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Evaluating Risk
By Adam Reeves

FAIS Licence No. 41200: Authorised as a financial services provider under the Financial Advisory and Intermediary Act, 2002
Registration No.  2010/011223/07 VAT registration No.  4650254479

INTRODUCTION

isk news stories always make front-page news.  Losing money, whether it is a bank’s capital or a client’s (Retirement Fund), is a very serious issue and one that taxpayers and members alike must concern themselves with.  After all, they are the ones who have to ultimately foot the bill.

Why are we looking at it in more depth now? JP Morgan Chase! In summary:

  • Big losses at JP Morgan Chase (JPM) have put the spotlight on risk processes used.
  • JPM are reputed to have some of the smartest people around.
  • JPM didn’t suffer the same issues as other banks during the sub-prime crash of 2008/9.
  • JPM has current losses (published) of ±$2bn (ZAR 16.7bn) with many market commentators calling this “the start” of the losses on this position, and some even estimate it to be as much as $5bn (ZAR41.75bn).
  • JPM and many other market participants use a risk analysis system called Value-at-Risk (“VaR”).
  • The VaR risk system is being held up to scrutiny as it estimated losses of only $67m (ZAR559.45m).
  • Satyajit Das: the bank’s restatement of its VaR risk from $67 million to $129 million (an increase of 93%) and the reinstatement of an older risk model are also significant, suggesting a failure of risk modeling.
  • Jamie Dimon JP Morgan Chase CEO is quoted as saying: “I know it was done with the intention to hedge tail risk… it was unbelievably ineffective…”.
  • Risk Santelli (American on-air editor for CNBC Business News network) and Tim Backshall (Founding Partner Capital Context – ex risk manager and author on risk): “VaR is useless”.
  • Regulators including the Financial Services Authority (FSA) are understood to be looking at the use of "Value at Risk" (VaR) models used by the giant US investment bank in monitoring its multi-billion-dollar Chief Investment Office (CIO).

OUR FINDINGS ON RISK

  • VaR is not the optimal risk measure. VaR will give you the minimum loss incurred in x% worst cases of the portfolio over a specified time period (in other words, the best worst case scenario). VaR is indifferent as to how serious the losses are beyond that % threshold.
  • VaR is not sub additive (portfolio diversification should always lead to risk reduction). With VaR it may produce an increase in the risk values even when partial risks are triggered by mutually exclusive events).
  • Expected Tail Loss (“ETL”) takes the average loss of all of the events past the threshold used by VaR.
  • VaR assumes returns are normally distributed, which is not the case as numerous studies have shown. ETL takes into account the skewness and kurtosis of returns.
    • Skewness: whether the distribution of returns are more positively shaped (leaning to the right) or negatively shaped (leaning to the left)
    • Kurtosis: how steep or flat the distribution of returns is.
  • ETL can give markedly different results from VaR. In our opinion, ETL gives a truer risk number.

WHAT IS RISK?

At Templewood Consulting we define Risk as the “unexpected loss of capital.”

If we invested in a risky asset, we should expect some amount of risk. When we create whole portfolio, risk is not falling short of an expected return but the extreme loss of capital. If we defined the amount of risk that we were happy to take on our portfolio at 5% of capital, and we did lose 5% of capital due to negative market movements, we wouldn’t be too unhappy as we knew pretty much what was at stake trying to get to our desired return. If however we lost 30%, we would be very unhappy as we could only tolerate 5% and would be wondering what went wrong and why didn’t our investment consultant do their job properly and help us construct our portfolio correctly?

When looking at complex issues such as risk, one has to use models to simplify and calculate what is happening. This is true in many industries and many sciences. It is the basis of understanding complex theories and mechanisms.

Simple models once used to minimize computing complexity will eventually be out of date. Assumptions made to try and explain new concepts are then taken at face value and used in old systems throughout the industry. Even the practitioners of some systems agree that the basic assumptions made can limit the accuracy of results and even lead to a misunderstanding of the conclusions. As the power of computing has increased at an astonishing rate, can the same be said of people’s use of basic models?

