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### What is convexity?

Posted: October 20th, 2007, 3:50 pm

### 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