 fa
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### What is convexity?

CorrectionI slipped up in my previous posting. The third paragraph is not true. The slope coefficient dB/dY is a component of the formula for modified duration: D=(dB/dY)(1/B). It is the slope coefficient of tD (true duration) that lies between the slope coefficients of the modified durations D and D*. The statement in the paragraph does not apply to the durations themselves. My notation isn’t too hot either. Sorry.AdditionAnother way of looking at the formula for tD is as follows. Assume a fall in the yield from (1+Y) to (1+Y*) such that (1+Y)=(1+Y*+a). Further assume that there is one value of Y* but n values of Y and a, ie, (1+Y_i)=(1+Y*+a_i). It follows that the formula given in the previous posting can be rewritten as deltaB/B=product(a_i)/(1+Y*)^nor(deltaB/B)(1/a_1)=tD=product(a_i)/(1+Y*)^n where a_1=orthodox deltaY=(Y_1-Y*).Every element in the top line of this last formula for tD reaches into negative or complex territory and has been ignored by orthodox analysis to date.The adjustment to the yield can be rewritten multiplicatively instead of additively, ie, (1+Y_i)=(1+Y*+a_i)=(1+Y*)(1+m_i) where m is the multiplicative adjustment. The latter equation can be manipulated into product(a_i)/(1+Y*)^n=product(m_i). The left hand side of this equation is deltaB/b and so deltaB/B=product(m_i). An accurate, precise figure for a change in the value of a portfolio of bonds that does not require convexity is equal to the product of the multiplicative increments to ALL possible yields, including the highly negative and complex yields. The last expression is very general; it applies to many more financial situations than just fixed income.The conclusion is that convexity is a patch to cope with the inaccuracy that results from ignoring the bulk of the solutions for the yield in the core pricing equation.QuestionHow can you make this stochastic? The above deals with a single, instantaneous move (time does not change). The shift in the principal, real yield needs to be made random (taking into account what happens, or doesn’t happen, elsewhere in the plane). Then there needs to be a sequence of such moves with time changing too, and the number of zeros in the plane steadily diminishing. If the time increment is short then the polynomials will be of large order with many zeros padding the coefficients. My math is not up to it. fa
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### What is convexity?

Further thoughtsThought one – On realityObjections can be raised about the use of complex rates of interest. Not real. Don’t exist. But there appears to be no other way of calculating an accurate yield sensitivity that does not require convexity and the other terms of a Taylor series expansion. If the ‘complex plane’ approach works, then it must possess an element of reality. Many readers will be physicists and electrical / electronics engineers; surely there will be no objection from that quarter.Thought two – On practicalityTake a portfolio of vanilla bonds; say 100 of them, of all tenors, coupons, issue dates and market conventions. Compare two programming problems.A - Program the orthodox duration and convexity of each bond and find the weighted average of all of them.B - Take the vector of cash flows for the entire portfolio as the coefficients of a single polynomial, call it V; it will consist of irregular, variable coefficients, and will be padded with zeros on dates where no cash flow takes place. Then find product(abs(roots(V)/(1+Y*)-1) ), or whatever is needed in the math program of choice. It calculates the accurate interest sensitivity in one fell swoop.Which program is easier to code?Thought three – On eleganceCompare the inaccurate formulas for duration and convexity with the accurate formula deltaB/B=product(m_i) embodied in the algorithm above. Would Dirac approve?Thought four – On past and future workIf all solutions for the yield in the constellation S are necessary to calculate an accurate interest rate sensitivity in the context of a one-shot move, is there not a question mark to be raised over the current stochastic approaches that follow the multiple period course of the one real yield (that actually, behind the scenes, stands for all yields at once). Is there not a ‘stochastic convexity’ issue? I don’t know what it all means. As I said before, my math is not up to it. Just raising the question.I’m not the first one to raise the point. It has been raised before in the context of capital budgeting – see Robert Dorfman (1981) The meaning of internal rates of return, The Journal of Finance, Vol. 36, No. 5. He examines the case where the proceeds of an investment are serially reinvested in projects of the same type, and the process is continued indefinitely. He shows that the growth path of a dollar placed in such an investment depends on all the roots of the internal rate of return equation, complex as well as real. muniangel
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### What is convexity?

normally,convexity can be used in some perference in economic theories,but in finance ,does it work? xtang1
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### What is convexity?