It is commonly known that the VaR model has got a few flaws in its design and yet we don’t come across too many disclaimers from market practitioners warning clients of the dangers of relying on the misuse or reliance on such models. The fact that Trustees are being asked to make such important decisions as “how much risk can we tolerate in our fund” based on limited knowledge (theirs and their consultant’s) is grounds for some serious reflection. Do the consultants that use VaR fully understand risk and or are they just hoping that the markets behave themselves?

The Latin phrase “Ceteris paribus” (all other things being the same) when used in financial modeling is one of the most dangerous statements that can be made. When in the history of financial markets, has anything ever been constant?

We will be looking at the basic assumptions of the Value at Risk model in the point below titled “What is VaR?” and the problems that demonstrate themselves in the following sections.

OTHER RISK CONSIDERATIONS

It’s not only the risk system used that should concern members and investors, but also who uses these systems and what independent checks and balances are routinely performed?

Consider:

  • An Investment Consulting company that uses a VaR risk system to monitor Asset Managers and Hedge Fund Managers performances and risk numbers.
  • It uses the same system to consolidate the risk reports used by clients.
  • It uses the same system to construct portfolios and choose managers.
  • It charges a very “reasonable” fee for risk consulting (standardized risk monitoring results in standardized reports, all of which are scalable and cheaper to produce).

What would happen if there were major flaws in that risk system?
Who would pick it up? Who would question the findings?
Is it prudent for a consultant to have this much power?
Can one be a gamekeeper and poacher?

JP MORGAN CHASE AND THE VARIOUS RELATIONSHIPS WITH VAR

Whilst the concept of VaR had been around since the late 1980’s it was JP Morgan’s CEO in 1994 that actually pioneered the use of Value at Risk (“VaR”) for use in mainstream risk management when looking at his firm’s overall risk.  JP Morgan (“JPM”) published the methodology and gave free access to estimates of the necessary underlying parameters.  This was the first time VaR had been exposed beyond a relatively small group of quants.

Back in December 2008 when JP Morgan Chase boss Jamie Dimon was being routinely feted as the great survivor of the financial crisis, he popped up on CNBC television.  "The pedestal is a terrible place to be," the banker declared. "I almost want to get knocked off the pedestal so I don't have to hear this anymore."

Well, he's managed that.  On 17 May 2012, Dimon dropped the bombshell that a bank renowned for its financial rectitude had somehow blown $2bn.  Not only that – it had racked up the losses at the London wing of its Chief Investment Office (CIO), the very part of the business that's meant to hedge all the risks that go with JP Morgan's thumping $2.3 trillion balance sheet.

The CIO was mandated to make investments that hedged the risks inherent in the bank's colossal balance sheet, the left over mortgage and sub-prime exposure left on JPM’s balance sheet after the 2008/9 crisis among other risks.  But, somewhere along the line, the boundaries between hedging and proprietary trading seem to have blurred.  Since 2007, the value of securities held by the CIO has rocketed from $76.5bn to top $350bn.

Rumours have been running for months, so much so that Dimon was quizzed on the issue at JP Morgan's first-quarter results on April 13.  Then, in a phrase that has come back to haunt him, he dismissed the issue as a "complete tempest in a teapot", stressing that, as a big bank with $1.1 trillion of deposits, it was no surprise JP Morgan had a big CIO portfolio.  "Our job is to invest that portfolio wisely and intelligently over a long period of time to earn income and to offset other exposures we have," he said at the time.

One theory behind the $2bn losses, which Dimon admits could increase, is that as the market got wind of JP Morgan's huge positions, hedge funds bet against them.

On 2011 figures, the two wings (JPM Bank & CIO) had virtually the same value at risk: $57m on most days of the year at the CIO versus $58m at the investment bank.  Now Dimon has admitted that in the first quarter of 2012, the figure for the CIO is actually $129m, not $67m.

So where did the loss of $2bn come from?

JP Morgan is facing regulatory scrutiny over the use of risk models that failed to identity significant losses building up on its balance sheet.  Regulators including the Financial Services Authority (FSA) are understood to be looking at the use of "value at risk" (VAR) models used by the giant US investment bank in monitoring its multi-billion-dollar Chief Investment Office (CIO).  The bank introduced a new VAR system, chief executive Jamie Dimon has admitted, which failed to identify the true scale of the risks being built up on the bank's balance sheet through risky credit hedging positions.  The discussions will focus on whether the $2bn trading loss was merely a function of poor decision making or if it is symptomatic of a wider breakdown in the checks and balances that banks are required to have in place.

