In this first workout. We are asked to calculate the approximate new price of the following Bond if interest rates were to increase by 50 basis points. We've got the settlement date of the bond. If you like the issue date on the purchase date of the 1st of January of 2014 maturity seven years later on the 1st of January 2047.

A coupon rate of 2.5% a yields maturity of 1.75% and a frequency of one which means we have one cash flow per year or annual coupon payments. The par value of a thousand and that expected increase in the yield of maturity of 0.5% For us to calculate the new price of the bond, so to do that, we need to identify modified duration and we're going to use the M duration function to get there. So we need the settlement dates and the maturity date then the coupon rates and yields maturity. And then finally the frequency of once per year. That will give us the modified duration which tells us that this bond is expected to move by 6.41% given a 1% change in interest rates in the opposite direction. So if we look into calculate the new price of the bond based on the modified duration calculation, we need first of all the current price or to get the current price the bond. We're going to use a negative present value function where we need our discount rates that yield of 1.75. We need the number of periods these seven years. We then need the amount of cash flow that we have for each payment. That is the coupon rate multiplied by our thousand of par value and then we need our future value which is the thousand of par value itself. This will give us the price for the bond of 1049.

So they expected change in percentage terms in the price of the bond is going to be negative of the duration. Since there's an inverse relationship between interest rates and price changes multiplied by 100 multiplied by the change in yields that we're modeling. So that will give us negative 3.2 telling us that the price of this bond is going to fall by 3.2% We can then translate that into value terms. We can take the current price and multiply it by that change in yield divided by a hundred turn it into a percent. To give us a 33.62 decrease in the price of the bond. Which will therefore give us the new price being the old Price Plus the change in the price. So 115.39.

We do have a slight problem though to demonstrate the weakness of modified duration in that it is only an approximation because we can calculate the price of this Bond just using our first principles so we can just pick up the formula that we use to calculate the price of the bond to begin with.

And then adjust this discount rates. By the change in the yield that we're modeling. So we're adding on 0.5 to the discount rates and we'll get a new price for the bond. Now this new price of the bond 1016.03 is the correct price of the brand because we present value the expected future cash flows at the discount rate. That is now half a percent higher. Using modified duration to get to the new price of the bond. We've come up with too low a price because modified duration assumes. We have a linear relationship between interest rates and bond prices.

To get to a better approximation. We need to go on to work out too, but then will allow us to calculate the convexity adjustment for that previous workout.

So using the bond from the previous example, whereas the calculate convexity based on a 1% change in yield to maturity. And we're also asked to calculate the convexity adjustment in percentage terms and then the expected new bond price including or incorporating that convexity adjustment. To calculate the convexity number or we need to do first of all is to Model A change in interest rates up and down from the original price. So let's put the original formula and we have the bonds price.

And then we can adjust that discount rate. First of all.

Up by 1% Okay, so I'm going to add on to that the C30 cell that I have here on the screen that will give me a new price for the bond if interest rates increase.

and then if we go down to c33 and adjust that discount rate I can deduct from the yield. The change in yield that we're modeling for this convexity calculation. So we're going to get two new prices for the bond. We've got a lower price of interest rates increase and a higher price if interest rates decrease.

From this we can now go on to calculate convexity. And I convexity formula says we need to take the lower price when interest rates increase and add to that the higher price when interest rates decrease and deduct from that two times the original bonds price the 1000 and 1049. We then need to take that which is our numerator and divide all of its. By two times the original price Times by the change in yield.

squared this will give us a convexity number of 20. 4.61 this convexity number can be somewhat hard to interpret but if we take that convexity number multiply it by 100 and multiply it by the change in yield that we're interested in for our bond from the previous example of half a percent. squared this will then tell us that the adjustment that we need to make in percentage terms to the price of the bond is going to be 0.062% This does tell us that the convexa2 adjustment is a relatively small adjustment to the price of the bond. And if you want to turn that into dollar terms, we can take that percentage amount divided by a hundred to turn it into a decimal form and multiply it by the original price of the bond.

Telling us that we need to adjust the price of the bond by 0.65 and then we can get the new price of the bond with the convexity adjustment. Then the number that we got with modified duration from the previous workouts. The 1015.39 plus the convexity adjustment of 0.65 to give us 1016.04. This 1016.04 that we now get with the modified duration calculation plus a convexity adjustment gets us almost exactly for the 1016.03 that we get as the present value of the bond using discounting calculations. So the price of the bond using discounting calculations is very very similar to the price of the bond with modified duration and the convexity adjustment. The remaining difference comes down to the fact that convexity is not a perfect adjustment, but gets us very close.