded quarterly. To derive the formulae for the second approach, let tk denote the actual term (in years) from the value date to the maturity date corresponding with each of the quarterly swap rates rk. 25) ) k–1 where k denotes each of the quarters and dfk is the discount factor for the contract with maturity tk.

18) where r3 denotes the average three-year rate for which we need to solve iteratively. 18) as follows: 18 YIELD CURVE MODELING 0 = –CF0 + (r3 × df1) + ( r3 × df2) + (r3 × df3) where CF0 = PV. This can be interpreted as the investor that invests an initial amount of CF0 and receives three cash flows calculated from the average rate r3. The same concept is used when we value the fixed leg of an interest rate swap. 19) where df1 denotes the one-year discount factor. This means by investing a nominal amount of 1, the investor is paid back the nominal plus interest of r1 after one year.

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