Immunization & Convexity

Does anybody know why the convexity of assets must be greater than liabilities if multiple, but not with a single liability? I get that the purpose is to reduce structural risks for a single liability but don’t understand the reason conceptually.

With single liability you want convexity of assets to be not less than convexity of liabilities, nevertheless you want to reduce structural risk (BPV of assets equal of BPV of liabilities) if you have assets with greather convexity the BPV of assets will be greather than the liabilities of for example interest rates decrease.

I appreciate the response, but not quite following.

You want BPV of assets to = BPV of liabilities for multiple liabilities (which is based on modified duration), but for single liabilities, you want to match the Macaulay durations.

And for single, since already (Macaulay) duration matched to protect from parallel changes, my understanding is that you want to minimize convexity to protect against any structural risks, so you want convexity less than the liability (I think?) … but not positive

The convexity for assets can’t be less, a parallel shift of interest rates downward would appreciate the liabilities more than the assets

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That would make sense. But then both single and multiple immunization, the rule would be the same: Ca > Cl, and minimized after.

But the text repeatedly mentions the 3rd rule for single immunization is “minimizes the portfolio convexity statistic” … or “Structural risk to immunization arises from some non-parallel shifts and twists to the yield curve. This risk is reduced by minimizing the dispersion of cash flows in the portfolio, which can be accomplished by minimizing the convexity statistic for the portfolio”

Compared to multiple immunization, where CFAI specifically calls out the 3rd rule as “the dispersion of cash flows and the convexity of assets are greater than those of the liabilities”

@S2000magician, any chance you can provide some insight here?

By selecting the portfolio with matching duration and the lowest convexity, you are minimising the dispersion of cash flows around the Macaulay duration, making the portfolio closer to the zero coupon bond that ideally will provide perfect immunisation. convexity of the asset is not and cannot be lower than that of the liability. Remember that a zero coupon bond would be the ideal immunisation strategy so we find an asset with matching duration to the liability and with lowest convexity because this convexity is higher than that of a zero coupon bond and this will provide the best immunisation. Hope this helps.

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Very helpful. Thank you both @Tomiisinoyee and @Julio77

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