On VaR, the source admitted: "These are complex questions that will need to be looked at."

Mr. Dimon has already conceded there was a problem with the system that had been in place. Reports suggested it both underestimated and understated the firm's risk. It was only when a previous model was reintroduced that the true scale of the problems at the bank was revealed.

The trading losses suffered by JP Morgan Chase have surged , surpassing the bank’s initial $2 billion estimate by at least $1 billion, according to people with knowledge of the losses.  When Jamie Dimon, JP Morgan’s chief executive, announced the losses mid-May, he indicated they could double within the next few quarters.

Risk Santelli (American on-air editor for CNBC Business News network) and Tim Backshall (Founding Partner Capital Context – ex risk manager and author on risk) critically address the complicity of banks, regulators, analysts, and The Fed in giving 'banks the benefit of the doubt' with regard to their use of the bottomless pit of capital they implicitly have. But what is more important for the hordes of sell-side analysts and buy-side people to understand is just what this JPM debacle exposes about bank risk.

Satyajit Das, one of the most respected financial and risk author in the world today, writes in Topiary Lessons – “effective hedging?”:

Given JPM vaunted risk management credentials and boasts of a “fortress like” balance sheet, it is surprising that the problems of the hedge were not identified earlier.  In general, most banks stress test hedges to ensure their efficacy prior to implementation and monitor them closely.

While the US$2 billion loss is grievous, the bank’s restatement of its VaR risk from $67 million to $129 million (an increase of 93%) and reinstatement of an older risk model is also significant, suggesting a failure of risk modeling.

The fate of specific actors is more difficult to predict. Mr. Dimon’s language in describing the losses was expressive:  “… Errors, sloppiness, and bad judgment… Badly executed, badly monitored. I’m not going to repeat it 800 times… “I know it was done with the intention to hedge tail risk… it was unbelievably ineffective…”

We have been working with assistance from academia (a South African University and one in the United Kingdom) on a more refined way at looking at risk and one we are sure will be taken up with the industry eventually.  Let’s hope it’s not too late or too expensive for the clients of other industry participants to catch up.

WHAT IS VALUE AT RISK (VaR)?

Please refer to Annexure A for a detailed description of VaR.

Investopedia defines VaR as “a technique used to estimate the probability of portfolio losses based on statistical analysis of historic price trends and volatilities.”

Value at Risk is a special type of downside risk measure. Rather than produce a single statistic or express absolute certainty, it makes a probabilistic estimate. With a given confidence level, it asks, "What is our minimum loss incurred in x% worst cases of our portfolio over a specified time period?"  There are three methods by which VaR can be calculated:

  • the historical simulation,
  • the variance-covariance method and
  • the Monte Carlo simulation.

1.            Historical method

Whilst this is the easiest to calculate, it is also the most flawed. Every market has different drivers and different implied fundamental factors driving market sentiment (e.g. inflation, emerging markets, commodities, carry trade etc.).

The historical simulation requires more computational data; it also assumes that "past is prologue".

Purely using historic figures and been described as “driving along a road only using your rear view mirror to help you with the road ahead!” – good to see where you have gone, but not so affective in telling you about upcoming obstacles.

2.            Variance-Covariance method

The Variance-covariance method is easiest because you need to estimate only two factors: average return and standard deviation.  However, it assumes returns are well behaved according to the symmetrical normal curve and that historical patterns will repeat into the future.

3.            Monte Carlo method

When using the Monte Carlo method, the model uses a matrix of covariances (the relationship between asset prices to each other. High covariance will give you a low diversification. This is complicated when there are a lot of assets, all with different relationships to each other).

No one can predict the severities of a crisis and the way that markets react are completely irrational and therefore quite hard to predict. Ask any statistician and they will tell you that the hardest thing to model is the covariance that assets have under times of crisis. When a stock market crashes, money goes into bonds and cash, goes from weak currencies back into “safer” currencies. When the financial crisis hit in 2008/9, virtually all assets lost value. That is what made it a crisis of such a monumental scale.

