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# Question on SaR

#### maoxindi

##### New Member
I have a question on study note.
On Page 26: Another Example: Alternative approach

You can see another application of the alternative approach (i.e., page 26 that leads to $18.1) in this question 12.2 at http://www.bionicturtle.com/forum/threads/p2-t8-12-surplus-at-risk-sar.5488/, replicated here: 12.2. Public Employee Retirement Fund (PERF) has$600 million in assets and $600 million in liabilities, for a current surplus of zero. The annual expected return on assets is 8.0% with 18.0% volatility per annum; the annual expected return on liabilities is 6.0% with 14.0% volatility per annum. Both are normally distributed. The correlation between assets and liabilities is 0.60. What is the 95% absolute surplus at risk (absolute SaR); i.e., the worst expected SHORTFALL, or loss relative to current surplus of zero, with 95% confidence? (Note: this is similar to a previous FRM question. The method to derive SaR is different, but the assumptions given are different) I hope that helps, thank you for your patience! #### maoxindi ##### New Member Which approach should I use during the exam? In addition, do I need to remember critical numbers (eg. 1.64 for 95% one tail) during Part II exam? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi maoxindi, • I've got them both in there b/c the first is used by Jorion in the assignment, but the second has appeared on an exam; i.e., I can't say, they are similar approaches, they do not contradict each other, it is just that they use different assumptions (the latter assumes a correlation between assets & liabilities) • Yes, absolutely be ready to use -2.33 at 99% and -1.645 @ 95%. Almost guaranteed need for them. #### lighta ##### New Member I may have some misunderstanding of these approaches and have some questions : p. 25 First approach: Surplus at risk = 21.87% *$1000 (the assets value). Why use assets value but not $100 (the surplus value)? p.26 Alternate approach : Surplus at risk = 18.1 (but why not volatility of surplus * normal deviate = 13.8*1.64 = 22.63? Also why absolute SaR use$20 as the drift but not use expected growth in surplus i.e. 4.6?

