However, fewer portfolios will be both duration and convexity matched. Thus, the match on both measures will likely result in higher cost portfolio.

Immunization strategies are widely used in order to mitigate the effect of interest rate changes. Extensive research has been done on designing immunized portfolios. We now discuss some implications of this research. The duration on a portfolio of bonds is a weighted average of the duration of the individual assets that make up the portfolio.7 Let Xj be the proportion of bond i in the portfolio, Di be the duration of asset i, and Dp be the duration on the portfolio with N bonds.

There are obviously an enormous number of ways to construct a portfolio of a particular duration. For example, assume that a bond portfolio with a duration of 10 years is required. Further assume that four bonds are being considered with a duration of 6, 8,10, and 12 years. Simply holding the bond with a duration of 10 years would meet the constraint. Alternatively one-sixth of the money could be invested in the bond with 6 years' duration, one-fourth in the bond with 8 years' duration, and the remaining seven-twelfths in the 12-year bond. This results in a duration of 10 years since

Two different strategies have been explored: a barbell strategy and a focused strategy. The focused strategy finds a portfolio of bonds with each bond having a duration close to the duration of the liability. For example, if the liability is 10 years, then the bonds might have a duration between 9 and 11 years. The bond portfolio is focused around the duration of the liability. The barbell strategy uses bonds with very different durations, for example, 5 and 15 years. The 10-year duration would be met by one-half in the 5-year duration bonds and one-half in the 15-year duration bonds. The advantage of a barbell strategy is that there is no necessity to construct individual bond portfolios to meet each liability. Instead, liabilities of different duration can be met by selecting different mixtures of the 5-and 15-year duration portfolios.

These two strategies have been explored to determine which one better meets the goal of having the asset and liability mix equally sensitive to changes in interest rates. The empirical evidence gives some support to the focused strategy. The reason seems to be as follows. All duration measures are approximations of the effect of the true shift in interest rate patterns. When individual assets and liabilities have similar durations, these errors are similar. When the individual assets in a portfolio have different durations from the liabilities even though the portfolio has the same duration, the error patterns can be very different. This latter pattern is what occurs with a barbell strategy. Thus inaccuracies in the duration estimate explain in part the evidence tending to support focusing.

Before closing this discussion, one more facet of immunization should be discussed. Immunization is often presented as a passive strategy, and therefore one by which a set of bonds is purchased and held to maturity. This impression is incorrect. Duration is calculated for a particular yield curve. As the yield curve shifts, duration changes and the assets and liabilities may no longer have the same duration. If the differences become large enough,

7This is a property of most duration measure. For the duration measure discussed here, it only holds if the yield curve is flat.

n i=i restructuring is required. Furthermore, even if the yield curve stays constant, the duration of the assets and the liabilities will move apart unless both assets and liabilities have the same cash flow pattern. This also requires restructuring. Thus immunization is an active strategy.

What are the risks of an immunized strategy? The principal one is the selection of the wrong duration measure. Each duration measure is derived assuming a different pattern of shifts in the yield curve. A portfolio is usually immunized using one measure but not another. For example using a measure that accurately measures price change for parallel shifts in a flat yield curve will not accurately measure price change if the yield curve steepens (long rates increase more than short).

Lest the reader become overly concerned, it is worth repeating that even the simplest measure discussed in this chapter works very well. The second risk of immunization concerns major yield shifts when the portfolio is not immunized. As discussed previously, either the passage of time or small changes in the yield curve will result in the portfolio not being immunized. Cash flows from the portfolio are used to purchase bonds to rebalance the portfolio so the duration of assets is closer to the duration of liabilities. Bond sales and purchases (rebalancing) could also be used to immunize the portfolio exactly. However, bond swaps are costly, so that a manager will let the duration of the assets drift away from the duration of the liabilities and not be immunized at all points in time.8 The risk is that just before the manager engages in a bond swap to adjust the duration in order to immunize the portfolio, the interest rates may change dramatically.

A cash flow matched portfolio is, of course, immunized. Since its immunization comes from matched cash flows rather than the accuracy of a measure, it is generally less risky. Thus, the immunized portfolio has to be less costly than the cash flow matched portfolio for an organization to immunize. It is often optional to cash flow match part of the portfolio and immunize the remainder.

In this section, we have presented techniques for protecting against interest rate shifts. In the next section we discuss techniques for constructing portfolios when performance over a one-year period is being evaluated.

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