# Fixed Income PM (1): Shrewsbury Case Scenario

Extract of question:

Silver and Shrewsbury begin discussing a client that sponsors a US DB plan. The client wants to immunize the liabilities such that changes in interest rates under various scenarios will not cause a deterioration in funded status. Key data for the plan assets and liabilities are provided in Exhibit 2. Silver’s forecast is that interest rates will rise in a non-parallel fashion. In fact, he expects a bear steepening of the curve as inflation accelerates because of rising wages.

Exhibit 2

Defined Benefit Plan Characteristics

Description Assets Liabilities
Market Value in USD 517,342,000
Liability, PBO* in USD 500,000,000
Macaulay Modified Duration 12.66 13.10
Convexity 21.40 22.51
Dispersion 6.48 6.70
Cash Flow Yield (%) 4.90 4.50
PV01 654,281 684,276

Q. Based on the data in Exhibit 2, will the client discussed most likely be able to immunize its DB plan given the interest rate scenario described by Silver?

A. Yes
B. No, because of the differences in money duration
C. No, because of the differences in convexity and dispersion

C is correct. The money duration of the assets and liabilities are equal: 517,342,000 × 12.66 = 6,548,381,000, and 500,000,000 × 13.10 = 6,548,381,000. For parallel changes, the equal money durations and PV01 imply that assets and liabilities would move in tandem. Silver expects a bear steepener; that is, long rates will rise faster than short rates. In a bear steepener, long rates rise faster than short rates in a non-parallel fashion. Given that the assets have lower convexity and dispersion than the liabilities, they will underperform; that is, the liabilities would change by a greater amount than the assets.

A is incorrect because Silver expects a bear steepener; that is, long rates will rise faster than short rates. In a bear steepener, long rates rise faster than short rates in a non-parallel fashion. Given that the assets have lower convexity and dispersion than the liabilities, they will underperform.

B is incorrect because the differences in convexity and dispersion are unfavorable; that is, they are lower for the assets than for the liabilities. If the opposite were the case, then the liabilities would be immunized.

Isn’t PV01 the money duration? And the PV01 is larger for liabilities than for assets.

I get it that money duration is calculated as ModDur x Value of Asset/liability, but why does PV01 show a different figure for Asset and liabilities?

Yes, and yes.

Perhaps they’re typos?

Don’t fret it. The real exam will not have such errors.

I dont get the answer to this question. why cant we immunize this portfolio?

Can anyone please explain why C?
Duration of asset portfolio is lower than the duration of liability portfolio, in a scenario where interest rates are expected to rise, shouldn’t the asset portfolio outperform? (i.e. fall less)