var的计算逻辑及方法

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VALUE AT RISK (VAR)

What is the most I can lose on this investment? This is a question that almost

every investor who has invested or is considering investing in a risky asset asks at some

point in time. Value at Risk tries to provide an answer, at least within a reasonable bound.

In fact, it is misleading to consider Value at Risk, or VaR as it is widely known, to be an

We then consider the various estimation issues and questions that have come up in the

context of measuring VAR and how analysts and researchers have tried to deal with

them. Next, we evaluate variations that have been developed on the common measure, in

some cases to deal with different types of risk and in other cases, as a response to the

limitations of VaR. In the final section, we evaluate how VaR fits into and contrasts with

the other risk assessment measures we developed in the last two chapters.

What is Value at Risk?

In its most general form, the Value at Risk measures the potential loss in value of

a risky asset or portfolio over a defined period for a given confidence interval. Thus, if

the VaR on an asset is $ 100 million at a one-week, 95% confidence level, there is a only

used most often by commercial and investment banks to capture the potential loss in

value of their traded portfolios from adverse market movements over a specified period;

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this can then be compared to their available capital and cash reserves to ensure that the

losses can be covered without putting the firms at risk.

Taking a closer look at Value at Risk, there are clearly key aspects that mirror our

discussion of simulations in the last chapter:

1. To estimate the probability of the loss, with a confidence interval, we need to define

the probability distributions of individual risks, the correlation across these risks and

the effect of such risks on value. In fact, simulations are widely used to measure the

VaR for asset portfolio.

period over which risk is assessed and a confidence interval. The VaR can be

specified for an individual asset, a portfolio of assets or for an entire firm.

4. While the VaR at investment banks is specified in terms of market risks – interest rate

changes, equity market volatility and economic growth – there is no reason why the

risks cannot be defined more broadly or narrowly in specific contexts. Thus, we could

compute the VaR for a large investment project for a firm in terms of competitive and

firm-specific risks and the VaR for a gold mining company in terms of gold price

risk.

In the sections that follow, we will begin by looking at the history of the development of

this measure, ways in which the VaR can be computed, limitations of and variations on

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though their efforts were directed towards a different end – devising optimal portfolios

for equity investors. In particular, the focus on market risks and the effects of the co-

movements in these risks are central to how VaR is computed.

The impetus for the use of VaR measures, though, came from the crises that beset

financial service firms over time and the regulatory responses to these crises. The first

regulatory capital requirements for banks were enacted in the aftermath of the Great

Depression and the bank failures of the era, when the Securities Exchange Act

established the Securities Exchange Commission (SEC) and required banks to keep their

different capital requirements for each, ranging from 0% for short term treasuries to 30%

for equities. Banks were required to report on their capital calculations in quarterly

statements that were titled Financial and Operating Combined Uniform Single (FOCUS)

reports.

The first regulatory measures that evoke Value at Risk, though, were initiated in

1980, when the SEC tied the capital requirements of financial service firms to the losses

that would be incurred, with 95% confidence over a thirty-day interval, in different

security classes; historical returns were used to compute these potential losses. Although

the measures were described as haircuts and not as Value or Capital at Risk, it was clear

the SEC was requiring financial service firms to embark on the process of estimating one-

banks were becoming larger and more volatile, creating a need for more sophisticated and

timely risk control measures. Ken Garbade at Banker’s Trust, in internal documents,

presented sophisticated measures of Value at Risk in 1986 for the firm’s fixed income

portfolios, based upon the covariance in yields on bonds of different maturities. By the

early 1990s, many financial service firms had developed rudimentary measures of Value

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at Risk, with wide variations on how it was measured. In the aftermath of numerous

disastrous losses associated with the use of derivatives and leverage between 1993 and

1995, culminating with the failure of Barings, the British investment bank, as a result of

unauthorized trading in Nikkei futures and options by Nick Leeson, a young trader in

Singapore, firms were ready for more comprehensive risk measures. In 1995, J.P.

Morgan provided public access to data on the variances of and covariances across various

security and asset classes, that it had used internally for almost a decade to manage risk,

and allowed software makers to develop software to measure risk. It titled the service

Measuring Value at Risk

There are three basic approaches that are used to compute Value at Risk, though

there are numerous variations within each approach. The measure can be computed

analytically by making assumptions about return distributions for market risks, and by

using the variances in and covariances across these risks. It can also be estimated by

simple to compute if we can derive a probability distribution of potential values. That is

basically what we do in the variance-covariance method, an approach that has the benefit

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For a comprehensive overview of Value at Risk and its measures, look at the Jorion, P., 2001, Value at

Risk: The New Benchmark for Managing Financial Risk, McGraw Hill. For a listing of every possible

reference to the measure, try www.GloriaMundi.org.

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of simplicity but is limited by the difficulties associated with deriving probability

distributions.

Consider a very simple example. Assume that you are assessing the VaR for a

single asset, where the potential values are normally distributed with a mean of $ 120

fact that the assets in the portfolio often move together. As we noted in our discussion of

portfolio theory in chapter 4, the central inputs to estimating the variance of a portfolio

are the covariances of the pairs of assets in the portfolio; in a portfolio of 100 assets, there

will be 49,500 covariances that need to be estimated, in addition to the 100 individual

asset variances. Clearly, this is not practical for large portfolios with shifting asset

positions.

It is to simplify this process that we map the risk in the individual investments in

the portfolio to more general market risks, when we compute Value at Risk, and then

estimate the measure based on these market risk exposures. There are generally four steps

involved in this process:

The first coupon matches up to a one-year zero coupon bond with a face value of C,

the second coupon with a two-year zero coupon bond with a face value of C and so

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The 95% confidence intervals translate into 1.96 standard deviations on either side of the mean. With a

90% confidence interval, we would use 1.65 standard deviations and a 99% confidence interval would

require 2.33 standard deviations.