In this workout. We are asked to demonstrate that the following Bond will be able to be used to meet a liability of 50,000 in five years time if interest rates are currently 4% the bond that we're given as a settlement date of the 1st of January of 2030 and a maturity date of the first of January 2036 We've got six years therefore to maturity current yield of 4% The bond has a coupon rate of 9.35% and here we've given you the nominal value of a bond that we're going to need to be able to meet this 50,000 liability. in five years time The first thing that we need to be able to do is to calculate the present value of the bonds cash flows or the price of the bond. So using the PV function we take the discount rate of 4% the number of periods of six The cash flow per year of the coupon rate multiplied by the nominal value and then in the future, we're going to get the nominal value only in that last year to give us a to give us a price of the bond. Or 41,096. The duration of the bond is going to be useful for us here as well to demonstrate why this bond is going to be beneficial for an immunized approach. So we're going to use the duration function. Which needs the settlement date or purchase date the maturity date the coupon rate in percentage terms the yield and then our frequency is just going to be one. Because we have an annual coupon amount the key reason why we're using this particular bond is because it has a duration of five years that perfectly matches the time until the liability Falls due the second thing that might not be so clear at the moment is why we've got 32,095 of nominal value. Well, that is because We need to set the nominal value of the bond such that the market value of our portfolio matches the current value the present value of the liability. So if we were to take the liability amount in five years time and Present Value it at our current interest rates of 4% over five years we would end up with the present value of the liability. being the same at 41,096 as the value of the bond that we're holding that 41, And 96 up there in row 14. So essentially we have to hold sufficient nominal value of a bond such that the market value of the bond matches the current market value of the liabilities.

With this now will in place we can prove that buying 32,095 of nominal value will allow us to meet this liability of 50,000 in five years. To do that. We're going to look at a number of different interest rate scenarios. 4% to begin with they're in column D. And then we're going to think about what happens if interest rates change if they go down to 3% or up to 5% Before we start working through those numbers though. We need to have a look at the cash flows on this Bond. So the cash flows on the bond are going to be the coupon rates. The 9.35% multiplied by the nominal value of the bond that we hold. To give us just over 3,000 of a cash flow every year.

That will be received every year. until maturity And then in the maturity Year, we will also receive the nominal value back as well. The 32,095.

So we've now got our cash flows identified for us. Now if we go ahead and think about what those cash flows are going to be worth at Time 5 when the liability Falls due we should be able to prove that we're going to be able to meet that liability. If we take this cash flow that we've received at Time 1 we're going to be able to reinvest this for four more years up until time five when the liability Falls due. So we're going to need to multiply this by one plus our interest rate, which here is 4% And then take that to the power of how many years left. There are until the liability Falls due which we do by taking the liability date itself and subtracting away from that. the number of the time period that we are in we've reinvested this 3000. Just over 3,000 our 4% for four years, which means we end up with 3,510.6. in four years time now if we just adjust this formula a little bit so that For the cash flow. We always want to be looking at column C for the interest rate. I always want to be looking at row 23 only. For the number of years until the liability Falls do that will never change at five and for the number of the year. We always want to look at column B, but we want the rotate change. So now what we should be able to do is copy this formula down for the first five years. Up until the liability Falls due for payment. Every year we receive the same cash flow, but we get less reinvestment income as we invest it for a few a number of time periods until the liability Falls due. The year five cash flow is received at the end of the fifth year. So there's no time left to reinvest that until the liability Falls due.

We don't ever want to wait until year 6 to receive the cash flows on the bond. What we're going to need to do is take those years six cash flows and Present Value them back to where we're standing one year earlier at our interest rate or 4% So we're present valuing the remaining cash flows at time five of this Bond. But at time five there's one year left to go. So if we present value those cash flows, we'll get to the 33,746. So if we then add up all of these cash flows We will be able to demonstrate that we can exactly cover our liability the cash flows on this Bond will equate to being worth 50,000 at Time 4.

If we've set this formula up correctly and having locked onto the appropriate elements. So I should now be able to copy this across. For the scenario in column e if interest rates decrease down to 3% If interest rates decrease we get less reinvestment income than we would have got our 4% and we get that for every single coupon that we would have received except for the year 5 coupon where there's no time left to reinvest it. But then for the sale proceeds of the bond, we are applying a lower discount rate to present value the remaining cash flows on the bond, which results in US. Having $50,000 at time five, even though interest rates have fallen. Now we've got a little bit more than 50,000 because of the convexity effect. We've only used duration in our calculations. And we've calculated the actual value of the bond using properties counting calculations. So there's a bit of convexity benefit that we're feeling here, but we can demonstrate that we have at least 50,000 to be able to meet that liability at Time 5, and then equally if interest rates go up.

We're going to get the first five years. more reinvestment income at higher interest rates But when we get to the sale proceeds, we're going to present value those cash flows at a higher discount rates giving us lower sale proceeds than we would have had at either 4% or 3% interest rates. However, it doesn't matter for us. We still have enough money to cover that liability of 50,000. So it doesn't matter whether interest rates go down. We can still cover the 50,000 liability or interest rates go up. We can still cover that 50,000 liability at time five.