In the **Analytical Approach** we have a direct formula for estimating VaR:

where

- \(a(1-c)\) is the inverse cumulative distribution function of the Gaussian distribution, which gives us the percentile \(x_{(1-c)\%}\).
- \(\delta t\) is the time horizon
- \(\Delta\) is number of stocks of that asset
- \(S\) is the current price of that asset
- \(M\) is the number of portfolio assets

We multiply these parameters by the covariance matrix. This formula is applicable to the single-stock and multi-stock portfolio.

## getVar()

```
public class Analytical extends RiskMeasure
```

In `Analytical.java`

, we define our implementation of `getVar()`

- a concrete version of the abstract method in `RiskMeasure.java`

.

### Parameters

VaR takes two parameters, *Confidence* and *Time Horizon*.
Suppose we take the confidence level \(c = 99\%\).
This means we are \(99\%\) sure that we wonâ€™t lose more than \(V\), our estimate of Value at Risk, within our *Time Horizon*, which is usually one day.

```
double Confidence = Double.parseDouble(hashParam.get("Confidence"));
double TimeHorizon = Math.sqrt(Integer.parseInt(hashParam.get("TimeHorizonDays")));
```

These parameters are stored in a static `HashMap<String,String>`

instance, `hashParam`

.
We must parse to convert to numeric values.

### Normal Distribution

Analytically, we look at estimating VaR in terms of the standard Gaussian.

\[\alpha(1-c)\]```
NormalDistribution distribution = new NormalDistribution(0, 1);
double riskPercentile = -distribution.inverseCumulativeProbability(1 - Confidence);
```

Note: `NormalDistribution`

is part of the Apache Commons Math Library

### Market Data

In this block of code we retrieve:

- a vector of current prices for all assets in our portfolio
- Greek letter \(\Delta\) - the number of shares of a stock in our portfolio
- a matrix of percentage changes:
`double[][] matrixPcntChanges`

```
double[] currentPrices = new double[countAsset];
double[] stockDelta = new double[countAsset];
double[][] matrixPcntChanges = new double[countAsset][size];
try {
for (int i = 0; i < countAsset; i++) {
String sym = strSymbols[i];
Stock stock = stockHashMap.get(sym);
currentPrices[i] = stock.getQuote().getPreviousClose().doubleValue();
stockDelta[i] = new Double(hashStockDeltas.get(sym));
// get percentage changes of stock
double[] percentageChanges = PercentageChange.getArray(stock.getHistory());
matrixPcntChanges[i] = percentageChanges;
}
} catch (Exception e) {
e.printStackTrace();
}
```

Next we retrieve a covariance matrix using the above matrix.
We pass a string to `getType()`

to specify how we are estimating variance.

```
double[][] covarianceMatrix = new VolatilityFactory()
.getType(volatilityMeasure)
.getCovarianceMatrix(matrixPcntChanges);
```

## Estimating VaR

Lastly, we compute the linear combination of the product of Deltas, prices and covariances.

```
double sum = 0.0;
for (int i = 0; i < countAsset; i++)
for (int j = 0; j < countAsset; j++)
sum += stockDelta[i]
* stockDelta[j]
* currentPrices[i]
* currentPrices[j]
* covarianceMatrix[i][j];
```

Then we multiply the square root of the above with the timehorizon and riskPercentile.

```
double VaR = Math.sqrt(TimeHorizon)
* riskPercentile
* Math.sqrt(sum);
return VaR;
```

## Output

Assuming we have a portfolio consisting of 100 shares in *GOOG*, 200 in *MSFT*, and 100 in *AAPL*, with 95% confidence level and a time horizon of 1 day:

```
Analytical EW
VaR: 3558.909656
Analytical EWMA
VaR: 2555.420454
```

These results are computed using 5 years of historical data