Taking the results from Annexure A
Monthly VaR (95% confidence)
Variance-covariance method:           19.5%
Monte Carlo simulation:                     15.0%

So what is the correct number to use?

WHAT ARE THE LIMITATIONS WITH VaR?

Both the historical and Monte Carlo simulation methods have their advocates; but the historical method requires crunching historical data, and the Monte Carlo simulation method is complex.  The Variance-covariance method is easiest because you need to estimate only two factors: average return and standard deviation.  However, it assumes returns are well behaved according to the symmetrical normal curve and that historical patterns will repeat into the future.  MARKETS ARE NOT WELL BEHAVED.

Looking at 1-month returns of the JSE/FTSE All Share (J203) graph we can see a feature called a “fat tail”. The returns are NOT normally distributed (close up until 20% down). There are more downside observations after -20%. In fact there are an additional 50% more downside movements than what a VaR calculation would have made.


Source: Salient Quants

VaR is not the worst case; it is the best of a series of worst cases.

The historical simulation improves on the accuracy of the VaR calculation, but requires more computational data; IT ASSUMES THAT "PAST IS PROLOGUE".

The main disadvantage of Monte Carlo Simulations VaR is the computer power that is required to perform all the simulations, and thus the time it takes to run the simulations. If we have a portfolio of 1,000 assets and want to run 1,000 simulations on each asset, we will need to run 1 million simulations (without accounting for any eventual simulations that may be required to price some of these assets – like for options and mortgages, for instance). MOREOVER, ALL THESE SIMULATIONS INCREASE THE LIKELIHOOD OF MODEL RISK.

Simulations are not necessarily predictive of the future. Importantly, adding more trials does not necessarily improve the simulation.

Related, and perhaps another form of model risk, the whole thing is built on the robustness of the embedded covariance matrix (i.e., which gives the covariances between factors). If that is not applicable going forward and/or under stressful conditions, the simulation is flawed

CONCLUSION ON VaR

VaR essentially asks, “What is the minimum loss incurred in the x% worst cases of our portfolio?”

Due to the “minimum loss” in its definition, VaR is not a sub-additive measure.  Simply being the threshold of the possible x% losses, VaR is indifferent of how serious the losses beyond that threshold actually are.  Little imagination is needed to construct portfolios with identical VaR and dramatically different levels of risk in the same x% worst-case sample.

The global risk of a portfolio will be the sum of the risks of its parts ONLY if the sources of these risks conspire to act together.  In all other cases, the global risk of a portfolio will be less than the sum of its partial risks thanks to risk diversification.  This is essentially how a risk measure should behave under the composition / addition of portfolios.  It is the key test for checking whether a measurement of a portfolio’s risk is consistent with those of its parts.

EXPECTED TAIL LOSS (“ETL”)

The question that should be asked when looking at the risk of a portfolio is “What is the expected loss incurred in the x% worst cases of our portfolio?”

The average expected loss beyond VaR is also known as Conditional Value at Risk (CVaR) or Average Value at Risk (AVaR).  It can be interpreted as the expected shortfall assuming VaR as a benchmark. ETL does not have the deficiencies of VaR as it is a true downside risk measure that can recognize diversification opportunities and has good optimality properties.  ETL is a sub additive measure and therefore can be used to aggregate or decompose risk at the portfolio or strategy levels.  This is why ETL is the risk measure to be used as the basis for risk budgeting.

Modified Value at Risk: The Modified Value at Risk is calculated in the same manner as Value at Risk but doesn't assume a normal distribution of returns.  In contrast it "corrects" the Value at Risk using the calculated skewness and kutosis.

Skewness: the degree of asymmetry of a distribution around its mean. Positive skewness indicates a distribution with an asymmetric tail extending toward more positive values. Negative skewness indicates a distribution with an asymmetric tail extending toward more negative values.

Kurtosis: the relative peakedness or flatness of a distribution compared with the normal distribution. Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution of the distribution of returns.

What we are now focusing on is the average of a range of potential losses.  VaR gives the best case of the potential losses (which can be different depending on what model has been used).

