As the ten OAT have the same anniversary dates, in other words the different cash-flows are paid at the same dates (25-Apr-08, 25-Apr-09, 25-Apr-10,), then we used a direct method of calculation of the spot rates, through two steps. Using a polynomial interpolation, we can recover the yield curve. We use a cubic interpolation of the term structure of zero-coupon rates. The interpolated discount rate R(0,t) is defined by R(0,t) = at3+bt²+ct+d with te[t1,t4] and we impose that R(0,t) is on the curve. Given a, b, c and d for each segment, we can compute all the intermediate rates (Cf appendix 1) and draw the term structure of discount rates. For the linear interpolation, we use the following formula : R(0,t) = [(t2-t) R(0,t1) + (t1-t) R(0,t2)] / (t2-t1). The strength of this method is to give a better approximation than the linear interpolation. The limitations of this method are the approximation of intermediate rates and the imposition of real prices as fair prices: the polynomial interpolation is indeed a direct method.
[...] a23+b2²+c2+d a - R = = a43+b4²+c4+d b - 0.000075 R = = a73+b7²+c7+d ( c = = 0.000450 R = = a93+b9²+c9+d d Given c and d for each segment, we can compute all the intermediate rates (Cf appendix and draw the term structure of discount rates. For the linear interpolation, we use the following formula : = R(0,t1) + / (t2-t1). The strength of this method is to give a better approximation than the linear interpolation. The limitations of this method are the approximation of intermediate rates and the imposition of real prices as fair prices: the polynomial interpolation is indeed a direct method. [...]
[...] A solution for accounting for non-parallel shifts is to regroup different risk factors to reduce the dimensionality of the problem: for example, a short, a medium and a long maturity factors. One classic model is Nelson and Siegel's, which writes discount rates as a function of three parameters: a level parameter, a slope parameter and a curvature parameter Explain the strategies that an active portfolio manager who anticipates a fall in interest rates could follow An anticipation of a fall in interest rates implies an increase in the price of the bond or futures contract due to the inverse relationship between the price and the yield. [...]
[...] To do so, he has to be short on bonds with low duration and long on bonds with high duration. His portfolio will be more sensitive to the variation of the interest rates and he will consequently optimize his capital or relative capital gain from the increase of the value of his portfolio. The gain can be measured twofold: The relative gain: dP/P = - Modified Duration*dy The absolute gain: dP = $ Duration*dy The investment choice of the manager depends on his will to optimize his absolute gain or his relative gain. [...]
[...] Calculate on April the price, the yield-to-maturity and the effective annual yield of a hypothetical OAT with a semi-annual coupon rate and semi-annual compounding We assume that the maturity of the OAT is on April 25, 2017; therefore, as we have a semi-annual coupon rate and semi-annual compounding, we will have 10x2 or 20 periods of six months until maturity. And for each of these periods of time, the coupon payment will amount to / 2). The calculation of the 20 annualized discount rates will be based on the discount rates obtained at Question and corresponding to the maturities between April and April (in other words, we know R0,t, with t ( Regarding the intermediary periods 9.5 the linear method will give us a direct calculation of the different spot rates R R Thus, for a rate with maturity such that t1 < t < t we calculate R0,t : R0,t = [ x R0,t1 + x R0,t2 ] / (t2-t1) ( As an example, with 1 < t = 1.5 < we obtain: R = [ 1.5 ) x R0,1 + ( 1.5 x R0,2 ] / The 20 annualized discount rates obtained will enable the calculation of the price of the OAT; and since the OAT has a semi-annual coupon rate and semi-annual compounding, we will divide each discount rate per two in order to refer to a semi-annual compounding. [...]
[...] The purpose of this strategy is to adjust the weights of the wings and the body so as to obtain a cash and $duration neutrality. $duration neutrality, in other words no sensitivity to changes in the yield curve, is generated through the use of the ladder property. This property ensures a quasi- perfect interest rate hedge against small parallel shifts of the yield curve. In this case, the strategy is structured so as to have a positive convexity and therefore generate positive gains [...]
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