An Overview of Value-at-Risk: Part III – Monte Carlo Simulations VaR

by Romain Berry
J.P. Morgan Investment Analytics & Consulting
romain.p.berry@jpmorgan.com

This article is the fourth in a series of articles exploring risk management for institutional investors.

The last (and most complex) of the three main methodologies used to compute the Value-at-Risk (VaR) of a portfolio of financial instruments employs Monte Carlo Simulations. Monte Carlo Simulations correspond to an algorithm that generates random numbers that are used to compute a formula that does not have a closed (analytical) form – this means that we need to proceed to some trial and error in picking up random numbers/events and assess what the formula yields to approximate the solution. Drawing random numbers over a large number of times (a few hundred to a few million depending on the problem at stake) will give a good indication of what the output of the formula should be. It is believed actually that the name of this method stems from the fact that the uncle of one of the researchers (the Polish mathematician Stanislaw Ulam) who popularized this algorithm used to gamble in the Monte Carlo casino and/or that the randomness involved in this recurring methodology can be compared to the game of roulette.

In this article, we present the algorithm, and apply it to compute the VaR for a sample stock. We also discuss the pros and cons of the Monte Carlo Simulations methodology compared to Analytical VaR and Historical Simulations VaR.

Methodology

Computing VaR with Monte Carlo Simulations follows a similar algorithm to the one we used for Historical Simulations in our previous issue. The main difference lies in the first step of the algorithm – instead of picking up a return (or a price) in the historical series of the asset and assuming that this return (or price) can re-occur in the next time interval, we generate a random number that will be used to estimate the return (or price) of the asset at the end of the analysis horizon.

Step 1 – Determine the length T of the analysis horizon and divide it equally into a large number N of small time increments Δt (i.e. Δt = T/N).

For illustration, we will compute a monthly VaR consisting of twenty-two trading days. Therefore N = 22 days and Δt = 1 day. In order to calculate daily VaR, one may divide each day per the number of minutes or seconds comprised in one day – the more, the merrier. The main guideline here is to ensure that Δt is large enough to approximate the continuous pricing we find in the financial markets. This process is called discretization, whereby we approximate a continuous phenomenon by a large number of discrete intervals.

Step 2 – Draw a random number from a random number generator and update the price of the asset at the end of the first time increment.

It is possible to generate random returns or prices. In most cases, the generator of random numbers will follow a specific theoretical distribution. This may be a weakness of the Monte Carlo Simulations compared to Historical Simulations, which uses the empirical distribution. When simulating random numbers, we generally use the normal distribution.

In this paper, we use the standard stock price model to simulate the path of a stock price from the ith day as defined by:

where

• R_i is the return of the stock on the ith day
• S_i is the stock price on the ith day
• S_i+1 is the stock price on the i+1th day
• μ is the sample mean of the stock price
• δt is the timestep
• σ is the sample volatility (standard deviation) of the stock price
• φ is a random number generated from a normal distribution

At the end of this step/day (δt = 1 day), we have drawn a random number and determined S_i+1 by applying (1) since all other parameters can be determined or estimated.

Step 3 – Repeat Step 2 until reaching the end of the analysis horizon T by walking along the N time intervals.

At the next step/day (δt = 2), we draw another random number and apply (1) to determine S_i+2 from S_i+1. We repeat this procedure until we reach T and can determine S_i+T. In our example, S_i+22 represents the estimated (terminal) stock price in one month time of the sample share.

Step 4 – Repeat Steps 2 and 3 a large number M of times to generate M different paths for the stock over T.

So far, we have generated one path for this stock (from i to i+22). Running Monte Carlo Simulations means that we build a large number M of paths to take account of a broader universe of possible ways the stock price can take over a period of one month from its current value (S_i) to an estimated terminal price S_i+T. Indeed, there is no unique way for the stock to go from S_i to S_i+T. Moreover, S_i+T is only one possible terminal price for the stock amongst an infinity. Indeed, for a stock price being defined on (a set of positive numbers), there is an infinity of possible paths from S_i to S_i+T (see footnote 1).

It is an industry standard to run at least 10,000 simulations even if 1,000 simulations provide an efficient estimator of the terminal price of most assets. In this paper, we ran 1,000 simulations for illustration purposes.

Step 5 – Rank the M terminal stock prices from the smallest to the largest, read the simulated value in this series that corresponds to the desired (1-α)% confidence level (95% or 99% generally) and deduce the relevant VaR, which is the difference between S_i and the αth lowest terminal stock price.

Let us assume that we want the VaR with a 99% confidence interval. In order to obtain it, we will need first to rank the M terminal stock prices from the lowest to the highest. Then we read the 1% lowest percentile in this series. This estimated terminal price, S_i+T^1% means that there is a 1% chance that the current stock price Si could fall to S_i+T^1% or less over the period in consideration and under normal market conditions. If S_i+T^1% is smaller than S_i (which is the case most of the time), then S_i - S_i+T^1% will corresponds to a loss. This loss represents the VaR with a 99% confidence interval.

Applications

Let us compute VaR using Monte Carlo Simulations for one share to illustrate the algorithm.

We apply the algorithm to compute the monthly VaR for one stock. Historical prices are charted in Exhibit 1. We will only consider the share price and thus work with the assumption we have only one share in our portfolio. Therefore the value of the portfolio corresponds to the value of one share.

Exhibit 1: Historical prices for one stock from 01/22/08 to 01/20/09

From the series of historical prices, we calculated the sample return mean (-0.17%) and sample return standard deviation (5.51%). The current price (S_i) at the end of the 20th of January 2009 was $18.09. We want to compute the monthly VaR on the 20th of January 2009. This means we will jump in the future by 22 trading days and look at the estimated prices for the stock on the 19th of February 2009.

Since we decided to use the standard stock price model to draw 1,000 paths until T (19th of February 2009), we will need to estimate the expected return (also called drift rate) and the volatility of the share on that day.

We can estimate the drift by

(2)

The volatility of the share can be estimated by

(3)

Note that since we chose δt = 1 day, these two estimators will equal the sample mean and sample standard deviation.

Based on these two estimators, we generate S_i+1 from S_i by re-arranging (1) as

and simulate 1,000 paths for the share.

Exhibit 2: Reading VaR for one share (click to enlarge)

Disadvantages

The main disadvantage of Monte Carlo Simulations VaR is the computer power that is required to perform all the simulations, and thus the time it takes to run the simulations. If we have a portfolio of 1,000 assets and want to run 1,000 simulations on each asset, we will need to run 1 million simulations (without accounting for any eventual simulations that may be required to price some of these assets – like for options and mortgages, for instance). Moreover, all these simulations increase the likelihood of model risk.

Consequently, another drawback is the cost associated with developing a VaR engine that can perform Monte Carlo Simulations. Buying a commercial solution off-the-shelf or outsourcing to an experienced third party are two options worth considering. The latter approach will reinforce the independence of the computations and therefore reliance of its accuracy and non-manipulation.

Conclusion

Estimating the VaR for a portfolio of assets using Monte Carlo Simulations has become the standard in the industry. Its strengths overcome its weaknesses by far.

Despite the time and effort required to estimate the VaR for a portfolio, this task only represents half of the time a risk manager should spend on VaR. Indeed, the other half should be spent on checking that the model(s) used to calculate VaR is (are) still appropriate for the assets that composed the portfolio and still provide credible estimate of VaR (back testing), and on analyzing how the portfolio reacts to extreme events which occur every now and then in the financial markets (stress testing).