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VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
VaR Methodologies Jp Morgan
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VaR Methodologies Jp Morgan

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  • 1. RISK MANAGEMENT SPOTLIGHT VALUE-AT-RISK: AN OVERVIEW OF ANALYTICAL VAR by Romain Berry J.P. Morgan Investment Analytics and Consulting romain.p.berry@jpmorgan.com In the last issue, we discussed the principles of a sound risk management function to efficiently manage and monitor the financial risks within an organization. To many risk managers, the heart of a robust risk management department lies in risk measurement through various complex mathematical models. But even one who is a strong believer in quantitative risk management would have to admit that a risk management function that heavily relies on these sophisticated models cannot add value beyond the limits of understanding and expertise that the managers themselves have towards these very models. Risk managers relying exclusively on models are exposing their organization to events similar to that of the sub-prime crisis, whereby some extremely complex models failed to accurately estimate the probability of default of the most senior tranches of CDOs1. Irrespective of how you put it, there is some sort of human or operational risk in every team within any given organ- ization. Models are valuable tools but merely represent a means to manage the financial risks of an organization. This article aims at giving an overview of one of the most wide- that should always be kept in mind when handling VaR. spread models in use in most of risk management departments VaR involves two arbitrarily chosen parameters: the holding across the financial industry: Value-at-Risk (or VaR)2. VaR calcu- period and the confidence level. The holding period corre- lates the worst expected loss over a given horizon at a given sponds to the horizon of the risk analysis. In other words, when confidence level under normal market conditions. VaR esti- computing a daily VaR, we are interested in estimating the mates can be calculated for various types of risk: market, credit, worst expected loss that may occur by the end of the next operational, etc. We will only focus on market risk in this article. trading day at a certain confidence level under normal market Market risk arises from mismatched positions in a portfolio that conditions. The usual holding periods are one day or one is marked-to-market periodically (generally daily) based on month. The holding period can depend on the fund’s invest- uncertain movements in prices, rates, volatilities and other rele- ment and/or reporting horizons, and/or on the local regulatory vant market parameters. In such a context, VaR provides a requirements. The confidence level is intuitively a reliability single number summarizing the organization’s exposure to measure that expresses the accuracy of the result. The higher market risk and the likelihood of an unfavorable move. There the confidence level, the more likely we expect VaR to approach are mainly three designated methodologies to compute VaR: its true value or to be within a pre-specified interval. It is there- Analytical (also called Parametric), Historical Simulations, and fore no surprise that most regulators require a 95% or 99% Monte Carlo Simulations. For now, we will focus only on the confidence interval to compute VaR. Analytical form of VaR. The two other methodologies will be treated separately in the upcoming issues of this newsletter. PART 2: FORMALIZATION AND APPLICATIONS Part 1 of this article defines what VaR is and what it is not, and describes the main parameters. Then, in Part 2, we mathemati- Analytical VaR is also called Parametric VaR because one of its cally express VaR, work through a few examples and play with fundamental assumptions is that the return distribution varying the parameters. Part 3 and 4 briefly touch upon two crit- belongs to a family of parametric distributions such as the ical but complex steps to computing VaR: mapping positions to normal or the lognormal distributions. Analytical VaR can risk factors and selecting the volatility model of a portfolio. simply be expressed as: Finally, in Part 5, we discuss the pros and cons of Analytical VaR. (1) PART 1: DEFINITION OF ANALYTICAL VAR where α • VaRα is the estimated VaR at the confidence level VaR is a predictive (ex-ante) tool used to prevent portfolio managers from exceeding risk tolerances that have been devel- 100 × (1 – α)%. oped in the portfolio policies. It can be measured at the port- • xα is the left-tail α percentile of a normal distribution folio, sector, asset class, and security level. Multiple VaR is described in the expression methodologies are available and each has its own benefits and where R is the expected return. In order for VaR to be mean- drawbacks. To illustrate, suppose a $100 million portfolio has a ingful, we generally choose a confidence level of 95% or monthly VaR of $8.3 million with a 99% confidence level. VaR 99%. xα is generally negative. simply means that there is a 1% chance for losses greater than $8.3 million in any given month of a defined holding