FIN 3820 Module 4 Midterm Project: Describing and Evaluating Value at Risk Example

A Useful Number With Dangerous Habits: Describing and Evaluating Value at Risk

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B.S.B.A. in Finance, Thomas Edison State University

FIN-3820: Risk Management

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Month Day, Year

A Useful Number With Dangerous Habits: Describing and Evaluating Value at Risk

Introduction

Value at risk became the standard way financial institutions summarize market risk in the 1990s, after J.P. Morgan published its RiskMetrics methodology, and banking regulators adopted it for capital requirements soon after. Its appeal is that it reduces a complex portfolio's risk to one number in dollars: the most the portfolio should lose over a set horizon at a chosen confidence level. That simplicity is also its danger. This paper describes how VaR is calculated and used and then evaluates when it informs decisions and when it misleads.

What VaR Measures

VaR has three parameters: a holding period, a confidence level and a currency amount. A one-day 99 percent VaR of $300,000 means that the institution expects to lose more than $300,000 on about one trading day in a hundred. Hull (2023) emphasizes that VaR is a statement about the left tail of the distribution of portfolio changes, and that it is silent about how far the tail extends. Institutions use VaR to set trading limits, allocate capital among business lines, report risk to boards and, for larger banks, calculate regulatory capital.

VaR also depends on the horizon. Under the usual assumption that daily changes are independent, a ten-day VaR is the one-day figure multiplied by the square root of ten, so the bank's one-day 99 percent VaR of about $299,800 implies a ten-day figure of roughly $948,000, the horizon regulators have historically used for market risk capital. The square-root rule is itself an assumption: if losses cluster, as they do in a crisis, losses over ten days can be worse than the scaled figure suggests.

Two Approaches to Calculating VaR

Model-building

The model-building, or variance-covariance, approach assumes that changes in risk factors are normally distributed and uses estimates of volatilities and correlations to compute the portfolio's standard deviation. For the community bank's $15 million portfolio of a stock fund and a bond fund, with an EWMA volatility of 1.315 percent for stocks, 0.45 percent for bonds and a correlation of −0.2, the one-day 99 percent VaR is about $299,800. The approach is fast and easy to explain, and it makes the role of correlation explicit. Its weaknesses are the normality assumption, which understates fat tails, and its poor handling of nonlinear instruments such as options.

Historical simulation

Historical simulation applies the actual daily changes in risk factors over a past period, often 500 days, to today's portfolio and reads VaR from the resulting distribution of losses: at 99 percent, roughly the fifth-worst loss in 500 scenarios. It needs no distributional assumption and handles options naturally. Its weakness is that it can only reflect the history in its window. If the 500 days were calm, VaR will be low just before a crisis; if they include a crash, VaR will remain high long after.

For the bank's simple, linear portfolio, the two methods give similar results in normal markets; they diverge when the historical window contains extreme days.

Backtesting

A VaR model is only useful if it is accurate, and backtesting checks this by counting exceptions, days when actual losses exceeded VaR. Over 250 days, a 99 percent VaR should be exceeded about 2.5 times. Kupiec (1995) developed a likelihood ratio test of whether the number of exceptions is consistent with the stated confidence level; too many suggests the model understates risk, too few suggests it is too conservative. Basel's traffic-light approach applies the same idea: zero to four exceptions in a year is green, five to nine yellow, with a higher capital multiplier, and ten or more red. Backtesting has a weakness of its own: it tests the frequency of exceptions, not their size, so a model can pass while badly underestimating losses in the tail.

VaR in Regulation

The 1996 market risk amendment to the Basel Accord allowed banks to use their own VaR models to set capital for trading risk, with a multiplier of at least three applied to ten-day 99 percent VaR. After the 2007-2009 crisis showed that VaR based on calm periods had badly understated risk, Basel 2.5 added a stressed VaR, calculated using a period of significant market stress. The later Fundamental Review of the Trading Book replaced VaR with expected shortfall at a 97.5 percent confidence level, partly because expected shortfall averages the losses beyond the cutoff and combines sensibly across portfolios, two of the properties Artzner et al. (1999) identified as essential for a coherent risk measure.

The Basel Committee's later reforms responded to these weaknesses. The Fundamental Review of the Trading Book replaced VaR with expected shortfall at a 97.5 percent confidence level for internal model capital, because expected shortfall averages the losses beyond the threshold and therefore reflects the size of tail losses that VaR ignores. Expected shortfall is also subadditive, so it does not penalize diversification, a property VaR can lack. Its weakness is that it is harder to backtest, which is why regulators kept VaR-based backtesting alongside it.

The 2012 Chief Investment Office Losses

JPMorgan Chase's 2012 trading losses in its Chief Investment Office, about $6.2 billion on a large credit derivatives position, illustrate how VaR can mislead when it is trusted too much. In early 2012 the office adopted a new VaR model that roughly halved its reported VaR, allowing the position to grow within its limit; the model contained spreadsheet errors and was not adequately reviewed. The bank's own task force later concluded that the model change, weak oversight and the size and complexity of the position contributed to the losses (JPMorgan Chase & Co., 2013). The lesson is not that VaR is useless but that a VaR number can create false comfort when its model is not validated and when it becomes a limit to be managed rather than a measure to be understood.

Evaluation

Strengths

VaR provides a common language for risk across very different positions, makes the effect of diversification visible, supports limits and capital allocation and can be checked against outcomes through backtesting. For a community bank's simple portfolio, it is an efficient summary.

Weaknesses

It says nothing about losses beyond its threshold; its normal version understates fat tails; its historical version depends on its window; it may not reward diversification consistently; and it can be gamed or mis-specified, as the 2012 case showed. Correlations estimated in calm periods tend to rise in crises, so VaR understates risk precisely when it matters most.

When VaR Helps and When It Misleads

VaR helps when used as one input among several: alongside expected shortfall, stress tests and scenario analysis, with models independently validated and limits set with judgment. It misleads when treated as the worst case, when its assumptions go unexamined and when traders or managers are rewarded for keeping it low rather than for understanding their risk.

For the community bank, the practical question is not whether VaR is perfect but whether management understands what it leaves out. A risk committee that sees VaR beside a stress loss from a 2008-style scenario and a 2022-style rate shock has a much fuller picture than one that sees a single number. The recommendations below follow that principle.

Recommendations for the Community Bank

1. Report both VaR and expected shortfall for the investment portfolio each month.

2. Calculate VaR by both methods and investigate large differences.

3. Backtest monthly and report exceptions to the risk committee.

4. Run quarterly stress scenarios, including a repeat of 2008 and a rapid rate rise like 2022.

5. Have the VaR model validated by someone independent of the treasury function.

Conclusion

Value at risk is a valuable summary of market risk and a weak predictor of catastrophe. Its usefulness depends on the discipline around it: validated models, honest backtesting, complementary tail measures and managers who treat it as a measure to understand rather than a number to hit.

References

Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228. https://doi.org/10.1111/1467-9965.00068

Hull, J. C. (2023). Risk management and financial institutions (6th ed.). Wiley.

JPMorgan Chase & Co. (2013). Report of JPMorgan Chase & Co. Management Task Force regarding 2012 CIO losses.

Kupiec, P. H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73-84. https://doi.org/10.3905/jod.1995.407942