An immunization strategy also referred to as classic immunization can be used to meet a single liability at some point in the future. The way that this liability is covered is by ensuring that the bonds that are purchased today have the same duration as that of the liability and the duration of the liability will be the time until the liability falls due such that even if there's a change in interest rates, we will still be able to meet that single liability in the future. Let's have a look at an example to put that into some context. Let's assume that we've got a liability in four and a half years time. And they're so happens to be a bond out there that's got a maturity of five years a coupon rate of 5.86% And we're going to look at maybe a par value of $100. If we were to look at that bond and calculate its duration macawly duration to be specific. We see that its duration would be 4.5 which would match the duration of the liability 4.5.

Now you'll notice from this that the maturity of the bond happens after the liability falls due. So what we're going to need to do is sell the bond on the date when the liability falls due to give us the cash proceeds that we need to be able to meet the liability.

But that won't be the only source of cash that we have to make the liability because this bond is also going to be paying us a coupon at the end of each of the next four years of 5.86% And those coupons can be reinvested up until that maturity date to earn us some reinvestment income. What immunization says is so long as we match the duration of the bond to our liability as we have in this example. We shouldn't be exposed to any changes in interest rates. We have here some examples of what might happen over time for this particular Bond. So we've bought $100 of par value of this bond. And as a result, we're going to receive. $5.86 of cash flow every single year up until the maturity of the bond and then also in the fifth year would also receive the $100 apar value if we held onto the bond until the maturity date. Now we're trying to meet a liability here after four and a half years. So we're not going to just wait to receive all of those cash flows in the second column of our grid. Instead what we're going to do is see what cash will have available at time four and a half.

The logic behind immunization says that it doesn't matter what happens to interest rates. So, let's see what happens if we have interest rates of 3% If we have interest rates of 3% the purple numbers on the middle of this table, we'd be able to take the $5.86 of a coupon that we receive after one year and reinvest it until the time when the liability falls due so that would be reinvesting it for a further three and a half years.

So if we can earn interest that 3% for three and a half years from time one to time four and a half that would mean that we would be investing the 5.86 after one year, but then getting that money back three and a half years later at time 4.5 and what we would get back if we were earning interest at 3% would be $6.49. The cash flow that we get at the end of the second year. We can only invest for two and a half years from time to time four and a half. Meaning that we get a bit less reinvestment income and would only end up with $3.61. The third year cash flow. We can only reinvest for a year and a half to time four and a half. So again less reinvestment income again to give us $6.12 and then the year 4 cash flow. We can only reinvest for half a year to time four and a half. So we end up with $5.96. We still get a bit of interesting income. When we get to time four and a half though, there's still half a year left to the maturity of the bond and we don't want to wait another six months to receive the 105.86. We're going to need to sell the bond. With six months to go to maturity to get enough cash to meet our liability. So we're going to need to present value the bond to identify what its price would be on that sale date six months prior to maturity. So we take the 105.86 and Present Value over six months at our 3% interest rates. We would get a sale price for the bond of 104.30.

If we were to add up all of those cash flows all of the coupons plus the reinvestment income on those coupons and the sale prices on the bond. We would get to 129.17.

Let's move on to the red numbers then but in the red numbers, what we're going to say is we're going to reinvest the coupons not at 3% but at 4% so we're going to earn slightly more interest income. So at the end of the three and a half years having reinvested the first year's coupon, we would end up with $6.72 not the $6.49 that we got when interest rates were only 3% that's gonna be the case for every single coupon that we've receive and can reinvest. We're going to earn a bit more reinvestment income at 4% rather than 3% but when we get to sell the bond, that's six months left to go to maturity. We're going to be present valuing at a higher discount rate for percent. So we're going to end up with a lower present value lower sale proceeds on the bond only 103.80. However, and this is really the strength of the immunization strategy we end up with the same total amount of money, the 129.17.

if we match duration Then the extra interest income exactly offsets against the lower sale proceeds. And that will always be the case for this strategy. Because the duration on a bond has to be less than it's time to maturity. So we're always going to have to sell the bond to get cash to meet the liability on the date when that liability falls you. Just for demonstration purposes if interest rates were 5% would earn even more interest income. But further down in the green numbers, we're going to get an even lower sale price, but the two changes still balance out to give us cash of 129.17 at time 4.5.

So what immunization here is showing us? Is that no matter how interest rates might change. We're still going to end up with the same amount of money at the time when the liability falls due.