July 15th, 2007, 8:46 pm

drdSorry about the delayed reply -- been busy. Im not sure which of the previous posts you regard as nonsense, some of them or all of them? But I dont think my main point about convexity is nonsense. Heres another attempt to explain it. The bond pricing equation is merely a special case of the time value of money (TVM) equation. The TVM equation is probably the simplest and most commonly used equation in finance. Yet, for a long time, (since Macauley and Hicks?) there has not been a decent answer to one of the simplest questions you can ask of it, namely, what is the effect on present value of a change in the yield?A numerical answer is trivial. In the context of the bond pricing equation, simply put two different yields into it, and get two different bond prices. But a numerical answer is not insightful. Insight requires algebra. The traditional answer is to apply calculus -- employ the first two elements of a Taylor series expansion in the form of duration and convexity. But the resulting equations are messy and inaccurate (they represent, after all, a truncated series expansion). Most important, the messy results provide little insight.Nor do I think it enough to say that duration is a linear approximation to the curvilinear relationship between bond price and yield, and that convexity adds an element of curvature to the analysis. It's possible to go deeper than that.When faced with a bond, or portfolio of bonds, in the market, what is known? We know the likely stream of future cash flows and the current market value of the stream. In which case, we can calculate the yield to maturity (YTM). Or, more precisely, and this is a significant comment, we can calculate all n values of the YTM. One particular value of financial importance may be in my mind when I ask for the YTM of a cash flow, but most decent math programs will happily spit out all the roots of a polynomial when asked.I trained as an economist and was always taught that data is valuable and should not be discarded lightly. A thirty year mortgage at inception has 360 monthly payments and, therefore, 360 yields. Why throw away 359 of them? A 30 year bond with semi-annual coupons on the day of issue has 60 yields. Why ignore 59?As a result of this last thought, after calculating all the solutions to a bond pricing equation, do not focus on the single YTM and ignore the (n-1) other solutions. Instead, retain them all. Select the single orthodox root, call it (1+r), and connect it with straight lines to the other (n-1) roots. Then calculate the product of the (n-1) distances. Divide the result by (1+r) to the power n. The answer is modified duration.Here is the crucial point. Repeat the above procedure with the following single modification. Move one root, and one root only, the traditional value of (1+r), to the new location, to wherever you think the new yield should be. Call it (1+r*). Now apply the same procedure again. Join (1+r*) by straight lines to all the other roots of the TVM equation (that have not shifted). Calculate the product of the (n-1) distances. Divide the result by (1+r*) to the power n. In other words, everywhere we had r before, replace it by r*. The result is a value of duration that is precise. By duration, I mean the coefficient that stands between the change in the yield and the percentage change in the bond. And by precise I mean it is as precise as the calculation of the roots. The new duration is usually accurate to within $1 in a trillion. Convexity is not needed. The difference between the two outcomes, modified duration and new, precise duration, could be described as convexity. Convexity is where this thread started. What is convexity? Convexity is the result of failing to perform a true other-things-being-equal calculation. Since the time of Macauley and Hicks we have not taken into account the fact that we do not have one yield to play with in the TVM equation; we have many. The symbol r in the orthodox TVM equation stands for all of them at once. Move it and you move all of them. To solve the original Macauley and Hicks problem about duration, suitable expressions are needed that distinguish between all possible yields. Then we move one yield and hold the others steady. My earlier entries contain such expressions. I believe that they are insightful. Apart from anything else, the expressions hold true for any cash flow, any yield curve, and any shift in the yield curve. The traditional criticisms of duration fall away.There are many thoughts and questions that follow from the analysis but for the moment I think one will suffice. I conjecture that it is impossible to find a simple, accurate, algebraic expression for the impact of a change in yield on the present value of the TVM equation without employing all the other roots of the equation. Analysis about the real, positive solutions alone will not serve. A comprehensive and accurate answer to a simple question about the most elementary financial equation requires knowledge about the complex solutions as well as the real, about the negative as well as the positive.