Would appreciate very much of your help.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi lighta, good questions
1. because that's what Jorion does in his example on Chapter 17: his 17.2.3 example has A - L = S of 1,000 - 900 = 100, and then his assumption (emphasis mine) of "volatility of the surplus, scaled by assets, is 5 percent" would, I think, typically lead us to $100 surplus * 9.4% * 2.33 = ~22 million, but he arrives at$220 million. So, I've always inferred (with less than 100% confidence) that the "scaled by assets" is meaningful and just means "volatility of assets." Here is possible motivation: the surplus can be zero. Then how would we express it's volatility in dollars? (for that matter, surpluses near zero are non-trivial)
2. Yes, you could do that. As prior discussions share, I included this page only because this approach appeared on a prior FRM exam. Meanwhile, I have every year requested to GARP a standardized definition of SaR. The page previously included three approaches, including I did previously include your "pure relative VaR" but three definitions were confusing. In this case ....
we can refer to values unequivocally:
• surplus starts the year at 20 and
• is expected to grow to 24,
• but the worst 95% expected outcome is a surplus which ends the year at $1.90 but this allows for definitional choices: • shortfall at 95% =$1.90; note how we might read this to be a shortfall of -1.90 but it really is +1.90!
• a relative SaR = 13.8*1.64 = $22.63; a reference to which i removed from the page due to confusion (and lacking any standardization from GARP, but i will try again in a few weeks with the 2014 draft syllabus feedback). I agree with you: this is a sensible VaR, as it refers to "loss relative to future expected value" • shortfall relative to initial surplus = 20 - 1.90 = +18.1; notice this really should be called absolute SaR! You can see why I want GARP to standardize on SaR: technically, there are several approaches. In the meantime, the burden is on precise questions formats, e.g., http://www.bionicturtle.com/forum/threads/p2-t8-12-surplus-at-risk-sar.5488/ meanwhile, your "relative SaR" -- i.e., dollar volatility of assets [as proxy for surplus] * deviate -- is fine because that is what Jorion calls the SaR: his SaR is$220 = $1,000 assets * 9.4% * 2.33. What I am proposing (as feedback) to GARP is to introduce a distinction between: • surplus as risk (jorion's example) with implication of a relative VaR • shortfall at risk with implications of an absolute VaR value relative to zero (assets = liabilities) #### lighta ##### New Member Thanks so much for your very clear explanation. #### michael4129 ##### New Member Hi David, Thanks for the explanation in this example, but one more question: Shouldn't there be a + instead of a - in the portfolio variance formula of the alterative approach: =D2^2*D7^2+D3^2*H7^2+2*D2*D3*D7*H7*D10 instead of =D2^2*D7^2+D3^2*H7^2-2*D2*D3*D7*H7*D10, which would make variance of surplus 242.28 instead of 190.44? Thanks! Michael #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi Michael, I don't think so because Surplus (S) = Assets (A) - Liabilities (L) and we seek variance (S) = variance (A - L) = variance (A) + variance (L) - 2*COV(A,L). Thanks, #### michael4129 ##### New Member Thanks David. Clear now! Since we have no aggregated portfolio of assets like in the previous examples of the script, but a "netting position" of asset and liabilites it makes sence to do a substraction here. #### Pflik ##### Active Member Hi lighta, good questions 1. because that's what Jorion does in his example on Chapter 17: his 17.2.3 example has A - L = S of 1,000 - 900 = 100, and then his assumption (emphasis mine) of "volatility of the surplus, scaled by assets, is 5 percent" would, I think, typically lead us to$100 surplus * 9.4% * 2.33 = ~22 million, but he arrives at $220 million. So, I've always inferred (with less than 100% confidence) that the "scaled by assets" is meaningful and just means "volatility of assets." Here is possible motivation: the surplus can be zero. Then how would we express it's volatility in dollars? (for that matter, surpluses near zero are non-trivial) I was reading this and I was also wondering why the assets are used to calculate the SAR. But given this answer I think I'll just have to memorize to use the assets. #### southeuro ##### Member quick question: once we find the volatility of surplus and multiply this with the deviate, do we then multiply this with the increase in the surplus or the total surplus (including the increased portion)? Thanks #### ShaktiRathore ##### Well-Known Member Subscriber Hi, In case of var we take volatility of x ,multiply by deviate and initial value of x to get value of var dollars.analogos to it i think we should multiply vol*dev of surplus with initial total surplus value to get sar. Thanks Last edited: #### hamu4ok ##### Active Member Hi Michael, I don't think so because Surplus (S) = Assets (A) - Liabilities (L) and we seek variance (S) = variance (A - L) = variance (A) + variance (L) - 2*COV(A,L). Thanks, Hi David, Quick question. I do understand how to get relative SaR, which is quite easy (volatility of surplus as calculated as above * deviate * Assets Size) how about absolute SaR? for it we need expected value of surplus ("mean") = WA*RoA - WL*RoL, where WA - weight of assets =A/A=1, WL=weight of liabilities (L/A)? and SaR = (MEANs - deviate*ST.DEVs)* Assets? #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber Hi @hamu4ok It's a great question, I wish I could tell you the FRM had a settled definition, but as I've asked more than once, I can tell you they don't. Conceptually, you are correct: Absolute SAR, by consistency, should net the expected drift (growth) in the surplus, as you seem to imply. But there remains still a choice. To illustrate, here is what I mean; but please bear in mind (as it is exam week), for practical purposes, you probably only care about relative SaR (anything beyond a simple relative SaR is highly unlikely to appear on the exam; or it would require very careful definitions anyway): Assume: • Assets (A) =$1,100 with ROA = 10% and volatility(A) = 10%
• Libabilities (L) = $1,000 with ROL = 8% and volatility (L) = 8%; correlation between A&L = 0.30 • Volatility of surplus = sqrt[1100^2*10%^2 + 1000^2*8%^2 - 2*1100*1000*10%*8%*0.30] ~=$115, such that 95% relative SaR (aka, "SaR")= $115*1.645 =$189; i.e., the most testable concept, full stop. Beyond this is theory .... to your question now:
• The current surplus = 1,100 - 1,000 = $100 and the expected change in the surplus = (1,100*10%) - (1,000*8%) = +$30.
• One concept of absolute SaR is -30 + 189 = $159. This is consistent with our absolute MVaR and represents something like "the worst expected decrease in the surplus; in this case, a 159 drop from +100 to -59" • Another concept is to equate absolute SaR with the actual shortfall, in this case$59. This is akin to treating the current surplus in the negated drift. It's not consistent, but it's represents the answer to "what is the worst expected contribution we will need to make to restore any shortfall."
you can see how your concept of subtracting the mean is correct, but you probably don't need "WA*RoA - WL*RoL" as this is effectively a long-short portfolio anyway (long assets, short liabilities). I hope that at least explains why you don't need to worry about "absolute SaR" for the exam! Thanks,

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#### frmexam

##### New Member
David - shouldn't volatility of surplus be sqrt(variance of Assets minus Liabilities)
which means from your above calc
Volatility of surplus = sqrt[1100^2*10%^2 + 1000^2*8%^2 + 2*1100*1000*10%*8%*0.30]
should be
Volatility of surplus = sqrt[1100^2*10%^2 + 1000^2*8%^2 - 2*1100*1000*10%*8%*0.30]
I replaced with + with - in vol calculation

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Yes @frmexam sorry, my typo (fixed), you are correct; looks like the 115 is still okay. Thanks!

#### evelyn.peng

##### Active Member
Hi maoxindi,
• I've got them both in there b/c the first is used by Jorion in the assignment, but the second has appeared on an exam; i.e., I can't say, they are similar approaches, they do not contradict each other, it is just that they use different assumptions (the latter assumes a correlation between assets & liabilities)
• Yes, absolutely be ready to use -2.33 at 99% and -1.645 @ 95%. Almost guaranteed need for them.
Hi David,
On the issue of the z-value to use to calculate SaR:
GARP 2019 Practice Exam question number 24. In which the question asked for the "lower bound of the 95% confidence interval for the expected end-of-year surplus". The answer key uses 1.96 instead of the 1.645. I think the 1.96 makes sense as we're calculating an upper bound and lower bound as it is two-tailed; compared to a 1 tailed SaR which Jorion shows in his text.
I wanted to confirm with you to see if my thinking is correct?
Thank you,
Evelyn

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @evelyn.peng This question has a history with variations and has been discussed here at
... where you can see that yes I definitely do agree with you per my point 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,

.... although I think it's a tiny bit unfair of GARP: I think it should ask a surplus at risk (SaR) question if it's going to refer to surplus, given the confusion that seems to ensue!