In this first workout, we are asked to identify which of the following bonds want a or Bond B is more sensitive to a change in interest rates, assuming a 1% change in yields. In this instance interest rates and yields or yield to maturity can be used fairly interchangeably in this circumstance. Here at Bond a has two years to maturity and Bombay has 10 years to maturity. But in the other aspects that we have here, these bonds are the same so they both have a 5% coupon rate and they both have a 5% yield to maturity. So let's go ahead and calculate the current price of the bond with the current yield of 5% for both Bond a and bond B. So if we use the present value function, we're going to need to use a negative sign in front to flip the sign around to give us a positive outcome. But we're going to take the discount rate being the yield of 5% We're then going to have the number of periods being the years to maturity and then the payment per period is going to be the coupon rate multiplied by our par value, which here we're going to assume is a hundred dollars and then we need the power value as well of 100.

So the price of bond a not a huge surprise given that the coupon rate matches the yield or if you like the required return of investors to give us the bond being priced at its par value. That is also the case for Bond B, since it also has a matching yield and coupon rate. However, if we go down to consider a change in interest rates so we're going to think about how our interest rates might change first of all by considering if yields increase by one percent So what I'm going to do is to copy this whole formula. And then just paste it in but I'm going to go back and amend the rate to add 1% to it. And you can see that as interest rates go up as the yield on this Bond now goes up to six percent. The coupon rate isn't enough on its own. So the bonds price Falls to below the par value giving the investor a capital gain as well. However, if we copy this to the right, we'll see that for Bond B. The price has fallen by substantially more in absolute terms, but also in percentage terms as well. So if we take the new price and divide it by the historic price and take one away, then we can see the percentage change in the value of that Bond. So you can see that when interest rates have risen and the bond price has fallen for these two bonds that are the same except for their time to maturity. There's a greater sensitivity for Bond B, which had that longer time to maturity. For completeness. Let's just look at what would happen if there is a decrease in rates. So again, let's just put that formula in there, but then deduct our 1% change in interest rates.

This will give us a higher price for the bond. But there's much greater sensitivity for Bond B again because it has that longer time to maturity. So what we can say is for bonds with longer times maturity. or for longer dated bonds There is more interest rate sensitivity. And longer dated bonds price is more sensitive to a change in interest rates than a shorted data Bond if everything else is the same. Let's go onto a workout 2. And again here we've been asked to identify which of the following bonds is more sensitive to a change in interest rates given a 1% change in yields. But here we have two new bonds Bond C and bond D. These two bonds both have 10 years to maturity. They both also have a 5% yield, but now they have different coupon rates. Soap on C has a 2% coupon rate and bond D has a 10% coupon rate. Let's go on again to see what impact a change in interest rates has on these bond prices. So first of all, we're going to need to calculate the bond prices themselves. So again a negative PV function. First of all, we need the discount rate 5% We then need the number of periods time to maturity being 10 years. We then need for Bond see the payment amount being the coupon rate times our par value. And again, we're going to use the hundred dollars and the hundred dollars of par value as well to give us the price for Bond C.

As we've calculated previously now not a huge surprise that Bond D has a much higher price because it has a much higher coupon rate. But what we're interested in here is how these prices change in percentage terms as interest rates change. So we're first of all going to consider what happens if interest rates go up by one percent. So let's grab that formula, but then increase the yield Or the discount rates by 1% The price of bonsee goes down. Not a shoe surprise when interest rates go up and the same as the case for Bondi as well the price Falls as interest rates for ice. However, we can see that if we calculate the percentage change in price. So new price divided by all price minus one. You can see there is a much greater sensitivity of bond Seas price as interest rates Changed by the same amount. So there's a 1% change in interest rate. But Bond C that let's not forget has the lower coupon rate is more sensitive to a change in interest rates. Let's confirm. This is the case for a decrease in yields. So if we put the original formula in and then deduct 1% from the yield we're going to higher price for Bond c and a higher price for Bond d as well, but if we take this new price and divide it by the original price take one away. You can see that there is a much greater sensitivity of this change in interest rates to bond C that has the lower coupon. lower coupon bonds will have more interest rates sensitivity or in other words. The bond price is more sensitive to a change in interest rates if we have a lower coupon rate.