Estimating market risk measures

[Imad]

Hi David

Hope you are well. Appreciate if you could explain below example taken from Qbank. I couldn't understand the relation btwn PVBP and VAR?

Thanks
Imad


Question 11 - #29487
The price value of a basis point (PVBP) of a $20 million bond portfolio is $25,000. Interest rate changes over the next one year are summarized below:

Change in Interest rates​

Probability​

>+2.50% ​

1%​

+2.00-2.49% ​

4%​

0.00-1.99% ​

50%​

-0.99-0.00% ​

45%​

<-1.00% ​

5%​

Compute VAR for the bond portfolio at 95 percent confidence level.
A)
$2,500,000.
B)
$2,750,000.
C)
$5,000,000.
D)
$12,500.
The correct answer was C) $5,000,000.
At 5% probability level change in interest rates is 2.00% or higher.Change in Portfolio value for 200 bps change in interest rate = 200*$25,000VAR = $5,000,000.

 

[David Harper CFA FRM]

Hi Imad,

The question assumes that an adverse change to the value of the (long) bond portfolio is a increase in the interest rate (higher rate --> lower bond price), such that the worst expected 95/5% outcome is the +2% increase in the rate (e.g., the 99% worst would be the +2.5% increase). I find it easier to be flexible w.r.t 5% or 95% and just focus on: we want the adverse tail (causing a drop in value) and we can perceive the adverse tail as either 95% or 5%, which is here an (presumed!) increase in the rate (although notice a better question would specify the portfolio is long; if it were short, the worst expected 5%/95% would be a 1% rate drop.)

I don't know why the probabilities don't appear to sum to 1.0 (part of the definition of a prob distribution is summation to 1.0)

finally, PVBP (which is more commonly referred to as DV01 in the FRM, but they are the same) is the dollar increase associated with a one basis point decline (so it's approx equal to dollar decrease associated with a one basis point increase). There are 200 basis point in +2%. So answer is +2% * 100 bps/% * $25,000 change/bps. Thanks,

 

[Aleksander Hansen]

Just another example of why Bionic Turtle is superior in terms of practice questions:
1) Correct answer
2) More insightful and well thought through questions
3) Rapid response, with an intuitive, easy-to-understand answer.

 

[Karim_B]

Hi @David Harper CFA FRM
Someone in the whatsapp group was asking how to solve a question from another provider which I thought I should be able to answer, but wasn't.

I thought we'd need to build up to 5% probability in the tail when rates rise and bond prices fall to find the VaR, but since we're given ranges of interest rates, I'm not sure which values to use in the calculation.

What would your answer be please?

"The price value of a basis point (PVBP) of a $20 million bond portfolio is $25,000. Interest rate changes over the next one year are summarized below:



Compute VaR for the bond portfolio at 95% confidence level."

Thanks
Karim

 

[David Harper CFA FRM]

HI @Karim_B Interesting! I agree that it's ambiguous, by which I mean you could justify at least two approaches. You are right that we want the 5.0% probability threshold. Assuming this is a long bond position (right? if this is a short, the VaR is on the other side where rates drop), then the 5.0% probability tail falls between the bins "2.00-2.49%" and "0.00-1.99%." So it would be understandable to retrieve +2.25% = average (2.0 to 2.49%) as the worst expected yield shock.

I suspect the question is looking for 2.00% as the worst expected yield shock because it's on the border (between bins), such that the answer is 200 * $25,000 = $5.0 million. But I agree with you that more precision in the question would be appropriate. To support my contention, btw, you could even go fancy pants and apply Allen/Hull's hybrid logic (which assumes the loss observation--in this case the interest rate change--itself is a random event with a probability mass centered where the observation is observed); that approach would assume the 4.0% probability p.m.f. weight (i.e., 5.0% CDF) extends to the midpoint of the "0.00 - 1.99%" bin, which retrieves 1.0%. Not that I am defending that choice. Rather, I think it's pretty clear the question is looking for the 2.0% that falls right "on the border," which is to me easily the most natural choice. I do think it's helpful to see why 2.0% rate shock is the natural choice: we want the beginning of the 5.0% probability tail (or end of the 95.0% body, if you like), this location would naturally fall at the edge of the "2.00 - 2.49%" bin.

For a similar reason, a justifiable solution is also 199 * $25,000 = $4,975,000; i.e., just inside the 95.0% body. This would be consistent with Dowd's approach actually. So if I were reviewing this question (eg, for GARP), I would probably cite the ambiguity vis à vis assumptions given. I hope that helps!