Assume a portfolio that is being used to immunize a single liability (sometimes called a bullet) due in five years, with the portfolio and liability D initially matched at the 5.0 medium (M) duration of the liability, the initial value of the portfolio at the discounted (at portfolio IRR) PV of the liability, and portfolio convexity being greater than convexity of the single cash flow liability.

Assume that the portfolio is made up of two bonds with a shorter (S) and longer (L) duration than the liability duration (M). This describes a barbell portfolio strategy, concentrating the assets in longer and shorter duration around the liability’s single (bullet) duration.

If the yield curve shifts up or down in parallel fashion, the portfolio results will slightly exceed the amount required to pay the future liability. Duration matching alone would have led to meeting the future liability need, but the additional positive convexity of the assets will lead them to outperform duration results alone for large parallel shifts in the curve.

- For a large parallel increase in the curve (
**Parallel Yield Shift Up**), the immediate decrease in portfolio value will be less than the decrease in the PVL due to the positive convexity effect. With the parallel increase, the new portfolio IRR will increase by basically the same amount as the increase in discount rate for the PVL. In other words, the future rate of increase in A and L are still the same, but starting from a new PVA that is relatively higher than the PVL, the FVA will exceed the FVL. - For a large parallel decrease in the curve (
**Parallel Yield Shift Down**), the immediate increase in portfolio value will exceed the increase in the PVL due to the positive convexity effect. With the parallel decrease, the new portfolio IRR will decrease by basically the same amount as the decrease in discount rate for the PVL. In other words, the future rate of increase in A and L are still the same, but starting from a new PVA that is relatively higher than the new PVL, the FVA will exceed the FVL.

*Parallel Yield Shift Up*

*Parallel Yield Shift Down*

The parallel shift analysis indicates that the duration (rather than cash flow) matching immunization strategy does have **structural risk**. The structural risk is due to creating portfolio duration with a different allocation of asset durations (L and H) versus the allocation of liability durations (M only). That can lead to differing performance of the assets and liabilities as the yield curve shifts. Fortunately, most interest rate changes can be described as roughly parallel, and by building the portfolio with an asset dispersion (hence, convexity) that exceeds the single liability payout date, the portfolio benefits from the structural risk.

The parallel shift analysis:

- Indicates that immunization can be described as
*zero replication*. A single zero-coupon bond could have been used for a no-risk, perfect cash flow match. The changes in portfolio value and IRR have replicated (or done better due to positive convexity) the changes in yield and value of that replicating zero-coupon bond. In general, if the change in portfolio IRR matches change in yield of the replicating zero, the risks for the strategy are low. - Does not indicate the strategy is always structurally risk free. Other kinds of yield curve reshaping may or may not cause the strategy to fail in meeting the future payout. These other reshapings are discussed shortly.
- Does indicate the parallel shift assumption is sufficient to lead the strategy to succeed—but it is not a necessary assumption because the strategy may still be successful in other conditions. It is
*sufficient,*but not*necessary*. - Does not mean the strategy is buy and hold. Coupon-bearing bond duration declines more slowly than maturity, while the bullet liability duration will decline linearly with the approaching pay date. To maintain the immunization, the portfolio assets must be continually rebalanced to continually match portfolio to liability duration as time or market conditions change; otherwise, the strategy is at risk.

If the curve either steepens or flattens, the analysis becomes more complex and the structural risk increases. Assume for this discussion that rates do not change for the M duration liability, but move in roughly opposite directions for the L and H duration assets.

- Steepening twist (
**Steepening Twist**): Yield L decreases while yield H increase relative to yield M. The portfolio market value will decrease because the decline in value of the longer duration bond will exceed the increase in the value of the shorter duration bond. PVL will be unchanged with no change in yield M. PVA is now below PVL. That, by itself, does not indicate the strategy will fail. If portfolio IRR increases sufficiently, the required FV might still be reached. (Recall that portfolio IRR would tend to increase above a single point M YTM with a steeper curve). This indicates that a steepening curve may create structural risk. *Steepening Twist*- Flattening twist (
**Flattening Twist**): Yield L increases while yield H decrease relative to yield M. The portfolio market value will increase because the increase in value of the longer duration bond will exceed the decrease in the value of the shorter duration bond. PVL will be unchanged with no change in yield M. PVA is now above PVL. That, by itself, does not indicate the strategy will succeed. If portfolio IRR decreases sufficiently, the required FV of assets to meet the payout may not be reached. (Recall that portfolio IRR would tend to decrease below a single point M YTM with a flatter curve). This indicates that a flattening curve may create structural risk. *Flattening Twist*- Positive butterfly twist (
**Positive Butterfly Twist**): Yield L and H increase while yield M decreases. The portfolio market value will decrease as both yield L and H increase. PVL will increase as yield M decreases. PVA is now below PVL. That is certainly detrimental, but it is possible the strategy could succeed if the portfolio IRR increases enough versus the decrease in liability discount rate. This indicates that the positive butterfly may create significant structural risk. *Positive Butterfly Twist*- Negative butterfly twist (
**Negative Butterfly Twist**): Yield L and H decrease while yield M increases. The portfolio market value will increase as both yield L and H decrease. PVL will decrease as yield M increases. PVA is now clearly above PVL. That is certainly favorable, but does not guarantee the strategy will succeed if the portfolio IRR decreases too much in relation to the increase in liability discount rate. It again indicates the possibility of significant structural risk. *Negative Butterfly Twist*

The risk in immunization is higher when the change in portfolio IRR does not match the change in yield of the replicating zero or is insufficient to fund the liability at the new level of rates. This structural risk can be reduced by reducing the dispersion of asset cash flows around the liability cash flow. This is not surprising because if you make dispersion 0, you have a zero-coupon bond and a perfect cash flow match to the single liability. Now, recall the earlier equation for determining convexity from duration and dispersion; reducing dispersion is directly related to reducing convexity. This leads to the rules for immunizing a single liability:

- Initial portfolio market value (PVA) equals (or exceeds) PVL. (There are exceptions to this for more complex situations where the initial portfolio IRR differs from the initial discount rate of the liability.)
- Portfolio Macaulay duration matches the due date of the liability (D = D
_{L}). - Minimize portfolio convexity (to minimize dispersion of asset cash flows around the liability and reduce risk to curve reshaping).
- Regularly rebalance the portfolio to maintain the duration match as time and yields change. (But also consider the tradeoff between higher transaction costs from more frequent rebalancing versus the risk of allowing durations to drift apart.)