What is convexity?
Posted: October 20th, 2007, 3:50 pm
bilbo1408Good question. I dont know. Two comments.First, I suspect my approach gives the same result as summing all the terms of the Taylor series.Secondly, Ill tell you a little about how I came to the result. It had been on my mind for some years that the zeros of the TVM polynomial had to have some financial meaning, or utility, or both. Around 1989 I started to work on the problem in my spare time. Id left the academic world and was under no pressure to publish so it was tempting to try to answer a blue sky question.At first I didnt know where the zeros of a typical financial polynomial were located. Lacking modern software, I did many laborious calculations before realising they were located around the unit circle. For ten years I played around with simple models until I had some intuition about the magnitudes of various combinations of the distances between roots (absolute differences between interest rates). It was an interesting journey. For example, the elementary distinction between number as a point in complex space, and number as a distance between two points in space, eluded me and held me up for some years.Eventually, in late 1999, I found a combination of distances between roots that equalled modified duration to 12 decimal places in numeric terms. But I didnt know why and I didnt know if it was general. I needed a proof. A scan of the classic books on polynomials (Marsden, Polya and Szego for example) yielded no hint. Then, by accident, months later, I came across a theorem by Roger Cotes that led to a proof of the result.Intuition suggested that other results might be available. Some more guesswork led to the precise result for duration. Once again I needed a general proof. That took many more months. Interestingly, the precise formula is easier to obtain than the orthodox formula. All this skips over many false starts and misunderstandings. Its been a bit of a struggle over many years with rare moments of elation along the way.The point of telling the story is that my way into the analysis had nothing to do with MacLaurin or Taylor series. It was unconventional, starting with the stubborn belief that all that information in the complex plane had to have some meaning or use. The unorthodox approach means I cant answer your question. Id like to know the answer myself.
What is convexity?
Posted: October 22nd, 2007, 4:40 pm
It appears you have spent a good bit more time thinking about this than I have. I have learned that local quadratic approximation can be estimated using a Maclaurin polynomial such that f(x) ~ f(0) + f ' (0)x + (f '' (0)/2!)x^2 which is a good estimator in the neighborhood. However, this approximation does not improve much on the convexity calculation. Therefore, it is natural to ask whether one can improve on the accuracy of a local quadratic approximation by using a polynomial of degree 3. Specifically, one might look for a polynomial of degree 3 with the property that its value and the values of its first three derivatives match those of f at a point; and if this provides an improvement in accuracy, why not go on to polynomials of even higher degree? So, for your example stemming from the 360 yields of a 30 year mortgage, I am suggesting that your method is very similar to "finding" a 360 degree polynomial whose value and value of its first 360 derivatives match those of your bond/yield curve function at point f(x). The problem here is that I offer no method of "finding" that polynomial. I imagine that it would not be a simple task. Nevertheless, this has turned into a great discussion, and I invite someone to correct me......I am not certain that what I have suggested here is 100% correct.
What is convexity?
Posted: August 11th, 2008, 6:17 pm
convexity is the mismatch between your upside and your downside
What is convexity?
Posted: March 28th, 2009, 3:19 am
how to split the convexity of MBS into "pure" interest rate component and default component?when rates change, it causes changes in default rates too. changes in default rates change the interest income and principal amortization.... i.e. there there's higher order effects caused by defaults embedded in convexity. how to attribute a portion of convexity to defaults?
What is convexity?
