Fixed income analysis and risk management

Résumé du document

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.

Extraits

[...] Conclusion : the strategies based on the level of interest rates present some weaknesses since they assume that there is only one translation move, i.e. either an upward or a downward parallel shift, and the focus is made on the yield to maturity Timing bets on specific changes in the yield curve The yield curve is potentially affected by many other movements than parallel shifts. These include, in particular, pure slope and curvature movements, as well as combinations of level, slope and curvature movements Bullet strategy This strategy consists in the construction of a portfolio concentrated in investments in particular maturity of the yield curve. [...]

[...] In fact, the longer the maturity and the lower the coupon rate, the higher the modified duration of a bond. 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. [...]

[...] Vasicek model: 3. Cubic splines: Question 2 bis Question 3 Bibliography 1. CFA Book, Level 1 (2008), Volume www.investopedia.com 3. [...]

[...] Fixed income analysis and risk management 1. Regarding the prices of 10 OAT on the last coupon date (April 25 2007), their maturities, coupon frequencies and coupon rates, we can determine the appropriate discount factors and discount rates 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: - First, we determined the discount factors by solving the following system: OAT 101.0359 = 105.25 x as there is only one coupon + principal payment due on 25-Apr-08 OAT 99.7459 = 4 x + 104 x as there are two coupon payments + principal payment until maturity OAT 103.7541 = 5.50 x + 5.50 x + 105.50 x . [...]

[...] Moreover, we have 20 periods, which give us the price as follows: POAT = ( / R0,t with t = { Here are the results: We obtain a price of for our hypothetical OAT. To determine the yield-to-maturity, we will use the Excel function Rate with the following details: Number of periods = 20, coupon rate = Present value = - Future value = 100%. The Yield-to-Maturity obtained is such that: = ( / YTMa ) with t = { and YTMa = YTM/2, since we are in the case of semi-annual coupon payments. [...]