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Risk Management in Banking and Insurance · Market Risk Management

Value at Risk (VaR) Methods: Calculation and Limits

Updated 11 October 2026 · Fact-checked

Value at Risk (VaR) is the maximum loss a portfolio is not expected to exceed over a set holding period at a given confidence level. You compute it by the variance-covariance method (σ × z × value), historical simulation (rank past returns) or Monte Carlo simulation (random scenarios).

Understand Value at Risk (VaR) Methods

Value at Risk (VaR) puts a single rupee figure on market risk. It answers: how much can I lose over a given period, with a given level of confidence, under normal market conditions?

A statement such as "1-day 99% VaR is ₹5 crore" means that on 99 days out of 100 you expect the loss to be ₹5 crore or less. On about 1 day in 100 the loss may be larger. VaR does not say how much larger.

Three inputs define every VaR figure: the holding period (1 day, 10 days), the confidence level (95%, 99%) and the portfolio value and its volatility. A longer holding period or a higher confidence level gives a bigger VaR.

There are three methods. Variance-covariance (parametric) assumes returns are normally distributed and uses standard deviation. Historical simulation reorders actual past returns and reads off the loss at the chosen percentile. Monte Carlo simulation generates thousands of random scenarios from a model and reads off the percentile. Historical and Monte Carlo do not need the normal assumption in the same way, and they handle options and non-linear positions better.

VaR has limits. It ignores losses beyond the cut-off, depends on past data, and can fail in a crisis. Banks therefore backtest VaR by comparing it with actual daily profit or loss and counting the exceptions, and they supplement it with stress testing and expected shortfall.

Key rules to remember

Parametric VaR (1 day)
VaR = Portfolio value × z × σ
σ is daily standard deviation of returns. Mean return is taken as zero unless the question gives it.
Common z-values (one-tailed, normal)
95% → 1.645; 99% → 2.33
Use the z-value the question supplies if it gives one.
Holding period scaling
VaR (N days) = VaR (1 day) × √N
Valid under the square-root-of-time rule: independent returns with constant volatility.
Two-asset portfolio standard deviation
σp = √(w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2)
Use when VaR is needed for a portfolio. Low correlation ρ gives diversification benefit.
Historical simulation percentile
Loss at (1 − confidence) percentile of ranked returns
With 100 observations at 95%, VaR is the 5th worst outcome; at 99% it is the 1st worst (or interpolate, as the question directs).
Undiversified vs diversified VaR
Diversified VaR ≤ Sum of individual VaRs
Equality holds only when correlation is +1.
Backtesting exception
Exception = day when actual loss > VaR
Expected exceptions = observations × (1 − confidence). Under the Basel framework the number of exceptions over 250 days drives the multiplier on capital.

How to solve Value at Risk (VaR) Methods questions

Use this sequence for any VaR question, whichever method is asked.

  1. 1Note the method asked, the confidence level, the holding period and the portfolio value.
  2. 2For variance-covariance, find the portfolio standard deviation. For two assets, use the weights, volatilities and correlation.
  3. 3Pick the z-value for the confidence level (1.645 for 95%, 2.33 for 99%, unless the question gives others).
  4. 4Compute 1-day VaR = value × z × σ.
  5. 5If the holding period is N days, multiply by √N.
  6. 6For historical simulation, rank the returns or P&L from worst to best and pick the observation at the (1 − confidence) position. Convert to rupees.
  7. 7For Monte Carlo, explain the steps: model the risk factors, simulate many paths, revalue the portfolio, rank outcomes and read the percentile.
  8. 8State the result in words: 'with X% confidence, loss will not exceed ₹__ over __ days', and add one limitation or comment if the question asks for interpretation.

Quickest way: Quick parametric VaR

When to use it: Use when the question gives volatility and asks for a VaR figure in a few marks.

  1. Write value, σ, z and days in one line.
  2. Multiply value × σ first to get the 1-day ₹ volatility.
  3. Multiply by z, then by √N.
  4. Check the units: σ as a decimal (1.5% = 0.015).
  5. Write the one-line interpretation to secure the final mark.

Common mistakes in Value at Risk (VaR) Methods

  • Using the wrong z-value, such as 1.645 for 99% confidence

    Students mix up the 95% and 99% values.

    Fix: Write 95% → 1.645 and 99% → 2.33 at the top of your answer before computing.

  • Multiplying by N instead of √N when scaling the holding period

    Students assume risk grows in a straight line with time.

