# Value at risk

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## Definition

In economics and finance, the Value at risk, or VaR, is a measure used to estimate how the value of an asset or of a portfolio of assets could decrease over a certain time period (usually over 1 day or 10 days) under usual conditions. It is typically used by security houses or investment banks to measure the market risk or volatility risk of their asset portfolios, but is actually a very general concept that has broad application. VaR has three parameters: the time horizon (period) we are going to analyze (i. e. the length of time over which we plan to hold the assets in the portfolio - the “holding period“), the confidence level at which we plan to make the estimate, and the unit of the currency which will be used to denominate the value at risk. The typical holding period is 1 day, although 10 days are useless, for example, required to compute capital requirements under the European Capital Adequacy Directive (CAD). For some problems, even a holding period of 1 year is inappropriate. Popular confidence levels usually are 99% and 95%.

As an example, an investment bank might report that its portfolio has a 1-day VaR of \$5 million at the 95% confidence level. This implies that (provided usual conditions will prevail over the 1 day) the bank can expect that, with a probability of 95%, the value of its portfolio will decrease by 5 million or less during 1 day, or in other words: it can expect that with a probability of 5% (i. e. 100%-95%) the value of its portfolio will decrease by more than 5 million during 1 day. Stated yet differently, the bank can expect that the value of its portfolio will decrease by 5 million or less on 95 out of 100 usual trading days, in other words by more than 5 million on 5 out of every 100 usual trading days.

VaR (1 day; 95%) measures what will be the maximum loss (i. e. decrease in portfolio value) over 1 day, if one assumes that the 1 day will not be one of the 5% days that are the worst under normal conditions. It thus measures how much one could lose, but it also provides an indication of how much money might be put aside as a cushion for days when losses are unexpectedly large. Thus VaR is not only a risk measurement tool, but also facilitates risk management.

## Common VaR calculation models

In the following, return means percentage change in value.

A variety of models exist for estimating VaR. Each model has its own set of assumptions, but the most common assumption is that historical market data is our best estimator for future changes. Common models include:

• (1) variance-covariance (VCV), assuming that risk factor returns are always (jointly) normally distributed and that the change in portfolio value is linearly dependent on all risk factor returns,
• (2) the historical simulation, assuming that asset returns in the future will have the same distribution as they had in the past (historical market data),
• (3) Monte Carlo simulation, where future asset returns are more or less randomly simulated

The variance-covariance, or delta-normal, model was popularized by J.P. Morgan Chase (formerly J.P. Morgan) in the early 1990's. In the following, we will take the simple case, where the only risk factor for the portfolio is the value of the assets themselves. The following two assumptions enable to translate the VaR estimation problem into a linear algebraic problem:

(1) The portfolio is composed of assets whose deltas are linear, more exactly: the change in the value of the portfolio is linearly dependent on (i.e. is a linear combination of) all the changes in the values of the assets, so that also the portfolio return is linearly dependent on all the asset returns.

(2) The asset returns are jointly normally distributed.

The implication of (1) and (2) is that the portfolio return is normally distributed because it always holds that a linear combination of normally distributed variables is itself normally distributed.

We will use the following notation:

• [itex]\ _i [itex] means “of the return on asset i“ (for σ and [itex]\mu[itex]) and "of asset i" (otherwise)
• [itex]\ _p [itex] means “of the return on the portfolio”(for σ and [itex]\mu[itex]) and "of the portfolio" (otherwise)
• all returns are returns over the holding period
• there are N assets
• [itex]\mu[itex]= expected value, i. e. mean
• σ = standard deviation
• V = initial value (in currency units)
• [itex] \omega_i \ = \ V_i \ / \ V_p [itex]
• [itex]\boldsymbol{\omega}[itex]= vector of all [itex] \omega_i [itex] (T means transposed)
• [itex]\boldsymbol{\Sigma}[itex]= covariance matrix = matrix of covariances between all N asset returns, i. e. an NxN matrix

The calculation goes as follows.

(i) [itex] \mu_p\ = \sum_{i=1}^N \omega_i \mu_i,[itex]

(ii) [itex] \sigma_p\ = \sqrt{\boldsymbol{\omega}^T \boldsymbol{\Sigma}\boldsymbol{\omega}} [itex]

The normality assumption allows us to z-scale the calculated portfolio standard deviation to the appropriate confidence level. So for the 95% confidence level VaR we get:

(iii) [itex]\ VaR \ = \ - \ V_p \ ( \mu_p \ + \ 1.645 \sigma_p \ ) [itex]

The benefits of the variance-covariance model are the use of a more compact and maintainable data set which can often be bought from third parties, and the speed of calculation using optimized linear algebra libraries. Drawbacks include the assumption that the portfolio is composed of assets whose delta is linear, and the assumption of a normal distribution of asset returns (i. e. market price returns).

Historical simulation is the simplest and most transparent method of calculation. This involves running the current portfolio across a set of historical price changes to yield a distribution of changes in portfolio value, and computing a percentile (the VaR). The benefits of this method are its simplicity to implement, and the fact that it does not assume a normal distribution of asset returns. Drawbacks are the requirement for a large market database, and the computationally intensive calculation.

Monte Carlo simulation usually involve principal components analysis of the VCV matrix, followed by random simulation of the components. Benefits are the ability to handle any underlying distribution, plus a more accurate assessment when non-linear risk factors are present in the portfolio (e.g. options). Drawbacks include the inherently opaque nature of Monte Carlo calculations, and the computationally intensive process.

## Caveats

Unfortunately, VaR is not the panacea of risk measurement methodologies. A subtle technical problem is that VaR is not sub-additive. That is, it's possible to construct two portfolios, A and B, in such a way that VaR(A + B) > VaR(A) + VaR(B). This is unexpected because we expect portfolio diversification to reduce risk. See http://www.fenews.com/fen40/risk-reward/risk-reward.htm for more on this topic. Fortunately, Expected Value at Risk, or EVaR, modifies the VaR methodology slightly in a way that solves this difficulty.

• Crouhy M., D. Galai, and R. Mark, Risk Management, McGraw-Hill, 2001.
• Dowd, Kevin, Measuring Market Risk, John Wiley & Sons, 2002.
• Holton, Glyn A., Value-at-Risk: Theory and Practice, Academic Press, 2003.
• Hull, John C., Options, Futures, and Other Derivatives, 5th ed., Prentice Hall, 2002.
• Jorion, Philippe, Value at Risk: The New Benchmark for Managing Financial Risk, 2nd ed., McGraw-Hill Trade, 2001.

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