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Weekly Trivia Contest - Week of March 31st - Win Prizes!!! (VaR hodgepodge)

Nicole Seaman

Director of FRM Operations
Staff member
Subscriber
Head on over to our Facebook page to enter our Trivia Contest! You will be entered to win a $15 gift card of your choice from Starbucks, Amazon or iTunes (iTunes is US only)!

(All participants will be entered into our random drawing regardless of correct or incorrect answers. There will be two winners drawn at random.)




If you do not have Facebook, you can enter right here in our forum. Just answer the following questions:


1. [This classic value-at-risk question includes several key ideas. It is based on practice question 2010.P1.16, which is harder because it actually asks for a comparison between diversified and undiversified VaR!].

Question: Consider an equally-weighted portfolio with two bonds:

  • Bond A: $50.00 price (exposure), 4.0% yield, 3.0 year duration, 3.0% annual yield volatility;
  • Bond B: $50.00 price (exposure), 5.0% yield, 9.0 year duration, 7.0% annual yield volatility;
The correlation between the bond returns is 0.30. Assume there are 250 trading days in a year. Which is nearest the diversified 95.0% value at risk (VaR) over the next ten (10) trading days?

a. $8.15
b. $9.47
c. $10.90
d. $12.63

2. [This variation on practice question 2011.P1.24 is a nice way to quiz scaling VaR]

Question: Assume that portfolio daily returns are independently and identically normally distributed. Sam Neil, a new quantitative analyst, has been asked by the portfolio manager to calculate the portfolio Value-at-Risk (VaR) measure for 10, 15, 20 and 25 day periods. The portfolio manager notices something amiss with Sam’s calculations displayed below. Which one of following VARs on this portfolio is inconsistent with the others?

a. 1-day 99% VaR = $141.43
b. 10-day 95% VaR = $316.23
c. 20-day 99% VaR = $527.49
d. 25-day 95% VaR = $500.00

3. [variation on 2012.P1.1]

Question: You have been asked to estimate the VaR of a straddle option position in Big Pharma Inc. The company’s stock is trading at $100.00 and the stock has a daily volatility of 1.0%. The delta of the call option is +0.70. Using the delta-normal method, the VaR at the 95% confidence level of the straddle over a 1-day holding period is closest to which of the following choices?

a. Zero
b. $1.48
c. $1.97
d. $3.45

4. [variation on 2013.P2.2]

Question: The annual mean and volatility of a portfolio are 9.0% and 20.0%, respectively. The current value of the portfolio is USD 1,000,000. How does the 1-year 95% VaR that is calculated using a normal distribution assumption (normal VaR) differ from with the 1-year 95% VaR that is calculated using the lognormal distribution assumption (lognormal VaR)?

a. $26,409
b. $57,580
c. $83,333
d. $94,361

5. [variation on 2014.2.4]

Question: A risk manager is analyzing a 1-day 97.0% VaR model (an uncommon confidence level, you probably noticed). Assuming 250 days in a year, what is the maximum number of daily losses exceeding the 1-day 95% 97.0% VaR that is acceptable in a 1-year backtest to conclude, at a 99.0% confidence level, that the model is calibrated correctly?

a. 7
b. 9
c. 12
d. 14
 
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David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
FYI, for these questions, I went back though the last five years of GARP's practice exams, selected prototypical value at risk (VaR) questions, then changed the inputs/assumptions so the answer are unique. I think they represent a good sample of the FRM's basic VaR question type. (Interestingly, question #1 above is simpler than the actual; the actual question asks for the difference between a diversified and undiversified VaR).
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
@Nicole Manley I think we'll accept even partial attempts? ... since maybe this week the trivia was too much "work" to be fun (I like my VaR problems but i'm not the most fun person :rolleyes: ) ... actually I did sort of enjoy coloring the hodgepodge letters to match the rainbow pencil, that was a highlight for me
 

Nicole Seaman

Director of FRM Operations
Staff member
Subscriber
@Alex_1 You are this week's trivia winner! Thank you for participating even though it was a more difficult trivia quiz :)
 

Atin

Consultant
Hi David, Nicole

Can you please post the answers to the above questions? I attempted these questions but missed on sharing my answers :( Also, is Q1 part of L1 2014? VaR Mapping, as AIMs show, is part of L2. Just want to make sure if I am missing something!

Thanks much..
 

Nicole Seaman

Director of FRM Operations
Staff member
Subscriber
Hello @Atin,

Thank you for reminding me! I forgot to post the answers yesterday. I am giving you a gold star for reminding me to post the answers :) Here they are:

1. Answer: C. $10.90
2. Answer: C. 20-day 99% VaR should be $632.50
3. Answer: C. $1.97
4. Answer: A. $26,409
5. Answer: D. 14
 

Thierry S

New Member
Hi Melanie, David,

I am struggling with question 3. I can't get the right answer. Could you please give some hints/details?

