Duration Intro and Calculation

in this video we're going to introduce a topic that's important to bond analysis called duration and if you're watching this video you may want to download the template that I'm using here it's got some formulas and some examples that we walk through that it might be nice to have on a printout instead of just a video if you want that I've got it up on a PDF file under in Google Docs and this is the link to it we're going to start with three basic bond interest rate relationships that you probably learned when you first started talking about bonds and the first of those relationships is the inverse relationship between market rates of interests and bond prices simply stated bond prices go up when interest rates go down and bottom prices go down when interest rates go up is when you're buying a bond you're buying a fixed cash flow stream so as market rates of interest go up the present value of that cash flow stream goes down as market interest rates go down the present value of that cash flow string goes up so one of the fundamental concepts of bonds is that they're going to move in the opposite rate of interest rates you want to hold bonds in a declining interest rate environment bonds are less attractive in a rising interest rate environment now it's more than just the inverse relationship there's two other important concepts one all else equal long-term bonds are more sensitive to interest rate changes than short-term bonds so a 20-year bond is going to go up more when interest rates drop it's going to go down more when interest rates increase then maybe say a five-year bond our third relationship all else equal low coupon bonds are more sensitive to interest rate changes than high coupon bonds so if you're holding a zero coupon bond that's going to be the most sensitive to interest rate changes on the other hand if you're holding a high coupon bond say 9 10 11 percent coupon rate it's not going to be as sensitive to interest rate changes now notice in our statements here say all else equal it's a classic professor speak that doesn't always hold up in various scenarios for example let's say we're looking at a 20-year nine percent coupon bond and a 10-year three percent coupon bond now reset that all else equal a long-term bond is going to be more sensitive which would mean the 20-year bond will be more sensitive to interest rate changes we also said that low coupon bonds are more sensitive so is the longer time going to be a big factor or is the lower coupon going to be the driving factor how do we know which of these bonds is going to be more sensitive to interest rate changes and the answer to that is the topic of our video today duration instead of focusing on interest are the coupon rate as one item and the time to maturity is the second item we can just say whichever bond has the longer duration is going to be more sensitive to interest rate changes in duration taking into account the time to maturity as well as the coupon rate for the bond now another issue with duration that sometimes important is when we think of bonds and interest rate changes typically we're referring to this price risk price risk refers to the idea as we stated earlier that bond prices are inversely related to the market rates of interest as interest rates go up bond prices come down however if you're going to hold your bond until maturity you might be more concerned with the reinvestment rate risk remember if interest rates go up you're gonna receive a coupon payment every six months or every year depending on if the bond semiannual or annual coupon and you might decide to reinvest that coupon payment instead of going out and spending it you might reinvest it for a longer term and if interest rates have gone up you'll be able to earn a higher rate of return on that investment so when we look at interest rate changes there's actually two separate things going on here price risk and reinvestment risk let's look at an example the 10-year 10% coupon bond now if I hold that bond for just one year then price risk is going to be more relevant to me if I only hold this bond for one year I'm not going to have very much time to reinvest my coupons so I'm not really gonna care much what happens to the reinvestment rate but when I sell that bond it's still gonna have nine years remaining until maturity so I'm gonna have quite a bit of price risk on the other hand what if I hold that same bond for nine years if I hold that bond for nine years now when I sell it there's only one year left to maturity it's a short-term bond so it's not going to be sensitive very sensitive to what happened to interest rates but I'm gonna have had several years to reinvest those coupon payments so the reinvestment rate risk would be more important to me so if I'm looking at a short holding period then I need to focus more on price risk on the other hand if I'm looking at a longer holding period then reinvestment rate risk becomes more relevant now another question to think about or another issue to think about is these two risks move in the opposite direction if interest rates go up that hurts me from a price risk perspective it helps me from a reinvestment risk perspective if interest rates go down that helps me from a price risk perspective and hurts me from a reinvestment rate risk perspective so to some extent these risks are going to offset one another if I hold the bond for just the right amount of time price risk and reinvestment risk will be equally important and because they move in opposite directions they'll kind of cancel each other out so it won't be as critical to me what happens to interest rate changes this is a way to reduce the interest rate risk associated with holding bonds and here's where duration comes into play again going back to what we said duration can tell us kind of how sensitive these bonds will be it also is the holding period or price risk offsets reinvestment rate risk so duration is two things one helps measure the price sensitivity of a bond taking into account both the time to