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# GARP.FRM.PQ.P22019 Practice Test - Q62 - calculation error and back-asswards explanation

##### New Member
A financial risk consultant assumes that the joint distribution of returns is multivariate normal and calculates the
following risk measures for a two-asset portfolio managed by a mid-size insurance company:
Asset Position (JPY) Individual VaR (JPY) Marginal VaR
Financial 20,000,000 4,787,400 0.316
Real-Estate 20,000,000 7,299,300 0.562
Portfolio 40,000,000 11,562,450

Question: What is the closest to the correct estimate for the component VaR of the financial asset?
A. JPY 4,787,000
B. JPY 6,322,000
C. JPY 7,299,000
D. JPY 11,240,000

Official explanation:

Firstly, if you punch all the numbers into the calculator, you get 6,320,000, so there is a 2000 calculation error.
Secondly, why are they bothering with the beta????? Marginal var is the measure of how much component var will increase for a unit of asset value increase. So the answer, by definition of marginal VaR is as simple as:

CVaR=MVaR*AssetValue=0.316*20,000,000=6,320,000 (1)

In fact, if you write out the two "official" formulas above on a piece of paper, you will see that (Portfolio VaR) cancels out, and you are left with:

(Marginal VaRf*Portfolio Value)*(Asset Weight) (2)

Asset weight is nothing but AssetValue/(Portfolio Value), so (Portfolio Value) also cancels out in (2), and you are left with (1).

Why are they taking the scenic route? Am I missing something?

