Value-at-Risk (VaR) Shortcomings and Qualities

MANAGEMENT OF FINANCIAL RISK
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Q. Discuss the concept and theory of Value at Risk (VaR) and its shortcomings.
The concept of Value at Risk (VaR) is a representative measure of the highest loss that an investor is likely to incur in the near future. Hence, it measures the largest possible loss that a portfolio is likely to suffer in the future for a prolonged holding period at a specific confidence level (Ghasemi et al., 2018). Therefore, VaR is a measure of the prevailing market risk and is regarded as a measure that is equal to one standard deviation of a portfolio’s distribution of returns.
Figure 1: Probability distribution of portfolio value
Source: Zabolotskyy & Bilyy (2014).
Calculation and Measure of Value at Risk (VaR)
Value-at-Risk (VaR) is a consolidated technique of measuring risk that is often used as a portfolio risk performance benchmark since it measures the asset risk exposure level (Akram & Khalil, 2018).
Characteristics of VaR
The traditional Value-at-Risk (VaR) has the following qualities
1. It is a probability tool used for measuring the potential losses that a portfolio can incur at different confidence levels
2. It is a consistent measure used to calculate the financial risk based on the possible dollar loss benchmark. It helps financial analysts calculate comparative performances across diverse portfolios and assets.
3. VaR is determined based on a common timeframe, and hence it permits quantification of possible losses within that specific time horizon (Ghasemi et al., 2018).
The calculation of VaR is based on a confidence level that underlies the industry requirements and reporting standards. However, the timeframe is determined by the type of portfolio, asset, or business line. For instance, VaR for common stock can be calculated based on any timeframe. Secondly, a portfolio VaR is calculated based on a turnover timeframe while the time horizon portfolio remains constant. Still, VaR can be based on employee valuation periods, critical decisions, regulatory and external quality assessment needs. The calculation of VaR comparison for dual-portfolio requires two constant variables: timeframe and confidence level (Global portfolio et al., 2021).
Methods of Calculating VaR
The reliability of VaR depends on the calculation method. Still, time horizon and data availability determine consistency and reliability.
Historical Method
The historical method is the most fundamental method that has several shortfalls such as large amounts of time series data and over-reliance on historical data for risk calculation that is unsuitable for future predictions.
Historical Method: Tabulated VaR Data
The historical technique relies on a set of 30 values that represent stock returns. To calculate VaR, the values of returns are arranged in ascending order and then counted. It requires that 5% of daily VaR be taken for at least 5% of the portfolio/asset returns out of a possible set of 30 values (Konovalova et al., 2016). However, 1% daily VaR requires counting at least 1% of the returns out of a possible set of 30 values. In a nutshell, the VaR counting technique measures the risk of a portfolio by taking the 1.5th and 0.3rd values for the second-lowest and the lowest return respectively.
Table 1: Historical Method VaR Distribution table
Bin