maturity and the coupon rate second and gives us the holding period where price risk and reinvestment risk offset one another so now we want to look at calculating duration and try not to cringe when you see this next little slide here here is our formula for calculating duration duration is the sum from 1 T equals 1 to n where n is the years to maturity C is the coupon payment so we need the present value of the coupon payment divided by the bond price times T for each year until maturity plus the number of years to maturity times the present value of the maturity price or par value divided by the bond price big messy formula let's go ahead and put it into action with an example we want to find the duration of a 10-year 5% coupon bond when the market rate of interest is 7% first thing we need to do here is go ahead and calculate the bond price so if we get a financial calculator and replug in 10 for n 7 for interest rate 50 for our payment 1000 for future value and then solved for present value that's going to give us our bond price and I already did that the answer for that is eight hundred fifty nine dollars and 53 cents so when you find the value for the bond price you should come up with eight hundred fifty nine dollars and 53 cents note reassume annual coupon payments in this calculation so once we have the bond price now we go back that was one piece of information we needed the bond price we also need the coupon payment love Reba five percent coupon bond that's going to give us a coupon payment of $50 market rate of interest is seven percent that's our K the par value realizes in par value is one thousand dollars unless it's otherwise stated and our n is the ten years so maturity so now we have all the pieces of information and probably the easiest way to do this is with a table now if you want to probably easiest to set this up in Excel and then you can very easily have Excel calculate all these values for you and it'll save you a lot of time but we start with the first column as the years so we have years one to ten notice we have ten twice down here at the bottom because 10 is our last coupon payment and then also we receive our maturity value in year ten so then we have our cash flows coupon payments $50 a year and years one through ten par value or maturity value gets returned at the end of year 10 next we need the present value of the cash flows present value of $50 discounted back one year at seven percent is forty six dollars and 73 cents present value of $50 our second coupon payment discounted back two years at seven percent remember seven percent was our market rate of interest is going to give us forty three dollars and sixty seven since all the way down and our par value $1,000 discounted back 10 years at 7% it's gonna give us 508 dollars and 35 cents if we sum those all up that's gonna give us the present value of all the coupon payments and par value which as we know is the bond price same thing we did back here with our five key approach so you can either do it on your financial calculator or you can just add up the present value of all your cash flows and that's going to be your bond price as well next column we need is the present value divided by the bond price so the present value of that cash flow divided by the bond price so forty six point seven three divided by our eight hundred fifty nine dollars and 53 cents gives us point zero five four four forty three dollars and sixty seven cents divided by eight hundred fifty nine dollars and 53 cents gives us 0.2 0.0 eight so on all the way down through all years of the bond finally we have our last column which is the year times the present value of that cash flow as a proportion of the overall bond price so we take 0.5 four four zero point zero five four four times one I just rounded these off to three decimal places here gear two point zero five zero eight times 0.1 oh two year three point zero four seven five times three gives us point one four to keep going all the way down the list add them up and that's how we get our duration so this seven point nine three five that's our final answer that's the duration of this bond seven point nine three five and oftentimes you'll hear people refer to duration in years seven point nine three five years now if you know this is a rather long way to calculate duration if you're doing it with Excel it's not too bad once you setup the template you can modify for different coupon rates different market rates of interest and get your answer very quickly however if you're trying to do it by hand can you imagine how much time it's going to take to work through the duration for a 30-year bond you're going to have to calculate all those present values 31 separate present values get the bond price take all of those 31 values as a proportion of the bond price and then again all those 31 times the number of years sum them up you're probably looking at 20 to 30 minute calculation just to come up with duration so oftentimes I use another approach to calculate duration this one looks a little bit Messier when you look at the formula but actually it works a little bit easier for longer-term bonds and for this we start with the coupon payment we've got a bunch of things dealing with discount rate and the number of years to maturity so one plus the market rate of interest raised to the n plus one power minus one plus the market rate of interest minus the market rate of interest times the number of years to maturity divided by market rate of interest squared times 1 plus the market rate of interest to the nth power add in the maturity value the $1,000 times the number of years to maturity 1 plus K to the nth power divide all that by the bond price so once we have that formula again let's go to our example and again I did these calculations ahead of time to save a little bit of time on this video I strongly encourage you to walk through these calculations on your own get some practice doing it and you'll get a better feel for how to apply the formula but we start out with our coupon payment remember