Duration and Convexity

in this video we're going to take a look at duration there's a previous video that I did that talked about how to calculate duration and introduce the idea of what duration was in this one we're going to start looking at how we can use duration to predict bond price changes we're going to introduce the concept of convexity and how that introduces a little bit of error in our prediction of bond price changes and we'll also introduce the idea of an immunization strategy now there are a couple of handouts in this video that I'm going to go through that it probably going to be pretty hard to read on just the video so I'd encourage you to go through and download the template from Google Docs it's just at tiny oral comm brokkr duration too and that way you can have that follow along might be helpful to work through some of the examples and also like I say some of the charts and tables that I'm going to introduce in here are a little bit small might not get captured quite as well in the video will be handy to have it with you that you can look at a little bit closer first thing we're going to look at is how to use duration to predict changes in bond prices and so we've got this nice little formula here that says the change in bond price is equal to negative duration divided by one plus K times the change in K where K is the market rate of interest now a couple of quick things on here first of all this change in bond price is a percentage change not a dollar value second thing is that notice we take duration divided by one plus the market rate of interest this is sometimes referred to as modified duration in the previous video that I went through calculating duration we had a couple of formulas to calculate duration and those calculated what is referred to as makkal a duration in order to convert it to the bond price change eat a little more accuracy if you use modified duration where you take duration divided by one plus the market rate of interest now in that previous video there was another formula called effective duration that were you that's already sort of a modified duration so you don't need to make this conversion if you're using the effective duration calculation and then re multiplied by the change in the market rate of interest now note this negative sign up here that's to capture the inverse relationship between market rates of interests and bond prices if interest rates go up bond prices should come down so this negative is going to flip around the interest rate change to get us back to the bond price change so let's go through an example you have a 10-year 8% coupon bond current market rate of interest is seven point two five percent now in this video to save a little bit of time it's already rather long video I don't want to calculate duration so I already did that and just provided the value so duration is given a seven point three two years if interest rates fall by 80 basis points what will happen to the bond price so let's go ahead and walk through that go ahead and rewrite the formula up here the change in the bond price is equal to negative duration divided by one plus the market rate of interest times the change in the market rate of interest now we just want to plug in our values and our example our duration is seven point three two years our market rate of interest is seven point two five percent and here you want to make sure you plug that in as a decimal so it's not one plus seven point two five we don't want to divide by eight point two five we want to divide by one point zero two five and then by the change in the interest rate and here we say interest rates are going to fall by 80 basis points that's point eight percent it's a fall and interest rate so our interest rate changes negative interest rates are declining now we just grab our calculator go through the calculations you should end up with a change in the bond price of five point four six percent now one thing to keep in mind is that's the change in the bond price in percentage terms if we want to figure out what the bond is going to be worth we have to first of all know what the starting value is so we can multiply by one plus the change in the bond price so real quick if you want to do that assume annual interest payments we had ten years to maturity we said the market rate of interest was seven point two five percent as our starting point remember we're solving for the original bond price so were solving for the present value was an eight percent coupon bond that means eighty dollars a year in coupon payment and lastly we have the par value returned to us at maturity so that's going to be our future value one thousand dollars let me quick go through that calculation that gave us a starting price of one thousand fifty two dollars and seven cents so reset given this change in interest rates bond price should increase by five point four six percent so that's just our starting point a lot to increase by five point four six percent and we're going to predict the new bond price to be 1,100 nine dollars and 51 cents so that just goes through our prediction using duration to predict the change in the bond price note that duration says the bond price will be rise by five point four six percent that's just what we did assuming the original acute are assuming annual coupon payments that bond was originally valued at 1050 207 which showed how that calculation was done so duration predicted that the new bond price would an interest rates drop by 80 basis points would be one thousand one hundred nine dollars and 51 cents however if we actually calculate the new bond price all we have to do is say well if interest rates change overnight there are still ten years remaining until maturity now the market rate of interest dropped by 80 basis points so instead of seven point two five percent it's now going to be six point four five percent we're solving for the present value bond price still has an eighty dollar coupon payment and a one thousand dollar future value so we grab our financial calculator plug these values in solve for the present value and what we're gonna come up with it's one thousand one hundred and eleven dollars and sixty nine cents notice that's not what Reproductive from duration duration said it should be one