Bond convexity

hi David a banach turtle with an illustration of how we calculate bond convexity which is a more advanced bond topic so I attached the spreadsheet next to the video on the website if you'd like to take a look at the calculations this is a little bit harder than bond duration and I'm going to look at convexity quickly from two different perspectives but first here I've got a 30-year zero coupon bond so I deliberately chose a long maturity bond because they have greater convexity and then I plotted the familiar price yield curve for the bond that is to say as we move to the right here this is the yield as yield increases the price of this 30-year zero coupon bond decreases that's in blue that's the price yield plot for the bond and here at five percent yield I plotted the tangency line which would be really the slope of that line would be dollar duration and it really highlights for us that the price yield curve for a bond is not linear so this curvature is really captured we capture most of it with the convexity measure so this high degree of curvature here reflects a bond that has high convexity why does this matter well think about if you were long this bond or owned this bond convexity is your friend in this case because here imagine starting at 5% if rates go down the price of the bond goes up but notice duration just tells us about the linear behavior of that and notice you're benefitting here with incremental upside due to the curvature as the yields go down on the other hand if yields go up notice the red line here describes where we would be under a linear relationship but as press yields go up bond price does go down but not as much as they went under the red line so actually convex these your friend even on the downside because it's mitigating the loss so being long the bond just generally - like the convexity now in order to do a more real realistic example I'm going to use a five-year bond with a 4% on my bond is shorter maturity now with a coupon so I'm not going to get as much convexity Beacon Steel of blue line still captures convexity and this is the first perspective that they put it very simply treats convexity as x squared or specifically convexity is the weighted average of maturity squares of the bond so here assumptions bond par of 104 dollar coupon paid semi-annually so that's two years every six months and I'm going to assume a yield of 5% so you can see here I'm plotting out the six month times six months one year one in 18 months two years and then each of the coupon cash flows two dollars is half of the four dollar coupon final cash flow is here in this column compute I compute the present value of those cash flows this is probably familiar because the sum of the present value of all the cash flows gets me the model price of the bond this bond has a price of ninety five dot sixty two cent at a five percent yield it is less than par which does is consistent with the fact that I have a coupon that's less than the yield and in this column here I've got the calculations that lead to the Macaulay duration which to say you can look this is pretty simple are pretty straightforward here I multiply the time or term one-year times the present value of the cash flow so basically I'm computing here the weighted average maturity of the bond but what are the weights well the weights the present values of the cash flows so if I sum all of those I get this number and then so I have here I have the sum of this column and if I divide that by the price of the bond you can see that's what I've got right here my point here is not to illustrate Macaulay duration I've got other videos for that but you can see if I divide this by the price of the bond I get four point five seven I'm going to remove that dollar sign because it's not really in dollars we sometimes say this is in years I get the Macaulay duration of four point five seven for this bond and it's less than the mature five years okay so here's the convexity column and and here's the key idea try not to worry about the plus 0.5 that falls out because convexity is a second derivative or dollar convexity is second derivative convexity is a function of the second derivative so this ends up being a function of the derivative but you can see what I've got here is it's a lot it's like duration in the sense that we do take the present value of the cash flow that really is the weighting function and we multiplied by the time here's the 1.25 or 18 months but we multiply by time again or almost time again you can see we've got this see 10 times see 10 plus 1/2 is basically squaring the term or squaring the maturity we're doing that for each of these and so that's why I say convexity is really x squared or specifically the weighted average of maturity squares of the bond so in this column we're effectively squaring all of these maturities because they each correspond to a cash flow if we summarize that here and then at the finally the convexity ends up being very similar there's a little bit of an adjustment factor here but basically we're taking this summation here which which effectively squared all these maturities one for each cash flow and divide it by the price of the bond and we end up getting the convexity in this case twenty three point one nine is the convexity and and really captures most of that curvature on the bond that's the first perspective again to treat convexity is the weighted average of maturity squared squaring time being the important concept now if we look at the second perspective quickly this focus is on convexity being a second derivative and here as I just zoomed out on our price line and if remind the duration or dollar duration is specifically the slope of that straight line dollar duration is the first derivative the slope of the point of tangency on the priced yield curve so if dollar duration is the first derivative dollar convexity is the second derivative it is the rate of change of dollar duration such that if the line has no convexity if it's a straight line there's no change in dollar duration and we would have convexity we would have dollar convexity and convexity of zero that's not the case here so in this case instead of doing that analytical approach like did before I'm straight-out repricing the bond here's the bond at 5% that bond price same as before is 95 62 and then here is my first derivative or dollar duration at 5% so really it's a really difficult to see here but at 5% it's the slope of that tangent Z line and you can see it's a big negative number and that and the formula here is really just a familiar rise over run we've got here c19 minus c18 is the change in bond price that's the rise over the change in the yield which is the run here and I get a big negative number for the negative slope of the point of tangency so here's the dollar duration at 5% and then I do the same thing the calculation explicitly of the slope rise over run at five plus one basis point five point O one percent there's the dollar ation if this were a straight line the slope would be constant it's not here's the slope at five point O one percent so here we have dollar duration at five percent here we have dollar duration at 5.0 one percent convexity is the rate of change of the dollar duration here it's the second derivative or rate of change of the first derivative and so in this case it's also just a rise of a run what's the change in my dollar duration as I move from 5% to 5.0 1% and here we get a big number that's dollar convexity such that if I divide that by the price I take the dollar convexity divided by the bond price here notice I get the convexity twenty three point one nine same number I got before and it's non zero because there is convexity or curvature in this line so that's the second way to look at it you can see sort of the brute force I went at the convexity directly up here if you'd like to take a look at the open the spreadsheet you can see how we apply the convexity as that second order approximation so this is David Harper the Bionic turtle thanks for your time [Music]

Channel: Bionic Turtle

Views: 71,044

Rating: 4.7064219 out of 5

Keywords: Bond, convexity, duration

Id: yOwRgWhIn_g

Length: 10min 2sec (602 seconds)

Published: Wed Jun 30 2010