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Value at Risk (VaR) is the maximum dollar portfolio amount that can be lost in a given period of time with a specified level of confidence--usually 5%. VaR has become a valuable tool that financial managers can use to measure market risk. The three basic VaR methodologies are Historical, Parametric and Simulation VaRs. Each has advantages and limitations, as well as ease of application under varied circumstances. In theory, the three should generally give equal values and any differences in their computed values are attributable to modeling issues and violations of assumption. Trends in VaR should be noted and explained. More problematic than the actual number and/or differences in the number is the array of possible realizations during the "other 5% of the time." If the other 5% of the time is well behaved, then those realizations should be anticipated and easily dealt with. On the other hand, if the other 5% of the time is not well behaved, then those realizations could be catastrophic and could lead to the demise of the enterprise.
Introduction
Risk, and how to measure it, are topics of great interest and increasing concern in the financial services industry these days. This comes as a consequence of macroeconomic turmoil and concomitant financial meltdowns in fixed income, mortgage and mortgage-backed securities over the past 12 months, as well as the spill-over into the equity markets, which has eroded personal and corporate balance sheets.
There are many sources of risk to an enterprise. Risk can be interest-rate related, exchange-rate related, political or geopolitical, weather risk, macroeconomic, model risk, accounting, fraud and malfeasance related, among others. Witness the financial crises the financial community has suffered postwar in general, but post 1970s in particular: oil price shocks (1973), Black Monday (Oct. 1987), the Mexican Peso crisis (1994), the Asian Crisis (1997), the Russian political crisis (1998), Long Term Capital Management (1998) the World Trade Center terrorist attack (2001) and others, all of which were the result of specific types of risk mentioned above. The importance of risk metrics increases with the size of the enterprises as well as increased risk associated with financial leverage and the use growth of derivative contracts.
Financial managers have seized on Value at Risk (VaR), a tool introduced in the 1980's, to monitor and manage market risk. VaR summarizes in one number the market risk of an enterprise--it is the maximum amount that an enterprise can lose in a given period of time with a given level of confidence. The purpose of VaR is to provide an analyst, an executive, a risk manager or a regulator with:
1. a guide as to the risk of market risk of an enterprise at any particular moment, as well as
2. the trends in the market risk of an entire enterprise over time.
VaR is a mathematical and statistical methodology with a number of assumptions--some quite restrictive and some less so. The accuracy of the VaR computation depends upon the degree to which the assumptions hold. If the assumptions do not hold, then the VaR will not be very accurate and the unknowing financial manager can be lulled into a false sense of security.
This paper discusses the intuition and methodology associated with several VaR metrics and identifies limitations and assumptions which underlie their accuracy. The broadcast and print media alike have wondered aloud how, given the level of expertise of so many large and sophisticated organizations, such large portfolio losses could have occurred where risk was presumably being adequately monitored. This paper helps identify reasons that could have happened and how, as the question has come to be framed, "so many smart people could get it wrong".
VaR--The Basic Goal
A discussion of Value at Risk (VaR) best begins with a definition reiterated from above:
Consider an investment portfolio consisting of stocks, bonds and other securities. Exhibit 1, below, displays a histogram of the one-day dollar returns of the current portfolio, using current portfolio weights over the last 500 days, with a normal, bell-shaped curve overlaid upon the histogram.
[GRAPHIC OMITTED]
Notice the vertical line dropped at -$16,449.63. This line separates the bottom 5% of the portfolio dollar returns per day, from the top 95% of the days. This value is the 5%, 1-day VaR, i.e. the portfolio loss in dollars, such that there is only a 5% probability that the portfolio could lose more than that amount in one day. Even though the cutoff is technically negative, the VaR estimate is expressed as a positive number.
