# Value at Risk (VaR)

Financial markets started to use the “Value at Risk” extensively since 1990’s. But the measures of Value at Risk (VaR) were active in different names since as early as 1920’s (Holton 2003). It is the measurement of the worst expected loss at a given confidence level under normal market conditions over a specific time interval.

**Value at Risk (VaR)**specifically for you

It can also be expressed as the lowest confidence level of the potential losses that can occur within a given portfolio during a specified time period. Value at Risk only presents the worst-case scenario (Harper n. d. ).

The two major parameters to be chosen for risk measurement are the time period and the confidence level. The time period can vary from a few hours to a few years. For example it can be stated that when a portfolio manager has a daily VaR at $1 million at 1%, it means that there is only 1 chance in 100 to incur a daily loss of more than $1 million under normal market conditions. The commonly used methods to estimate Value at Risk are: Variance – Covariance Method, Historical Performance and Monte Carlo Simulation (Benninga & Wiener 1998). Variance – Covariance Method:

This model was made popular by J. P. Morgan in early 1990’s. This approach is based on the assumption that the underlying market factors have a multivariate normal distribution. This assumption helps in determining the distribution of mark-to-market portfolio profits and losses. After finding the distribution of possible portfolio profits and losses, the standard mathematical properties of Normal distribution can be used to determine the loss that will be equaled or exceeded x percent of the time which is called Value at Risk (Linsmeier & Pearson 1996).

The following example can be taken to discuss the theory. A U. S. company entered a FX forward contract in the past. The difference between current date and date of delivery is 91 days. The contract requires the company to deliver $15 million in 91 days and in exchange it will receive £10 million. The facts taken into consideration are the spot exchange rate expressed in dollars per pound (S), 3 month pound interest rate (rGBP) and 3 month dollar interest rate (rUSD). The current mark to market values in dollars is calculated based on the following formula:

USD mark to market value= S x GBP 10million – USD 15 million 1+ rGBP (91/360) 1+ rUSD (91/360) Here the holding period is one day and the probability is 5%. The distribution of possible profit and loss on this portfolio has the mean of zero as the expected change in portfolio value over a short holding period is almost always close to zero. A standard property of the Normal distribution is that if a probability of 5% is used in determination of the Value at Risk then it will be equal to 1. 65 times the standard deviation of changes in the portfolio value.

Standard deviation is the measure of the spread or dispersion of the distribution and computing the value of the standard deviation of changes in the portfolio value is the main factor in this method (Linsmeier & Pearson 1996). Value at Risk = 1. 65 x standard deviation of change in portfolio value The first step in measurement of VaR through this method is to determine the basic market factors and standardized market positions through “Risk Mapping”. In this case the basic market factors are spot exchange rate and 3-month dollar and pound interest rates.

The associated standardized positions are spot pounds, dollar dominated 3 month zero coupon bond and a 3 month zero-coupon bond exposed only to changes in the pound interest rate. The next step is to estimate the parameters of distribution assuming that the percentage changes in the basic market factors have a multivariate Normal distribution with means of zero and thus capturing the variability of market factors by standard deviation and co-movement by the correlation coefficients.

The third step is to compute standard deviations and correlations of the changes in the values of standardized positions using the covariance matrix of changes in the basic market factors. The final step is to calculate the value of variance and standard deviation of the portfolio using standard mathematical results about the distributions of sums of Normal random variables. Standard deviation is the square root of variance. In our case its value is $ 52500 approximately. Now as the probability was taken as 5%, the formula comes to Value at Risk = 1. 65 x standard deviation of change in portfolio value = 1. 65 x $ 52,500 = $ 86,625

The benefit of this model is that it uses compact and maintainable data set often available from market and third parties and calculation is quite speedy using algebraic formulae. The drawback of this method is that it assumes the change of the portfolio value to be linearly dependent on all the changes in the values of assets and also that the asset returns normal distributed (Jorion 2006). Historical Performance: Historical Performance method is the simplest and most transparent method that takes into account relatively lesser number of assumptions about the statistical distribution of underlying market factors (Linsmeier & Pearson 1996).

