Zero Replication Immunization

When it comes to immunization against interest rate changes, nothing beats a zero-coupon bond.

So how might you replicate that position as closely as possible over the investment horizon, using a coupon bond?

No, that wouldn't work. The Macaulay duration would be too short since coupons will be paid before maturity.

Yes! This is __zero replication__ immunization, where you would set up the portfolio with the same Macaulay duration, and then continually update it in order to keep it matched. Sure, there's some rebalancing that must be done, but the end result is more effective immunization.

No. That's not the best goal. That would leave the investor completely exposed to potential interest rate drops at the higher end of the curve.

Consider now that you have a portfolio of four bonds that have been selected in certain weights to get you that nine-year duration needed. You want to keep it there. The portfolio's cash flows start soon and extend until 12 years from now. If the 8-year to 10-year rates dropped in the yield curve, what is the last thing you would want to see in the short-term and long-term rates?

But suppose that a zero-coupon bond for the investment horizon that you need just doesn't even exist. There's nothing there to replicate. Do you think you could still pursue zero replication?

You're right!

Actually, you could.

But how?

Precisely.

No. That would pull you away from your goal. Instead, just mimic the theoretical instrument.

The zero-coupon bond is a simple discount instrument, and its price grows over time to par. At any time, its market price could deviate from the smooth path due to some change in the yield curve. But that's fine: just discount the face value at any point in time, and see what it does. Then match the cash flow yield of the portfolio to the zero-coupon bond's yield to maturity. Then no matter what path prices take, market value will be preserved.

Exactly! Think about that for a minute. If those rates both fall, fine. It's a parallel shift, and the duration match is meant to handle that. If they move in opposite directions, it's still not so bad. A steepening or flattening of the curve will allow price changes to offset each other somewhat. But if the rate for the investment horizon drops, that theoretical zero-coupon bond moves up in value. But you don't have that. Instead, you have shorter- and longer-term cash flows. If those rates rise, then what you're holding drops in value, at least relatively. That's no good. This curve change is called a positive butterfly twist, and it is a structural risk to immunization.

No, that's okay. That's a parallel shift, which the duration match is meant to handle.

No. That will only cause smaller changes. This is a steepening or flattening of the curve, so at least the bond values in the portfolio will offset each other somewhat as they change.

So how might you want to structure your bond portfolio to be more structurally sound?

No. The bullet portfolio is definitely better.

Absolutely!

The barbell strategy causes the problem: maturities too far on either side of the investment horizon, placing variance in the cash flows. But the bullet portfolio sees the vast majority of cash flows weighted right near the horizon of choice, minimizing that variance.

After all, recall how convexity is calculated. $$\displaystyle \mbox{Convexity} = \frac{\mbox{MD}^2 + \mbox{MD} + \mbox{Dispersion}}{(1 + \mbox{Cash Flow Yield})^2} $$ Convexity is the enemy of immunization of a single liability. Given a certain investment horizon and yield curve, duration is what it is, and yield is what it is. Convexity rises and falls with dispersion, and so minimizing convexity means minimizing dispersion. You still can't beat a zero-coupon bond for immunization. So just get as close to it as you can.

To summarize: [[summary]]

Use a bond with the same maturity

Set up the duration match, and keep it matched

Carefully select bonds with shorter maturities to get reinvestment income

They both move up

They move down as well

They move in opposite directions

No

Yes

Mimic the theoretical instrument

Mimic an available zero-coupon bond with the nearest maturity

As a bullet portfolio

With a barbell strategy

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