The expected tail loss will give you a number that in our experience is markedly different from the VaR. A true reflection on what losses a portfolio could incur.

Surely that is the number that Trustees should be after?

CONCLUSION

We have used the recent losses at JP Morgan Chase to explain the limitations of using VaR as your sole risk measure.  There are numerous reasons  - at least $2billion of them (and counting) – why ETL will provide a more relevant and realistic risk number.

ANNEXURE A – WHAT IS VaR?

Investopedia defines VaR as “a technique used to estimate the probability of portfolio losses based on statistical analysis of historic price trends and volatilities.”

The idea behind VaR

The most popular and traditional measure of risk is volatility.  The main problem with volatility, however, is that it does not care about the direction of an investment's movement: a stock can be volatile because it suddenly jumps higher.  Of course, investors are not distressed by gains!

For investors, risk is about the odds of losing money, and VaR is based on that common-sense fact.  By assuming investors care about the odds of a really big loss, VaR answers the question, "What is my worst-case scenario?" or "How much could I lose in a really bad month (quarter, or other specific time period)?"

A VaR statistic has three components: a time period, a confidence level and a loss amount (or loss percentage). Keep these three parts in mind as we give some examples of variations of the question that VaR answers:

  1. What is the most I can - with a 95% or 99% level of confidence - expect to lose in dollars over the next month?
  2. What is the maximum percentage I can - with 95% or 99% confidence - expect to lose over the next year

You can see how the "VaR question" has three elements: a relatively high level of confidence (typically either 95% or 99%), a time period (a day, a month or a year) and an estimate of investment loss (expressed either in dollars or percentage terms).

Methods of Calculating VaR

Institutional investors use VaR to evaluate portfolio risk, but in this introduction we will use it to evaluate the risk of a single index that trades like a stock: the Nasdaq 100 Index, which trades under the ticker QQQ.  The QQQ is a very popular index of the largest non-financial stocks that trade on the Nasdaq exchange.

There are three methods of calculating VaR:

  1. the historical method,
  2. the variance-covariance method and
  3. the Monte Carlo simulation.

1.            Historical Method

The historical method simply re-organizes actual historical returns, putting them in order from worst to best. It then assumes that history will repeat itself, from a risk perspective.

The QQQ started trading in March 1999, and if we calculate each daily return, we produce a rich data set of almost 1,400 points. Let's put them in a histogram that compares the frequency of return "buckets". For example, at the highest point of the histogram (the highest bar), there were more than 250 days when the daily return was between 0% and 1%. At the far right, you can barely see a tiny bar at 13%; it represents the one single day (in Jan 2000) within a period of five-plus years when the daily return for the QQQ was a stunning 12.4%!

Notice the red bars that compose the "left tail" of the histogram. These are the lowest 5% of daily returns (since the returns are ordered from left to right, the worst are always the "left tail"). The red bars run from daily losses of 4% to 8%. Because these are the worst 5% of all daily returns, we can say with 95% confidence that the worst daily loss will not exceed 4%. Put another way, we expect with 95% confidence that our gain will exceed -4%. That is VaR in a nutshell. Let's re-phrase the statistic into both percentage and dollar terms:

  1. With 95% confidence, we expect that our worst daily loss will not exceed 4%.
  2. If we invest $100, we are 95% confident that our worst daily loss will not exceed $4 ($100 x -4%).

You can see that VaR indeed allows for an outcome that is worse than a return of -4%. It does not express absolute certainty but instead makes a probabilistic estimate. If we want to increase our confidence, we need only to "move to the left" on the same histogram, to where the first two red bars, at -8% and -7% represent the worst 1% of daily returns:

  • With 99% confidence, we expect that the worst daily loss will not exceed 7%.
  • Or, if we invest $100, we are 99% confident that our worst daily loss will not exceed $7.

