The value of a portfolio of financial assets is subject to many risks: credit risks, market risks, etc. **Value at Risk**, **VaR**, is a statistical estimate of the market risk of a portfolio. VaR attempts to answer the following question. Given a certain confidence level and a specified time horizon, what is the maximum potential loss of the portfolio?

Researchers and practitioners have proposed many methods of measuring market risk. In 1994 J. P. Morgan disclosed its RiskMetrics methodology and made its volatility and correlation data set publicly available. This quickly set the RiskMetrics variance-covariance method of calculating Value at Risk as an industry standard and it has become a benchmark in the measurement of financial market risks.

» FINCAD provides a toolbox of functions and several application workbooks that **calculate Value at Risk** based on this methodology. Click here to start your **free trial** of the latest version of FINCAD Analytics Suite!

## Variance-Covariance VaR

The methodology assumes that the market returns of the present value of future cash flows in a portfolio are normally distributed. Hence to compute the VaR of a portfolio, one only needs to know the volatility, i.e., the standard deviations of the market returns and the correlation between these returns.

Volatility and correlation data are only available at certain risk points. In the J.P. Morgan Riskmetrics data set, these points are classified by currency, asset class, and time. For example one risk point is USD spot equity and the class of USD bonds includes the risk points 1M, 3M, 6M, 12M, 2Y, 3Y, 4Y, 5Y, 7Y, 9Y, 15Y, 20Y and 30Y.

Given the risk points, an essential step in the VaR process involves mapping all financial instruments to the risk points. At this point, depending on how this mapping is done, the variance-covariance VaR method can take several forms. The most widely used (and simplest) method is called **Delta normal** or** linear VaR method**. In linear VaR, all instruments are mapped to the risk points using cash flows. The underlying assumption is that the change in the instrument's value is linearly related to the changes of some cash flows at the risk points. In the case of an option, this means that its change in value will be approximated using its delta.

## Five steps to calculating VaR

**Obtain the relevant volatility and correlation data for the positions in a portfolio.**

This data may be calculated or one may use publicly available volatility and correlation data. For example, the RiskMetrics data set is freely available and updated daily.

We note that in the RiskMetrics data sets, volatilities have been multiplied by the constant 1.65 (this corresponds to the 95-th percentile) and are in percentage values.

Let us consider a specific example of a portfolio of US money market securities and US treasury bonds. A US based investor holding this portfolio is subject to interest rate risk. Using the J. P. Morgan RiskMetrics data, the relevant risk points are the US money market risk points:

1M, 3M, 6M, 12M

and the US government bonds risk points:

2Y, 3Y, 4Y, 5Y, 7Y, 9Y, 15Y, 20Y, 30Y.

The volatility and correlation data from these risk points form the base for the

**VaR calculation**.**Identify cash flows.**

In the variance-covariance methodology, any financial instrument must be replicated by cash flows. The cash flows themselves must be related to the underlying risk points. For example, an interest rate forward position can be replicated by two cash flows, one at the effective date of the rate and the other at the maturity date. A bond is replicated by the present value of each cash flow on each of the coupon dates. An individual equity, if its volatility and correlation data are not available, is replicated as a cash flow of a relevant stock index (for which there is data) multiplied by the beta of the equity (the beta with respect to that index). An interesting class of instruments is options. In the linear VaR method, an option position is approximated as the product of the change of the underlying and the delta of the option.

The cash flows at this step are often referred to as raw cash flows.

**Map the cash flows to the risk points.**

As described, in step 2, any financial instrument is replicated by cash flows. In the case of interest rates, the raw cash flows may have any maturity date. As we saw, in the US example above, data was available only at the 1M, 3M, 6M, 12M, 2Y, 3Y, 4Y, 5Y, 7Y, 9Y, 15Y, 20Y, and 30Y risk points. Hence, interest rate raw cash flows must be mapped to the available risk points. This mapping is done in a risk preserving way using the volatility and correlation data.

We note that if a portfolio involves multiple currencies, one applies the cash flow mapping currency by currency.

**Rebase the volatility and correlation data.**

Suppose one would like to calculate the VaR in a currency other than the base currency of the volatility and correlation data. For example RiskMetrics data is based on the US dollar and a German based investor may want to calculate his VaR in Euros. In this case, it is useful to rebase the volatility and correlation data to the currency of the VaR calculation. FINCAD provides the function aaVAR_rebase for FX rebasing.

**Calculate VaR.**

After all the positions of a portfolio have been replicated by cash flows and all these replicating cash flows have been mapped on to the relevant risk points, one is ready to calculate VaR.

## Cash Flow Mapping

Within the value at risk (VaR) process, if is necessary to map interest rate cash flows to the available risk points. As an example, for US bonds, J.P. Morgan's RiskMetrics provides volatility and correlation data for the following maturities:

1 M, 3 M, 6 M, 12 M, 2Y, 3 Y, 4Y, 5 Y, 7 Y, 9 Y, 10 Y, 15 Y, 20 Y, 30 Y.

To calculate the VaR of a portfolio containing US dollar denominated cash flows, it is necessary to map them to these risk points.

Rebasing FX Volatility and Correlation

Suppose that the available volatility and correlation data is based in one currency and suppose one would like to use this data to calculate the value at risk (VaR) denominated in another currency. A common first step in the VaR process is to rebase this volatility and correlation data with respect to the new currency.

For example, J. P. Morgan's RiskMetrics volatility and correlation data is based on the USD. For a German based investor, to calculate VaR in EU using this data, it is useful to rebase the data based on the EU.

FINCAD provides a toolbox of functions and several application workbooks that calculate VaR based on this methodology. The functions work with the RiskMetrics data sets though it is not required that these data sets be used. To find out more information about VaR calculation with FINCAD solutions, contact a FINCAD Representative