period • P is the marked-to-market value of the portfolio. under normal market conditions. The Central Limit Theorem states that the sum of a large number It is worth noting that VaR is an estimate, not a uniquely defined of independent and identically distributed random variables value. Moreover, the trading positions under review are fixed will be approximately normally distributed (i.e., following a for the period in question. Finally, VaR does not address the Gaussian distribution, or bell-shaped curve) if the random vari- distribution of potential losses on those rare occasions when ables have a finite variance. But even if we have a large enough the VaR estimate is exceeded. We should also bear in mind sample of historical returns, is it realistic to assume that the these constraints when using VaR. The ease of using VaR is also returns of any given fund follow a normal distribution? Thus, we its pitfall. VaR summarizes within one number the risk exposure need to associate the return distribution to a standard normal of a portfolio. But it is valid only under a set of assumptions distribution which has a zero mean and a standard deviation of 1 CDO stands for Collaterized Debt Obligation. These instruments repackage a portfolio of average- or poor-quality debt into high-quality debt (generally rated AAA) by splitting a portfolio of corporate bonds or bank loans into four classes of securities, called tranches. 2 Pronounced V’ah’R. SEPTEMBER 2008 EDITION — 7
  • 2. RISK MANAGEMENT SPOTLIGHT one. Using a standard normal distribution enables us to replace replace in (4) the mean of the asset by the weighted mean of xα by zα through the following permutation: the portfolio, μp and the standard deviation (or volatility) of the asset by the volatility of the portfolio, σ p. The volatility of a (2) portfolio composed of two assets is given by: which yields: (6) (3) where zα is the left-tail α percentile of a standard normal distribution. • w1 is the weighting of the first asset Consequently, we can re-write (1) as: • w2 is the weighting of the second asset • σ1 is the standard deviation or volatility of the first asset (4) • σ2 is the standard deviation or volatility of the second asset EXAMPLE 1 – ANALYTICAL VAR OF A SINGLE ASSET • ρ1,2 is the correlation coefficient between the two assets Suppose we want to calculate the Analytical VaR at a 95% confi- And (4) can be re-written as: dence level and over a holding period of 1 day for an asset in which we have invested $1 million. We have estimated3 μ (7) (mean) and σ (standard deviation) to be 0.3% and 3% respec- Let us assume that we want to calculate Analytical VaR at a 95% tively. The Analytical VaR of that asset would be: confidence level over a one-day horizon on a portfolio composed of two assets with the following assumptions: • P = $100 million This means that there is a 5% chance that this asset may lose at • w1 = w2 = 50%6 least $46,347 at the end of the next trading day under normal • μ1 = 0.3% market conditions. • σ1 = 3% EXAMPLE 2 – CONVERSION OF THE CONFIDENCE • μ2 = 0.5% LEVEL4 • σ2 = 5% Assume now that we are interested in a 99% Analytical VaR of • ρ1,2 = 30% the same asset over the same one-day holding period. The corresponding VaR would simply be: (8) There is a 1% chance that this asset may experience a loss of at least $66,789 at the end of the next trading day. As you can see, the higher the confidence level, the higher the VaR as we EXAMPLE 5 – ANALYTICAL VAR OF A PORTFOLIO travel downwards along the tail of the distribution (further left COMPOSED OF N ASSETS on the x-axis). From the previous example, we can generalize these calcula- EXAMPLE 3 – CONVERSION OF THE HOLDING tions to a portfolio composed of n assets. In order to keep the mathematical formulation handy, we use matrix notation and PERIOD can re-write the volatility of the portfolio as: If we want to calculate a one-month (21 trading days on average) VaR of that asset using the same inputs, we can simply apply the square root of the time5: (5) (9) Applying this rule to our examples above yields the following where VaR for the two confidence levels: • w is the vector of the weights of the n assets • w’ is the transpose vector of w • Σ is the covariance matrix of the n assets Practically, we could design a spreadsheet in Excel (Exhibit 1) to EXAMPLE 4 – ANALYTICAL VAR OF A PORTFOLIO calculate Analytical VaR on the portfolio in Example 4. OF TWO ASSETS Let us assume now that we have a portfolio worth $100 million that is equally invested in two distinct assets. One of the main Note that these parameters have to be estimated. They are not the historical parame- 3 ters derived from the series. reasons to invest in two different assets would be to diversify 4 Note that zα is to be read in the statistical table of a standard normal distribution. the risk of the portfolio. Therefore, the main underlying ques- 5 This rule stems from the fact that the sum of n consecutive one-day log returns is the n- tion here is how one asset would behave if the other asset were day log return and the standard deviation of n-day returns is √n × standard deviation of to move against us. In other words, how will the correlation one-day returns. 6 These weights correspond to the weights of the two assets at the end of the holding between these two assets affect the VaR of the portfolio? As we period. Because of market movements, there is little likelihood that they will be the aggregate one level up the calculation of Analytical VaR, we same as the weights at the beginning of the holding period. SEPTEMBER 2008 EDITION — 8
