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

\[\mathit{VaR} = -\alpha(1-c)(\Delta t)^{1/2}\sqrt{\sum_{j=1}^M\sum_{i=1}^M\Delta_i S_i\cdot\Delta_j S_j\Sigma_{ij}} \\\]

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 won’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

Updated: