Effective convexity

Just as standard duration is not appropriate for bonds with embedded options, neither is traditional convexity. This is because traditional convexity, like traditional duration, fails to take into account the impact on a bond s future cash flows of a change in market interest rates. As discussed in chapter l4, the approximate convexity of any bond may be derived, following in Fabozzi (1997), using equation (11.14). 

If the prices used for P f, P, and Po are calculated assuming that the bond s remaining cash flows will not change when market rates do, the convexity computed is for an option-free bond. For bonds with embedded options, the prices used in the equation should be derived using a binomial model, in which the cash flows do change with interest rates. The result is effective or option-adjusted convexity. 

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The convexity is a more correct measure of the price sensitivity. It measures the curvature of the price-yield relationship and the degree in which it diverges from the straight-line estimation. Like the duration, the standard measure of convexity does not consider the changes of market interest rates on bond s prices. Therefore, the conventional measure of price sensitivity used for bonds with embedded options is the effective convexity. It is given by (11.2) ... 

The prices used in equation (4.4) to calculate convexity can be obtained by either assuming that when the yield changes the expected cash flows either do not change or they do change. In the former case, the resulting convexity is referred to as modified convexity. (Actually, in the industry, convexity is not qualified by the adjective modified. ) In contrast, effective convexity assumes that the cash flows do change when yields change. This is the same distinction made for duration. 

As with duration, there is little difference between modified convexity and effective convexity for option-free bonds. However, for bonds with embedded options there can be quite a difference between the calculated modified convexity and effective convexity measures. In fact, for all option-free bonds, either convexity measure will have a positive value. For bonds with embedded options, the calculated effective convexity measure can be negative when the calculated modified convexity measure is positive. 

The formula for calculating approximate convexity is (14.20). If P and T+ are obtained using a valuation model that incorporates the effect of a change in interest rates on the expected cash flows, the equation derives effective convexity. The effective convexity of a mortgage pass-through security is invariably negative. 

In transforming bis-ketone 45 to keto-epoxide 46, the elevated stereoselectivity was believed to be a consequence of tbe molecular shape — tbe sulfur ylide attacked preferentially from tbe convex face of the strongly puckered molecule of 45. Moreover, the pronounced chemoselectivity was attributed to tbe increased electropbilicity of the furanone versus the pyranone carbonyl, as a result of an inductive effect generated by tbe pair of spiroacetal oxygen substituents at tbe furanone a-position. ... 

The reduced oxidation near sample corners is related to these stress effects, either by retarded diffusion or modified interfacial reactionsManning described these stresses in terms of the conformational strain and distinguished between anion and cation diffusion, and concave and convex surfaces. He defined a radial vector M, describing the direction and extent of displacement of the oxide layer in order to remain in contact with the retreating metal surface, where ... 

We have seen that the output neuron in a binary-threshold perceptron without hidden layers can only specify on which side of a particular hyperplane the input lies. Its decision region consists simply of a half-plane bounded by a hyperplane. If one hidden layer is added, however, the neurons in the hidden layer effectively take an intersection (i.e. a Boolean AND operation) of the half-planes formed by the input neurons and can thus form arbitrary (possible unbounded) convex regions. ... 

At places where the front is concave toward the unburnt gas, the heat flux is locally convergent. The local flame temperature increases and the local propagation velocity also increases, see the red arrows in Figure 5.1.5. The converse holds for portions of the front that are convex. The effect of thermal diffusion is to stabilize a wrinkled flame. 

The lower symmetry of nanorods (in comparison to nanoshells) allows additional flexibility in terms of the tunability of their optical extinction properties. Not only can the properties be tuned by control of aspect ratio (Figure 7.4a) but there is also an effect of particle volume (Figure 7.4b), end cap profile (Figure 7.4c), convexity of waist (Figure 7.4d), convexity of ends (Figure 7.4e) and loss of rotational symmetry (Figure 7.4f). 

Unlike solid electrodes, the shape of the ITIES can be varied by application of an external pressure to the pipette. The shape of the meniscus formed at the pipette tip was studied in situ by video microscopy under controlled pressure [19]. When a negative pressure was applied, the ITIES shape was concave. As expected from the theory [25a], the diffusion current to a recessed ITIES was lower than in absence of negative external pressure. When a positive pressure was applied to the pipette, the solution meniscus became convex, and the diffusion current increased. The diffusion-limiting current increased with increasing height of the spherical segment (up to the complete sphere), as the theory predicts [25b]. Importantly, with no external pressure applied to the pipette, the micro-ITIES was found to be essentially flat. This observation was corroborated by numerous experiments performed with different concentrations of dissolved species and different pipette radii [19]. The measured diffusion current to such an interface agrees quantitatively with Eq. (6) if the outer pipette wall is silanized (see next section). The effective radius of a pipette can be calculated from Eq. (6) and compared to the value found microscopically [19]. 

Some examples of conjugate addition reactions of allylic silanes are given in Scheme 9.5. Entries 1 to 3 illustrate the synthesis of several (3-allyl ketones. Note that Entry 2 involves the creation of a quaternary carbon. Entry 4 was used in the synthesis of a terpenoid ketone, (+)-nootkatone. Entry 5 illustrates fluoride-mediated addition using tetrabutylammonium fluoride. These conditions were found to be especially effective for unsaturated esters. In Entry 6, the addition is from the convex face of the ring system. Entry 7 illustrates a ring closure by intramolecular conjugate addition. 

Now consider the influence of the inequality constraints on the optimization problem. The effect of inequality constraints is to reduce the size of the solution space that must be searched. However, the way in which the constraints bound the feasible region is important. Figure 3.10 illustrates the concept of convex and nonconvex regions. 

The velocity profile effect—For a convex flow velocity profile, for instance, in a steady bubbly flow, the centerline velocity is higher than the average velocity. With bubbles usually concentrating at the center, they attain a higher velocity than the liquid. 

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