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© Copyright 2000 Rogers Media. The following article first appeared in the March 2000 edition of
BENEFITS CANADA magazine.
The Value of VaR
Value at Risk aims to answer that all important question: "How much might I lose?"
It's a measurement tool gaining popularity among pension trustees.
BY PAUL CARTER
Investors use many different criteria to determine which investments deserve their attention, but in the
end, the analysis boils down to two basic concepts: risk versus return. The concept and calculation of
investment return is generally well understood. The same cannot be said for investment risk.
Historically, plan sponsors, investment managers and consultants have summarized an investment product's or
strategy's risk profile with standard deviation or variance statistics, which are not really measures of
risk per se, but rather of volatility. As volatility of returns and investment risk are not necessarily
synonymous (as upside volatility is in fact a good thing), much has been written about the superiority of
"downside risk" or "semi-variance" in explaining the true risk to one's capital.
Regardless of one's view on the statistical purity of semi-variance over variance, trying to explain to
someone that their investment portfolio had a standard deviation of 5.8%, a downside deviation of 4.3% and
a downside probability of 29.4% will often be met with a confused expression and one simple question:
"Worst-case scenario, how much money can I lose?"
As a result, the concept of Value at Risk (VaR)--which originated in the 1980s with banks that were
attempting to quantify potential losses on their daily trading portfolios--is becoming increasingly popular
among longer-term thinking fund trustees. VaR is an attempt to quantify one's investment risk in one
simple, easy-to-understand figure. Simply stated, it is the amount of money, stated as either a dollar
figure or percentage return, that an investor could lose x% of the time in a specified time period.
There are many different ways to calculate VaR. Unlike a rate of return or standard deviation calculation,
there are no industry standards governing calculation methodology. The Association for Investment
Management and Research Reporting Performance Presentation Standards Handbook, the performance
measurement bible for industry professionals, is silent on the issue. The Risk Standards Working Group, an
independent group of senior buy-side investment and risk professionals in the U.S., released the Risk
Standards for Institutional Investment Managers and Institutional Investors in 1996. However, VaR is
merely touched upon in these standards with the comment that "there are a number of approaches to computing
VaR. The results are quite sensitive to the assumptions made and model used, and both should be
understood." Generally speaking, however, there is a direct relationship between the complexity of the VaR
calculation (and therefore the cost), and the accuracy of the results.
AN EXAMPLE
As an example, consider an investment in a Standard & Poor's (S&P) 500 index fund. If one wants to
know what tomorrow's VaR is for a $1 million investment in this index, one needs to develop a sense of all
possible one-day returns for this index. One way to do this is to look at past data. Rates of return for
this index are widely available and through the Internet, investors can access the data going back 50
years.
From January 1950 to August 1999, there have been over 12,000 trading days. The average daily rate of
return over this time period was 0.04%, and the standard deviation of daily returns was 0.85%. Using these
statistics, and the properties of the normal distribution, one can say that 95% of the time, the daily rate
of return on the S&P 500 index over the last 50 years was at least -1.36%.
Therefore, roughly 5% of the time (approximately one day out of every month), the index fell more than
1.36%. Using the past 50 years as a proxy for what could happen in the future, then the one-day VaR for a
$1 million index investment is $13,600 ($1 million x 1.36%). In other words, there is a 95% chance that
tomorrow, you will not lose more than $13,600.
However, is what happened in the 1950s, 1960s and 1970s necessarily relevant?
The stock market has become more volatile in recent years than it was many years ago. Therefore, one may
want to redo this calculation using only the most recent five years of data. Since 1995, 95% of the time,
the daily rate of return on the index was at least -1.54%, meaning the daily VaR (at a 95% confidence
level) is actually $15,400, not $13,600.
If one is to look at just the past two years, the VaR would be $19,400. And if one looks at just what has
happened in 1999, the VaR would be $18,500.
One other issue that needs to be clarified is that of confidence level. All of the above VaRs are
one-day-in-a month events. When investors ask the question: "Worst-case scenario, how much money can I
lose?," they probably aren't too interested in what will happen, on average, once every month.
Therefore, using a 99.6% confidence level (a one-day-in-a-year event) might be more appealing than a 95%
confidence level. Using data since 1995, the one-day VaR (at a 99.6% confidence level) is $25,300 instead
of $15,400.
But institutional investors tend to look longer-term than just one day. A one-quarter, one-year or even
four-year time horizon is probably more appropriate. There are some relatively straightforward techniques
that can be used to calculate VaRs for longer time horizons from daily historical data.
ACTIVELY MANAGED PORTFOLIOS
The above is a basic example of VaR. Most pension funds aren't invested in just the S&P 500 index. How
can VaR be calculated for a portfolio that is actively managed? This is where things can get more complex.
First of all, using your investment managers' historical daily, monthly or quarterly returns series to
calculate your fund's VaR can be irrelevant because the composition of your portfolio today may be
significantly, if not completely, different from what it was just a couple of years ago. How much money you
could potentially lose tomorrow is dependent upon what stocks, bonds and other investments you hold right
now, not what you held last month or last year.
