Equivalent Bond Yield

The spot rates are annual rates and are reported as bond-equivalent yields. When present values are computed, we use the appropriate semiannual rates which are taken to be one half the annual rate. 

Equation (14.15) computes the yield for a bond making monthly coupon payments (as the majority of mortgage-backed bonds do, although some pay quarterly). For purposes of comparison, this figure must be converted to a bond-equivalent yield. Equation (14.16) derives the annualized bond-equivalent yield for a mortgage-backed bond paying monthly coupons. 

As noted in chapter 3, it is possible to calculate a bond s price given its yield and vice versa. As with a plain vanilla bond, a mortgage-backed bond s price is the sum of the present values of its projected cash flows. The discount rate used to derive the present values is the bond-equivalent yield converted to a monthly basis. 

The first component can be estimated by assuming a prepayment rate during the holding period the second entails assuming a reinvestment rate. For the third, two assumptions are necessary one concerning the bond s bond-equivalent yield at the end of the holding period, and another about the prepayment rate projected by the market at this point, which is a function of the projected yield. 

The monthly return can be converted to an annualized bond-equivalent yield using formulas (1.24a) or (b), as discussed in chapter 1. 

The theoretical 1-year spot rate is twice 2, or 0.06308, for an annualized bond-equivalent yield of 6.308 percent. This figure can now be used to calculate the theoretical 1.5-year spot rate. The cash flows for the 7 percent 1.5-year coupon Treasury are September 1, 1999 3.50... 

The market convention is sometimes simply to double the semiaimual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained in this manner is known as a bond equivalent yield. It was noted earlier that the one disadvantage of the YTM measure is that its calculation incorporates the unrealistic assumption that each coupon payment, as it becomes due, is reinvested at the rate rm. Another disadvantage is that it does not deal with the situation in which investors do not hold their bonds to maturity. In these cases, the redemption yield will not be as great. Investors might therefore be interested in other measures of return, such as the equivalent zero-coupon yield, considered a true yield. 

The presence of a transition metal is not necessarily required for hydrocarbon insertion. Alkyne incorporation has been reported for boracyclobutenes, as well as metallacyclobutene complexes of the transition elements. Boracyclobutene 51, a reactive intermediate prepared in situ (Section 2.12.9.2.1), inserts an additional equivalent of trimethylsilylacetylene into the B-C(sp2) bond to yield boracyclohexadiene 52 (Scheme 7). This isomerizes to the interesting bridged compound 53, an analogue of a nonclassical carbocation <1994AGE2306>. The related boracyclobutene 7 inserts the alkyne to yield a persistent boracyclohexadiene 54, but this product clearly arises from insertion into the boracyclobutene carbon-carbon bond rather than a boron-carbon bond <1994AGE1487>. 

Note Addition of a second equivalent ofphosphane is likely to break the S-Pd donor bond and yield the corresponding mononuclear complex. 

Simple terminal alkenes or tranx-disubstituted alkenes gave good yields of the formal addition product when three equivalents of a secondary amine (dimethyl- or diethyl-, but not diisopropylamine) and bis(benzonitrilc)palladium chloride were used at low temperature, after reduction of the C-Pd bond. The yields were modest to very low with (Z)- or cyclic alkenes and using primary amines9, l0. 

This sequence of events is in contrast with the reduction of 0(,j3-unsaturated ketones with diborane, a process that is known to start with the addition of diborane to the C=C and C=0 bonds that yields VII (see Scheme 8.2). This intermediate subsequently undergoes elimination to the alkene VIII wherein the C=C linkage appears shifted. Remember VI requires two equivalents of BH3 for its conversion to VIII, since the ketone does not contain the strategic components for its self-redox as compound I does.)... 

The best documented and established function of some carotenoids is their provitamin A activity, especially of P-carotene. One mole of P-carotene can theoretically be converted, by cleavage of C 15 = C 15 double bond, to yield two moles of retinal (Reaction 9.1). However, the physiological efficiency of this process appears to be only 50%. The observed average efficiency of intestinal P-carotene absorption is only two thirds of the total content. Thus, a factor of 1/6 is used to calculate the retinol equivalent (RE) from P-carotene, but only 1/12 from the other provitamin A carotenoids in food (Combs, 1992). In fruits and vegetables P-carotene content is used as a measure of the provitamin A content. 

All the protons of cyclooctatetraene (4mz, with = 2) resonate at a single frequency of t 4.309 68>. This magnetic equivalence is brought about by two processes, a ring inversion between two tub-shaped forms, and bond shifts, yielding the two structures 3 a and 3b, which, in the absence of substitution are equivalent. Studies of the n.m.r. temperature... 

After reductive opening of the epoxide the resulting titanocene-bound radical adds to the double bond to yield a primary radical. Trapping with a second equivalent of titanocene(III) chloride results in the formation of an organotitanium compound. After protic workup the corresponding hydrocarbon is isolated. Addition of iodine instead of protic workup results in the formation of an iodo titano-cene alkoxide that cyclizes to yield a tetrahydrofuran (Scheme 23) [33a]. 

Isolable transition metal complexes containing hydride and terminal oxo ligands are rare however, Tp Re( = 0)(H)X (X = Cl, H or OTf) and TpRe( = 0)(H)Cl have been synthesized, isolated and characterized. Reactions of Tp Re( = 0)(H) OTf (12) with unsaturated substrates (e.g., ethene, propene or acetaldehyde) result in insertion of C = C or C = 0 bonds into the Re-H bond to yield Tp Re( = 0)(R) (OTf) (R = ethyl or propyl) or Tp Re( = 0)(0Et)(0Tf) (Scheme 6). Oxidation of 12 with pyridine-iV-oxide or DMSO produces Tp Re( = 0)3, acid and free pyridine or dimethylsulfide, respectively. A likely mechanism involves initial oxidation of 12 to produce [Tp Re( = 0)2H][0Tf] (13) followed by the formation of Tp Re( = 0) (OH)(OTf) (14) via a 1,2-migration of the hydride to an oxo ligand (Scheme 6). Reaction of 14 with a second equivalent of oxidant in the presence of base yields Tp Re( = 0)3 (15). Direct deprotonation of 13 is noted as less likely than the pathway shown in Scheme 6 due to the lack of precedent for acidity of related rhenium hydride systems. 

Rule 4 A fall in the bond s yield will raise the bond s price by more than the fall in the bond s price that would result from an equivalent increase in the bond s yield. 

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