Value at Risk (VaR)

Market risk management departments measure the amount of risk in their positions with VaR and other measures, compare it with set limits, and report it to the board of directors¹.

What VaR shows

Market risk is the risk of loss when movements in markets such as interest rates, exchange rates, and stock prices reduce the value of the assets and liabilities you hold². The interest rates, stock prices, and other variables behind those movements are called risk factors (the variables that move the value of a position). VaR (value at risk) expresses the size of this market risk as a single loss amount. To calculate VaR, you set the following three things³.

Consider a hypothetical example in which you hold 600 million yen of asset A and 400 million yen of asset B, and the VaR with a 1-day holding period and a 99% confidence level is 20.27 million yen. This means the probability that tomorrow's loss exceeds 20.27 million yen is estimated at 1%³. On the other 99% of days, the loss stays within 20.27 million yen or there is a profit. You would expect the loss to exceed 20.27 million yen on about 1 in every 100 business days⁴.

A Bank of Japan training document calls VaR the "maximum loss" (our translation), and so does a paper from the Bank of Japan's Institute for Monetary and Economic Studies. But VaR is not an upper limit on losses⁵. VaR alone does not, in general, show the size of the losses larger than 20.27 million yen that occur with a probability of 1%⁶.

How the variance-covariance method works

The variance-covariance method assumes that profit and loss follow a normal distribution, and calculates VaR by multiplying the standard deviation of profit and loss by a factor set by the confidence level⁷. It assumes that risk factor moves follow a normal distribution and that the sensitivity of the position's value to those moves is constant, meaning the position's value changes in proportion to the moves⁷⁸.

With two assets, the calculation goes in this order.

1. For each asset, find the standard deviation of one-day profit and loss as the position × the daily volatility (the standard deviation of one-day returns).
2. Square each of the two standard deviations and add them, then add "2 × correlation × the product of the two standard deviations" to get the variance of total profit and loss⁹.
3. Take the square root of the variance to get the standard deviation of total profit and loss.
4. Multiply the standard deviation by the confidence-level factor.

$$\text{VaR} = \text{confidence-level factor} \times \text{standard deviation of profit and loss}$$

The confidence-level factor is the point on the standard normal distribution below which that probability lies: 1.645 for 95%, 1.960 for 97.5%, and 2.326 for 99%¹⁰. The Bank of Japan training document rounds the 99% factor to 2.33⁷. Correlation expresses, as a value from −1 to 1, how closely the two assets move in the same direction. Normal distributions and correlation are covered in Probability and Statistics Basics.

Figure 1 shows that, on the standard normal distribution, the probability of falling below the 99% factor of 2.326 is 1%.

Figure description

The curve is the probability density of a normal distribution with mean 0 and standard deviation 1, drawn with profit or loss on the horizontal axis. It is a bell shape that is symmetric around 0. A vertical dashed line at −2.326 marks the 99% factor of 2.326 (negative because it is on the loss side). The shaded area to the left of the line, on the loss side, is the 1% probability. The figure shows that the probability of losing more than 2.326 standard deviations below the mean is 1%, and that VaR is the size of the loss at that boundary.

Deep diveThe standard deviation of profit and loss for a two-asset portfolio

Let $w_A$ and $w_B$ be the positions in asset A and asset B, $\sigma_A$ and $\sigma_B$ their daily volatilities, and $\rho$ their correlation. From the formula for the variance of the sum of two random variables, the variance of one-day profit and loss is as follows⁹.

$$\sigma_P^2 = (w_A \sigma_A)^2 + (w_B \sigma_B)^2 + 2 \rho \, (w_A \sigma_A)(w_B \sigma_B)$$

Its square root $\sigma_P$ is the standard deviation of profit and loss, and multiplying it by the confidence-level factor $z$ gives VaR, $z \, \sigma_P$. With three or more assets, you add the terms for every pair of assets. The Bank of Japan training document shows this calculation as a multiplication of the correlation matrix and the variance-covariance matrix¹¹.

Worked example: VaR of asset A and asset B

When you hold 600 million yen of asset A (daily volatility 1.2%) and 400 million yen of asset B (daily volatility 0.8%), with a correlation of 0.3 between them, the VaR at a 99% confidence level in this example is 20.27 million yen. The calculation below works in million yen.

The factor in the table is 2.3263, the 99% factor to four decimal places¹². Because profit and loss are treated as normally distributed with a mean of 0, a profit of more than 20.27 million yen also occurs with the same 1% probability.

Calculating VaR for each asset separately gives 16.75 million yen for asset A (2.3263 × 7.20) and 7.44 million yen for asset B (2.3263 × 3.20). Together they come to 24.19 million yen, which is 3.93 million yen more than the total VaR of 20.27 million yen¹³. When the correlation is below 1, losses on the two assets do not necessarily fall on the same day, so the total VaR is smaller than the sum of the individual VaRs. In a worked example in the Bank of Japan training document, too, the VaR that takes correlation into account is smaller than the simple sum of the VaRs of an equity investment trust and government bonds¹¹.

