Percentage change in portfolio

CFA Level III
#1

A bond portfolio’s duration is 6.0, in below situations:

  1. given a parallel shift in the yield curve of 60 bps for treasury yields

  2. If the OAS for all bond sectors changes by 60 bps while tresury yields remain unchanged

The portfolio will change in 3.6% in value.

Why situation 1 and situation 2 will have the same effects?

#2

The won’t have the same effect, but the effects will be similar.

Where, out of curiosity, did you get this question?

The author made a common mistake: thinking that a change in the OAS is the same as a change in the (par) yield curve. In fact, the OAS is not a spread added to the par curve; it’s a spread added to the forward rates in a binomial interest rate tree. And a 60bp change in all of the forward rates is not the same as a 60bp change in all of the par rates.

I hate when authors do this: they’re trying to be too clever, and they really don’t understand the underlying ideas fully.

#3

What if the underlying bonds were optionless, or the yield curve was flat?

#4

In a binomial interest rate tree you won’t get the same effect changing the OAS as changing the par curve, even with option-free bonds and a flat yield curve.

#5

I’m only drawing this out in my head, but if the forward rates are the same, and cash flows are independant of interest rate movements, shouldn’t the spread applied to all the forward be the same as the ones applied to the par curve?

If not, then I guess the only condition is a zero standard deviation.

#6

It turns out the answer’s no.

Here’s a simple example:

  • 1-year par rate: 2%
  • 2-year par rate: 4%

Then the 1-year spot rate is 2% and the 2-year spot rate is 4.0408%.

Further, the 1-year forward rate starting today is 2%, and the 1-year forward rate starting in 1 year is 6.1224%.

Let’s add a spread of 100bp to the par curve:

  • 1-year par rate: 3%
  • 2-year par rate: 5%

Then the 1-year spot rate is 3% and the 2-year spot rate is 5.0510%.

Further, the 1-year forward rate starting today is 3%, and the 1-year forward rate starting in 1 year is 7.1429%.

So the spread added to the 1-year forward rate starting today is 100bp, but the spread added to the 1-year forward rate starting in 1 year is 102.04bp.

Granted, it’s not much of a difference from 100bp across the board, but it is a difference.

You can imagine if you had a 30-year yield curve: the discrepancy would increase the farther you go along the forward curve. If we extrapolate linearly (not a good idea, but it provides a glimpse): a 160bp change to the 1-year forward rate starting in 30 years. That’s quite a difference from 100bp across the board.

The upshot: changing the par curve with a parallel shift is not equivalent to changing the forward rates with a parallel shift, so a 60bp parallel shift in the par curve will not have the same effect as a 60bp change in OASs across the board.

#7

It is from Kaplan Schweser’s practice volume.

#8

Thanks for the illustration!

#9

Well, you can tell them from me that they’re wrong.

That’ll make their day.

#10

My pleasure.