2.            The Variance-Covariance Method

This method assumes that stock returns are normally distributed. In other words, it requires that we estimate only two factors - an expected (or average) return and a standard deviation - which allow us to plot a normal distribution curve. Here we plot the normal curve against the same actual return data:

The idea behind the Variance-Covariance is similar to the ideas behind the historical method - except that we use the familiar curve instead of actual data. The advantage of the normal curve is that we automatically know where the worst 5% and 1% lie on the curve. They are a function of our desired confidence and the standard deviation ():

The blue curve above is based on the actual daily standard deviation of the QQQ, which is 2.64%. The average daily return happened to be fairly close to zero, so we will assume an average return of zero for illustrative purposes. Here are the results of plugging the actual standard deviation into the formulas above:

3.            Monte Carlo Simulation

The third method involves developing a model for future stock price returns and running multiple hypothetical trials through the model. A Monte Carlo simulation refers to any method that randomly generates trials, but by itself does not tell us anything about the underlying methodology.

For most users, a Monte Carlo simulation amounts to a "black box" generator of random outcomes. Without going into further details, we ran a Monte Carlo simulation on the QQQ based on its historical trading pattern. In our simulation, 100 trials were conducted. If we ran it again, we would get a different result--although it is highly likely that the differences would be narrow. Here is the result arranged into a histogram (please note that while the previous graphs have shown daily returns, this graph displays monthly returns):

To summarize, we ran 100 hypothetical trials of monthly returns for the QQQ. Among them, two outcomes were between -15% and -20%; and three were between -20% and 25%. That means the worst five outcomes (that is, the worst 5%) were less than -15%. The Monte Carlo simulation therefore leads to the following VaR-type conclusion: with 95% confidence, we do not expect to lose more than 15% during any given month.

Converting One Time Period to Another
In the section above, we calculate VaR for the Nasdaq 100 index (ticker: QQQ) and establish that VAR answers a three-part question:
"What is the worst loss that I can expect during a specified time period with a certain confidence level?"

Since the time period is a variable, different calculations may specify different time periods - there is no "correct" time period.  Commercial banks, for example, typically calculate a daily VaR, asking themselves how much they can lose in a day; pension funds, on the other hand, often calculate a monthly VaR.

To recap briefly, let's look again at our calculations of three VaRs in the section above using three different methods for the same "QQQ" investment:

* We do not need a standard deviation for either the historical method (because it just reorders returns lowest-to-highest) or the Monte Carlo simulation (because it produces the final results for us).

Because of the time variable, users of VaR need to know how to convert one time period to another, and they can do so by relying on a classic idea in finance: the standard deviation of stock returns tends to increase with the square root of time.  If the standard deviation of daily returns is 2.64% and there are 20 trading days in a month (T = 20), then the monthly standard deviation is represented by the following:

To "scale" the daily standard deviation to a monthly standard deviation, we multiply it not by 20 but by the square root of 20.  Similarly, if we want to scale the daily standard deviation to an annual standard deviation, we multiply the daily standard deviation by the square root of 250 (assuming 250 trading days in a year).  Had we calculated a monthly standard deviation (which would be done by using month-to-month returns), we could convert to an annual standard deviation by multiplying the monthly standard deviation by the square root of 12.

Below we incorporate the time-conversion element into the Variance-Covariance method:

Now let's apply these formulas to the QQQ. Recall that the daily standard deviation for the QQQ since inception is 2.64%. But we want to calculate a monthly VaR, and assuming 20 trading days in a month, we multiply by the square root of 20:

* Important Note: These worst losses (-19.5% and -27.5%) are losses below the expected or average return. In this case, we keep it simple by assuming the daily expected return is zero. We rounded down, so the worst loss is also the net loss.

So, with the Variance-Covariance method, we can say with 95% confidence that we will not lose more than 19.5% in any given month.  The QQQ clearly is not the most conservative investment! You may note, however, that the above result is different from the one we got under the Monte Carlo simulation, which said our maximum monthly loss would be 15% (under the same 95% confidence level).

Conclusion

The Variance-Covariance method is easiest because you need to estimate only two factors: average return and standard deviation.  However, it assumes returns are well behaved according to the symmetrical normal curve and that historical patterns will repeat into the future.

The historical simulation improves on the accuracy of the VaR calculation, but requires more computational data; it also assumes that "past is prologue".  The Monte Carlo simulation is complex, but has the advantage of allowing users to tailor ideas about future patterns that depart from historical patterns.


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