it was a 5% coupon bond that we were looking at so we had a coupon payment of $50 our market rate of interest was 7% so one plus point zero seven to the n plus 1 power so we raise this to the 11th power - 1.07 - point O seven times n divided by market rate of interest squared times 1 plus the market rate of interest to the nth power + par value times n divided by one point zero seven to the tenth power again 1 plus K to the nth power now you might notice over here that's just a present value of the par value present value of the maturity value times 10 because we have 10 years to maturity so that just gives you a little idea what's going on there and then this was our bond price eight hundred fifty nine dollars and 53 cents now it's just a matter of working through the calculations so when I go through all these calculations I'm going to come up with two point 105 - 1.07 - point seven divided by point zero zero nine 64 multiply that by 50 here I've got my 1000 times 10 divided by 1.07 to the 10th power divided by the bond price and keep working through the calculations and ultimately I get down to my final answer duration is seven point nine three six years now note that's essentially the same thing we came up with in our previous approach our last approach we said the duration was seven point nine three five years now we've got seven point nine three six years it's just a rounding difference so either way works fine I like this approach better for longer term bonds if I have a two or three year bond it might be quicker to go through the table approach and work through this way but if I have a 30-year bond this is going to take me a few minutes to go through and save me quite a bit of time so is the formula that I stressed to my students in class that they should use there's one last way we're going to introduce for calculating duration and that's referred to as the effective duration and what this is designed to do is there are some bonds that may have what we call embedded options a call provision to put provision different things like that maybe a convertibility feature lots of different things going on so the formulas might not work instead we might want to just look at how sensitive that bond price is so what you'll want to do is calculate the bond price if the interest rate Falls a little bit calculate the bond price at the interest rate rises a little bit and divide by two times the original bond price times the change in the interest rate and this works well if you have more complex calculations for the bond prices or you can even use estimates of the bond prices just from watching it a couple days and seeing what happens to the bond price as interest rates fluctuate a little bit and my example we're going to assume a simple bond so this approach here we're not really adding much value in this example but I just want to show you how it works we want to go back and calculate the effective duration for a 10-year 5% coupon bond 7% market rate of interest assuming interest rates change by 75 basis points so this is our same example as before but now we're looking at given a certain percentage change in interest rate 75 basis points is 0.75 percent so we're looking at interest rates going from 7% down to six point two five percent and seven percent going up to seven point seven five percent that's that 75 basis points it's telling us so our original bond price was eight hundred fifty nine dollars and 53 cents I calculated the bond price at six point two five percent that was just five key approach so if you want to look at that we had 10 as our n 6.25 as our interest rate our present values what we're solving for payment was 50 because it's a 5% coupon bond over here so 50 payment and $1,000 par value or future value solve for the present value and we got that nine hundred nine dollars and eight cents did the same thing at seven point seven five percent our interest rate increase got eight hundred 1337 our change in interest rate is point zero seven five so now we just use the effective duration formula we want the bond price when the interest rate fell - the bond price when the interest rate increase so that's going to be the 909 o eight - eight hundred 1337 that's our numerator and we want to divide by two times the original bond price times the change in rate two times the original bond price 859 53 times the change in rate which was our 75 basis points 0.75 percent or in decimal format point zero zero seven five let me grab my calculator real quick 909 point oh eight - eight thirteen point three seven gives us a numerator of ninety five seventy one and then two times eight fifty nine point five three times point zero zero seven five kisses twelve point eight nine ninety five point seven one divided by twelve point eight nine gives us an effective duration of seven point four to five years now note that wasn't exactly what we got before before we had seven point nine three five or seven point nine three six but it gives us a pretty close approximation also there's another factor that leads to that difference when we calculate duration and a little bit later on a net another video I'll show you how we do duration to predict bond price changes we'll make an adjustment to make it modified duration and that modified duration will be a little bit smaller so this is effectively closer to the modified duration than the duration recalculated here which is sometimes referred to as the makkal a duration this should give you an introduction to the duration I know this was a rather long video but there were quite a few calculations again that template if you want the original template is available to you finding that at tiny Earl calm Brocker duration and you can have this template to walk through hope that helped you follow a little bit on what duration is and how to calculate it thank you

Channel: Kevin Bracker

Views: 84,017

Rating: 4.8939757 out of 5

Keywords: Educational, finance, investments, bonds, duration

Id: Rxham7bJ4e0

Length: 24min 33sec (1473 seconds)

Published: Sun Jul 10 2011