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#### David Harper CFA FRM

Staff member
Hi @ahead I agree; ie, you aren't missing anything. Actually, it's even worse: those numbers aren't internally consistent. The assumptions are not consistent with those marginal VaRs; in fact, those assumption imply marginal VaRs, respectively, of 0.2234 and 0.3547. And of course the component VaRs need to sum to the Portfolio VaR, but the component VaRs apparently sum to $17,560,000 (but they should sum to 11,562,450, assuming that is the diversified Portfolio VaR). I am so disappointed, I really thought I had already showed GARP how to do these correctly, we helped correct all the previous flaws in these Portfolio VaR (years of feedback ....). These portfolio VaR questions have never been wholly correct but I thought I gave them everything they need to understand. Internally consistency (i.e., building up the assumption set from correct dependencies) is critical because it ensures each of the VaR measures (e.g., component VaR) returns the same result whether you use any of the two/three alternative formulas. When the assumptions are just "dropped in," you can get different answers based on your approach, but that should be impossible. Ughh ... I have no words .... I'm just disappointed as we work hard to teach the fundamentals and ourselves uphold a standard. Last edited: #### ahead ##### New Member Hi David, thank you for the additional insight. Here is another example of taking a two-step detour - Q2 of the 20-question version of the 2019 practice test. They are rolling the value of a forward three quarters, and then rolling it back two quarters. Why not just roll the value of the contract by one quarter to begin with: 20.3*exp(0.04*0.25)=20.5040 Here is the original problem and explanation: Three months ago, Valtemp Construction Company (VCC) entered into a 9-month forward contract with Millenia Steel Products (MSP) to purchase 6,000 tons of stainless steel from MSP. At the time the forward was entered into, 6,000 tons of stainless steel was priced at EUR 20.3 million but is currently priced at EUR 23.5 million. The continuously compounded risk-free rate has remained stable at 4.0% per year and is not expected to change during the remainder of the contract period. Assuming the forward is fairly priced, what is the current potential credit risk exposure on the forward contract and who bears the risk? A. EUR 2.790 million; MSP bears the potential credit risk B. EUR 2.790 million; VCC bears the potential credit risk C. EUR 2.996 million; MSP bears the potential credit risk D. EUR 2.996 million; VCC bears the potential credit risk Correct Answer: D Explanation: D is correct. Given the risk-free rate of 4.0%, we can estimate the forward price (at maturity, in nine months) of the contract as: Forward price = Spot*exp(r*t) = 20.3*exp(0.04*0.75) = EUR 20.9182 million. Today, after 3 months (6 months remaining to maturity), the forward contract price estimate = 20.9182/exp(0.04*0.5) = EUR 20.5040 million. Note that, Forward Contract Value = Credit Risk Exposure. Therefore, given that the current (with 6 months remaining to maturity) underlying asset price of EUR 23.5 million, the long forward contract’s exposure value is given by: Current Potential Exposure Value of Forward Contract = (Market Price – Contract Price) = 23.5 – 20.5040 = EUR 2.996 million. Because the contract value of EUR 2.996 million is positive, the long counterparty (VCC) bears the credit risk exposure. Positive exposure = Max(value, 0), Negative exposure = Min(value, 0) and for long forward contracts: Contract Value = (Market Price – Contract Price). For forwards, while there is no current credit risk (loss), because payment is only made at expiration, there is always positive potential exposure so long as market price > contract price, and negative potential exposure if market price < contract price. At origination (time 0), there is neither current credit risk nor potential credit exposure (since market price = contract price). A and B are incorrect. They compute the contract price incorrectly by discounting the forward value over 3 months and not 6 months as follows: The forward contract price = 20.9182*exp(-0.04*0.25) = EUR 20.7101 million. Therefore, Current Value of Forward Contract = (Market Price – Contract Price) = 23.5 – 20.7101 = EUR 2.7899 million. C is incorrect (see explanation for D above). #### Attachments • 57.6 KB Views: 2 Last edited by a moderator: #### David Harper CFA FRM ##### David Harper CFA FRM (test) Staff member Hi @ahead Right, well, I think the sloppy and inaccurate terminology is the real culprit. The first line should read: Delivery price, K = Spot*exp(r*t) = 20.3*exp(0.04*0.75) = EUR 20.9182 million. And the worst thing about this question is how it is asking for a "current potential credit risk" which is not a term because "current potential" is oxymoronic: we either want the current exposure or, alternatively, we might want a potential future exposure (PFE). So "current potential" makes me cringe because it could lead the exam taker understandably to seek a forward value. So the question should read: "Assuming the spot price is currently EUR 23.5 million, what is the current credit exposure on the forward contract and who bears the risk?" ... note that even "risk exposure" is sloppy . As any part 2 candidate knows, we just want "current exposure" or it's okay to be a bit redundant as say "current counterparty exposure." We don't need "risk exposure," uggh. ... Also, this clause is misleading: "Assuming the forward is fairly priced." It suggests the 23.5 is a fairly price forward price, which is not the case, 23.5 is the current spot price (I infer from the solution!). The accumulation of incorrect terminology I deem to be the actual problem here, no wonder the solution is convoluted. It's very hard to get these Q&A right unless and until we define the terms correctly. So here is a properly re-phrased question that uses the correct (i.e. consistent with Hull is always a good test when asking about futures!) terminology that has been assigned to FRM candidates for over ten years, so we might expect GARP to be fluent in the same: "Three months ago, Valtemp Construction Company (VCC) entered into a 9-month forward contract with Millenia Steel Products (MSP) to purchase 6,000 tons of stainless steel from MSP. At the time the forward was entered into, 6,000 tons of stainless steel had a spot price of EUR 20.3 million but the current spot price is EUR 23.5 million. The continuously compounded risk-free rate has remained stable at 4.0% per year and is not expected to change during the remainder of the contract period. Assuming the forward is fairly priced, What is the current potential credit risk exposure on the forward contract and who bears the risk?" ... again, we really do not want "assuming the forward price is fairly priced," it can only mislead: the contract's delivery price is fixed and already established. The current spot price does not require an assumption. Given that question, then here is a solution which is consistent with Hull, who has been assigned to this topic from the beginning: At inception, the delivery price was, F(-0.25) = K = S(-0.25)*exp(r*T) = 20.3*exp(0.04*0.75) = EUR 20.9182 million. Value of the forward today, f = S(0) - K*exp(-rT) = 23.5 million - 20.9182 * exp(-0.040*0.5) = 2.996. Alternatively, Today's forward price per COC should be, F(0) = 23.5*exp(0.040*0.5) = 23.975; Value of the forward today, f = [F(0) - K]*exp(-rT) = ($23.975 - 20.9182)*exp(-0.04*0.5) = 2.996
I think you are totally right to be puzzled by the solution, notice this line in the solution:
.. the forward contract price estimate = 20.9182/exp(0.04*0.5) = EUR 20.5040 million.
... what exactly is a "forward contract price estimate"? This value, 20.5040 is the discounted delivery price, or if you like, the present value of the (fixed) delivery price. It is not an estimate. We don't refer to discounted present values (aka, discounted values) as estimates; estimates are values returned by estimators.

