Macaulay Duration

when interest rates rise the price of any bonds that we hold is going to fall so then the question becomes is there any metric that we can use to evaluate a bond and see how sensitive it's going to be the price of that bond and given that there's a change in the interest rate and luckily we have a tool called Macaulay duration that will perform just that function so Macaulay duration is actually it's going to be measured in years and what it is is it's the weighted average maturity of the cash flows from a bond so if we want to look at a bond from from an investor's perspective we can look and use Macaulay duration and say okay looking at this bond how long is it going to take us to receive these cash flows and so we can actually compare the duration in terms of years for different bonds in a bond with a higher number of years for its Macaulay duration is going to be more sensitive to interest rate changes not talking we're talking about the volatility of the bonds price is going to be higher with respect to changes in the interest rate so the way when we calculate Macaulay duration I've actually provided a formula here for the mathematically inclined of you so basically in the numerator we're going to have the sum of the present value of the time-weighted cash flows and then we're going to divide that by the price of the bond and the price of the bond of course is just the present value of the cash flows of the interest payments and then the face value of the bond so you might be seeing what we're actually dividing the present value of the cash flows by the present value of the cash flows what what is going on here well it's important to note in the numerator we're talking about the time-weighted present value the cash flows now I know that's kind of abstract this may be a little difficult to understand let's walk through an example and I'll become a little bit easier so let's say that you have a five-year bond and it has a $1000 face value and the coupon rate is 4% so since we have a $1000 face value and a coupon rate of 4% that means that this bond is going to be paying out interest of 40 dollars a year so we can go ahead and we can we can think of the cash flows of this bond in period one right in the first year we're going to have a cash flow of forty dollars for the interest on this bond from the coupon payment right now the market rate of interest is four point five percent so what can we do we can use that market rate of 4.5% in order to discount the cash flow right so we're just going to take 40 let me let me think of a place to put this here so to calculate this we're going to take 40 and then we're going to divide it by one plus our discount rate right which is going to be this 4.5% so we just look let me put in here hope okay one point view and there we go so when we calculate that out it's actually kind of come out and let me change to make it green to make this consistent to thirty-eight dollars and 28 cents and I've rounded here so I apologize if my number is slightly different than yours so all we've done is discount the cash flow here now what we're going to do and now we're going to multiply this this discounted cash flow year by the time period by this one right and that's going to give us thirty eight twenty eight again now you might think this is kind of mechanical why are we doing this well actually it's going to become obvious as we start going now into further time periods into the future so when we go for period two for example we're going to take that $40 cash flow we're going to discount it again now obviously this time it's being discounted two periods back because this is year two so we're not going to divide by this I'm not going to go through all the present value calculations that you could check out our other video if you need refreshing there but it's going to give $36.63 but now we're going to multiply it by the time period right that's what we're talking about when we set back with our formula that in the numerator we're talking about the time-weighted present value of the cash flows we're going to multiply each time we get the present value of the cash flow we're going to multiply it by the time so I'm going to take this two over here and I'm going to multiply that by this $36.63 and is going to give us 70 $3.26 likewise I'm going to continue to discount all the cash flows and bear in mind in gear 5 we're actually having a cash flow of 1040 because you're getting the face value of $1000 plus the $40 interest payment and so all I've done is I just discount these cash flows in each period and then when we look at these discounted cash flows here there's 3505 in period 3 we're going to multiply it by 3 and that's going to give us 105 dollars and sixteen cents now I actually I noticed that it's off by 1 cent there forgive me I used Microsoft Excel to do the calculations so the rounding is going to be off a little bit here but you get the idea so we go ahead and we take in each case we take the time period that's our T right that time period and we multiply it by the present value of the cash flow and that's going to give us the present value of the time-weighted cash flow okay now once we have that we can go and sum up all of these time-weighted cash flows the present value there's time-weighted cash flows that's going to give us four thousand five hundred and twenty three dollars and sixty one cents now to get it the price of the bond will the price of the bond is just the present value of the cash flows right not the time-weighted ones just the regular old-fashioned present value of the cash flows so we sum up all these numbers here and that gives us nine hundred and seventy eight dollars and five cents this makes sense that the bond is trading at a discount because the current rate of market or the market rate of interest is higher than what the coupon rate is right so people can get a higher rate of interest on the market than what our bonds pay so our bonds are trading at a discount for nine hundred seventy eight dollars and five cents so now in order to calculate the Macaulay duration what we're going to do is we're going to take the present value of the time-weighted cash flow and just divide it by the present value of the cash flows or the sum of the present value of the cash flows I should say which is the price of the bond right so we just take that four thousand five hundred and twenty three dollars and 61 cents and we divide it by nine hundred and seventy eight dollars and five cents now that's going to yield four point six three years remember we said that the Macaulay duration that macaulay done just breathing here is MACD the Macaulay duration is expressed in terms of years now what does that mean let's say that we had another bond and that we have this other bond that we were looking at and it had a Macaulay duration of 7.2 years that means that basically this person the investor who gets the 7.2 year bond is going to be exposed to more interest rate risk right more rate if there's a change in interest rates for example interest rates skyrocket up then that's going to affect the price of the bond with the higher duration more right because that's the longer the duration the longer it's going to take for the investor to receive all these cash flows right and so we can just look and say look there's going to be more volatility more interest rate risk the higher the duration that the bond has and now another nice thing we can do with Macaulay duration is we can use it to calculate a different type of duration which is called modified duration in which we're going to talk about in our next

Channel: Edspira

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Keywords: Duration, macaulay duration, bond, corporate finance, finance

Id: F7XnR7sKWiE

Length: 7min 49sec (469 seconds)

Published: Fri Apr 17 2015