  • 3. RISK MANAGEMENT SPOTLIGHT Exhibit 1 – Excel Spreadsheet to calculate Analytical VaR for PART 4: VOLATILITY MODELS a portfolio of two assets We can guess from the various expressions of Analytical VaR we Analytical VaR have used that its main driver is the expected volatility (of the asset or the portfolio) since we multiply it by a constant factor Expected greater than 1 (1.6449 for a 95% VaR, for instance) – as parameters opposed to the expected mean, which is simply added to the p 100,000,000 Asset 1 Asset 2 expected volatility. Hence, if we have used historical data to Standard derive the expected volatility, we could consider how today’s w1 50% Deviation 0.03 0.05 volatility is positively correlated with yesterday’s volatility. In w2 50% that case, we may try to estimate the conditional volatility of the Correlation asset or the portfolio. The two most common volatility models μ1 0.3% Matrix 1 0.3 used to compute VaR are the Exponential Weighted Moving σ1 Average (EWMA) and the Generalized Autoregressive 3% 0.3 1 Conditional Heteroscedasticity (GARCH). Again, in order to be μ2 0.5% exhaustive on this very important part in computing VaR, we σ2 5% will discuss these models in a future article. Covariance p1,2 30% Matrix PART 5: ADVANTAGES AND DISADVANTAGES OF Σ 0.00090 0.00045 ANALYTICAL VAR μp 0.40% 0.00045 0.00250 Analytical VaR is the simplest methodology to compute VaR and σp 3.28% is rather easy to implement for a fund. The input data is rather Exposures limited, and since there are no simulations involved, the Confidence computation time is minimal. Its simplicity is also its main level w1 0.5 0.5 drawback. First, Analytical VaR assumes not only that the histor- 95% -1.6449 ical returns follow a normal distribution, but also that the changes in price of the assets included in the portfolio follow a Σw 0.00068 normal distribution. And this very rarely survives the test of 0.00148 reality. Second, Analytical VaR does not cope very well with securities that have a non-linear payoff distribution like options σ 2=w’Σw 0.00108 or mortgage-backed securities. Finally, if our historical series exhibits heavy tails, then computing Analytical VaR using a σ 0.03279 normal distribution will underestimate VaR at high confidence levels and overestimate VaR at low confidence levels. VaR 4,993,012 .77 Grey = input cells CONCLUSION Source: J.P. Morgan Investment Analytics & Consulting. As we have demonstrated, Analytical VaR is easy to implement as long as we follow these steps. First, we need to collect It is easy from there to expand the calculation to a portfolio of n historical data on each security in the portfolio (we advise using assets. But be aware that you will soon reach the limits of Excel at least one year of historical data – except if one security has as we will have to calculate n(n-1)/2 terms for your covariance experienced high volatility, which would suggest a shorter matrix. period of time). Second, if the portfolio has a large number of underlying positions, then we would need to map them against PART 3: RISK MAPPING a more manageable set of risk factors. Third, we need to calcu- In order to cope with an increasing covariance matrix each time late the historical parameters (mean, standard deviation, etc.) you diversify your portfolio further, we can map each security of and need to estimate the expected prices, volatilities and corre- the portfolio to common fundamental risk factors and base our lations. Finally we apply (7) to find the Analytical VaR estimate calculations of Analytical VaR on these risk factors. This process of the portfolio. is called reverse engineering and aims at reducing the size of As always when building a model, it is important to make sure the covariance matrix and speeding up the computational time that it has been reviewed, fully tested and approved, that a User of transposing and multiplying matrices. We generally consider Guide (including any potential code) has been documented and four main risk factors: Spot FX, Equity, Zero-Coupon Bonds and will be updated if necessary, that a training has been designed Futures/Forward. The complexity of this process goes beyond and delivered to the members of the risk management team and the scope of this overview of Analytical VaR and will need to be to the recipients of the outputs of the risk management function, treated separately in a future article. and finally that a capable person has been allocated the over- sight of the model, its current use, and regular refinement. Opinions and estimates offered in this Investment Analytics and Consulting newsletter constitute our judgment and are subject to change without notice, as are statements of financial market trends, which are based on current market conditions. We believe the information provided here is reliable, but do not warrant its accuracy or complete- ness. References to specific asset classes, financial markets, and investment strategies are for information purposes only and are not intended to be, and should not be interpreted as, recommendations or a substitute for obtaining your own investment advice. This document contains information that is the property of JPMorgan Chase & Co. It may not be copied, published, or used in whole or in part for any purposes other than expressly authorized by JPMorgan Chase & Co. www.jpmorgan.com/visit/iac SEPTEMBER 2008 EDITION — 9 Digitally signed by Sreehari Menon Signature Not Verified Sreehari Menon DN: cn=Sreehari Menon, c=IN Date: 2010.09.25 07:21:45 Z
