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# Survey of Forensic Science FOR 101

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This 3 page Class Notes was uploaded by Mr. Ernie Ryan on Wednesday September 30, 2015. The Class Notes belongs to FOR 101 at Pace University - New York taught by Staff in Fall. Since its upload, it has received 34 views. For similar materials see /class/217111/for-101-pace-university-new-york in Forensic Science at Pace University - New York.

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Date Created: 09/30/15

Bootstrapping the Yield Curve The yield curve can be obtained empirically by examining the payoffs associated with a bond simultaneously with the bond s purchase price Let D1 be the discount function for time t that is D1 11y0tt This means that a cash ow paid at time t will be discounted by multiplying it by the discount function D1 PV CFt D1 CFt 1y0tt A little algebra produces the following spot rate yot 1Dt t 1 Thus one can obtain the spot rates yoyt from the bond s current purchase price P0 and expected future cash ows from coupon payments and face value CFL Thus consider a 1000 face value bond making a single interest payment at an annual rate of 5 Suppose this bond is currently selling for 102 meaning 102 or 1020 and that it matures in one year when its coupon payment is made The oneyear spot rate implied by this bond is determined as follows 1020 50 1000 D110501y0711 D1 10201050 9714286 1 19714286 1 Yo1 0294 Thus the oneyear spot rate is 294 However a difficulty arises when the bond has more than one cash ow As spot rates may vary over time there may be a spot rate for each period hence a spot rate for each cash ow Consider a 1000 face value twoyear bond making interest payments at an annual rate of 5 Suppose this bond is currently selling for 10175 meaning 10175 or 10175 and that it matures in two years when its second coupon payment is made The twoyear spot rate implied by this bond is bootstrapped from the oneyear bond as follows 10175 50 9714286 50 1000 D2 D2 10175 50 971428650 10009227891 19227891 2 1 y02 0410 Bootstrapping simply means to make use of the rate the oneyear rate or information that is already known to obtain the desired result the twoyear rate The threeyear spot rate y073 implied by the threeyear bond is bootstrapped from the oneyear and twoyear bonds as follows 10150 50 9714286 50 9227891 50 1000 D3 D3 1015 50 9714286 50 922789150 10008764658 18764658 3 1 y03 0449 More generally this bootstrapping process is applied as follows PVcFED1cFED2cFED3 cF Dn1cFF Dn EcF1y0t cF F ly071quot PV ECFL D1 cF F 1y0tquot ECFL ly0lt cF F ly071quot Dn P0 EcF Dt cF F Bootstrapping requires that there be one bond maturing in each year t so that its D1 can be used to determine bootstrap the D1 for the bond maturing in one year subsequent Thus one starts by determining D1 D2 and so on until all D values have been determined These expressions are used to bootstrap spot rates from bond prices maturities and coupon rates in Table 2 and in Figure l mapping out the yield curve Any iyear forward rate ymyt from year ti to year t is determined from D1 Dti1l 1 If we accept the Pure Expectations Theory for the term structure of interest rates we can obtain forward rates from spot rates For example Based on this theory the two tear spot rate is a function of the oneyear spot rate and the oneyear forward rate on a loan originating in one year as follows y2 0410 1y0711y172 1 102941y172 1 We can use this relationship to solve for the oneyear forward rate on a loan originating in one year as follows 104102 102941y172 104102 1 052731 10294 y12 Similarly we can solve for the one year forward rate on a loan originating in two years forward rate yzy3 as follows 104103 102941029410527311 y l 04493 1 0294l 052731 The twoyear forward rate on a loan originating in one year forward rate y173 is determined as follows y 1 052731 104103 102941y1732 3 y W4052737 10294 which is identical to ym 10527311052744 1052737 102 400 Nh 14 14 14 14 14 14 14 14 14 104 104 104 Dt 09714286 09227891 08764658 08323484 07927128 07549645 071 90138 06847751 064455 0612399 0581851 8 05528283 05193962 04923187 04666528 L Rate 294 410 449 469 476 480 483 485 500 503 505 506 517 519 521 Ask Maturit Cou on Price 1 2 500 101 3 500 101 4 500 101 5 500 101 6 500 101 7 500 101 8 500 101 9 525 102 10 525 102 11 525 102 12 525 102 13 550 14 550 15 550 600 500 w400 35300 0 1339 200 100 000 0 16 575 105 34 04331835 537l Table 2 Bootstrapping Spot Rates Figure 1 Mapping the Yield Curve

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