    Fix: Volatility scales with the square root of time. Multiply 1-day VaR by √N.

  • Adding individual VaRs to get portfolio VaR

    It looks simpler.

    Fix: Compute portfolio σ with correlation first. Simple addition is right only when ρ = +1.

  • Picking the wrong position in historical simulation

    Students count from the best return, or confuse 95% with 5%.

    Fix: Sort from worst loss. At 95% confidence with 100 observations, take the 5th worst.

  • Saying VaR is the maximum possible loss

    The word 'maximum' is used loosely.

    Fix: Say VaR is the loss not expected to be exceeded at the stated confidence. Losses beyond it can occur and VaR gives no size for them.

  • Counting backtesting exceptions against the wrong benchmark

    Students forget the expected number.

    Fix: Expected exceptions = days × (1 − confidence). Far more exceptions mean the model understates risk.

Worked examples

Example 1

A bank's trading portfolio is worth ₹200 crore. Daily return standard deviation is 1.2%. Compute the 1-day 99% VaR and the 10-day 99% VaR using the variance-covariance method (z = 2.33, mean return zero).

Show the solution
  1. 1-day VaR = 200 × 2.33 × 0.012.
  2. 200 × 0.012 = ₹2.4 crore.
  3. ₹2.4 crore × 2.33 = ₹5.592 crore.
  4. 10-day VaR = 5.592 × √10.
  5. √10 = 3.1623, so 5.592 × 3.1623 = ₹17.68 crore (approx.).

Answer: 1-day 99% VaR = ₹5.592 crore. 10-day 99% VaR ≈ ₹17.68 crore. With 99% confidence, the loss over 10 days should not exceed about ₹17.68 crore.

Example 2

A portfolio has two positions: ₹60 lakh in Asset A (daily σ 2%) and ₹40 lakh in Asset B (daily σ 3%). Correlation is 0.5. Compute the 1-day 95% VaR (z = 1.645) and compare it with the sum of individual VaRs.

Show the solution
  1. Position σ in rupees: A = 60 × 0.02 = ₹1.2 lakh; B = 40 × 0.03 = ₹1.2 lakh.
  2. Portfolio variance = 1.2² + 1.2² + 2 × 0.5 × 1.2 × 1.2 = 1.44 + 1.44 + 1.44 = 4.32.
  3. Portfolio σ = √4.32 = ₹2.0785 lakh.
  4. Portfolio VaR = 1.645 × 2.0785 = ₹3.419 lakh (approx.).
  5. Individual VaRs: 1.645 × 1.2 = ₹1.974 lakh each; sum = ₹3.948 lakh.
  6. Diversification benefit = 3.948 − 3.419 = ₹0.529 lakh (approx.).

Answer: Portfolio 1-day 95% VaR ≈ ₹3.42 lakh, against ₹3.95 lakh as the simple sum. Correlation below +1 gives a diversification benefit of about ₹0.53 lakh.

Exam tips

  • Show the z-value and the formula before substituting. Marks are given for method even if arithmetic slips.
  • In theory questions, compare the three methods on assumptions, data needs, handling of options and computing effort. A short comparison of this kind scores well.
  • Always add a one-line interpretation of the VaR figure and one limitation, such as no information on losses beyond VaR.
  • For backtesting, state expected exceptions and what too many exceptions imply about the model.
  • Use the z-value given in the question. Do not replace it with your own.

Practice questions from Market Risk Management

Value at Risk (VaR) Methods in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Value at Risk (VaR) Methods: frequently asked questions

What is the difference between historical simulation and Monte Carlo VaR?

Historical simulation uses actual past returns as the set of scenarios. Monte Carlo generates new scenarios by random draws from a model of how risk factors move. Monte Carlo is more flexible but needs a model and more computing.

Why is VaR scaled by the square root of time?

If daily returns are independent with constant volatility, variance adds up over days, so standard deviation grows with √N. The rule fails when returns are autocorrelated or volatility changes.

What are the main limitations of VaR?

It says nothing about losses beyond the cut-off, may understate tail risk when returns are not normal, and relies on past data that may not match future markets. It also differs across methods and can be hard to apply to illiquid positions.

What is backtesting of VaR?

Backtesting compares the daily VaR estimate with the actual profit or loss. You count days when the loss exceeded VaR. Too many exceptions show the model is weak, and under the Basel framework they raise the capital multiplier.