  • 4. RISK MANAGEMENT AN OVERVIEW OF VALUE-AT-RISK: PART II – HISTORICAL SIMULATIONS VAR by Romain Berry J.P. Morgan Investment Analytics and Consulting romain.p.berry@jpmorgan.com This article is the third in a series of articles exploring risk management for institutional investors. In the previous issue, we looked at Analytical Value-at-Risk, whose cornerstone is the Variance-Covariance matrix. In this article, we continue to explore VaR as an indicator to measure the market risk of a portfolio of financial instruments, but we touch on a very different methodology. We indicated in the previous article that the main benefits of from a Local Valuation method in which we only use the Analytical VaR were that it requires very few parameters, is information about the initial price and the exposure at the easy to implement and is quick to run computations (with an origin to deduce VaR. appropriate mapping of the risk factors). Its main drawbacks Step 1 – Calculate the returns (or price changes) of all the lie in the significant (and inconsistent across asset classes assets in the portfolio between each time interval. and markets) assumption that price changes in the financial The first step lies in setting the time interval and then calcu- markets follow a normal distribution, and that this method- lating the returns of each asset between two successive ology may be computer-intensive since we need to calculate periods of time. Generally, we use a daily horizon to calcu- the n(n-1)/2 terms of the Variance-Covariance matrix (in the late the returns, but we could use monthly returns if we were case where we do not proceed to a risk mapping of the to compute the VaR of a portfolio invested in alternative various instruments that composed the portfolio). With the investments (Hedge Funds, Private Equity, Venture Capital increasing power of our computers, the second limitation and Real Estate) where the reporting period is either will barely force you to move away from spreadsheets to monthly or quarterly. Historical Simulations VaR requires a programming. But the first assumption in the case of a port- long history of returns in order to get a meaningful VaR. folio containing a non-negligible portion of derivatives Indeed, computing a VaR on a portfolio of Hedge Funds with (minimum 10%-15% depending on the complexity and only a year of return history will not provide a good VaR esti- exposure or leverage) may result in the Analytical VaR being mate. seriously underestimated because these derivatives have non-linear payoffs. Step 2 – Apply the price changes calculated to the current mark-to-market value of the assets and re-value your port- One solution to circumvent that theoretical constraint is folio. merely to work only with the empirical distribution of the returns to arrive at Historical Simulations VaR. Indeed, is it Once we have calculated the returns of all the assets from not more logical to work with the empirical distribution that today back to the first day of the period of time that is being captures the actual behavior of the portfolio and encom- considered – let us assume one year comprised of 265 days passes all the correlations between the assets composing – we now consider that these returns may occur tomorrow the portfolio? The answer to this question is not so clear-cut. with the same likelihood. For instance, we start by looking Computing VaR using Historical Simulations seems more at the returns of every asset yesterday and apply these intuitive initially but has its own pitfalls as we will see. But returns to the value of these assets today. That gives us new first, how do we compute VaR using Historical Simulations? values for all these assets and consequently a new value of the portfolio. Then, we go back in time by one more time HISTORICAL SIMULATIONS VAR METHODOLOGY interval to two days ago. We take the returns that have been The fundamental assumption of the Historical Simulations calculated for every asset on that day and assume that methodology is that you look back at the past performance those returns may occur tomorrow with the same likelihood of your portfolio and make the assumption – there is no as the returns that occurred yesterday. We re-value every escape from making assumptions with VaR modeling – that asset with these new price changes and then the portfolio the past is a good indicator of the near-future or, in other itself. And we continue until we have reached the beginning words, that the recent past will reproduce itself in the near- of the period. In this example, we will have had 264 simula- future. As you might guess, this assumption will reach its tions. limits for instruments trading in very volatile markets or Step 3 – Sort the series of the portfolio-simulated P&L from during troubled times as we have experienced this year. the lowest to the highest value. The below algorithm illustrates the straightforwardness of After applying these price changes to the assets 264 times, this methodology. It is called Full Valuation because we will we end up with 264 simulated values for the portfolio and re-price the asset or the portfolio after every run. This differs thus P&Ls. Since VaR calculates the worst expected loss DECEMBER 2008 EDITION — 8
