Let's discuss cash flow immunization.
Imagine a company has liabilities over the next 4 years in different amounts, for example here. The company wants to buy some bonds of different maturities so that the cash flows the company expects to receive from the bonds exactly match the liabilities that the company has to pay. In this case let's just assume that all cash flows are on an annual basis.
So in this case you can see that if you buy several different bonds of different maturities, the cash flow from year 1 which is the 20 million dollars will be covered in part by all 4 different bonds. First the one year bond will pay off its face value plus its coupon, and then there will be a coupon from 2 year bond and 3 year and 4 year bond. Now the 2 year cash flow liability will be covered by 3 different cash flows, and so on, until the last cash flow in year 4 is only covered by the face value and coupon of the 4 year bond.
Now, in order to figure out how much of each bond you need to buy in order to precisely match all the liabilities, assuming that you know the coupon rates of the target bonds, the easiest thing to do is to start from the end, because you know for year 4 cash flow the only thing that is covering that 30 million dollar liability is the face value plus coupon payment of the 4 year bond. So that should be an easy enough problem to solve, to figure out how much of this 4 year bond you need to buy in order to have a sufficient cash flow to cover the 30 million dollar liability in year 4.
The next step would be to figure out how much of the 3 year bond you need to buy, in order to cover the remaining part of the year 3 liability. Remember that part of this liability will be covered by the already known coupon from year 4 bond, because you already know how much of the 4 year bond you bought. So again, you should just figure out how much of the 3 year bond you need in order to pay out the remaining bit of the year 3 liability.
And at this point you know how much of the 3 year and the 4 year bond you need to buy, and you need to go again backwards in time and figure out how much of the 2 year bond you need, because you know the total cash flow is 25 million, and you know the coupon you will receive from the 3 and 4 year bond, because you already know how many of these you are going to buy, so again you can solve for the 2 year bond.
And finally you move back to year 1, knowing the coupon for 2, 3, and 4 year bonds, and the size of your liability, you can figure out how much of the 1 year bond you need to buy to match the remaining part of the liability of year 1. So you can see that the nature of this problem is that you can solve it by going from the end where we know that we need only 1 bond to pay off this liability, and recursively go back in time until we figure out how much of each bond we need to buy in order to immunize all four of the cash flows.
Now let's examine how we would approach this problem in the context of the last portion of major assignment 3.
We have a situation where a company has a particular sequence of expected liabilities over 4 years, and it is planning to use 4 different bonds with different maturities, 1-4 years, in order to try to immunize itself against these expected liability cash flows.
As we discussed, we are going to start from the back end, from year 4 liability, which we are going to match with the 4 year maturity bond. Our liability remaining to cover is just going to be the 12 million dollars that we expect to need to pay in 4 years, there it is, 12 million dollars. So given that we need a cash flow of 12 million dollars, how many of these bonds do we need to buy?
We are expecting to cover this liability with the face value repayment plus the coupon payment on these bonds, so we will take this 12 million liability and divide it by the face value plus the face value times the coupon rate. Because we are receiving $1000 from each bond, plus we are receiving 8.69% of a thousand which happens to be 86 bucks, so you can see once we run the calculations we need to buy 11,040 and a bit of these 4 year bonds.
How much is the market price of these 11,040 bonds? Now we can look at our price column, you can see that each bond costs 98.88 percent of par value, so in order to calculate the price we take the target number of bonds, multiply by the par value, and also multiply by the price expressed as percentage of par value, so we need to pay 10.9 million or so because these bonds cost 98% of face value.
From here we can move on to year 3 cash flow. What is the liability that is remaining to cover by using our 3 year bond? We start with the initial value of the liability, which is 10 million dollars, but we need to subtract the expected coupon amount that we will received from the 11,040 4 year bonds that we will own at the time. So let's subtract the 11,040 quantity, times the face value (actually this face value, even though they are all the same, it doesn't matter), and times the coupon amount.
So at this point, we only need to cover 9 million or so dollars of liability, because approximately 1 million of this will be covered by the coupon that we are receiving from the 4 year bonds. And from here on we basically do the same thing, find the target number of bonds that we need to cover this liability, and the market value, and recursively move up to year 2 and year 1 liability, until you figure out the total number of bonds and their market value that you need to buy.