Distribution of durations of assets vs. liabilities when immunizing a multiple liabilities in the case of a parallel shift in the YC

CFA Level III Portfolio Management
#1

This is from a past CFA exam: 2017 AM session Question 9C:

Three conditions must be satisfied to assume multiple liability immunization in case of a parallel rate shift:

  1. PV of the assets = PV of liabilities
  2. Composite duration of assets = Composite duration of liabilities
  3. Distribution of the durations of individual portfolio assets must have a wider range than the distributions of assets:

I understand 1 and 2 are necessary but why 3? Also, does that change if it’s a non parallel shift?

Thanks!

#2

The idea is to try to minimize the combination of reinvestment risk and price risk.

If the shortest bond you own has a maturity after the first liability you need to fund, then you’ll have to sell that bond for an unknown price, leading to (possibly considerable) price risk.

If the longest bond you own has a maturity before the last liability you need to fund, then you’ll have to reinvest the last coupon plus par, leading to (possibly considerable) reinvestment risk.

Ideally, you would like the first bond to come due the same day that the first liability does, and the last bond to come due the same day that the last liability does. When that is impossible (which is, for all practical purposes, always), you get the maturity of the first bond to be a little less than that of the first liability, and the maturity of the last bond to be a little more than that of the last liability.

Finally, as maturity goes, so goes duration (for the most part).

#3

Only one doubt: if the maturity of the last bond exceeds the maturity on the last liability, how can I meet the liability payout? I should sell the bond before its expiration, coming back to price risk…isn’t it?

#4

I understand that the shortest bond should have a maturity before the first liability but I don’t understand why the longest bond should mature after the last liability. Wouldn’t you want the longest bond to mature before the liability is due? Also, in the answer, they picked a portfolio (portfolio B) where the maturity of the shortest bond is after the first liability that is due a year from now:

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#5

Yes, there’s some price risk. As I said, you’re trying to minimize price/reinvestment risk; you can’t eliminate it.

If all of your bonds mature before their respective liabilities come due, you may have difficulty matching the duration of the liabilities, which is a given.

#6

Reread the last sentence in the first paragraph of the vignette: they’re explicit about minimizing reinvestment risk, not price risk.

The generalities of the theory have to give way to the specifics of the vignette.

1 Like
#7

Thanks Bill! understood, well we need to be very careful because at first glance I would select bond C

#8

Of course, many candidates will claim that that’s a trick question.

And, of course, it isn’t.

Read the <blessed> vignette!

#9

The devil is in the details…I think this is a good lesson on how to read the material, sincerely I would have not linked that sentence with the question B. Thanks, Bill!

#10

Thanks for being helpful as always.

#11

Just curious about the definition of composite duration.
The average duration is the market value weighted average of even bond’s duration.
And the composite duration is the cashflow based duration?

I remember that they said that in an upward trending yield curve, these two might be different?

#12

That’s correct.