thousand one hundred nine fifty one and in reality was a couple bucks higher now that's not a big error but it still is a little bit of an error so what went wrong this hour duration messed up did we screw up a calculation somewhere the answer is no instead this is what we're talking about when we're talking about convexity duration is not a 100 percent accurate predictor of bond prices it's an approximation that approximation is better for small interest rate changes here you can see this is one of those reasons why I said you might want to print this out some of the text in here is kind of small to read but here we're looking at a 30-year 14% coupon bond this is just an example to show you convexity so we're looking at a longer-term high coupon bond here and what we see with the convexity is this is the actual bond price this reddish pinkish line here is what's actually happening when interest rates go down the bond price goes up when interest rates go up the bond price goes down this bluish line here is the duration prediction duration says as interest rates go down the bond price will go up interest rates go up the bond price will go down but no duration predicts a linear relationship this is a straight line here this is a curved line the actual bond price is convex duration predicts linear actual bond prices convex what that means is we're going to get some errors note this is our starting point here zero percent if we have a 1% change not a very big change in the bond price you can see the duration prediction is real close to the actual bond price for small changes in interest rate duration is a very good approximation however what happens when we get to big swings when interest rates rise by 4% our interest rates fall by 4% we start to see that the duration misses by a lot more so one of the issues with convexity is that duration works better as a predictor of bond price changes for small interest rate changes than it does for large interest rate changes now another thing to note when we look at duration and the concept of convexity is that not all bonds have the same degree of convexity for example this graph here that I'm looking at is a 30-year bond notice quite a bit of convexity out of the extremes here's another graph with a 5-year bond now notice when we look at this one remember this is a just a five-year bond still same 14% coupon rate but notice the convexity here the errors are much small even out on the big swings in prices so one of the things we note is that all else equal shorter term bonds are less convex duration will be a better predictor for short term bonds and longer term bonds are more convex duration will be less accurate predictor for long term bonds another factor that plays a part is the coupon rate now both of these bonds are used a high coupon rate but convexity also differs based on the coupon rate all else equal the lower the coupon rate the more convexity the bond price would be so duration is a less accurate predictor for a zero coupon bond then it would be for my 14% coupon bonds that I'm looking at here so a quick summary of convexity remember actual bond prices are convex they're not linear so that's going to introduce error from our duration prediction how much error depends on the bond all else equal the higher the duration of the bond and remember long time to maturity means high duration low coupon means hydration so a longer time to maturity and are a lower coupon payment is going to mean there's going to be more air the bond is more convex and so duration will be less accurate duration works best for short term high coupon payment month it works worse for long term low coupon payment bonds another issue that we talked about the bigger the change in interest rates the more error there will be duration is a great tool for estimating small changes in interest rates and seeing what impact they're going to have on bond prices it's not as good of a tool for looking at big percentage changes if you want to see what happens if market rates of interest changed by 50 basis points or 100 basis points duration is going to give you a pretty good approximation if you want to see what happens when interest rates changed by 400 basis points durations not going to be near as a cure now one other issue with convexity that I want to talk about a little bit is what happens when there's a call provision so let me go ahead and just draw a quick graph here pretend those are straight lines I have a little bit of artistic challenges and a normal convex bond looks something like that you can see that it's not a linear relationship as interest rates decline bond prices go up more rapidly its interest rates go up bond prices fall at a slightly lower rate however let's think about what happens when there's a call provision in a call provision typically the issuer is gonna choose to call the bond if interest rates fall just like refinancing a mortgage nobody wants to go out and refinance when interest rates go up they want to refinance when interest rates come down and the call provision gives the bond issuer a way to do that they can buy back their bond for a fixed price and then re-entry issue bonds and lower interest rates however remember a call provision means the issuer gets to buy it back at a fixed price so the bond price isn't going to keep going up up and up when interest rates to climb because investors are going to start to figure out this bond isn't going to mature instead it's going to get called so what we're gonna see is relationship that looks more like this because the call price is going to cap off the value of the bond as interest rates fall we won't see that full convex relationship and said we'll see it kinked curve where it's concave at low interest rates call provision is going to prevent it from keeping to go up and then convex at higher interest rates so convexity just works a little bit differently with callable bonds something to be aware of now we're going to switch gears instead of talking about convexity we're going to talk about immunization one of the things duration can be used for us to talk about how bond prices change when interest rates change how sensitive is price - interest rate changes another thing that it can do those help us balance off price risk and Reinvestment