This question is badly written. Some might think I quibble, but they wouldn't know what they are talking about. The actual question is not terribly difficult, but it's cloaked in lazy and sloppy language, so the question seems harder than it really is. The candidate, who in most cases has barely mastered the material, is left with cognitive dissonance as the question equivocates with imprecise current and future concepts. The writer of the question is not crystal clear on the definitions, so there is a bit of ambiguity in the question. It's all the question writer's fault.

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##### New Member
Hi David,

Thank you for these clarifications, it makes a lot more sense now. Could you just clarify one more thing - how do you really feel about the way this question is written?

One more - in Q41 of the practice test and the equivalent Q14 of the pre-study test the loan principal is irrelevant, as it cancels out as part of the calculation. I can understand that unnecessary information can sometimes be included in the prompt to confuse the test taker or make the question sound more realistic, but I think its irrelevancy should be called out as part of the solution, rather than carried through each calculation.

#### David Harper CFA FRM

##### David Harper CFA FRM (test)
Staff member
Hi @ahead You want to know how I really feel? okay because I thought I was exercising restraint and good manners Feel free to PM to hear how I really feel!

In regard to the RAROC Q&A that duplicates the loan principal, it honestly does not trouble me. I'm happy GARP finally got this question correct after so many revisions. See below for a copy of how I have similarly explained a RAROC solution. This is another calculation that GARP was getting wrong a lot (both inconsistencies and mistakes) and we had to send a lot of feedback, as recently as last year So the question itself has a checkered history, believe it or not. Its previous version contained in fact three mistakes: paying members can observe our commentary here at https://trtl.bz/2HHXMnv. Over time, there have methodological issues also; e.g., should deposits equal the loan amount, or should deposits be less than the loan amount; should it be a pre-tax RAROC or post-tax RAROC?

So, I think the real reason everything is specified as a percentage of loan amount is to remove any of those doubts and ensure consistency. e.g., in Crouhy's examples, you'd typically find the operating cost as an absolute number rather than % of loan. My point is that, until this latest version, most of those questions did not "ripple" all of the assumptions through the loan amount. All prior questions mixed % of loan with dollar values. If redundancy removes doubts for the reader, it doesn't trouble me at all. Most of those solution redundancies are given simply because that's how the assumption are stated (as a % of loan). Redundancy is a tiny price to pay for finally getting RAROC to a point where candidates could know which version to expect. Thanks!

From solution to my own Question 706.2 @ https://www.bionicturtle.com/forum/threads/p2-t7-706-economic-capital.10722/
The numerator is after-tax risk-adjusted expected return and the denominator is the economic capital of $360 million (=$3.0 BB * 12.0%) such that:
• Expected Revenue of $3.0 BB * 10.0% =$300.0 million
• plus Return on Economic Capital (ROEC) of $360.0 million * 3.0% =$10.80 million
• minus Interest Expense of $3.0 BB * 5.0% = 150.0 million • minus Expected Loss (EL) of$3.0 BB * 3.0% = $90.0 million • minus Operating (aka, Indirect) Costs of$15.0 million
• Equals $300.0 + 10.8 - 150.0 - 90.0 - 15.0 =$55.80 pre-tax return, which implies:
• An after-tax return of $55.80*(1-30%) =$39.060
• And therefore RAROC = 39.060/360.0 = 10.850%.
ARAROC = 10.850% - 1.30 * (8.0% - 3.0%) = 4.35% which is greater than the risk-free rate. Alternatively, under the previous (but toward the equivalent conclusion) approach to ARAROC, we can compute ARAROC = (10.850% - 3.0%)/1.30 = 6.04% which is greater than the equity risk premium of 5.0%.

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