  • 5. RISK MANAGEMENT over a given horizon at a given confidence level under APPLICATIONS OF HISTORICAL SIMULATIONS VAR normal market conditions, we need to sort these 264 values Let us compute VaR using historical simulations for one asset from the lowest to the highest as VaR focuses on the tail of and then for a portfolio of assets to illustrate the algorithm. the distribution. Example 1 – Historical Simulations VaR for one asset Step 4 – Read the simulated value that corresponds to the desired confidence level. The first step is to calculate the return of the asset price between each time interval. This is done in column D in The last step is to determine the confidence level we are Table 1. Then we create a column of simulated prices based interested in – let us choose 99% for this example. One can on the current market value of the asset (1,000,000 as read the corresponding value in the series of the sorted shown in cell C3) and each return which this asset has expe- simulated P&Ls of the portfolio at the desired confidence rienced over the period under consideration. Thus, we have level and then take it away from the mean of the series of 100 x (-1.93%) = -19,313.95. In Step 3, we simply sort all simulated P&Ls. In other words, the VaR at 99% confidence the simulated values of the asset (based on the past level is the mean of the simulated P&Ls minus the 1% returns). Finally, in Step 4, we read the simulated value in lowest value in the series of the simulated values. This can column G which corresponds to the 1% worst loss. As there be formulated as follows: is no value that corresponds to 99%, we interpolate the VaR1-α = μ(R) – Rα (1) surrounding values around 99.24% and 98.86%. That gives where us -54,711.55. • VaR1-α is the estimated VaR at the confi- Table 1 – Calculating Historical Simulations VaR for one asset dence level 100 × ( 1-α )%. • μ(R) is the mean of the series of simu- lated returns or P&Ls of the portfolio. • Rα is the α th worst return of the series of simulated P&Ls of the portfolio or in other words the return of the series of simulated P&Ls that corresponds to the level of significance α . We may need to proceed to some interpo- lation since there will be no chance to get a value at 99% in our example. Indeed, if we use 265 days, each return calculated at every time interval will have a weight of 1/264 = 0.00379. If we want to look at the value that has a cumulative weight of 99%, we will see that there is no value that matches exactly 1% (since we have divided the series into 264 time intervals and not a multiple of 100). Considering that there is very little chance that the tail of the empirical distribution is linear, Asset Price for one asset proceeding to a linear interpolation to get 140 the 99% VaR between the two successive 130 120 time intervals that surround the 99th 110 percentile will result in an estimation of 100 the actual VaR. This would be a pity 90 80 considering we did all that we could to use 7 7 8 8 8 8 8 8 8 8 08 8 8 /0 /0 /0 /0 /0 /0 0 0 /0 /0 /0 /0 the empirical distribution of returns, 1/ 1/ 1/ /1 /1 1 1 1 1 1 1 /1 /1 1/ 2/ 3/ 4/ 5/ 6/ 7/ 8/ 9/ 11 12 10 11 wouldn’t it? Nevertheless, even a linear interpolation may give you a good esti- Histogram of Returns for one asset mate of your VaR. For those who are more eager to obtain the exact VaR, the Extreme Value Theory (EVT) could be the right tool 99% VaR for you. We will explain in another article 55,745 5.57% how to use EVT when computing VaR. It is rather mathematically demanding and would require us to spend more time to explain this method. -6% -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% 6% 7% DECEMBER 2008 EDITION — 9
  • 6. RISK MANAGEMENT This number does not take into account the mean, which is As you can see, we simply add a couple of columns to repli- 1,033.21. As the 99% VaR is the distance from the mean of cate the intermediary steps for the second asset. In this the first percentile (1% worst loss), we need to subtract the example, each asset represents 50% of the portfolio. After number we just calculated from the mean to obtain the each run, we re-value the portfolio by simply adding up the actual 99% VaR. In this example, the VaR of this asset is simulated P&L of each asset. This gives us the simulated P&Ls thus 1,033.21 – (-54,711.55) = 55,744.76. In order to for the portfolio (column J). express VaR in percentage, we can divide the 99% VaR This straightforward step of simply re-composing the portfolio amount by the current value of the asset (1,000,000), which after every run is one