QuoteOriginally posted by: piterbargConvexity, to an interest rate quant, is a situation when a payment measure is diffeernt from the martingale measure of the underlying rate-Vis that the same thing as saying that there is drift?Thanks. johnself11
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### What is convexity?

ok i'm not a quant and i think this question has gotten too theoretical...forget formulas and martingales for a moment... convexity is an attrbute of a term interest rate security which measures how the securites' price changes given a change in its duration.... it is the second-order derivative of the price/yield function of a bond, analogous to the gamma of a contigent premium (aka option) security.... the longer the duration of a bond, the more more convexity.... spotula
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### What is convexity?

Last edited by spotula on January 4th, 2008, 11:00 pm, edited 1 time in total. drd
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### What is convexity? aemmedar
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Joined: July 14th, 2002, 3:00 am

### What is convexity?

EXPECTATION OF CONSTANT RATES WITH NO RISK1y zc rate in t=0 is 10%; investors expect that 1y zc rate in t=1 will be 10% with no riskP(1) = 1/1,10P(2) =1/(1,10)2 Spot(1)=spot(2)=10%The term structure of spot rates will be flat at 10%EXPECTATION OF CONSTANT RATES WITH RISK1y zc rate in t=0 is 10%; investors expect that 1y zc rate in t=1 will be 10% with risk (12% with p=1/2 and 8% with (1-p)=1/2)P(1) = 1/1,10P(2) = 0,5*1/(1,12) + 0,5*1/(1,08)=0,82672Spot(1) = 10%spot(2) =((1/0,82672)^0,5)-1=9,982%even if spot(1) = 10% and expected 1y rate in 1y is 10%, the 2y spot rate will be less than 10%. In a volatility context, the expectation of constant rates will imply a downward sloping term structure. The difference between the spot rate in presence of volatility (9,982%) and the corresponding rate in absence of risk (10%) will be a convexity effect (1,8BP). More maturity will imply more convexity (Vasicek Fong say Convexity = duration + duration ^2 + M^2 ), since convexity is a favorable factor, more price and less returnthe convexity effect is a special case of jensen's inequality1/1.10 (.5*1/1,12+,5*1,08)>(1/1,10^2)see tuckman fixed income securities chapter 10 ashutoshkamat
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### What is convexity?

Thanks fa
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Joined: December 10th, 2003, 11:07 am

drdSorry about the delayed reply -- been busy. Im not sure which of the previous posts you regard as nonsense, some of them or all of them? But I dont think my main point about convexity is nonsense. Heres 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. bilbo1408 Posts: 179 Joined: August 3rd, 2007, 12:50 pm ### What is convexity? Last edited by bilbo1408 on December 13th, 2007, 11:00 pm, edited 1 time in total. esamsirachid Posts: 11 Joined: August 25th, 2005, 5:46 pm ### What is convexity? Hi,il whant to use future on Euribor3M in stripping yield curve but i finded that i need to pricing Future contract using interest rate model (short) if you use Bloomberg you can see by making Eus <GO> that the model used is HWdo you know any method to find convexity_adjustment used to pricing Future price Thanks esamsirachid Posts: 11 Joined: August 25th, 2005, 5:46 pm ### What is convexity? Hi,I whant to stripping yield curve using Future on Euribor3M do you know any methodology to extraction convexity -adjustment from those price using Market model Thanks gumpleon Posts: 14 Joined: January 8th, 2007, 4:40 pm ### What is convexity? QuoteOriginally posted by: drdConvexity, at its root, means curvature, and ONLY curvature. In the case of non-contingent IR products, it has NOTHING to do with volatility or anything else, as such. It is purely a property of the pricing formulas. However, as there are many aspects in finance and trading that lead to curvature, the word is used in many (and very different) contexts. Here are some of the more important variations on the use of the word convexity for non-contingent products (options are explained elsewhere):i agree with your argument: convexity is a simple rather than a complex concept.the difference between convexity and curvature is that "convexity" is a scale-free quantity while curvature is not. without causing much confusion, we can always think of it as some sort of curvature measure. bilbo1408 Posts: 179 Joined: August 3rd, 2007, 12:50 pm ### What is convexity? "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."Is this stemming from some sort of McLaurin or Taylor series polynomial property?
Last edited by bilbo1408 on October 18th, 2007, 10:00 pm, edited 1 time in total.  