VaRs are typically computed for 1, 10 or 20 days using 1% or 5% levels of probability. Monthly (20 trading days) VaRs are common for portfolios where daily performance is not readily available. For example, many hedge fund investors are only provided monthly returns by their managers without transparency into the underlying portfolio securities weights and returns, all of which might otherwise facilitate daily analysis of performance. Here, the hedge fund investor would be forced to make use of a monthly VaR. On the other hand, the hedge fund manager that knows the composition of the underlying portfolio would use a 1-day VAR.
VaR methodologies produce a number, hopefully an accurate number, which executives and risk managers can be comfortable with and have a sense of the risk of the enterprise at any specific moment in time. In all likelihood, that number is approximately accurate. The true number might be different, but within a margin of error, and probably a small difference relative to the capital of the firm. Managers can monitor the VaR number in a time series context. VaR numbers exhibiting a drift, however slight, up or down, for example the dotted lines shown in Exhibit 2, may be important in terms of the firm increasing or decreasing in risk. On the other hand, trendless variation in VaR, e.g., the solid line in Exhibit 2, can be considered statistical noise.
[GRAPHIC OMITTED]
While the VaR number is important, and trends in VaR are equally so, what is problematic is what happens "the other 5% of the time," that is, those periods when realized dollar loss is greater than the VaR estimate. It is not enough to know that a loss will be sustained more than $x, 5% of the time. It is more important to know, or at least worry about, the magnitude of the losses in those 5% of the instances. At the very minimum, one should be able to answer the question as to whether the tail regions, i.e. the other 5% of the time, are predictable or erratic. Often times it is quite difficult and at other times simply impossible, to know. The issue is akin to the insurance industry. While it might be known that 2% of the written hurricane policies produce a claim in a given year, it is the variation in those 2% of the claims which could be catastrophic. This issue, as well, will be elaborated upon below.
Three VaR Methodologies
There are three basic VaR methodologies, each with varying degrees of complexity:
1. Historical and Historical/Parametric VaR
2. Parametric VaR
3. Simulated Historical and Simulated Parametric VaR
Theoretically, any of the three methods could be applied to any portfolio. The reality, however, is that some methods are more easily and more appropriately applied in particular situations than others. The choice of method is largely affected by the availability and type of data. For securities with a long and liquid history, the historical approach is the preferred method. For more complicated securities, derivative instruments in particular and where the security is thinly traded, the parametric approach might be preferred because it can capture the effect of the underlying variables. For proposals of a newly designed product, especially one with asymmetric return streams, the simulation methods might be preferred.
Also in theory, the estimates using all methodologies should be the same or close. The reality here is that statistical issues and model intricacies will render different estimates. The causes of any differences should be investigated and understood.
Each method relies on a set of historical data as inputs for the computations and the interpretation of the estimate is the same regardless of the method employed. Each method is largely an exercise in statistics, economics, and portfolio theory, and makes moderate to intensive use of computer hardware and software, depending upon thenumber and size of portfolios. The difference between the methods lies in what is done with the historical data to arrive at a VaR. Where possible, most risk managers compute at least two of the four VaRs for at least two time horizons. One VaR is the primary metric and the second is used as a check.
VaR Computational Methods
This section describes the steps involved in the computation of each basic VaR methodology and identifies the advantages and disadvantages associated with each. Each of the basic methodologies can be enhanced to make more accurate replications of reality. Recall that the three VaR methodologies are:
1. Historical VaR and Historical Parametric VaR
2. Parametric VaR
3. Simulated Historical and Simulated Parametric VaR
Three assumptions which underlying the basic VaR models are:
1. Stationary mean
2. Stationary volatility
3. Stationary correlation structure between securities
The assumptions require that the mean return stream of the securities, the volatility of the return stream and the intercorrelation structure between securities have no drift over time. If these assumptions do not hold, then biased, or at least inaccurate VaR estimates, will be computed. In these cases, the basic models need to be enhanced to correct for the bias that would result in the VaR estimates. Later, extensions of each VaR method will be discussed to show how each can deal with relaxed assumptions to arrive at more accurate VaR estimates.
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