Opinion

Felix Salmon

How has VaR changed over time?

By Felix Salmon
August 6, 2009

Whenever I write about banks’ rising Value-at-Risk, a bunch of commenters tells me that duh of course VaR is rising, because VaR is a function of volatility, and volatility has gone up. So here’s my question: can someone come up with a baseline VaR chart, for a hypothetical bank which had, say, a fixed $1 million investment in the S&P 500. What would its quarterly Value-at-Risk have looked like over the past couple of years?

Would the decrease in volatility this year have shown up as decreased VaR in say the second quarter? Or do the volatility calculations go back so far that only now are we losing the Great Moderation datapoints and using volatility numbers only from the era of increased volatility?

Armed with that kind of baseline chart, we’ll be able to tell much more easily, for any given bank, whether it’s actually increasing the size of its bets, or whether increased VaR is simply a statistical necessity given the recent history of volatility. Does such a chart exist?

Update: Phorgy comes through. All banks calculate VaR differently, but this is a really useful resource. Basically Var increased enormously in 2008, and after that slowed down sharply in 2009, possibly even dropping significantly. Which means, I think, that any significant increases in VaR in 2009 can’t be blamed on increased volatility.

Comments
23 comments so far | RSS Comments RSS

My comment was certainly not meant as a “duh of course” comment. It’s just that I’ve seen a few write ups on this issue, both here and elsewhere, with no mention of the period-covered issue. I haven’t taken a look at VIX or S&P vol lately but obviously if the current volatility, although decreasing, is still higher than the data points exiting the calculations (from X years back) then yes, the 1$ million investment VaR is still rising.

Maybe the volatility data entering the VaR numbers now (or in Q2) are even lower than the ones a few years back but hey, just wondering out loud. In any event, I thought nobody gave VaR a second look anymore so interpreting increasing risk through VaR would be kind of pointless.

Also, following the limited Goldman info I have I thought their profits came from being the only underwriter left in town, gaining from certain bailouts, HFT and continued profiting from selling customers one thing and having the prop desk do the opposite. But what do I know, maybe the VaR number is an accurate measure of risk and the risk is the main reason for its current profits.

Posted by Gerter | Report as abusive
 

Felix,

Assuming a Black-Scholes type framework where the size of the investment doesn’t change (as you describe), the VaR will be a nearly-linear function of volatility.

Let volatility be S, the daily VaR will be roughly:

$1,000,000 * (1 – e^(-2.32 * S))

Since e^x has a slope of 1 around 0 (which is where our daily S will be close to), this VaR is not a perfectly linear function of S, but it’s close.

I have this in excel based on some numbers from Yahoo Finance if you’d like me to email it.

Mike N.

Posted by Mike Nute | Report as abusive
 

Of course, the upshot of this though is that it depends entirely on how volatility is calculated and whether the VaR model of the bank uses a lognormal distribution. Presumably the banks are a little smarter than that but, then again, maybe they aren’t.

Posted by Mike Nute | Report as abusive
 

Also, not to be overly technical, but VaR is necessarily specified at a particular return-period. If I’m not mistaken most of the banks use a 100-day return period (so, for a daily VaR, that would be a 1% excedance probability). That’s what I assumed above.

Posted by Mike Nute | Report as abusive
 

For many bank risk models, the VaR of an equity position would really just be proportional to its volatility, e.g.

VaR(S&P500) = 2.33*Volatility(S&P500)

I use a slightly more sophisticated (“Stable”) model that attempts to incorporate fat tails and non-symmetric return distributions.

I recently gave a seminar at UCLA and just uploaded portion of one slide showing daily VaR estimates for the S&P 500 going back to January 1930 through May 2009.

http://phorgyphynance.wordpress.com/2009  /08/06/80-years-of-daily-sp-500-value-a t-risk-estimates/

During “normal” times, the two curves agree, but during periods of extreme stress (or exuberance) the tails contribute more. This most recent crisis had larger tail contributions than any time period in the history of the S&P 500.

 

ya my charts agree with phorgy’s. I calculated volatility using a 90-trading-session trailing average (forgot to correct for weekends).

Posted by Mike Nute | Report as abusive
 

Glad to see you’re finally taking note that VaR is all but useless.

Anyway, I leave the charts and equations to others – am too dim for that. But the wonderfully amusing thing about VaR is that almost every bank calculates it in a different manner. Some use 1-day 95%, others 1-day 99%. And historical data input varies from just the preceding year to the preceding 5 years.

