 # R24 P163: "Partial PVBP" Epiphany

This formula from Reading 24, page 163 (underneath Exhibit 33) seems to be giving everyone fits:

Predicted change = Portfolio par amount × Partial PVBP × (–Curve shift)

First off, the confusion surrounding this topic is (as are most things) directly related to poor wording on the part of the CFA curriculum writers.

I came across the partial PVBV topic and didn’t fully understand its calculation or the logic behind the formula. After working through it, I could never get the decimal places to line up with the answers in the books either. After searching AF, it seems that there are only unanswered questions, nothing helpful. Meaning I’m not going crazy and others don’t understand this either (which makes me feel a little more sane haha).

So here goes: two issues seem to keep popping up:

1. Do you scale the Partial PVBP calculation by 100 to get the right answer?
2. If “PVBP” (and therefore Partial PVBP) is a figure & Portfolio Par Amount is a figure, aren’t you essentially squaring \$s?

Here’s what made sense to me- if it makes sense to others, please respond and/or critique.

1. The issue is not with the Partial PVBP number. On an earlier post, S2000 Magician guessed that it might be that Partial PVBP was supposed to be stated in % terms, but he was waiting for clarification from CFAI. Let’s take Exhibit 33 for example. The 2Y Maturity bond experiences an 18.3 Key Rate Curve Shift ( bps ) on a Portfolio par amount of 60,000 ( thousands) and has a Partial PVBP of 0.0056. If you multiply those #s out as per the equation below the exhibit (Predicted change = Portfolio par amount × Partial PVBP × (–Curve shift)), you get a number that is supposed to be in " thousands:" “-6,148.” However, Exhibit 33’s answer is “-61.5.” So why the difference of a factor of 100? Here’s the disgusting answer: You have to convert the “Key Rate Curve Shift” from bps to %. Look at the difference in wording of the formula in the Reading vs the formula in the End Of Chapter Solution to Q20. “Predicted change = Portfolio par amount × Partial PVBP × (–Curve shift)” VERSUS"Predicted change = Portfolio par amount × partial PVBP × (curve shift in bps)/100" (Forget the fact that the EOC Solution formula forgot to include a “negative” symbol for the curve shift portion). Once you convert the Key Rate Curve Shift to % from bps- i.e. divide by 100- all of the calculations will work.

2. Portfolio Amount is a figure. But PVBP (and Partial PVBP by extension) is not actually a " figure." It is a scaling number that you multiply by every \$1 of par value to get the change in price. Expand the calculations for the “PVBP” part of “Partial PVBP” to understand why you’re not squaring \$s. PVBP is Money Duration / 100. Money Duration is Modified (or Effective) Duration times the Portfolio Value per \$1 of Par divided by 100. Which means that the actual notation is “Portfolio Value in \$s divided by 1 unit of Par Value in \$s.” So the s cancel each other out. Put another way, if you rearrange the above equation to _Partial PVBP = Predicted change / (Portfolio par amount × (–Curve shift)_ and substitute "_0.0001 x (Effective duration x (Portfolio value / Portfolio par amount)"_ for _PVBP,_ the "Portfolio par amount" expressions cancel each other out and you are left with only 1 " figure" in the equation. Hence you are not squaring \$s.

I hope this helps and if anyone see anything wrong with mmy explanation / conclusions, please comment away!