Question

1. Compare paramatric, historical, and monte carlo simulation methods in identifying VaR (value at risk)

2. What are the pros and cons of those?

3. Identify some weights on historical losses that you think should make more sense in current trading war environment.

Answer 1

01.

VaR of a portfolio involves determining a probability distribution for the change in the value of the portfolio over the time period (known as the holding period). The value of the portfolio of financial instruments, at time t depends on the k risk factors (market variables). These risk factors could be exchange rates, interest rates, stock prices, etc. Thus, the estimation VaR is done via estimation of the distribution of the underlying risk factors. The general techniques commonly used include analytic techniques:

A.Parametric Simulation (Delta-Normal method)

B.Monte Carlo simulation

C.Historical Simulation

Parametric Simulation Method:

The delta-normal method is a parametric, analytic technique where the distributional assumption made is that the daily geometric returns of the market variables are multivariate normally distributed with mean return zero. Historical data is used to measure the major parameters: means, standard deviations, correlations. When the market value of the portfolio is a linear function of the underlying parameters, the distribution of the profits is normal as well.

The delta-normal method can be subject to a number criticism. A first problem is the existence of fat tails in the distribution of returns on most financial assets. The distribution of daily returns of any risk factor would in reality typically show significant amount of positive kurtosis. This leads to fatter tails and extreme outcomes occurring much more frequently than would be predicted by the normal distribution assumption, which would lead to an underestimation of VaR (since VaR is concerned with the tails of the distribution)

Monte Carlo simulation:

Monte Carlo simulation techniques are by far the most flexible and powerful, since they are able to take into account all non-linearities of the portfolio value with respect to its underlying risk factor, and to incorporate all desirable distributional properties, such as fat tails and time varying volatilities. Also, Monte Carlo simulations can be extended to apply over longer holding periods, making it possible to use these techniques for measuring credit risk. However, these techniques are also by far the most expensive computationally. The key difference between historical simulation and simulation Monte Carlo is that the historical simulation model carries out the simulation using the real observed changes in the market place over the last X periods (using historical market price data) to generate Y hypothetical portfolio profits or losses, whereas in the Monte Carlo simulation a random number generator is used to produce tens of thousands of hypothetical changes in the market. These are then used to construct thousands of hypothetical profits and losses on the current portfolio, and the subsequent distribution of possible portfolio profit or loss.

Historical Simulation:

Historical simulation method provides a straightforward implementation of full valuation. The key assumption in historical simulation is that the set of possible future scenarios is fully represented by what happened over a specific historical window. This methodology involves collecting the set of risk factor changes over a historical window: for example, daily changes over the last five years. The set of scenarios thus obtained is assumed to be a good representation of all possibilities that could happen between today and tomorrow. The instruments in the portfolio are then repeatedly re-valued against each of the scenarios. This produces a distribution of portfolio values, or equivalently, a distribution of changes in portfolio value from today's value. Usually, some of these changes will involve profits and some will involve losses. Ordering the changes in portfolio value from worst to best, the 99% VaR, for example, is computed as the loss such that 1% of the profits or losses are below it, and 99% are above it.