Posted: August 8th, 2009, 10:02 pm
No response, apart from bilbo1408s posts. Perhaps the notation is bad (I should learn Latex), or the explanations unclear. The intention was to stimulate comment since the analysis in my previous posts shows that convexity is redundant in at least one respect. Furthermore, I believe the analysis poses many interesting questions. So Ill have another go.Assume the simplest possible cash flow, a bond with one period to maturity and YTM = r. It has a final cash flow, C, that is the final coupon plus the face value, i.e., B=C/(1+r). Assume the YTM falls to R, i.e., B*=C/(1+R). Combine both equations by eliminating C and manipulate the result to get an expression for the proportionate change in the value of the bond:abs(dB/B)=abs(dr)/(1+R) where d before a variable means deltaThe relationship between r and R can be written in two ways. First, (1+r)=(1+R+dr) and secondly (1+r)=(1+R)(1+m). These two equations can be combined to giveabs(dr)/(1+R)=abs(m). Therefore abs(dB/B)=abs(m).Now assume many periods and cash flows, i.e., B=sum(Ci/(1+r)^i) for i=1 to n and B*=sum(Ci/(1+R)^i) for i=1 to n. An i immediately following a variable is a subscript i. Consider all n solutions for r from the first equation and only the new YTM, R, from the second equation, i.e., (1+ri)=(1+R)(1+mi)=(1+R+(ri-R)). It is not obvious, nevertheless it is true, that in this case of many cash flows:abs(dB/B)=product(abs(mi))=product(abs(ri-R))/(1+R)^n for i=1 to n.Assume the orthodox YTM is labeled r1. The rhs of the last equation contains an element that is the orthodox change in the rate of interest abs(r1-R)=abs(dr). This means the rhs can be broken into two parts:abs(dB/B)=[product(abs(ri-R)/(1+R)^n]. abs(r1-R) for i=2 to n in the square brackets because i=1 outside the square brackets. Call this the crucial equation.The crucial equation contains two familiar elements, namely, the proportionate change in the price of the bond on the lhs, and the change in the YTM on the rhs. Sandwiched between them is an expression for the impact of the change in the YTM on the price of the bond that is accurate and requires no embellishment. The history of the concept of duration is essentially a long search for such an expression.Before making some comments on this result, its worthwhile saying how the jump is made from the single period equation to its many-period generalization. Any polynomial can be rearranged into the form:-1+sum(ai/z^i)+1/z^n=0 where the ai are coefficients from i=1 to n, and, in this particular case, z=(1+r).The polynomial can be factorized into product(z-zi)=0 from i=1 to n. It can be shown thatabs(sum(ai))=product(abs(ri)).It is this last equation that provides the route to the multi-period result above.Now return to some comments on the result. First, in the narrow sense that the concept of convexity provides a fix to modified duration, convexity is made redundant by the new duration because the new formula gives accurate results. It takes us around the bend. This is not to say that that the concept of convexity in any of its forms is not useful; it undoubtedly is, as is shown in previous posts by other authors. Its just that the title of this thread is What is convexity?, and this new analysis of duration provides some insight into one definition of convexity. Secondly, the new expression for duration sits between two orthodox, everyday concepts. The expression itself, however, is neither orthodox nor everyday, because it contains every YTM previously ignored or thrown away by most economic & financial analysis to date. The new formula (that encompasses modified duration plus convexity plus all the other terms of the Taylor series expansion that are necessary to give accuracy) requires an excursion into the complex plane. The useful employment of complex interest rates is, I suggest, quite challenging to the mind.Thirdly, assume that the YTM does not change, i.e., r1-R=0. The second element on the rhs of the crucial equation is zero; therefore the change in the price of the bond on the lhs is also zero. This is trivial. The middle element, the new formula for duration, however, is not zero. It has value, because it is composed of all possible values of r, i.e., R=r1, as well as the remaining (n-1) unorthodox values of ri. It is possible to show that, in this particular case, the new duration is equal to modified duration, therefore it encompasses modified duration. This shows why modified duration provides results that are nearly true when the change in the YTM is small, and why they are progressively inaccurate as the change in the YTM grows in size. As the change in the YTM and the consequent change in bond price grow in size, the result from modified duration stays the same and requires convexity, etc, to help fill the gap, while the new formula for duration provides a value for the change in the bond price that shifts and retains its accuracy.It is a long and tedious proof that the new formula is equal to the old in the special case outlined above. This post is long already so it is omitted. I find it amusing that modified duration was developed to measure the impact of a change in the rate of interest on the bond price, and the only time it is equivalent to a formula that actually works is when the rate of interest doesn't change.Finally, one might ask the inevitable question so what? Readers who are familiar with the orthodox concept of duration might be dismissive of the new because the orthodox concept has been sidelined by modern financial analysis. The unfamiliarity of the new might only add to the skepticism.In answer to the so what? question, consider the following thought. As discussed above, in the special case that the YTM does not change, when r1 = R, the middle element in the crucial equation is mathematically identical to modified duration. That is to say:Modified duration = [product(abs(ri-R)/(1+R)^n] for i=2 to n where R=r1.The two are mathematically equivalent, yet their equations are profoundly different and the conceptual routes to them quite separate. Additionally, the new one is accurate in situations where the old one is not, i.e., in all practical situations where the YTM actually changes. It is ironic that a trip into the complex plane is necessary to answer what many might judge to be an elementary question about a real event what is the impact on the bond price of a change in the YTM? Could the same trip benefit more advanced finance? Financial analysis took off down the stochastic route decades ago. Can this analysis in the complex plane be made stochastic? Would it yield further insights? Someone with more math (and Latex) than I possess must take it from here. A start might be made with Bogomolny, Bohigas and Leboeuf (1996), Journal of Statistical Physics, Quantum Chaotic Dynamics and Random Polynomials.