Hi there,

Anyone know how to calculate partials? Working through Blue Box #2 of Reading 20 and am having a hard time figuring out how to calculate the curve shift for the 20 year security. There’s a bridge concept on page 164 that describes how to do it via linear interpolation and I’m not getting the right answer as the solution.

Question:

Assume Haskell revises his yield curve forecast as shown in Exhibit 34: Yields for the 2-year through 10-year maturities each decline by 5 bps, and the yield for the 30-year maturity increases by 23 bps.

Exhibit 34. A Steeper and Less Curved Yield Curve **Yield Curve Shift**** Maturity ** Beginning Curve** Ending Curve **

**Curve Shift**2 Year 0.816 0.767 –5.0 3 Year 0.987 0.937 –5.0 5 Year 1.345 1.296 –5.0 7 Year 1.649 1.600 –5.0 10 Year 1.935 1.885 –5.0 30 Year 2.762 2.991 23.0 2s–30s Spread 1.946 2.224 28.0 2/10/30 Butterfly spread 0.292 0.012

Using the data from Exhibit 34, we compare the partial durations of the two portfolios Haskell is considering:

**Key Rate PVBPs**** Total ** 1 Year** 2 Year

**5 Year**

**3 Year****20 Year ****

**10 Year****30 Year**Pro forma portfolio (1) 0.0587 0 0.0056 0.0073 0.0126 0.0127 0.0014 0.0191 More barbelled portfolio (2) 0.0585 0 0.0096 0.0040 0.0074 0.0119 0.0018 0.0238

Which portfolio would Haskell prefer to own under this scenario?

Solution:

If the curve becomes steeper and less curved, intuitively Haskell should prefer Portfolio 1. Portfolio 1 has significantly more partial duration at the intermediate maturities of 3 and 5 years, as well as substantially less partial duration at the shorter (2-year) and longer (30-year) maturities compared with Portfolio 2. Portfolio 1 thus would be expected to outperform Portfolio 2 under a scenario in which the yield curve steepens and its curvature decreases. This is confirmed as shown in the following table, which estimates the portfolio change using the portfolio’s key rate durations:

Market Value Changes Estimated Using Partial Key Rate Durations

**Pro Forma Portfolio (1)****More Barbelled Portfolio (2) ** Maturity** Actual Shift (bps)**

**Portfolio Par Amount ($000)**2 Year –5.0 60,000 0.0056 16.8 0.0096 28.8 3 Year –5.0 60,000 0.0073 21.9 0.0040 12.0 5 Year –5.0 60,000 0.0126 37.8 0.0074 22.2 7 Year –5.0 60,000 0 — 0 — 10 Year –5.0 60,000 0.0127 38.1 0.0119 35.7

**Change ($000)****Partial PVBP****Predicted Change ($000)****Partial PVBP***20 Year*30 Year 23.0 60,000 0.0191 −263.5 0.0238 −328.4 Total −156.5 −239.5

**9.0**60,000**0.0014**−7.5**0.0018**−9.7