How historical simulation and the Monte Carlo method work

Historical simulation and the Monte Carlo method both generate many one-day profit and loss figures, rank them from the largest loss, and take the loss at the rank that matches the confidence level as VaR¹⁴. The two differ in the data they use to generate the profit and loss figures.

Historical simulation assumes that the risk factor moves on each day of the past observation period are equally likely to happen tomorrow¹⁵. It applies each past day's moves to the current position, producing as many profit and loss figures as there are days in the observation period¹⁵. With a 250-day observation period there are 250 figures, and the largest 1% of losses is 250 × 1% = 2.5 figures. This site's simulator takes the third-largest loss as the 99% VaR, which is 19.14 million yen for the example position¹⁶.

The Monte Carlo method uses random numbers to generate risk factor moves from an assumed probability distribution, and revalues the position to produce profit and loss figures¹⁴. With 10,000 scenarios, the 100th-largest loss is the 99% VaR, which is 20.23 million yen in the example. If the position's value changes in proportion to the moves and a normal distribution is assumed, the Monte Carlo VaR in theory matches the variance-covariance VaR⁸. The example's 20.23 million yen differs from 20.27 million yen because only a limited number of scenarios are generated with random numbers.

Strengths and weaknesses of the three methods

The three methods differ in how they set the distribution of profit and loss, which changes the computing effort and the shapes of price moves they can capture¹⁷. Every method has limitations, so you choose one that fits the risk characteristics of the position¹⁸.

Fat-tailed means that large losses happen more often than a normal distribution assumes. Because the variance-covariance method and the Monte Carlo method assume a shape for the distribution, VaR will be off if the assumption differs from the actual distribution.

What changes VaR

VaR rises as the confidence level rises, and falls as the correlation between the two assets falls (when you have bought both assets). Changing one condition at a time in the asset A and asset B example, with everything else unchanged, gives the following variance-covariance VaR.

The 24.19 million yen at a correlation of 1 equals the sum of the individual VaRs. The lower the correlation, the more often a loss on one asset is offset by a gain on the other, so the total VaR gets smaller.

VaR from historical simulation and the Monte Carlo method also depends on the data used. If you change the simulator's random seed (the starting value that fixes the sequence of random numbers) from 1 to 5, VaR varies as follows, even though the data come from the same distribution.

With 250 days, historical simulation VaR is set by the few days with the largest losses, so it ranges from 16.53 million yen to 21.66 million yen. With 10,000 scenarios, the Monte Carlo VaR ranges from 19.99 million yen to 20.89 million yen, clustering near the variance-covariance VaR of 20.27 million yen. Measurements on actual market data also show that the shorter the observation period, the smaller the sample and the more historical simulation VaR varies¹⁹.

How VaR is used and what it does not tell you

At some financial institutions, the board of directors allocates capital to cover market risk and manages overall market risk with a VaR limit²⁰. There are also examples that combine VaR limits with DV01 limits and limits on outstanding positions for each type of risk, such as the interest rate risk of bonds and equity risk²⁰. The risk management department reports risk amounts and any limit breaches to the board of directors¹.

VaR alone does not, in general, show how large a loss becomes once it exceeds VaR. Even with the same 99% VaR of 20.27 million yen, VaR cannot tell you whether losses on the remaining 1% of days stop at 21 million yen or reach 50 million yen⁶. A measure of the average loss beyond VaR is covered in Expected Shortfall.

VaR is estimated with statistical methods from past data, and it does not prepare you for stress events that are not in the observation period²¹. If the market shifts into a regime unlike anything seen before, VaR can underestimate future losses²¹. It is therefore supplemented by stress testing, which measures losses under large past or assumed price moves²¹. Backtesting checks whether losses exceeding VaR occur no more often than the confidence level implies⁴.

Notes

1. Bank of Japan, "市場リスク管理態勢の整備" (September 2019), "６．リスク管理部門の役割" (6. Role of the risk management department), page 34, and "市場リスクの把握と報告" (Identifying and reporting market risk), page 35 (in Japanese). ↩ ↩²

2. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "市場リスク管理態勢の整備" (Developing a market risk management framework) (September 2019), "１．市場リスクとは" (1. What market risk is), page 3 (in Japanese). Citing the definition in the Financial Inspection Manual, the document counts as market risk not only the risk of loss from changes in the value of assets and liabilities but also the risk of loss from changes in the earnings they generate. ↩

3. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "Ⅱ．市場リスクの把握と管理" (II. Understanding and managing market risk) (June 2012), "３．ＶａＲによるリスク量の把握" (3. Measuring the amount of risk with VaR), page 19, and "市場ＶａＲ（概念図）" (Market VaR (conceptual diagram)), page 21 (in Japanese). For a 99% VaR, the diagram gives the probability that a future loss exceeds VaR as 1%. ↩ ↩²

4. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), "４．バックテストによるＶａＲの検証" (4. Verifying VaR by backtesting), page 39 (in Japanese). For VaR at a 99% confidence level, it gives the probability of a loss exceeding VaR as once in 100 times. ↩ ↩² ↩³

5. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), page 19, and Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応" (Measuring value at risk with the historical method), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), page 3 (in Japanese). Both call VaR the maximum loss. ↩

6. Yasuhiro Yamai and Toshinao Yoshiba, "バリュー・アット・リスクのリスク指標としての妥当性について" (On the validity of value at risk as a risk measure), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (April 2001), pages 38–39 (in Japanese). For profit and loss that do not follow a normal distribution, the paper states that computing VaR alone does not, in general, give information on the size of losses beyond VaR. For a normal distribution, it states that computing VaR gives that information. ↩ ↩² ↩³

7. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), "Ａ．分散共分散法" (A. The variance-covariance method) and "分散共分散法" (The variance-covariance method), pages 23–24 (in Japanese). It gives the 99% VaR as 2.33 multiplied by the sensitivity (delta) and the standard deviation. ↩ ↩² ↩³

8. Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応," Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), "イ．分散共分散法" (a. The variance-covariance method) and "ロ．モンテカルロ法" (b. The Monte Carlo method), page 4 (in Japanese). ↩ ↩²

9. National Institute of Advanced Industrial Science and Technology (AIST), "期待値・分散の定義および演算" (Definitions and operations of expected value and variance), the "平均値（期待値）・分散の演算" (Operations on the mean (expected value) and variance) section (in Japanese). It shows that the variance of the sum of two random variables is the sum of their variances plus twice their covariance. The covariance is the correlation multiplied by the two standard deviations. ↩ ↩²

10. National Institute of Standards and Technology, "Cumulative Distribution Function of the Standard Normal Distribution," section 1.3.6.7.1 of the NIST/SEMATECH e-Handbook, the table of critical values. It gives 1.645, 1.960, and 2.326 for cumulative probabilities of 0.950, 0.975, and 0.990. ↩

11. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), "分散共分散法（デルタ法）の計算例" (Worked example of the variance-covariance (delta) method), page 30 (in Japanese). Against a simple sum of 10.99 for the stand-alone VaRs of an equity investment trust and government bonds, the VaR that takes correlation into account is 8.35. ↩ ↩²

12. 2.3263 comes from the "Area under the Normal Curve from 0 to X" table in the NIST source cited above. The area is 0.48983 up to 2.32 and 0.49010 up to 2.33, and interpolating linearly between them to the point where the area is 0.49 gives 2.3263. The simulator calculates with the unrounded factor, and the values on screen are rounded to the nearest 10,000 yen. ↩

13. This is the unrounded difference (2.3263 × 10.40 = 24.1935 minus 2.3263 × 8.7123 = 20.2674, which is 3.9261) rounded to the nearest 10,000 yen. The difference between the rounded values in the tables is 3.92 million yen. ↩

14. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), pages 31–32 and 36–37 (in Japanese). It finds VaR from the percentile that matches the confidence level. The steps are in Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応" (November 2004), pages 5–6 (in Japanese). ↩ ↩²

15. Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応," Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), "（1）ヒストリカル法によるVaRの算出手順" ((1) Steps for calculating VaR by the historical method), pages 5–6 (in Japanese). ↩ ↩²

16. Which loss is taken as VaR depends on the convention. The simulator ranks losses from the largest and uses the loss at the rank given by the number of days × (1 − confidence level), rounded up (the third for 250 days). The "標本分位点法" (sample quantile method) in the Institute for Monetary and Economic Studies paper (Ando, November 2004) uses the (number of days + 1) × (1 − confidence level)-th figure, and when that is not a whole number, interpolates linearly between the two figures on either side (page 6). ↩

17. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), pages 23, 31, 33 (points to note 3), and 36 (in Japanese). The points that the variance-covariance method needs no simulation and does not suit options come from Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測" (November 2004), page 4 (in Japanese). ↩

18. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), "市場ＶａＲの計測手法" (Methods for measuring market VaR), page 22 (in Japanese). ↩

19. Yoshitaka Ando, "ヒストリカル法によるバリュー・アット・リスクの計測：市場価格変動の非定常性への実務的対応," Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (November 2004), the explanation of "図表 4" (Figure 4), pages 9–10 (in Japanese). It compares VaR measured with observation periods of 250, 500, and 750 business days. ↩

20. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "市場リスク管理態勢の整備" (September 2019), "市場リスク全体枠：設定の考え方" (Overall market risk limit: how to set it), page 16, and "留意事項：各種リスク枠の設定" (Points to note: setting the various risk limits), page 19 (in Japanese). The document uses the name BPV for the measure that corresponds to DV01. ↩ ↩²

21. Bank of Japan, "Ⅱ．市場リスクの把握と管理" (June 2012), "５．ＶａＲの補完とストレステスト" (5. Supplementing VaR, and stress testing), pages 46–47, the examples of stress scenarios on page 48, and "留意事項④" (Points to note 4), page 38 (in Japanese). Page 48 lists scenarios that use the price moves of past shocks as they are and scenarios that assume price moves that could happen in the future. ↩ ↩² ↩³