Check out GS’s 10-Q, which says. 1) we weight VaR to more recent movements; 2) VaR works only if markets behave themselves; 3)1-day VaR is useless for anything that cannot be sold or hedged within a day. Quoted excerpts below – from page 122:

“We use historical data to estimate our VaR and, to better reflect current asset volatilities, we
generally weight historical data to give greater importance to more recent observations. Given its
reliance on historical data, VaR is most effective in estimating risk exposures in markets in which
there are no sudden fundamental changes or shifts in market conditions. An inherent limitation of VaR
is that the distribution of past changes in market risk factors may not produce accurate predictions of
future market risk. Moreover, VaR calculated for a one-day time horizon does not fully capture
the market risk of positions that cannot be liquidated or offset with hedges within one day.”

Posted by Murray | Report as abusive
 

Nute/Phorgy:

Replace 2.33 with 1.95. The numbers Goldman use are the 95% percentile, 1-day.

http://blogs.reuters.com/felix-salmon/20 09/07/15/chart-of-the-day-goldman-var/

(Click through to the .pdf).

Yes. That number they report is the least far down the tail you can go with still calling it a VaR – remember banks go out to 99.99..% in the tail for VaR, Goldman isn’t a bank yet with it’s risk management!

I was digging into this and didn’t go further. I’ll get a chart later if I find anything interesting.

 

Hi Mike,

Felix was asking about what a bank’s VaR would look like if it invested in the S&P 500. The chart I provided answers this question.

A deeper (and important) question to ask (which it seems you are looking into) would be about the relationship between this S&P 500 VaR and the bank’s actually VaR. The bank’s P&L is obviously not a linear function of the S&P 500, so the relation should not be 1-to-1, but the sense that Felix seems to be looking for is how they should be related.

I agree that a bank’s VaR will increase as S&P 500 VaR increases, and conversely though the “Great Moderation” we saw VaR decrease. Since bank capital requirements are based on VaR, it is important to understand this. Decreasing VaR means increasing leverage. Increasing VaR means deleveraging.

Here is something I wrote on March 16, 2006 (when we were still in the Great Moderation):

http://phorgyphynance.wordpress.com/2008  /04/05/the-risks-of-risk-management-rev isited/

“When I look out at the world, one of the major risks to the markets that I see is, ironically, risk management. I suspect that one of the primary employers of junior quants in the last 5 years has been in risk analytics. If there is any truth to that, it means there is literally an army of quants who have not lived through a business cycle building risk systems on markets that no one really understands, e.g. CDS/CDOs.

[snip]

If things are at all like what I have seen, then we’ve got a bunch of fairly clueless risk managers out there with an army of fairly green quants developing sophisticated risk models that are probably pretty useless in a crisis. Nonetheless, there seems to be this completely ludicrous false sense of security.

Across the boards, vols seem to be historically low which would mean that most VaR engines are saying “smooth sailing”. What happens if vol increases? Everyone’s VaR model is going to start sending out little red flags. Assets are going to start getting reallocated. Since everyone has almost identical VaR models, the signals will be pretty much identical at all firms. I know it is not an original argument, but this could easily lead to a negative feedback. A small red flag due to increased VaR could signal everyone to make very similar reallocations. If everyone does it at the same time, the market will obviously be affected. In essence, the impact of risk management could actually increase systemic risk in the markets and amplify vol movements.”

 

Phorgy, that’s fantastic, if you have something just for the past few years, so we can zoom in on the right-hand side of the chart, that would be amazing…

Posted by Felix Salmon | Report as abusive
 

Back in 2008, there were several articles on this subject that described the time horizon used to calculate the volatility input into the VaR calculation. Here’s my blog post about it. Hope the link is okay. Bottom line is that it seems that the firms that got into the most hot water were using 4 years of data.

http://www.donfishback.com/blog/2008/10/ 27/ironic-the-same-assumptions-that-allo wed-reckless-leverage-have-reversed/

 

Phorgy,

Sorry I thought this post was about Goldman’s high VaR numbers instead of banks more generally. Read it too fast before running out the door. :)

Awesome chart. Remember that chart is 1 day, and most banks may (if memory serves) report 10 days, which will be higher. Multiply by sqrt(days) to get a back-of-the-envelope transformation.

