Macaulay Duration

let's understand the concept of Macaulay duration we know that a simple and period coupon bond with face value s is essentially a series of cash flows with final cash flow occurring at the end of period n question is what is the weighted average maturity of these cash flows and the answer is provided by Macaulay duration Macaulay duration is the weighted average term to maturity of a bonds cash flows let's take an example suppose that we have a bond with face value of $1,000 this bond pays semiannual coupons with annual coupon rate of 8% so that each coupon payment is 8% of the face value of a thousand divided by two which is the number of times the coupon is paid out in a year and this equals $40.00 the maturity of this bond is in two years and let's say the yield on this bond is 10% per annum with semi annual compounding since the bond pays semiannual coupons given the time in years the first coupon payment is made in half a year the second coupon payment comes in a year's time the third coupon payment is made in a year and a half and the final coupon is paid out along with a face value of the bond at the end of year two we know that each coupon payment is of 40 dollars but the final cash flow includes the $40 coupon plus $1,000 a face you which equals a thousand and forty we want to calculate the present value of these cash flows given the yield of ten percent per annum with semiannual compounding so first cash flow of forty dollars is discounted at ten percent with semiannual compounding over a period of half a year and this equals thirty eight point zero nine dollars similarly second cash flow forty dollars is also discounted at ten percent per annum that's semiannual compounding over a term of one year and this equals thirty six point two eight dollars discounting third cash flow forty dollars at ten percent per annum with semiannual compounding over the time period of a year and a half gives us thirty four point five five dollars and finally discounting the final cash flow of a thousand forty dollars at ten percent per annum with semiannual compounding over time period of two years gives us eight fifty five point six one dollars summing up these present values of cash flows gives us the current price of the bond of nine sixty four point five three dollars now let's calculate the weight of each cash flow which is essentially the present value of each cash flow divided by the current price of the bond so the weight of first cash flow is thirty eight point oh nine divided by nine sixty four point five three which equals zero point zero three nine five similarly the weight of the second cash flow is zero point zero three seven six the weight of the third cash flow is zero point zero three five eight and the wake of the final cash flow is zero point eight eight seven one these weights must sum up to one finally we will multiply the weight of each cash flow by the time when each cash flow is paid out so for first cash flow so we will get 0.02 and we're rounding to second decimal place here similarly for second cashflow we get 0.04 for third we get 0.05 and for fourth and final cash flow we get one point seven seven summing these up we get the weighted average term to maturity of one point eight eight years so Macaulay duration of this bond is one point eight eight years typically bond traders use duration as a measure to approximate change in bond price due to change in interest rates so in this example if rates go up by one percent bond price is expected to come down by one point eight eight percent but note that Macaulay duration is not the best measure of a bonds price sensitivity we will see in another video that modified duration is a better measure to approximate price changes Macaulay duration is better used to measure weighted average term to maturity of a bonds cash flows it follows from this that if we have an N period zero coupon bond then the Macaulay duration of this bond denoted by D with a subscript M a C equals N and that's because a zero coupon bond does not realize any cash flows until maturity so the weighted average term to maturity of a zero coupon bonds cash flows is the maturity itself so finally we can generalize the argument as follows if we have an end period bond with face value equal to F an annual coupon rate equals to C where coupon is paid n times a year and yield to maturity of the bond denoted by Y is also compounded M times a year then Macaulay duration of a bond equals one plus yield to maturity divided by M and this term is insured divided by the yield to maturity minus one plus the yield to maturity divided by M plus and over m times the annual coupon rate minus the yield to maturity and this whole term is divided by the annual coupon rate times one plus the yield to maturity divided by M to the power n minus one plus the yield to maturity so we have learned how to calculate Macaulay duration of a bond if there are any questions or comments please feel free to post thank you

Channel: finCampus Lecture Hall

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Keywords: Duration, Fixed Income, Macaulay Duration, Modified Duration, Interest Rate Changes, Income (Quotation Subject)

Id: dWFdP7fmEhM

Length: 7min 21sec (441 seconds)

Published: Sat May 25 2013