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# Estimating market risk measures

##### Member
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

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.

#### Aleksander Hansen

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

#### Karim_B

##### Active Member
Subscriber
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.

"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

##### David Harper CFA FRM
Staff member
Subscriber
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!

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