of the reasons behind the popularity of yields 5.57%. this methodology. Indeed, we do not need to handle sizeable Example 2 – Historical Simulations VaR for one portfolio Variance-Covariance matrices. We apply the calculated returns Computing VaR on one asset is relatively easy, but how do of every asset to their current price and re-value the portfolio. the historical simulations account for any correlations As we have noted, correlations are embedded in the price between assets if the portfolio holds more than one asset? changes. In this example, the 99% VaR of the first asset is The answer is also simple: correlations are already 55,744.76 (or 5.57%) and the 99% VaR of the second asset is embedded in the price changes of the assets. Therefore, 54,209.71 (or 5.42%). We know that VaR is a sub-additive risk there is no need to calculate a Variance-Covariance matrix measure – if we add the VaR of two assets, we will not get the when running historical simulations. Let us look at another VaR of the portfolio. In this case, the 99% VaR of the portfolio example with a portfolio composed of two assets. only represents 3.67% of the current marked-to-market value of the portfolio. That difference represents the diversification Table 2 – Calculating Historical Simulations VaR for a portfolio of two assets Portfolio Unit price Portfolio Histogram of Returns 270 250 230 99% VaR 73,422 210 3.67% 190 170 7 7 08 08 08 08 08 08 08 8 08 8 08 /0 /0 0 /0 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ 1/ /1 /1 /1 1/ 2/ 3/ 4/ 5/ 6/ 7/ 8/ 9/ / -5% -4% -3% -2% -1% 0% 1% 2% 3% 4% 5% 11 12 10 11 DECEMBER 2008 EDITION — 10
  • 7. RISK MANAGEMENT effect. Having a portfolio invested in these two assets makes Lastly, a minimum of history is required to use this method- the risk lower than investing in any of these two assets solely. ology. Using a period of time that is too short (less than 3-6 The reason is that the gains on one asset sometimes offset months of daily returns) may lead to a biased and inaccurate the losses on the other asset (rows 10, 12, 13, 17-20, 23, estimation of VaR. As a rule of thumb, we should utilize at 26-28, 30, 32 in Table 2. Over the 265 days, this happened least four years of data in order to run 1,000 historical simula- 127 times with different magnitude. But in the end, this bene- tions. That said, round numbers like 1,000 may have fited the overall risk profile of the portfolio as the 99% VaR of absolutely no relevance whatsoever to your exact portfolio. the portfolio is only 3.67%. Security prices, like commodities, move through economic cycles; for example, natural gas prices are usually more ADVANTAGES OF HISTORICAL SIMULATIONS VAR volatile in the winter than in the summer. Depending on the Computing VaR using the Historical Simulations methodology composition of the portfolio and on the objectives you are has several advantages. First, there is no need to formulate attempting to achieve when computing VaR, you may need to any assumption about the return distribution of the assets in think like an economist in addition to a risk manager in order the portfolio. Second, there is also no need to estimate the to take into account the various idiosyncrasies of each instru- volatilities and correlations between the various assets. ment and market. Also, bear in mind that VaR estimates need Indeed, as we showed with these two simple examples, they to rely on a stable set of assumptions in order to keep a are implicitly captured by the actual daily realizations of the consistent and comparable meaning when they are monitored assets. Third, the fat tails of the distribution and other over a certain period of time. extreme events are captured as long as they are contained in In order to increase the accuracy of Historical Simulations the dataset. Fourth, the aggregation across markets is VaR, one can also decide to weight more heavily the recent straightforward. observations compared to the furthest since the latter may not give much information about where the prices would go DISADVANTAGES OF HISTORICAL SIMULATIONS today. We will cover these more advanced VaR models in VAR another article. The Historical Simulations VaR methodology may be very intu- itive and easy to understand, but it still has a few drawbacks. CONCLUSION First, it relies completely on a particular historical dataset and Despite these disadvantages, many financial institutions have its idiosyncrasies. For instance, if we run a Historical chosen historical simulations as their favored methodology to Simulations VaR in a bull market, VaR may be underestimated. compute VaR. To many, working with the actual empirical Similarly, if we run a Historical Simulations VaR just after a distribution is the “real deal.” crash, the falling returns which the portfolio has experienced However, obtaining an accurate and reliable VaR estimate has recently may distort VaR. Second, it cannot accommodate little value without a proper back testing and stress testing changes in the market structure, such as the introduction of program. VaR is simply a number whose value relies on a the Euro in January 1999. Third, this methodology may not sound methodology, a set of realistic assumptions and a always be computationally