For question 5, is this correct? We can approximate the binomial with the normal distribution (for backtesting purpose). Then cutoff = z_99% * SQRT[ p*(1-p)*n] + p*n = 13.78
where
z_99% (one sided) = 2.33
p (binomial) = 3%
1-p = 97%
n = 250

Thanks.
 
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David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Here are more detailed explanations for the answers
Each of the following should correspond to the underlying Excel, one tab for each problem, which is available here @ http://trtl.bz/1e6vTkK

1.
VaR1 = $50*3%*3*1.645*SQRT(10/250) = $1.48;
VaR2 = $50*7%*9*1.645*SQRT(10/250) = $10.36;
SQRT(VaR1^2 + VaR2^2 + 2*VaR1*VaR2*correlation) = sqrt[1.48^2 + 10.36^ + 2*1.48*10.36*0.30] = $10.90;

or,
Portfolio volatility(%), per annum = sqrt[50%^2*(3%*3)^2 + 50%^2*(7%*9)^2 + 2*50%*50%(3%*3)*(7%*9)*0.30] = 33.1293%;
Portfolio volatility($), per annum = 33.1293% * ($50 + $50) = $33.1293
10-day 95% VaR = $33.1293 * 1.645 * sqrt(10/250) = $10.90

Note
: please make sure you totally understand this problem, it is rich with testable ideas:
  • Duration for linear approximation of price change; e.g., 3%*3 yrs and 7%*9 yrs
  • Two-asset variance/volatility under mean-variance
  • time scaling VaR under i.i.d. assumption; e.g., sqrt(10/250)

2.
For (a), (c) and (d), the 1-day 95.0% happens to be $100.0 for this example. There are myriad approaches including just starting with the 1-day VaR to reveal which does not scale:

If 1-day 99% VaR = 141.43, then 10-day 95% VaR = $141.43 * sqrt(10) * 1.645/2.33 = $315.76, which is close to given answer
and 20-day 99% VaR = $141.43 * sqrt(20) = $632.50, which is not close
and 25-day 95% VaR = $141.43 *sqrt(25) * 1.645/2.33 = $499.26, which is close such that (a), (c) and (d) appear to be consistent.

3.
A straddle is long a call and put with the identical strike prices and maturities: this is the condition for put-call parity and, further, this means we can infer the put's delta from the call's delta. As the call's delta, N(d1), is equal to 0.70, the put's delta must be equal to N(d1) - 1 = 0.70 - 1.0 = -0.30.
The straddle's 95% 1-day VaR = (100 * 3% * 0.7 * 1.645) + (100 * 3% * -0.30 * 1.645) = (0.70 - 0.30) * (100 * 3% * 1.645) = $1.973824

4.
Normal VaR = -9% + (20%*1.645) = 23.8971%;
Lognormal VaR = 1 - exp(9% - 20%*1.645) = 21.2562%
Difference = 23.8971% - 21.2562% = 2.6409% or $26,409

5.
Standard deviation = sqrt[97% * (1 - 97%) * 250] = 2.697221
Two-tailed normal deviate @ 99.0% = 2.58; i.e., 99.0% is not one-sided VaR but rather a typical two-sided significance
Mean (expected) number of exceedences for a 97.0% VaR model = 3.0% * 250 = 7.50
Cutoff = mean + deviate*[Std Dev] = 7.50 + 2.58*2.70 = 14.47

@Thierry S Your formula is basically correct, except that unless instructed otherwise, you want to assume the confidence interval (which informs the cutoff) is based on a two-tailed (two-sided) hypothesis test. The implict null hypothesis is "The VaR model is correct" (which is two-tailed corresponding to an "=". This null is not "<=" or ">="). So your 99% two-tailed value needs to be 2.58 (just as the 95% two-tailed value used by GARP in Practice Question 2014.P2.4 is 1.96 not 1.65). The backtest can be tricky because:
  • While the VaR is always a one-tailed metric; e.g., 95% confidence locates 5% entirely in the one loss tail,
  • The significance test is typically two-tailed (not always, it depends on the null! But, IMO, best to assume two-tailed and let the language "persuade" you into a one-tailed test)
 
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Thierry S

New Member
Thanks a lot David.

For question 3, my understanding is : straddle's 95% 1-day VaR = straddle's delta * stock 95% 1-day VaR
straddle's delta = +0.7 - 0.3 = 0.4
stock 95% 1-day VaR = z_95% * stock's volatility * stock price = 1.645 * 1% *$100

Where do you get the 3% from?

