Immunization, Convexity, and Structural Risk in Liability-driven Investing

How to manage the duration and convexity of assets in a liability-driven investing context will be extensively tested on the CFA L3 exam. This article covers all the nuances around immunization and the structural risks inherent in the different approaches. 

Immunization with flat interest rate structures

The goal of an immunized portfolio is to earn the target rate of return required to meet the future value necessary to fund the liability. This target rate or IRR is different than the portfolio’s present YTM unless the term structure of interest rates is flat (because there is a return effect due to the passage of time). The implication is that:

  • For an upward-sloping yield curve, the immunization target rate of return will be less than the YTM because of the lower reinvestment return
  • For a downward-sloping yield curve this will result in an immunization target rate of return greater than the YTM because of the higher reinvestment return

How does convexity impact immunization?

If liabilities and assets are duration matched but not convexity matched, any economic surplus will be exposed to variation in value resulting from the convexity mismatch. Thus in a scenario with upward sloping interest rates and a large parallel shift in interest rates the convexity effect will cause the portfolio to outperform relative to the liability. This is because the initial decline in the portfolio’s value is overestimated vs. using duration alone. You might see this concept tested as a stand-alone CFA Level 3 exam question.

This is best seen visually:

Convexity and Interest rate Changes - CFA exam

Of course the opposite is true as well in cases of a downward shift in interest rates, we'll underestimate the price increase for that specific duration on the yield curve.

Why do we care about this?

Convexity Introduces Additional Considerations for Hedging Durations 

As investors concerned about hedging the specific risk of liabilities on the yield curve we want to know how well our asset choices will match the duration changes of the liabilities for any given shifts or twists of the yield curve.

Let’s take this scenario and unpack it for different scenarios in terms of duration and convexity assuming that we match the duration of our liability with two bonds, one that has a shorter and one that has a longer duration than the liability.

Imagine this is our starting point:

Immunization with parallel interest rates - CFA exam


We are hedging our medium duration liability with a short and long duration bond respectively.

Scenario 1 and 2: Parallel shifts up or down in the yield curve

As we just discussed if the yield curve shifts UP in a parallel fashion then the positive convexity of the assets will outperform our assumption of duration matching. 

But if the yield curve shifts DOWN in a parallel fashion the increase in portfolio value will roughly match the increase in portfolio liabilities. 

That's the base case situation. For the CFA L3 exam it gets more interesting if we think about what risks are introduced in a given immunization strategy for non-parallel changes (twists) in the yield curve. 

Scenario 3 and 4: Interest Rates stay the same but the yield curve STEEPENS or Flattens

Steepening Yield Curve: If the yield curve steepens that means interest rates are rising for longer duration bonds and falling for lower interest rates. In this case the decline in value of the long duration bond will be greater than the increase in value of the shorter duration bond. Our liability (a single point) will be unchanged in value, but our assets just decreased. Thus a steepening yield curve creates structural risk to our immunization strategy. 

Flattening Yield Curve: Our asset value will increase because the increase in value from the longer duration bond will be greater than the decrease in value from the shorter duration bond. The PV of our assets will now be greater than the PV of our liabilities.

Visually these two scenarios look like this:

Immunization and Structural Risk - Steepening and Flattening Yield Curve CFA exam


Scenario 5 and 6: Positive and Negative Butterfly Twists

For a POSITIVE butterfly Twist This is where lower and higher duration yields rise while those in the middle fall. The result is that our asset value will fall overall (as the sensitivity of the longer duration bond is higher) while the value of our liabilities will increase. Overall the structural risk of our hedge has increased.

For a NEGATIVE butterfly Twist – Yields on the lower and higher duration end decrease while yields in the middle of the curve increase. This will increase the value of our assets while simultaneously decreasing the value of the liabilities


Positive Butterfly Twists and Immunization CFA
Negative butterfly twists and immunization - CFA Exam

Summarizing Immunization and Convexity for the CFA L3 Exam

In summary, our strategy for managing risk will be as follows:

  • Set the initial PV of Assets to be greater than or equal to the initial PV of Liabilities
  • Match the Durations of assets to liabilities
  • Minimize portfolio convexity (i.e. minimize the dispersion of cash flows around the liability)
  • Regularly rebalance the portfolio as duration or yields change. In this process you must consider the risk of NOT rebalancing against the costs of doing so

The CFA L3 exam might ask you which scenario exposes an investor to more structural risk or it might drill down into a specific situation. You should be able to manipulate the yield curve and extrapolate what will happen to the present value of the assets vs. the present value of the liabilities. But don't forget that just because structural risk increases that doesn't mean overall returns will suffer or that one strategy is better or worse than the other. It still depends on the IRR of the overall portfolio and how that is changed.