efficient when the portfolio rigorous discipline when conducting the exercise. The real contains complex securities or a very large number of instru- benefit of VaR lies in its essential property of capturing with ments. Mapping the instruments to fundamental risk factors one single number the risk profile of a complex or diversified is the most efficient way to reduce the computational time to portfolio. VaR remains a tool that should be validated through calculate VaR by preserving the behavior of the portfolio successive reconciliation with realized P&Ls (back testing) almost intact. Fourth, Historical Simulations VaR cannot and used to gain insight into what would happen to the port- handle sensitivity analyses easily. folio if one or more assets would move adversely to the investment strategy (stress testing). DECEMBER 2008 EDITION — 11 Digitally signed by Sreehari Menon Signature Not Verified Sreehari Menon DN: cn=Sreehari Menon, c=IN Date: 2010.09.25 07:23:34 Z
  • 8. RISK MANAGEMENT AN OVERVIEW OF VALUE-AT-RISK: PART III – MONTE CARLO SIMULATIONS VAR by Romain Berry J.P. Morgan Investment Analytics and 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 pricing we find in the financial markets. This process is called used to compute the Value-at-Risk (VaR) of a portfolio of discretization, whereby we approximate a continuous financial instruments employs Monte Carlo Simulations. phenomenon by a large number of discrete intervals. Monte Carlo Simulations correspond to an algorithm that Step 2 – Draw a random number from a random number generates random numbers that are used to compute a generator and update the price of the asset at the end of the formula that does not have a closed (analytical) form – this first time increment. means that we need to proceed to some trial and error in It is possible to generate random returns or prices. In most picking up random numbers/events and assess what the cases, the generator of random numbers will follow a formula yields to approximate the solution. Drawing random specific theoretical distribution. This may be a weakness of numbers over a large number of times (a few hundred to a the Monte Carlo Simulations compared to Historical few million depending on the problem at stake) will give a Simulations, which uses the empirical distribution. When good indication of what the output of the formula should be. simulating random numbers, we generally use the normal It is believed actually that the name of this method stems distribution. from the fact that the uncle of one of the researchers (the Polish mathematician Stanislaw Ulam) who popularized this In this paper, we use the standard stock price model to algorithm used to gamble in the Monte Carlo casino and/or simulate the path of a stock price from the ith day as defined that the randomness involved in this recurring methodology by: can be compared to the game of roulette. (1) In this article, we present the algorithm, and apply it to where compute the VaR for a sample stock. We also discuss the is the return of the stock on the ith day pros and cons of the Monte Carlo Simulations methodology is the stock price on the ith day compared to Analytical VaR and Historical Simulations VaR. is the stock price on the i+1th day METHODOLOGY is the sample mean of the stock price is the timestep Computing VaR with Monte Carlo Simulations follows a is the sample volatility (standard deviation) of the similar algorithm to the one we used for Historical stock price Simulations in our previous issue. The main difference lies is a random number generated from a normal in the first step of the algorithm – instead of picking up a distribution return (or a price) in the historical series of the asset and assuming that this return (or price) can re-occur in the next At the end of this step/day ( = 1 day), we have drawn a time interval, we generate a random number that will be random number and determined by applying (1) since used to estimate the return (or price) of the asset at the end all other parameters can be determined or estimated. of the analysis horizon. Step 3 – Repeat Step 2 until reaching the end of the analysis Step 1 – Determine the length T of the analysis horizon and horizon T by walking along the N time intervals. divide it equally into a large number N of small time At the next step/day ( = 2), we draw another random increments Δt (i.e. Δt = T/N). number and apply (1) to determine from . For illustration, we will compute a monthly VaR consisting of We repeat this procedure until we reach T and can twenty-two trading days. Therefore N = 22 days and Δt = 1 determine . In our example, represents the day. In order to calculate daily VaR, one may divide each day estimated (terminal) stock price in one month time of the per the number of minutes or seconds comprised in one day sample share. – the more, the merrier. The main guideline here is to ensure Step 4 – Repeat Steps 2 and 3 a large number M of times to that Δt is large enough to approximate the continuous generate M different paths for the stock over T. MARCH 2009 EDITION — 4
