# Bond Immunization

Bond immunization is closely related to defeasance which I discussed in posts Defeasance2.0 and Defeasance. The similarity between immunization and defeasance is that you are trying to hedge the interest rate risk involved with a given liability cash flow. The example on the spreadsheet “Immunize” may be over simplified, but I think it describes the concept sufficiently. Normally you would be using a portfolio of many bonds to immunize, but I have chosen to use one bond for the sake of ease. The objective of my example, is to match the duration of the assets (bonds) with the maturity of the liability cash flow. Assume you are an investment manager of an insurance company. You take in $1,000,000 from a client and promise a 6.75% return over the next five years, compounded semiannually. The cash flow of this liability will be a lump sum payment in five years. We’re going to assume a zero coupon bond, which would be the easiest way to hedge, is not available at the rate we need. We are working with a 25 basis point profit margin in this case, so we need to earn a 7.0% rate of return over the next five years. Again, for simplicity, we are going to assume that interest rates move up or down 300 basis points immediately, and stay there over the next five years and there are no transaction fees: So we have seven possible interest rate scenarios to assume could happen. In practice, with the possibility of almost infinite changes in interest rates over the next five years, the bond portfolio would need to be adjusted periodically, to maintain the correct duration. What would happen if we just purchased a 7.0% bond for five years, with these scenarios? Under this option, the bond matures in five years, so there is no market risk for the principal. The risk is the reinvestment of coupons at the prevailing rates. At the far right is the total semiannual rate of return for each level. As you can see, any drop in rates 200 basis points or more, would wipe out the 25 basis point profit margin. We are further assuming that the yield curve out in the five to seven year area is flat, so we purchase the same 7.0% bond at par, but now the maturity is six, not five years. You will notice on the left hand side below that the Macaulay Duration just so happens to be slightly over 5 years, which matches the maturity of the liability. At the end of five years, we need to sell the bond which will then have one year to maturity. As you can see, the gain or loss from selling the bond is approximately equal to the gain or loss of reinvestment. One offsets the other, so that the total semiannual rate of return is always very close to 7.0% and maintains the profit margin.