The idea of immunization is fairly straightforward: you have a known set of liabilities due at known future dates, and you want to create a portfolio that will fund those liabilities with a minimum of risk, at a reasonable price. Two types of immunization are discussed in the CFA curriculum: classical immunization and contingent immunization. We’ll cover the former in this article and the latter in the companion article.

Classical (Single-Period) Immunization

Classical immunization involves a single holding period (i.e., a single, known liability at a known future date); it has two goals:

- Provide sufficient money on the date that the liability is due to be able to pay the liability.
- Do not lose any value in the company if interest rates change.

To achieve the first goal, the present value of the funding bond portfolio must equal the present value of the liability, discounted at the expected rate of return on the portfolio.

To achieve the second goal, at a minimum the money duration of the bond portfolio must equal the money duration of the liability; equivalently, their basis point values (BPVs) must equal each other. This condition will ensure that the change in the value of the portfolio will equal the change in the (present) value of the liability for a single, instantaneous, relatively small, parallel shift in the yield curve. If the present value of the bond portfolio equals the present value of the liability, this condition is equivalent to saying that the effective duration of the bond portfolio must equal the effective duration of the liability.

Here’s where the curriculum and I disagree on a point: determining the duration of the liability. The curriculum refers only to the “duration” of the liability (without characterizing what sort of duration they mean), and it says that the duration of the liability is the time to maturity. So, without admitting to it, the authors are talking about the Macaulay duration of the liability. Unfortunately, they then equate the (Macaulay) duration of the liability to the (effective) duration of the funding portfolio, as the basis for immunization against interest rate risk.

Unfortunately, it doesn’t work. Macaulay duration is not a measure of interest rate risk. (If you’re a bit foggy on the differences amongst Macaulay, modified, and effective duration, look here.) Properly, we should compute the modified (or effective) duration of the liability (which will be slightly shorter than the Macaulay duration) and equate that to the effective duration of the funding portfolio. The differences will be slight, I grant you, but as long as we’re learning how to do this, we might as well learn to do it correctly.

The effective duration of normal (i.e., fixed coupon) bonds is generally shorter than their maturity – perhaps much shorter, depending on the coupon rate – so matching the duration of the portfolio to the duration of the liability generally requires purchasing a bond whose maturity is longer than that of the liability.

*Example*

You have a liability of $2,000,000 coming due in 5 years; going with CFA Institute’s convention, we’ll assume that the (effective) duration of the liability is 5 years (though it’s really a bit shorter than 5 years). You have two bonds that you can use to fund the liability: a 4-year, zero-coupon, $2,000,000 par bond (really, 2,000, $1,000-par bonds) and an 8-year, zero-coupon, $2,000,000 par bond (again, really 2,000, $1,000-par bonds). The current spot curve is:

Maturity |
Spot Rate |

1 | 2.000% |

2 | 3.000% |

3 | 3.800% |

4 | 4.440% |

5 | 4.952% |

6 | 5.362% |

7 | 5.689% |

8 | 5.951% |

The 4-year ($2,000,000 par) bond sells at:

\[\frac{\$2,000,000}{1.04440^4}\ =\ \$1,680,980\]

and the 8-year ($2,000,000 par) bond sells at:

\[\frac{\$2,000,000}{1.05951^8}\ =\ \$1,259,475\]

The modified (and effective) duration of the 4-year bond is:

\[\frac{4\ years}{1.04440}\ =\ 3.83\ years\]

and the modified (and effective) duration of the 8-year bond is:

\[\frac{8\ years}{1.05951}\ =\ 7.55\ years\]

The present value of the liability is:

\[\frac{\$2,000,000}{1.04952^5}\ =\ \$1,570,639\]

The weighted average of the bonds’ present values has to equal the liability’s present value:

\[w_4\left(\$1,680,980\right)\ +\ w_8\left(\$1,259,475\right)\ =\ \$1,570,639\]

and the weighted average of the bonds’ durations has to equal the liability’s duration:

\[w_4\left(3.83\right)\ +\ w_8\left(7.55\right)\ =\ 5.00\]

