# Immunization (finance)

In finance, interest rate immunisation, as developed by Frank Redington is a strategy that ensures that a change in interest rates will not affect the value of a portfolio. Similarly, immunisation can be used to ensure that the value of a pension fund's or a firm's assets will increase or decrease in exactly the opposite amount of their liabilities, thus leaving the value of the pension fund's surplus or firm's equity unchanged, regardless of changes in the interest rate.

Interest rate immunisation can be accomplished by several methods, including cash flow matching, duration matching, and volatility and convexity matching. It can also be accomplished by trading in bond forwards, futures, or options.

Other types of financial risks, such as foreign exchange risk or stock market risk, can be immunised using similar strategies. If the immunisation is incomplete, these strategies are usually called hedging. If the immunisation is complete, these strategies are usually called arbitrage.

## Cash flow matching

Conceptually, the easiest form of immunisation is cash flow matching. For example, if a financial company is obliged to pay 100 dollars to someone in 10 years, it can protect itself by buying and holding a 10-year, zero-coupon bond that matures in 10 years and has a redemption value of \$100. Thus, the firm's expected cash inflows would exactly match its expected cash outflows, and a change in interest rates would not affect the firm's ability to pay its obligations. Nevertheless, a firm with many expected cash flows can find that cash flow matching can be difficult or expensive to achieve in practice. That meant that only institutional investors could afford it. But the latest advances in technology have relieved much of this difficulty. Dedicated portfolio theory is based on cash flow matching and is being used by personal financial advisors to construct retirement portfolios for private individuals. Withdrawals from the portfolio to pay living expenses represent the stream of expected future cash flows to be matched. Individual bonds with staggered maturities are purchased whose coupon interest payments and redemptions supply the cash flows to meet the withdrawals of the retirees.

## Duration matching

A more practical alternative immunisation method is duration matching. Here, the duration of the assets is matched with the duration of the liabilities. To make the match actually profitable under changing interest rates, the assets and liabilities are arranged so that the total convexity of the assets exceed the convexity of the liabilities. In other words, one can match the first derivatives (with respect to interest rate) of the price functions of the assets and liabilities and make sure that the second derivative of the asset price function is set to be greater than or equal to the second derivative of the liability price function.

## Calculating immunisation

Immunisation starts with the assumption that the yield curve is flat. It then assumes that interest rate changes are parallel shifts up or down in that yield curve. Let the net cash flow at time ${\displaystyle t}$ be denoted by ${\displaystyle R_{t}}$, i.e.:

${\displaystyle R_{t}=A_{t}-L_{t}{\text{ for }}t=1,2,3,\ldots ,n}$

where ${\displaystyle A_{t}}$ and ${\displaystyle L_{t}}$ represent cash inflows and outflows or liabilities respectively.

Assuming that the present value of cash inflows from the assets is equal to the present value of the cash outflows from the liabilities, then:

${\displaystyle P(i)=0}$  [1]

## Immunisation in practice

Immunisation can be done in a portfolio of a single asset type, such as government bonds, by creating long and short positions along the yield curve. It is usually possible to immunize a portfolio against the most prevalent risk factors. A principal component analysis of changes along the U.S. Government Treasury yield curve reveals that more than 90% of the yield curve shifts are parallel shifts, followed by a smaller percentage of slope shifts and a very small percentage of curvature shifts. Using that knowledge, an immunized portfolio can be created by creating long positions with durations at the long and short end of the curve, and a matching short position with a duration in the middle of the curve. These positions protect against parallel shifts and slope changes, in exchange for exposure to curvature changes.[citation needed]

## Difficulties

Immunisation, if possible and complete, can protect against term mismatch but not against other kinds of financial risk such as default by the borrower (i.e., the issuer of a bond). It might also be difficult to find assets with suitable cashflow structures that are necessary to ensure a particular level of overall volatility of assets to have a proper match with that of liabilities.

Once there is a change in interest rate, the entire portfolio has to be restructured to immunise it again. Such a process of continuous restructuring of portfolios makes immunisation a costly and tedious task.

Users of this technique include banks, insurance companies, pension funds and bond brokers; individual investors infrequently have the resources to properly immunise their portfolios.

The disadvantage associated with duration matching is that it assumes the durations of assets and liabilities remain unchanged, which is rarely the case.

## History

Immunisation was discovered independently by several researchers in the early 1940s and 1950s. This work was largely ignored before being re-introduced in the early 1970s, whereafter it gained popularity. See Dedicated Portfolio Theory#History for details.