The method works by using historical changes in market rates and prices to estimate potential future loss or profit with the portfolio and thereby calculating the Value at Risk. This can be illustrated based on the above example. Here we assume the holding period as 1 day, probability of 5% and computation to be based on 100 preceding business days from the current date. The current day will be the 100th day. The method involves five steps. The first step is to identify the basic market factors and to determine the formula to express mark to market value.

In our case the basic market factors are 3 month dollar interest rate, 3 month pound interest rate and spot exchange rate. The formula for mark to market value is derived as USD mark to market value= S x GBP 10million – USD 15 million 1+ rGBP (91/360) 1+ rUSD (91/360) Next the values of the identified basic market factors for previous 100 days are to be obtained. Daily change in these rates will be able to set the base for constructions of hypothetical values of market factors useful in the calculation of hypothetical profit and loss.

The daily Value at Risk number is a measure of the portfolio loss caused by such changes over a one day holding period. The next and most important step is to subject the current portfolio to the changes experienced in the previous 100 days to calculate daily hypothetical profits and losses. In this step 100 sets of hypothetical values for market factors are calculated based on daily historical percentage changes in the market factors combined with current market factors. These hypothetical values are then used to compute 100 hypothetical mark to market portfolio values.

Subtraction of current day mark to market portfolio value from each of the 100 hypothetical values gives 100 hypothetical daily profits and losses. Ordering mark to market profits and losses from the largest profit to the largest lost is the next step. Finally the loss, which equals or exceeds 5% of the time is selected. In the present example of 100 days calculation the fifth worst loss will be the value at risk. This method relies completely on the historical data. Thus it may not be able to predict most accurately if the period chosen is not a typical one and is posing any special market condition (Jorion 2006).

Monte Carlo Simulation: This method is quite similar to the Historical Performance Method. The major difference is that this method uses statistical distribution to capture the possible changes in the market factors instead of observing historical changes in market factors to calculate hypothetical profit and loss. The method involves five steps to estimate Value at Risk. The same example of single forward contract can be considered in this respect. The first step here is to identify the basic market factors and to determine the formula to express mark to market value similar to the Historical Performance Method.

The next step is to assume a specific distribution for changes in the basic market factors and to estimate the parameters of that distribution. For the present example the percentage change in the basic market factors having multivariate Normal distribution is assumed and estimates of standard deviation and correlates are used as in this case the parameters like means, standard deviations and correlations can be interpreted naturally and their estimation is easier. However, it can be said that Monte Carlo Simulation allows risk managers to choose the distribution according to their requirements.

But this flexibility also runs a risk of a bad choice that may not be suitable for the particular case (Jorion, 2006). Pseudo-random generator is used in the following step to generate more than 1000 or sometimes 10000 hypothetical values of changes in market factors. These are then used to calculate hypothetical mark to market portfolio values. Actual mark to market portfolio value on the current date is subtracted from each of the hypothetical values to get the hypothetical daily profits and losses.

The following step is to order the mark to market profits and losses from the largest profit to the largest loss and the Value at Risk is selected as the loss which equals or exceeds 5% of time. While comparing the different aspects of these three methods it can be said that Historical Performance is the simplest method for estimating Value at Risk. It is suitable for estimation for any kind of options of the portfolio. It is easy to compute and implement and can be explained without much effort.

The drawbacks of the method are that it can be misleading if the data used is not typical and represents a specific condition quite similar to Monte Carlo Simulation and Variance-Covariance methods. It is too much dependent on historical data. It is not possible to analyze alternative assumptions through this method. Monte Carlo Simulation and Variance-Covariance methods on the other hand can easily analyze alternative assumptions. Variance-Covariance method though can not examine distribution of market factors other than normal. Both of these methods are easy to implement but tougher to explain.

Variance-Covariance method is easy in computation but can not capture the risks of portfolio with options when the holding period is long. Monte Carlo Simulation on the other hand is not easy to compute but it can surely capture the risks regardless of any options (Linsmeier & Pearson 1996). Thus it can be said that all of the three methods have their own benefits and drawbacks and it is completely at the discretion of the risk manager to choose a method appropriate to the portfolio based on the factors to be considered and the holding time.