Convolution Approach for Value at Risk Estimation

Formally, Value at risk (VaR) measures the worst expected loss over a given horizon under normal market conditions at a given confidence level. Very often, daily data are used to compute VaR and scale up to the required time horizon with the square root of time adjustment. This gives rise to an important question when we perform VaR estimation: whether the values of VaR (i.e., “loss”) should be interpreted as (1) exactly on [Formula: see text]th day and (2) within i days. This research attempted to answer the above question using actual data of SPX and HSI. It was found that, in determining the proportionality of the values, (i.e., slopes) of VaR versus the holding period, the slopes for within i days are generally greater than those for exactly on [Formula: see text]th day. This has great implications to risk managers as it would be inappropriate to simply scale up the one-day volatility by multiplying the square root of time (or the number of days) ahead to determine the risk over a longer horizon of [Formula: see text] days. The evolution of log return distribution over time using actual data has also been performed empirically. It provides a better understanding than a series of VaR values. However, the number of samples in actual data is limited. There may not be enough data to draw reliable observation after it has been evolved a few times. Numerical simulation can help solve the problem by generating, say, one million log returns. It has been used to provide many insights as to how the distribution evolves over time and reveals an interesting dynamic of minimum cumulative returns. Numerical simulation is time consuming for performing evolution of distribution. The convolution approach provides an efficient way to analyze the evolution of the whole data distribution that encompasses VaR and others. The convolution approach with modified/scaled input distribution has been developed and it matches the results of numerical simulation perfectly for independent data for both exactly on [Formula: see text]th day and within i days. As the autocorrelation of the single index is mostly close to zero, results show that the convolution approach is able to match empirical results to a large extent.