Frequency

-4.60%

1

-2.98%

1

-1.36%

7

0.27%

10

1.89%

7

More

4

There are two key ways of interpreting historical VaR. The first technique is based on the minimum loss possibility. If VaR is 5%, the interpretation is that there exists a 5% probability that the portfolio might suffer losses equivalent to 3.99% of the portfolio value. Secondly, it can be interpreted as a confidence level. Using the same scenario, 5% VaR is interpreted to imply a 95% possibility that an asset/portfolio could suffer losses above 3.99% of the investment value (Lulaj et al, 2020). This is the most common technique even though it excludes higher loss percentages. Given this drawback, portfolio managers often rely on both 5% and 1% VaR measurement techniques.
Figure 2: Historical Return Distribution
Source: Zabolotskyy & Bilyy (2014).
VaR Calculation for a portfolio
The calculation of VaR for a multiple-asset portfolio is done on a per-asset basis in the same time horizon. Relative portfolio values are obtained and later, the return is merged and historically distributed.
Table 2: Weighted Portfolio Return for Stocks-A,B,C & D
Firstly, VaR is associated with excessive risk-taking and financial leverage. The second criticism is that it largely focuses on manageable risks in the middle thereby ignoring the tails. The third drawback is that VaR provides portfolio analysts with the incentive to calculate excessive, though remote risks (Akram & Khalil, 2018). Still, as a risk measurement instrument, VaR can be detrimental especially when it creates a false sense of investment security. VaR is catastrophic when it depicts the optimal tolerable loss since it means three-loss expectations on a 1% oper day basis. As a risk measurement tool, it can be misleading when it is relied upon for risk control and management purposes. Instead, portfolio managers should be concerned about losses exceeding VaR (Radoi & Olteanu, 2016). In addition, VaR presumes reasonable losses not exceeding the value of three VaR values.
It means that once losses hit the VaR level, portfolio and asset managers should start preparing for the worst-case scenario. Another common shortcoming of VaR is that it can be computed based on unverified assumptions (Ghasemi et al., 2018). Also, VaR can get complex and hence difficult to derive for large and highly diversified portfolios. It is difficult to calculate the return, volatility, and correlations in large asset portfolios. In such a scenario, VaR reliability as a risk measuring tool cannot be guaranteed. Economic analysts also believe that VaR is not additive since correlations for individual risk factors are used in VaR calculation, which makes it less additive (Radoi & Olteanu, 2016). The VaR of a portfolio comprising two assets: A and B is not equal to the sum of each asset’s VaR. Its other disadvantage is that it's as good as its inputs and assumptions. The danger of using VaR is that it relies on the classical variance-covariance VaR technique which presumes normal distribution of returns with excess kurtosis. Hence, basing VaR on unrealistic distributions can lead to risk underestimation. The last disadvantage is that diverse VaR techniques produce varying results (Lulaj et al, 2020). For instance, the classical variance-covariance, historical VaR, and the Monte Carlo technique all lead to different VaR for the same asset or portfolio.
Suppose the daily changes for a portfolio have the first-order autocorrelation parameter 0.12. The 10-day VaR, calculated by multiplying the one-day VaR by YTO, is £2 million. What is a better estimate of VaR that takes into account the autocorrelation?
(iii)The change in the value of a portfolio in three months is normally distributed with a mean of £500,000 and a standard deviation of £3 million. Calculate the VaR and the expected shortfall (ES) for a confidence level of 99.5% and a time horizon of three months. Note: N -1(0.995) = 2.576 (10 marks)
Answer
Using Kupiec’s two-tailed test,
We take p=0.01; m=15 and n=1000.
Kupiec’s test statistic, í2 ln [0.999985× 0.0115] + 2 ln[(1 í15/1000)985× (15/1000)15] = 2.19. The value falls below 3.84. Since 2.19 is less than 3.84, the model is rejected. To measure the change in a portfolio for a 3 month time horizon, the normal distribution is a mean of $500,000. The standard deviation is $3 million. Based on these statistics, we determine the VaR and expected shortfall (ES), assuming a confidence level of 99.5% and 3 months timeframe. The possible loss equals a mean of $500 million and a standard deviation of 3000. It means that, Ní1(0.995) =2.576. The 99.5% VaR in $’000s is í500+3000×2.576) =7,227.
Therefore, we have 99.5% confidence that the potential loss cannot exceed $7.227 million.
The expected shortfall (ES) for the portfolio is -176,9005.0230005002/576.22u±²Se
Thus the possibility that the Expected loss falls within the 0.5% distribution range is $9.176 million. The possibility that the resulting loss will exceed $10 million within a timeframe of one month is 5%. To calculate the one-month 99% VaR, we assume that the change the portfolio value is evenly distributed across the portfolio where mean distribution=0; We determine the 5% tail of return distribution. The return distribution starts at 1.645 standard deviations from the mean return. To derive the standard deviation, we calculate $10/1.645=$6.080 million. Next, we calculate the one-month 99% VaR which is $6.080×Ní1(0.99) = $14.143 million. The three-month change in the portfolio will be normally distributed with a mean = $500,000. The standard deviation for the portfolio will be $3 million.
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