I’ll spare you the linear algebra; the solution is:

\begin{align}w_4\ &=\ 0.7068\\

\\

w_8\ &=\ 0.3037

\end{align}

Therefore, the market value of the 4-year bond you need to purchase is:

\[0.7068\left(\$1,680,980\right)\ =\ \$1,188,188\]

which has a par value of:

\[\$1,188,188\ ×\ 1.04440^4\ =\ \$1,413,684\]

The market value of the 8-year bond you need to purchase is:

\[0.3037\left(\$1,259,475\right)\ =\ \$382,451\]

which has a par value of:

\[\$382,451\ ×\ 1.05951^8\ =\ \$607,319\]

Suppose that there is an instantaneous 50bp increase in interest rates, then the present value of the liability will be:

\[\frac{\$2,000,000}{1.05452^5}\ =\ \$1,533,755\]

the present value of the 4-year bond will be:

\[\frac{\$1,413,684}{1.04940^4}\ =\ \$1,165,704\]

the present value of the 8-year bond will be:

\[\frac{\$607,319}{1.06451^8}\ =\ \$368,315\]

and the total value of the portfolio will be:

\[\$1,165,704\ +\ \$368,315 =\ \$1,534,019\]

Note that the difference of $264 (= $1,534,019 − $1,533,755) is the result of the slightly different convexities of the portfolio and the liability.

By the way, if we had used 4.76 years (= 5 years ÷ 1.04952) as the duration of the liability (which is * not* what CFA Institute uses, but is the correct way to analyze the problem), the weights would be w

_{4}= 0.7446 and w

_{8}= 0.2533.

*Rebalancing*

Because the durations and values of the liability and the portfolio will change over time, the portfolio will occasionally need to be rebalanced. One common way to rebalance the portfolio is based on the * rebalancing ratio*: the dollar duration of the liability divided by the dollar duration of the portfolio.

*Example* (*cont.*)

Suppose that one year later the spot curve has shifted and flattened somewhat:

Maturity |
Spot Rate |

1 | 2.500% |

2 | 3.400% |

3 | 4.100% |

4 | 4.640% |

5 | 5.052% |

6 | 5.362% |

7 | 5.589% |

8 | 5.751% |

The present value of the liability is now:

\[\frac{\$2,000,000}{1.04640^4}\ =\ \$1,668,165\]

and, assuming that its duration is 4 years, its dollar duration is:

\[\$1,668,165\ ×\ 4.00\ years\ ×\ 1\%\ =\ 66,727\ dollar-years\]

The present value of the (original) 4-year bond is now:

\[\frac{\$1,413,684}{1.04100^3}\ =\ \$1,253,142\]

its effective duration is:

\[\frac{3\ years}{1.04100}\ =\ 2.88\ years\]

and its dollar duration is:

\[\$1,253,142\ ×\ 2.88\ years\ ×\ 1\%\ =\ 36,090\ dollar-years\]

The present value of the (original) 8-year bond is now:

\[\frac{\$607,319}{1.05589^7}\ =\ \$415,036\]

its effective duration is:

\[\frac{7\ years}{1.05589}\ =\ 6.63\ years\]

and its dollar duration is:

\[\$415,036\ ×\ 6.63\ years\ ×\ 1\%\ =\ 27,517\ dollar-years\]

The total dollar duration of the portfolio is, therefore:

\[36,090\ dollar-years\ +\ 27,517\ dollar-years\ =\ 63,607\ dollar-years\]

The rebalancing ratio is:

\[\frac{66,727}{63,607}\ =\ 1.0491\]

Thus, you need to purchase an additional 4.91% of the 3-year bond and the 7-year bond:

\begin{align}\$1,253,142\ ×\ 4.91\%\ &=\ \$61,468\\

\\

\$415,036\ ×\ 4.91\%\ &=\ \$20,378

\end{align}

Note that by using the rebalancing ratio, we’re not explicitly trying to match the present value and effective duration of the portfolio to those of the liability. This is a flaw in the rebalancing ratio approach that never seems to be mentioned anywhere. Well, except here, of course.

(To be fair, the differences are insignificant in this example. Nevertheless, it seems a little silly to use the rebalancing ratio when it’s just as easy to calculate the values that match the liability’s value and effective duration. Oh, well.)