  • 9. RISK MANAGEMENT So far, we have generated one path for this stock (from i to Exhibit 1: Historical prices for one stock from i+22). Running Monte Carlo Simulations means that we 01/22/08 to 01/20/09 build a large number M of paths to take account of a broader 50 universe of possible ways the stock price can take over a period of one month from its current value ( ) to an 45 estimated terminal price . Indeed, there is no unique 40 way for the stock to go from to . Moreover, is only one possible terminal price for the stock amongst an 35 infinity. Indeed, for a stock price being defined on + (set 30 of positive real numbers), there is an infinity of possible 25 paths from to (see footnote 1). 20 It is an industry standard to run at least 10,000 simulations even if 1,000 simulations provide an efficient estimator of 15 Jan 08 Mar 08 Apr 08 Jun 08 Jul 08 Sep 08 Nov 08 Dec 08 the terminal price of most assets. In this paper, we ran 1,000 simulations for illustration purposes. 20th of January 2009 was $18.09. We want to compute the Step 5 – Rank the M terminal stock prices from the smallest monthly VaR on the 20th of January 2009. This means we to the largest, read the simulated value in this series that will jump in the future by 22 trading days and look at the corresponds to the desired (1- )% confidence level (95% estimated prices for the stock on the 19th of February 2009. or 99% generally) and deduce the relevant VaR, which is the difference between and the th lowest terminal Since we decided to use the standard stock price model to stock price. draw 1,000 paths until T (19th of February 2009), we will need to estimate the expected return (also called drift rate) Let us assume that we want the VaR with a 99% confidence and the volatility of the share on that day. interval. In order to obtain it, we will need first to rank the M terminal stock prices from the lowest to the highest. We can estimate the drift by: Then we read the 1% lowest percentile in this series. This estimated terminal price, 1% means that there is a 1% (2) chance that the current stock price could fall to 1% or The volatility of the share can be estimated by: less over the period in consideration and under normal market conditions. If 1% is smaller than (which is the case most of the time), then - 1% will corresponds to (3) a loss. This loss represents the VaR with a 99% confidence Note that since we chose = 1 day, these two estimators interval. will equal the sample mean and sample standard deviation. APPLICATIONS Based on these two estimators, we generate from by re-arranging (1) as: 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 (4) stock. Historical prices are charted in Exhibit 1. We will only and simulate 1,000 paths for the share. consider the share price and thus work with the assumption The last step can be summarized in Exhibit 2. We sort the we have only one share in our portfolio. Therefore the value 1,000 terminal stock prices from the lowest to the highest of the portfolio corresponds to the value of one share. and read the price which corresponds to the desired From the series of historical prices, we calculated the confidence level. For instance, if we want to get the VaR at a sample return mean (-0.17%) and sample return standard 99% confidence level, we will read the 1% lowest stock deviation (5.51%). The current price ( ) at the end of the price, which is $15.7530. On January 20th, the stock price 1 This is the reason why we used the discretized form (1) of the standard stock price model so that Monte Carlo Simulations can be handled more easily without losing too much information. Thus, the higher N and M are, the more accurate the estimates of the terminal stock prices will be, but the longer the simulations will take to run. MARCH 2009 EDITION — 5
  • 10. RISK MANAGEMENT was $18.09. Therefore, there is a 1% likelihood that the The main benefit of running time-consuming Monte Carlo share falls to $15.7530 or below. If that happens, we will Simulations is that they can model instruments with non- experience a loss of at least $18.09 – $15.7530 = $2.5170. linear and path-dependent payoff functions, especially This loss is our monthly VaR estimate at a 99% confidence complex derivatives. Moreover, Monte Carlo Simulations level for one share calculated on the 20th of January 2009. VaR is not affected as much as Historical Simulations VaR by extreme events, and in reality provides in-depth details of these rare events that may occur beyond VaR. Finally, we Exhibit 2: Reading for one share may use any statistical distribution to simulate the returns as far as we feel comfortable with the underlying assumptions that justify the use of a particular distribution. 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 ADVANTAGES is (are) still appropriate for the assets that composed the Monte Carlo Simulations present some advantages over the portfolio and still provide credible estimate of VaR (back Analytical and Historical Simulations methodologies to testing), and on analyzing how the portfolio reacts to extreme compute VaR. events which occur every now and then in the financial markets (stress testing). MARCH 2009 EDITION — 6 Digitally signed by Sreehari Signature Not Verified Sreehari Menon Menon DN: cn=Sreehari Menon, c=IN Date: 2010.09.25 07:24:39 Z

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