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# Q24 Mock exam GARP 2020 - Surplus at risk

#### Fjrodriguez

##### New Member
Hi everybody,

Doing the question number 24 of GARP mock exam 2020, I found an issue.

Explanation:
C is correct. The lower bound of the 95% confidence interval is equal to: Expected Surplus
– (95% confidence factor * Volatility of Surplus).
Expected surplus:
VA*[1 + E(RA)] – VL*[1 + E(RL)] = 180*1.06 – 140*1.10 = USD 36.80 million.
For a 95% confidence interval, the appropriate z-value is 1.96. Therefore, the lower bound of the surplus at the 95% confidence level = 36.80 – 1.96*35.76 = USD -33.2896 million.

I dont really understand why they took as z-value 1,96 instead of 1,65 with a Surplus at risk 95%.

I thought that surplus at risk is calculated as Var (1 tail) because in other exercises as well as in kaplan books i have always seen like this. Indeed, this same exercise in Garp exam 2017 or 2016 (i cant remember exactly) used 1,65 as zvalue instead of 1,96, but since 2018 exam they use 1,96.

could anyone solve me this doubt? do anyone think the same as i do?

Thank you in advance.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Fjrodriguez You are correct, this question was revised three times (GARP's practice exams contain many errors). See post at https://www.bionicturtle.com/forum/threads/garp-p2-question-24.21932/post-71742; i.e.,
Thank you @Nicole Seaman but I can see that GARP has edited this question (again, for the third time) based on the revised version we just got from Lisa.

@jaivipin I have not had time to properly analyze this question (their versions keep changing!), but I will just quickly say: most of these SaR questions by GARP have asked about the surplus as risk (SaR) which is a situation-specific VaR and, in the case of looking for a 95% SaR, we would use 1.645 because VaR (and SaR, and LVaR etc etc) is always one-sided. The "at risk" implies a one-sided look at only the loss side of the distribution. When looking for a SaR, if the distribution is normal, then the deviate is indeed always 1.645.

However, this question explicitly asks "what is the lower bound of the 95% confidence interval for the expected end-of-year surplus that the advisor can report?" ... please notice:
• There is no request for the "surplus at risk" (this is a revision improvement)
• Rather, this question is simply asking for the lower bound of a confidence interval. That's a typical two-tailed confidence interval approach to hypothesis testing. It's not SaR, not VaR. Yes, I do think the question would be slightly better to read " ... of the 95% two-sided confidence interval" ... but by convention C.I.'s are presumed two-sided unless otherwise specified. Thanks,

Notice the question does NOT ask for surplus at risk (SaR); SaR is a VaR, VaR is always one-tailed. This question asks instead for the CI around the expected surplus; that's a 2-sided CI. Thanks,

#### Ana Hsu

##### New Member
Hi @Fjrodriguez You are correct, this question was revised three times (GARP's practice exams contain many errors). See post at https://www.bionicturtle.com/forum/threads/garp-p2-question-24.21932/post-71742; i.e.,

Notice the question does NOT ask for surplus at risk (SaR); SaR is a VaR, VaR is always one-tailed. This question asks instead for the CI around the expected surplus; that's a 2-sided CI. Thanks,

Hi, David! Thanks for the clarification.

May I ask what if the question uses the term “greater than or equal to”?
Should I use two-tailed calculation since it’s not asking about “SaR”?
But I’m a little bit confused because “greater than or equal to” sounds more like a one-tailed calculation.

For instance, if the question is:
With confidence level of 95%, the surplus value will be greater than or equal to=?

Thanks a lot!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Ana Hsu Yes, I agree that "greater than or equal to" generally implies a one-tailed hypothesis test. If I were writing your question, I would eliminate the value so that it asked: "With confidence level of 95%, the surplus value will be greater than or equal to= ?" That's a super minor edit but the reason is that "value" might sound a little bit like "value at risk" and you don't mean value, so there's no reason to perhaps confuse. The question can either be about (i) a one- or two-sided confidence interval around the surplus, or (ii) the SaR which is a VaR so is always one-sided. But i do agree that "greater than" implies, in general, a one-tailed calculation. Although I can't stress enough: it is the burden of the question to ask a clear question! Right? The reference here is GARP's question that, as I recall, required them to revise three times before it was clear. It's unfair to ask candidates to decipher what the question is asking. The question should be clear so that all sufficiently knowledgeable candidates agree on what is being asked. It's too much to ask candidates to try and decipher the meaning of a question that is insufficiently clear. Thanks,

#### Ana Hsu

##### New Member
Hi, David @David Harper CFA FRM

Thank you so much for the detailed explanation.
Totally agree with you and truely appreciate your effort in keeping having conversations with GARP to clarify the questions!

Ana

#### Prajakta A

##### New Member
Hi David,

Could you please help understand the use of 1.96 instead of 1.65 z-value in the question 24? Attached is the screenshot. Shouldn't we be using one tail test instead of two tail test for VaR?
The same issue is also observed with question 12 that tests the backtesting concept.
Really appreciate your immediate response.

Thanks in advance.

Edited by Nicole to include practice question:

An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 40 million:

To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of 1 year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.68. Assume that the volatility of surplus in dollar is USD 35.76 million, what is the lower bound of the 95% confidence interval for the expected end-of-year surplus that the advisor can report?

A. USD -76.4 million
B. USD -58.2 million
C. USD -33.3 million
D. USD -22.0 million

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Prajakta A I moved your question to a pre-existing thread on this question (actually there are multiple conversations about this recycled question; e.g., tag = https://www.bionicturtle.com/forum/tags/garp20-p2-24/). In super brief, you are correct that the quantile for a normal 95.0% VaR will always be 1.65 (aka, VaR is always one-sided) but neither this question 24 nor question 12 ultimately asks about a VaR. This question 24 asks for the lower bound of a "95% confidence interval for the expected (end of year) surplus." That's not the same as asking directly about surplus-at-risk (SaR). This question requires a careful reading! Question 12 is classic VaR backtest: the significance of a VaR backtest is different than the confidence of the; for example, we can conduct a backtest at the 95.0% confidence level of a 99.0% confident VaR. I hope that's helpful!

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