March 31st, 2005, 7:09 pm
There is another way of looking at convexity.Assume a vanilla bond with known price B, coupon C, and number of periods N. If the equation is solved for the YTM, (1+Y), there are actually N solutions, an entire constellation, call it S. Under normal circumstances, only one (real) value of S is calculated, i.e., the orthodox rate that is produced by an HP calculator or Excel. The other (N-1) values of (1+Y) in S are either negative real numbers or complex conjugate pairs positioned off the real number line. Associated with this set-up, there will also be a value for modified duration, D.If the YTM changes from (1+Y) to (1+Y*), there will not only be a new bond price B*, but also a whole new constellation of interest rates S*, and a new value for modified duration, D*.The true (accurate, precise) value for duration is somewhere between the two modified durations, D and D*. And the differences between the true value, and one or other of the modified durations on either side of it, can be regarded as kinds of convexity. Therefore these differences repay investigation.According to the formula for modified duration, D, the change in the price of a bond is dB/B=D.dY where d is the differential operator. It is only approximately true for small values of dY. Assume that there is a version of duration that gives accurate, precise results for any change in the yield; call this duration tD. In this, necessarily, discrete version, the change in the price of the bond is deltaB/B=tD.deltaY. The critical point is this: a formula for tD does exist, but it can only be obtained by going into the complex plane and considering the entire constellations of interest rates S and S*.Let the old constellation of values be (1+Yi)=Zi and the new constellation (1+Y*i)=Z*i. The distances in complex space between any two of these yields can be defined as abs(Zi-Z*i)=abs((1+Yi)-(1+Y*i))=abs(Yi-Y*i). That is to say, distances in complex space between the zeros of the bond pricing equation are differences between interest rates. For example, the absolute value of the orthodox difference deltaY is equal to abs(Z1-Z*1)=abs((1+Y1)-(1+Y*1))=abs(Y1-Y*1) where Z1 in S and Z*1 in S* are the orthodox yields.It can be shown that the following equation holds true: (modified duration * dY) = dB/B={product[for i=2 to N] of (abs(Zi-Z1))}*abs(Z*1-Z1)/(Z1^N). Compare it with the following tidier formula: deltaB/B={product[for i=1 to N] of (abs(Zi-Z*1))}/(Z*1)^N. The latter formula encompasses tD, and it shows that an accurate formula for duration involves every interest rate that is usually thrown away or ignored in the orthodox analysis. It is worth drawing figures in complex space to ‘see’ the formulas.In words, the formula for tD is found as follows: plot the zeros of the bond pricing equation in the plane, join them with straight lines to the new orthodox yield (1+Y*1), multiply the lengths together and divide the result by (1+Y*1)^N.Earlier, it was assumed that convexity can be defined as the difference between dB/B (an approximate value) and deltaB/B (an accurate value). The new formulas for these concepts can be compared because they are in the same language of ‘products of distances in the plane’. The comparison suggests that convexity is the result of a lack of clarity. The confusion is about which yields change position, and which stay in the same location, within the constellation, S, when the orthodox yield is adjusted in the bond pricing equation.