rate risk remember the reinvestment risk means if I have a 10-year bond and interest rates go up maybe my bond price is going to drop a little bit but I can reinvest those coupon payments at a higher interest rate so from a reinvestment rate perspective in increases in the interest rate are beneficial to me decreases in the interest rate meaning those I can't reinvest those coupons at as high of a rate of return price risk is the normal bond price interest rate relationship that we talked about when interest rates go up bond prices come down when interest rates go down bond prices go up so if I look at for example a ten-year bond if I only hold that bond for let's say one year then price risk is going to be more important because I'm not going to have much time to reinvest my coupons by the time I get my coupon payments I'm sawing off the bond but when I sell that bond it saw has nine years remaining till maturity so it's going to be very sensitive to interest rate changes instead if I hold that bond for let's say nine years now the reinvestment rate risk is going to be more important and the reason for that is when I sell that bond there's only one year left to maturity it's a short term bond every no short term bonds aren't very sensitive to interest rate changes so it's not going to be affected the price will not be affected much by what has happened to interest rates on the other hand I've now had several years to reinvest my coupon payments so if I've got a nine year holding period then the reinvestment rate risk is going to be more important to me if I've got a one year holding period the price risk is going to be more important you can see somewhere in between there they're gonna balance each other out because they counteract each other increases in interest rates are good for one and negative for the other duration helps us find that holding period the duration tells us the holding period where price risk and reinvestment rate risk offset each other so here I did an example and again this is going to be too small to really look at closely in this video but I'll just walk through it a little bit hopefully you look at it a little more depth but what I did was I started with a bond and I forgot to write down the time to maturity on this bond but trying to remember I think it was like a 30-year bond and it had a duration of 12 years so if we have a bond with a duration of 12 years what happens then as interest rates go up we're starting out with an eight and a half percent return as interest rates go up I can reinvest these coupon payments so what I'm doing is it's a seven percent coupon bond and that coupon payment that I receive in year one I get to reinvest now for eleven years because remember I'm gonna hold it for 12 years I sell it off so I reinvest that for 11 years and it's worth 189 next year I get another coupon payment reinvest that for 10 more years at nine and a half percent it's worth one hundred and seventy three when I sell the bond I can sell it for seven hundred and eighty two because interest rates went up the bond price went down it's not as valuable to me but when I add up all the cash flows the reinvested coupons and the price that I can sell it at after twelve years I'll have total cash flows of two thousand two hundred forty dollars and eighty-seven cents what about if interest rates dropped if interest rates drop notice I don't get as much for reinvesting my coupons now I'm only getting 155 for reinvesting at coupon eleven years compared to the one hundred and eighty nine I got before I only got a hundred and forty-four for reinvesting this second coupon for ten years it's so on but when it comes time to sell the bond price now I can sell it for more so some these all up and notice very close answers regardless of whether interest rates went up to nine and a half percent are down to seven and a half percent the total future value is the same so that's what you mean by immunization if I hold that bond to duration it doesn't matter what happens to interest rates I'm gonna have approximately the same amount at the end now one quick comment on immunization this is a quick example doesn't work quite that precisely because what we assumed here is interest rates change and then stayed constant after that in reality interest rates are always bouncing around so if we hold till duration we're not guaranteed to have that payoff but it gives us an idea of about how long we're going to have to hold the bond until we're not as concerned about interest rate risk so if we hold a bond to duration we're going to eliminate a lot of the interest rate risk last thing on this video what happens if I'm not looking at a bond but instead of bond portfolio well if I have 50 or 60 bonds in there how do I calculate the duration of that portfolio the answer is it's really easy the duration for a bond portfolio is just equal to the weighted average of the individual bonds duration so quick example I made up let's assume we hold three bonds market value of bond a is 1 million market value of bond B is 3 million bond C is 4 million that gives us a total portfolio value of 8 million duration is different for each of these bonds I calculate the weight 1 million divided by eight million tells me about 12 and a half percent of my portfolio is in bond a 3 million divided by 8 million means thirty seven and a half percent in bond B and 4 million divided by eight million means about 50% of my portfolios in bond C so then I just take the weight times the duration it's gonna give me this column I called weighted duration add them up duration for my portfolio is six point eight one two five years so duration for part four it's just a weighted average of the duration for each of the bonds in the portfolio that's all I've got on duration and convexity again so probably be more useful if you have the video template there so I encourage you to take a look at that it's up on Google Docs just real quick again that was at tiny Earl calm Brocker duration - and you'll find this template hope that helps thank you

Channel: Kevin Bracker

Views: 79,970

Rating: 4.9251337 out of 5

Keywords: educational, finance, investments, duration, convexity, fixed-income

Id: 44ppVHxWHAM

Length: 22min 57sec (1377 seconds)

Published: Fri Jul 22 2011