The common (albeit somewhat scary) answer is to calculate historical returns, standard deviations and
correlation coefficients for each security within your fund, and to then combine these various statistics
to come up with a hypothetical historical distribution of returns for your portfolio--and from this, your
portfolio's VaR.
There are some problems with this approach, however. First of all, consider the example of trying to
calculate the VaR of the S&P 500 index on Oct. 18, 1987, using the prior 10 years worth of data. You
would have concluded, among other things, that there was a one-in-one-trillion chance that the index would
fall by more than 6% on any given day. And yet, the next day, the S&P 500 plunged by more than 20%. One
week later, it fell another 8.3%.
Obviously, the normal distribution isn't much good at explaining extreme observations. Using it as an
assumption may result in underestimating the risk to a fund.
Second of all, the necessary data for VaR calculations may be scarce. Many stocks and bonds that you hold
in your portfolio today may not have been around just a few years ago. Internet stocks are an excellent
example. For many Internet stocks, trying to determine VaR can be difficult since they may have gone public
only a few months ago. Bonds also pose a problem because many bonds in your portfolio may only have been
issued last year, so historical data for them may be limited.
Another problem is the complexity of the calculations. If your pension fund retains 10 investment managers,
each of whom holds 100 different stocks and bonds, then 1,000 different standard deviations and almost a
million different correlation coefficients will have to be calculated to determine VaR. Even the most
expensive desktop computer will have problems with this volume of calculations. One way around this is to
place each investment into a "risk bucket," which explains how certain types of securities behave.
For example, if you hold an 8% Government of Canada bond maturing in 2009 as well as a 7.75% Government of
Canada bond maturing in 2008, you may want to consider these two bonds as the same investment, at least
with respect to their risk profile. Bucketing individual securities into categories such as this will
reduce the necessary computational overhead.
Finally, once VaR is calculated, it becomes quickly obsolete. As soon as one of your investment managers
buys or sells a stock or bond, or adjusts your fund's asset mix, your investment risk, and thus your VaR,
changes. As a result, many end users require the ability to calculate VaR on a daily, weekly or monthly
basis to keep on top of the changing risk profile of their funds.
As the data requirements and calculations can become very complex, the cost to the pension fund can become
very high. Several large U.S. pension funds have reportedly spent over $1 million setting up elaborate
computer-based VaR systems. However, risk measurement technologists and investment consultants are coming
out with more and more VaR programs that will, in time, increase the availability and reduce the costs of
VaR analysis for pension funds.
Despite the fact that many obstacles will have to be overcome before VaR becomes a common point of
discussion in pension committee meetings in Canada, it is continuing to gain acceptance in investment
circles. Because of the need for better risk measurement and risk management techniques, and the relatively
comprehensible nature of the VaR output, VaR could potentially supplant standard deviation as the preferred
risk calculation methodology.
One day soon, pension fund trustees, investment managers, consultants and custodians will be able to
quickly analyze how any investment transaction or contemplated investment transaction will affect a fund's
risk profile.
Paul Carter is a research analyst at Frank Russell Company in Toronto. He wrote this article while he was
with COMSTAT Capital Sciences Inc. in Vancouver.
*** ***
Communicating VaR
Once the assumptions have been chosen and the calculations completed, communicating Value at Risk (VaR) to
a pension committee could potentially open up a can of worms. Comparative performance and risk measurement,
be it against either an index-based benchmark or a universe of other funds or managers, can give committees
a false sense of security. "Your fund's standard deviation of returns of 6.9% places your fund in the least
volatile quartile of the universe of similarly managed funds." The words 'least volatile,' along with an
abstruse standard deviation figure can give people comfort that everything is under control.
Rephrasing the above sentence to "there is a one-in-20 chance that you're going to lose at least $6.5
million this year, and a one-in-100 chance that you're going to lose at least $18 million this year," might
get committee members sitting up straighter.
It's imperative that consultants and investment managers ensure that clients understand this risk and the
implications of this risk, so that the ability to effectively manage risk isn't impaired by an overreaction
to a large potential loss. It is also essential for clients to recognize that there is a VaR with every
investment option (including even treasury bills), and that a level of loss tolerance over different time
periods should be firmly established before return maximization is attempted.
*** ***
VaR at work
The following chart demonstrates how the use of different historical data can lead to significantly
different conclusions regarding one's Value at Risk. The 99% Value at Risk of a $1 million investment in
the S&P 500 index will be approximately $220,000 if the past 50 years of historical data is used for
calculation purposes. In other words, there is a one-in-100 chance that over the next year, an investor
will lose more than $220,000 on this investment.
However, if one uses the past five years of historical data, when markets have been positive for investors,
one will conclude that the one-year Value at Risk is only $120,000. Because of the sensitivity of the VaR
calculation to just this one input assumption, conclusions have to be considered very carefully, and should
always be subjected to sensitivity analysis.
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