Negative convexity

The price volatility characteristic of a callable bond is important to understand. The characteristic of a callable bond—that its price appreciation is less than its price decline when rates change by a large number of basis points—is referred to as negative convexity. But notice from Exhibit 4.16 that callable bonds do not exhibit this characteristic at every yield level. When yields are high (relative to the issue s coupon... 

EXHBIT 4.17 Negative Convexity Region of the Price/Yield Relationship for a Callable Bond... 

As can be seen from the exhibits, when a bond exhibits negative convexity, the bond compresses in price as rates decline. That is, at a certain yield level there is very little price appreciation when rates decline. When a bond enters this region, the bond is said to exhibit price compression. ... 

The convexity adjustment is —1.2% and therefore the bond exhibits the negative convexity property illustrated in Exhibit 4.18. The approximate percentage price change after adjusting for convexity is... 

Notice that the loss is greater than the gain—a property called negative convexity that we discussed earlier and illustrated in Exhibit 4.18. 

The convexity measure increases with the square of maturity it decreases as both coupon and yield rise. It is a function of modified duration, so index-linked bonds, which have greater duration than conventional bonds of similar maturities, also have greater convexity. For a conventional vanilla bond, convexity is almost always positive. Negative convexity occurs most frequently with callable bonds. 

As explained in chapter 1, the curve representing a plain vanilla bond s price-yield relationship is essentially convex. The price-yield curve for a bond with an embedded option changes shape as the bond s price approaches par, at which point the bond is said to exhibit negative convexity. This means that its price will rise by a smaller amount for a decline in yield than it will fall for a rise in yield of the same magnitude. FIGURE 11.13 summarizes the price-yield relationships for both negatively and positively convex bonds. 

CHANGE IN YIELD POSITIVE CONVEXm NEGATIVE CONVEXITY... 

One simple but useful way to demonstrate the convexity upward (downward) of a function is to show that it is the sum of convex upward (downward) functions. The proof of this property follows immediately from the definition of convexity. For functions of one variable convexity upward (downward) can also be demonstrated by showing that the second derivative is negative (positive) or zero over the interval of interest. Much of the usefulness of convex functions, for our purposes, stems from the following theorem ... 

If the curved surface is convex, as in the case of liquid drops, or the surface of mercury depressed in a capillary tube, p >p, but if it is concave, as in the case of a liquid ascending and wetting a capillary tube, r is negative and p [Pg.203]

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

Le Chatelier (1888) has discussed the general form of the solubility curve in the light of equation (5). If dA/dT is negative (which is usually the case) the curve begins asymptotically to the T axis, and is convex to it. It then passes through a point of inflexion, and is concave up to the maximum where A = 0, df/dT = 0. If A then becomes negative, the solubility... 

A partial pressure curve which is concave to the concentration axis, i.e. a positive curve, indicates the dissociation of a polymerised component, whilst a curve which is convex to the same axis, i.e., a negative curve, indicates the formation of a chemical compound of the two components. In the first case the concentration of the constituent passing into the vapour would be increased, in the second case reduced, by the assumed change. As examples, Dolezalek quotes ... 

The dimer chains of Ca -ATPase can also be observed by freeze-fracture electron microscopy [119,165,166,172-174], forming regular arrays of oblique parallel ridges on the concave P fracture faces of the membrane, with complementary grooves or furrows on the convex E fracture faces. Resolution of the surface projections of individual Ca -ATPase molecules within the crystalline arrays has also been achieved on freeze-dried rotary shadowed preparations of vanadate treated rabbit sarcoplasmic reticulum [163,166,173,175]. The unit cell dimensions derived from these preparations are a = 6.5 nm b = 10.7 nm and 7 = 85.5° [175], in reasonable agreement with earlier estimates on negatively stained preparations [88]. 

While the functionin (B.l) is convex for all x, they( ) in (C.l) is concave from X = 0 to the inflection point x, = l/ /2 a and convex from Xi to oo. This means that the discretization error is negative for intervals between 0 and Xi and positive between Xj and oo, such that a partial cancellation of the error is possible. 

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]. 

Formation of products and intermediate species, as well as disappearance of reactants during the photocatalytic reactions can be discerned by the evolution of positive (i.e., concave shape) bands and negative (i.e., convex shape) bands, respectively. 

That the electrically induced transverse polarization of the coleoptile is not exerted on the auxin molecule itself, is strongly supported by the fact 3>, that the light-induced lateral EMF is just reversed to the externally applied one, generating the correct curvature the convex side, i.e. the faster growing side, will become the positive side, regardless of the type of curvature (positive or negative, Fig. 13). 

Fig. 13. Electrical and curvature responses of avena coleoptiles to unilateral irradiation of two minutes. (A) intensity chosen to produce positive curvature (B) intensity chosen to produce negative curvature. Clearly the convex side of the coleoptile is electrically positive, regardless of the type of curvature. This indicates a strong correlation of bending and electrical potential gradient3)...

Bowl bending whereby the bowl is physically bent by the application of force to the outside of each bowl journal bearing. Positive bending causes an increase in bowl convexity while negative bending causes an increase in concavity. 

Next, let us examine the matter of a convex function. The concept of a convex function is illustrated in Figure 4.10 for a function of one variable. Also shown is a concave function, the negative of a convex function. (If /(x) is convex, -/(x) is concave.) A function /(x) defined on a convex set F is said to be a convex function if the following relation holds... 

It can be shown from a Taylor series expansion that if/(x) has continuous second partial derivatives, /(x) is concave if and only if its Hessian matrix is negative-semidefinite. For/(x) to be strictly concave, H must be negative-definite. For /(x) to be convex H(x) must be positive-semidefinite and for/(x) to be strictly convex, H(x) must be positive-definite. 

Because all eigenvalues are zero or negative, according to Table 4.1 both gx and g2 are concave and the region is convex. 

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