3.
A straddle is long a call and put with the identical strike prices and maturities: this is the condition for put-call parity and, further, this means we can infer the put's delta from the call's delta. As the call's delta, N(d1), is equal to 0.70, the put's delta must be equal to N(d1) - 1 = 0.70 - 1.0 = -0.30.
The straddle's 95% 1-day VaR = (100 * 3% * 0.7 * 1.645) + (100 * 3% * -0.30 * 1.645) = (0.70 - 0.30) * (100 * 3% * 1.645) = $1.973824
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Thierry S yes, absolutely correct. It is my mistake. I assumed daily volatility of 3.0% in the answer rather than 1.0% given, such that if daily vol is 1.0% (as given) then:
  • The straddle's 95% 1-day VaR = (100 * 1.0% * 0.7 * 1.645) + (100 * 1.0% * -0.30 * 1.645) = (0.70 - 0.30) * (100 * 1.0% * 1.645) = $0.6579, or more simply:
  • straddle's 95% 1-day VaR = straddle delta of 0.40 * (100 * 1.0% * 1.645) = $0.6579
Plus +1 star for catching my error :eek: Thank you!
 
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Roshan Ramdas

Active Member
Hi David,

First and foremost,...thank you for putting up these VAR questions. I am personally finding this topic challenging and getting some of the questions right helps build confidence.

Just had a query with regards to the solution in the 4th question

Based on the highlighted line, I assume that formulae being used for lognormal VAR is (T - Exp (Mean - Standard Deviate * Volatility) ?

Is my interpretation of the formulae correct please.

Normal VaR = -9% + (20%*1.645) = 23.8971%;
Lognormal VaR = 1 - exp(9% - 20%*1.645) = 21.2562%
Difference = 23.8971% - 21.2562% = 2.6409% or $26,409

Thank you,
Roshan
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Roshan Ramdas

Thanks for liking them!

Yes, I think you mean "lognormal VAR is (1 - Exp (Mean - Standard Deviate * Volatility)"
Per Dowd, that is the lognormal VaR; i.e.,

\(\text{VaR(lognormal) = }1-{{e}^{\mu -\sigma z}}\)


This approach assumes that the asset price is lognormally distributed, which in turn, has the advantage that price cannot drop below zero (cannot become negative). The normal VaR has an awkward feature in that it can imply a negative asset price:

  • In normal VaR, [vol*deviate*sqrt(t)] can exceed [drift*(t)], such that implied loss > 100% [sic], but
  • 1-exp(drift - vol*deviate) realistically will not exceed 100%
 
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FRMCAND

Member
Hi David,

I'm a little confused about question n° 5

Question: A risk manager is analyzing a 1-day 97.0% VaR model (an uncommon confidence level, you probably noticed). Assuming 250 days in a year, what is the maximum number of daily losses exceeding the 1-day 95% VaR that is acceptable in a 1-year backtest to conclude, at a 99.0% confidence level, that the model is calibrated correctly?

Do you refer backtesting to 1-day 97% VaR ?

Thank you in advance for your attention,
 

abhinav0131

New Member
Hi David

I am a FRM L-1 May'14 candidate, have come across the BT website very recently and feel that you guys are doing a great job in helping people acquire sufficient knowledge to crack the exams.

I am facing problem in interpreting Q5, basically I am not able to understand it at all, do you mind helping me out here pls. Is something like this in the core reading theory ?

Thanks in advance.

Abhinav
 

Alex_1

Active Member
Hi @Thierry S yes, absolutely correct. It is my mistake. I assumed daily volatility of 3.0% in the answer rather than 1.0% given, such that if daily vol is 1.0% (as given) then:
  • The straddle's 95% 1-day VaR = (100 * 1.0% * 0.7 * 1.645) + (100 * 1.0% * -0.30 * 1.645) = (0.70 - 0.30) * (100 * 3% * 1.645) = $0.6579, or more simply:
  • straddle's 95% 1-day VaR = straddle delta of 0.40 * (100 * 1.0% * 1.645) = $0.6579
Plus +1 star for catching my error :eek: Thank you!

Hi David, I think in your answer above it should be "(0.70 - 0.30) * (100 * 1% * 1.645) = $0.6579" and one of the possible answers should also reflect this number (i.e. 0.6579). But then again this is nit-picking from my part. :) Thanks.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @Alex_1 Yes, I totally agree, thanks! I fixed my explain text; but I didn't re-edit the original question (yet) as these questions don't yet appear anywhere else for paid customers, this is (so far) just fun trivia, because you know, this is how we have fun :rolleyes::):confused:! (eventually, I will located them in an Study Planner resource somewhere then I'll repair the question....)
 

Alex_1

Active Member
Thanks, yes at the moment I am also trying to convince myself of how much fun I am having while going through PQs, practice exams, notes, videos and switching back and forth between P1 and P2. :D
 
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