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.

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Sommaire de l'étude de cas

The prices of 10 OAT on the last coupon date: Maturities, coupon frequencies and rates.

Usage of polynomial interpolation and 4 spot rates.

Limitations of the method.

Calculation of the value as on April 25 2007.

The yield-to-maturity of OAT 10.

The role of duration and convexity in fixed-income asset management.

The strategies that an active portfolio manager who anticipates a fall in interest rates could follow.

Market timing strategy: Trading on interest rates predictions.

Timing bets on specific changes in the yield curve.

Appendices.

Bibliography.

Extraits de l'étude de cas

[...] 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. [...]

[...] This is partially due to the fact that its coupons have a lower value ( 3.75 and are paid annually. In addition, the relative convexity of the OAT ( 65.80 ) is also higher than that of the hypothetical semi-annual coupon bond ( 61.67 which implies that it has a larger exposure to high changes in interest rates. Critically appraise the role of duration and convexity in fixed-income asset management Duration and convexity have an important role in fixed-asset management since portfolio managers seek to control or hedge the change in the value of a bond position to changes in risk factors. [...]

[...] However, this approach is based on very restrictive assumptions, such as small changes in yield and a flat yield curve affected with only a parallel shift. Consequently, duration hedging only works for small yield changes because the price of a bond as a function of yield is not linear: when yield changes, the $duration changes. That is the reason why, in the case of large shifts in the yield structure, portfolio managers should account for the convexity of the bond, which is the second derivative of the bond price function with respect to the yield to maturity. [...]

[...] We merge two polynomial functions, one for the short end (from 1 to 5 years) and one for the long end (from 5 to 10 years) of the curve. We minimize the gap between the theoretical prices and the actual prices under the two following constraints: the theoretical prices and their derivatives at the end of the first period and at the beginning of the second period are equal. For = at3+bt2+ct+d where a = b = c = - 0.0018807 and For = a't3+b't2+c't+d' where a' = b' = - c' = 0.0148126 and d' = 2bis. [...]