 

>Phorgy, that’s fantastic, if you have something just for the past few years, so we can zoom in on the right-hand side of the chart, that would be amazing…

Here you go…

http://phorgyphynance.wordpress.com/2009  /08/06/80-years-of-daily-sp-500-value-a t-risk-estimates/

I added a new chart with 10 years of daily VaR estimates from Aug 1999 to Aug 2009.

 

What I would like to see is if there is some way to chart the VaR over time using the implied volatility in the options at the same point in time rather than a trailing actual volatility in the stock prices.

This would be computationally intensive for a single stock and obviously many times moreso for the S&P index, but it is possible. If I had daily call option data (strike price, strike date, and price would be sufficient) for all the stocks, I think I could do it.

What it would tell you is the market’s perception of actual volatility (and thus VaR) for the index. Of course, if the banks are using “dumb” volatility metrics like a trailing average, which it appears they are, then maybe that’s not of much interest. I’d certainly bet that the banks have internal models that are much better than this.

Posted by Mike Nute | Report as abusive
 

See Goldman’s discussion of Var in the Market Risk discussion section of the 10 Q around p 121-125. They use weighted historical simulation which presumably would result in risk reflecting more recent volatlity. But this is a methodology that is inconsistent with BIS guidelines and other bank reg requirements so any comparisons are fraught with peril.

In addition they disclose those positions that are NOT included in the VAR calculaton,(p124) which I assume their peers would include, which are sizeable.

The total reported var is 221 million. The risk-10%senstivity (whatever that means)is 1.2 BILLION.
If this sensitivity number can be interpreted as a var equivalent, then including it in any var analysis is a must.

Either way, VAR is of very limited analytical value in GS case since it is so opaquely calculated and disclosed, does not impact its capital requirements (as a result of the exemption), as it does for its peers, and VAR does not reflect intraday risk (, where I believe most of GS exceptional profits originate).

If the excluded market risk is the source of its FI profits, then those numbers should be disclosed and the exception should be revoked immediatedly to require GS to include them in the VAR.

At a minimum GS should be required to disclose their sanctioned VAR vs the VAR they would report if they were not exempt. I’m sure they calculate and monitor it. Its kind of outrageous the SEC or the FED doesn’t require it as a condition of the exemption.

Posted by mcnet | Report as abusive
 

Hi Felix,

I just noticed your “Update” so I added an “Update” on my post to address your “Update”. Meta meta…

 

like the clever accountant or lawyer would say. What do you want it to be. VaR explained here: http://nickgogerty.typepad.com/designing _better_futures/2009/06/var-explained.ht ml

 

Finally at least you are trying to understand VAR. However, you quickly allow yourself to still jump to the conclusion you just seem to want to take, no matter what. That VAR shows Goldman increased their bets.
However, VAR for Goldman will be more than just S&P, more than just equities. You need to take interest rates, currency rates, credit rates, basically the whole thing into account. They will show different volatility and different extremes.

Posted by M | Report as abusive
 

1/ The investment bank sides of most banks are delta hedged in their equity positions. So the formulas I’m seeing are far too simplistic.

2/ Suppose the banking universe is closed (it’s not, but suppose it is), so there are some banks which are over-all buyers of volatility/correlation and others which are sellers. (They are delta hedged but not hedged in terms of gamma, vega, etc.) When there is increased volatility/correlation in the market, the Var of the buyers only goes down a little, because there Var gets taken from the calm days, and there’s always some calm days in the historical sample. The Var of the sellers, however, explodes. So over-all in the banking sector the Var explodes.

3/ The banking universe is not closed and as a rule most banks are sellers of catastrophe. If the markets go down a little, they make money; but if they go down a lot, they lose a lot. When catastrophe is not in the historical sample, the Var stays reasonable. But when catastrophe makes it into the historical sample, it gets reflected in the Var, and Var explodes.

Posted by a | Report as abusive
 

Remember that for GS, there is also a move from SEC approved models to Fed approved models as a result of them becoming a BHC that will result in higher VaR readings, ceteris paribus.

 

FYI. I added another 10-year chart that uses a weighting scheme that I actually use in my work.

 

You can check daily VaR numbers for major equity indices here:

http://www.finanalytica.com/en/page/100/ VaR_daily_statistics

Besides you can compare fat-tailed vs. normal VaR.

Posted by sixx | Report as abusive
 

In a similar vein, here’s something I wrote at the end of 2008 based on the DJIA over the last 100 or so years.

http://www.riskmetrics.com/publications/ research_monthly/20081100

 

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