Modified Duration

in this video we will discuss the concept of modified duration let's consider a bond with face value of thousand dollars the current price of this bond is 964 point five four dollars the bond matures in two years time coupons on this bond are paid semi-annually at an annual coupon rate of eight percent and the yield on this bond is ten percent per annum with semiannual compounding and finally Macaulay duration of this bond is one point eighty eight years bond traders often use Macaulay duration as a measure of bonds price sensitivity to changes in interest rates but we know that Macaulay duration is better used to estimate weighted average term to maturity of the bonds cash flows however modified duration denoted by D with a subscript mov is a better measure of approximate sensitivity of bond price to interest rate changes and is calculated as follows bond price if he'll declines - the price of the bond if yield increases divided by two times the initial price of the bond times change in yield denoted by Delta Y so in case of our bond if yields were to increase from 10% to 11% the new price of the bond equals nine forty seven point four two dollars we assumed in this video that viewers are comfortable with pricing a bond which is covered in the video on bond pricing on the other hand if yield declines from 10% to nine percent the new price of the bond will equal nine 82.0 six dollars so we know what new bond prices would be if yields were to go up or down by one percent so in this example modified duration equals the price of the bond if yield were to go down by one percent minus the price of the bond if you were to go up by one percent divided by two times the initial price of nine sixty four point five four times the change in yield of one percent note that we got these new prices after changing yields by one percent each direction so in this example modified duration equals one point eight this means that if you goes up by one percent bond price will come down by one point eight percent and if you goes down by one percent bond price will go up by one point eight percent another quick way to calculate modified duration is by taking the Macaulay duration and dividing it by one plus the yield to maturity divided by M where m is the number of times yield is compounded in a year and in our example this equals Macaulay duration of one point eight eight years divided by one plus the yield of ten percent divided by 2 since its semi annually compounded and this equals one point eight which matches the modified duration we got earlier this modified duration is a measure of bonds price sensitivity to changes in interest rates that change in bond price can be estimated using modified duration as follows- of modified duration times the initial price of the bond times change in yield so when yield goes up four 10% to 11% the change in bond price as estimated by modified duration equals negative of modified duration which is 1.8 times the initial price of the bond which is 960 4.5 4 times the changing yield of 1% and this equals negative 17 point 3 2 dollars so the new price of the bond equals original price of nine sixty four point five four dollars - seventeen point three $2 which equals nine forty seven point two two dollars this new price estimated using modified duration is slightly less than the actual new price when yield is 11% but as we can see modified duration does a fairly good job at approximating the price change similarly when healed decreases from 10% to 9% the change in price equals negative modified duration of one point eight times the initial price of nine sixty four point five four times the change in yield of minus one percent since you decline from 10% to nine percent and this equals seventeen point three two dollars so the new price equals the original price of nine sixty four point five four dollars plus seventeen point three two dollars and this equals nine eighty one point eight six dollarz again we can see that duration has done a fairly good job in estimating the price change but this 980 1.8 $6 estimated using modified duration is again less than the actual new price of 982 point zero six dollars when yield declines from 10% to 9% so in both cases we can see that modified duration is a good proxy to approximate changes in bond price due to interest rate changes however in both cases modified duration has also underestimated price of the bond we can explain the reason for this graphically note that this yellow curved line shows the inverse relationship between the price of a bond and its yield when we calculate duration we are essentially estimating the gradient of a line tangent to this price yield curve at a given yield which in our case is 10% so any change in price resulting from changing yield would be approximated by this tangent line so when yield decreased from 10% to 9% the resulting new price was estimated by the straight tangent line which gave us the price of 980 1.86 dollars but that lies below the price yield curve which at yield of 9% would have given us the price of 980 2.06 this difference in two prices is called the measurement ever since this tangent line always lies below the price yield curve it will always underestimate advant price due to change in interest rates finally let's look at how eurasian performs when yield changes by a large magnitude suppose that u changes by 300 basis points so that yield increases from 10% to 13% in that case the bond price will go down to nine one four point three six dollars however new price approximated using modified duration equals the original price of nine sixty four point five four dollars minus modified duration of one point eight times the initial price of nine sixty four point five four times the change in yield of three percent which equals nine one two point five nine dollars on the other hand when yield decreases from 10 percent to seven percent the price of the bond at this new yield of seven percent equals one zero one eight point three seven dollars whereas the new price approximated using modified duration equals the original price of nine sixty four point five four dollars minus modified duration of one point eight times the original price of nine sixty four point five four times the changing yield of minus three percent since the you declined in this case from 10 percent to seven percent the change in yield is negative three percent and this new price equals one zero one six point four nine dollars there are two things we're noticing here in both cases whether the yields went up or down the new price estimated using modified duration has been below the actual price at that new yield and secondly the measurement error calculated by subtracting the duration adjusted new price from the actual new price at the new yield in both cases is greater than when it was when yields only changed by 1% so we can see that measurement error gets larger with magnitude of change in yield so it can be concluded that modified duration is a reasonable proxy to estimate changes in bond price due to small changes in interest rates but it does not do an equally good job as the magnitude of rate changes gets larger if there are any questions or comments please feel free to post

Channel: finCampus Lecture Hall

Views: 31,389

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Keywords: Modified Duration, Macaulay Duration, Bond Price, Bond Convexity, Fixed Income, Yield to Maturity, YTM, Bond (finance), Finance (Industry)

Id: hZMo7jVhfaY

Length: 11min 53sec (713 seconds)

Published: Sat May 25 2013