Bonds are frequently purchased to fund future liabilities because of the relative certainty of the cash flows which are set contractually. However, the certainty as to the value of the terminal value of those future cash flows depends upon two factors:

the rate at which the future coupons can be reinvested remaining unchanged;

if the bond has a maturity longer than the desired holding period, the level of interest rates is the same at the beginning and end of the holding period.

Thus one of the major reasons for not achieving the expected terminal value is interest rate risk. This can be decomposed into two parts: price risk and reinvestment risk. Price risk is the risk of secondary market price fluctuations resulting from fluctuations in interest rates. Reinvestment risk is the risk that coupon proceeds will have to be reinvested at rates different to the rate prevailing when the bond was purchased. Price risk will apply to all holdings that have maturities longer than the desired holding period. Reinvestment risk applies to all holdings, irrespective of the holding period.

It is to be expected that price and reinvestment risk off-set each other, at least to some extent. As interest rates rise the interest on reinvested coupons will rise, but the price of the bond will fall. As interest rates fall the price of the bond will rise, but the interest on interest will fall.

Immunization is a strategy for ensuring that the expected future value of a bond holding matches the future value of a liability. It entails matching the modified duration of the bond portfolio to the duration of the liability and equating the present value of the liabilities to that of the portfolio. If the liability does not require intermediate future cash flows, the duration of the liability will be equal to its maturity.

Before we proceed to explain how futures can be used in the immunization process, it will be instructive to explain the classical immunization process using cash bonds.

Classical immunization

Redington (1952) noted that the duration of a bond represented that period when the effects of the price risk element of interest rate risk and the reinvestment risk exactly off-set each other and he introduced the term immunization. A portfolio is said to be ‘immunized’ when the effect of having to reinvest coupons at lower (higher) rates is off-set by the effects of those lower (higher) rates on the market value of the bond.

Imagine, for example, that immediately after the purchase of a bond, yields rise and stay at the new level for the remainder of the holding period. Also, consider two alternative portfolios. The first consists of a bond with a maturity equal to the holding period; the second has a maturity longer than the holding period, but with a modified duration equal to the holding period.

In the case of the first portfolio, coupons would be reinvested at higher than anticipated rates, and as the bond is redeemed at the end of the holding period, the terminal value of the investment will be greater than originally expected. It is a simple extension of this example to show that if rates had fallen immediately after purchase, the terminal value would have been less than originally expected.

In the case of the second portfolio, it was the duration — not the maturity — of the bond that matched the holding period. Therefore the effect of, say, a rise in yields causing a capital loss when the bond is sold is off-set by the higher interest on interest received on the reinvested coupons.

Thus in order to immunize a portfolio against a single future liability, the general principle is that: the modified duration of the portfolio must match the holding period, and the present value of the portfolio must equal the present value of the liability. The duration of a portfolio is simply the value weighted duration of the individual constituents of the portfolio.

As the duration of a bond will change as yields change, the immunization of the portfolio must be frequently monitored, and when the portfolio duration is significantly different from the remaining duration of the liability, the portfolio should be re-immunized. This is achieved by selling some of the existing bonds and buying others, so that the resulting duration of the portfolio matches that remaining life of the liability.

The immunization of portfolios by equating duration of assets and liabilities is only perfect if the assumption of a flat yield curve that only shifts in parallel is valid. Unfortunately, yield curves are rarely flat and only infrequently shift in parallel. Indeed parallel shifts in the term structure account for only about 75% of interest rate risk in the major bond markets.

More sophisticated techniques of interest rate risk assessment using, for example, factor models of term structure risk can be applied, with the interest rate sensitivity of the future again being that of the cheapest to deliver bond.

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