Modify duration estimates the change in the price of the bond for a given change in yield to maturity. And is often expressed as a percentage. The calculate modify duration as its name suggests. We need to modify duration or Macaulay duration. So we take our duration number and divide it by one plus the yield to maturity but that itself needs to be divided by the yields maturity itself needs to be divided by the number of payments per year. So if we have a semi-annually paid coupon on the bond, we would need to divide the yield by two before adding it to the one in the denominator in this formula. Modified duration can tell us about the risk of a bond because it estimates for us the change in the bonds price given a change in interest rates.

If you are trying to calculate the new price of a bond given a change in interest rates using modified duration, you need to ensure that you use the dirty or settlement price of the bond.

So let's have a look at a modified duration example here. We're looking at a bond that has got five years to maturity. It's got a coupon of 2% that is paid annually and a yield to maturity of 2% The Bond's price is gonna be it's par value since the yield equals the coupon. And it's Macaulay duration can be calculated as 4.81. To calculate modified duration. We then need to take the Macaulay duration of 4.81 and divide this by 1.02 that is one plus the yields maturity. We don't need to worry about the number of payments per year because they're going to be one since we have an annual coupon. This will give us a modified duration of 4.71 which we can interpret as telling us that if there is a 1% change in yields. There will be approximately a 4.71% change in the price of the bond in the opposite direction. If we do want to actually calculate the change in a Bond's price using modified duration. We then need to take the modified duration number as a whole number the 4.71 multiply it by the change in yield that we are modeling or estimating and then finally multiply that by the current bond price. There is a negative symbol at the beginning of this formula just to pick up the fact that we have an inverse relationship between One prices and use So if we were trying to model the change in the price of this Bond if interest rates were to rise by 0.5% or 50 basis points, we would then be able to say that the change in the price of a bond would be estimated using modified duration to be the negative 4.71 modified duration multiplied by the change in yields that we're interested in of half a percent in decimal form. So a rise of 0.005 so positive and multiply that by the current price of the bond To give a change in the bonds price of minus 2.31. Given the price is expressed as the price per hundred dollars of par value. This would be a fall in price of 2.36.