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Advanced Financial Management · Risk Management

Value at Risk (VaR) Methods for CA Final AFM

Updated 5 October 2026 · Fact-checked

Value at Risk (VaR) is the maximum loss a portfolio is expected not to exceed over a set period at a given confidence level. For the variance-covariance method, VaR = Z × σ × portfolio value × √t. Historical simulation ranks past returns; Monte Carlo ranks simulated returns.

Understand Value at Risk (VaR) Methods

Value at Risk (VaR) answers one question: how much can I lose over a given period, with a given level of confidence? A statement like "1-day 95% VaR is ₹2,00,000" means that on 95% of days the loss should be no more than ₹2,00,000. On the remaining 5% of days the loss may be larger.

Every VaR figure has three parts: the amount at risk, the holding period (1 day, 10 days) and the confidence level (95%, 99%). Change any one and the VaR changes. A longer period or a higher confidence level gives a bigger VaR.

There are three ways to measure it. The variance-covariance (parametric) method assumes returns are normally distributed and uses the standard deviation and a Z value. Historical simulation takes actual past returns, applies them to today's portfolio, ranks the outcomes and reads off the loss at the chosen percentile. Monte Carlo simulation generates many random scenarios from an assumed model and reads the percentile in the same way.

Each method has trade-offs. Variance-covariance is quick but depends on the normal assumption, which understates extreme losses (fat tails). Historical simulation makes no distribution assumption but assumes the past will repeat. Monte Carlo is flexible and handles options and non-linear positions, but it is computationally heavy and depends on the model you choose.

VaR has general limitations. It does not say how bad the loss can be beyond the cut-off. It is not a worst case. It can change a lot with the data window. Expected shortfall (also called conditional VaR) covers this gap: it is the average loss in the cases where the loss exceeds VaR.

Key rules to remember

Parametric VaR (single asset or portfolio)
VaR = Z × σ × V
σ is the standard deviation of returns for the holding period. V is the portfolio value. Z is the standard normal value for the confidence level.
Scaling to t days
σ(t days) = σ(1 day) × √t
Valid when daily returns are independent with the same σ. Square root of time rule, not t.
Common Z values (one-tailed)
90% → 1.28; 95% → 1.645; 99% → 2.33
Use the Z value given in the question. VaR is a one-tailed measure.
Two-asset portfolio standard deviation
σp = √(wA²σA² + wB²σB² + 2 wA wB ρAB σA σB)
Gives diversification benefit when ρ < 1. Covariance = ρ × σA × σB.
Portfolio VaR with diversification
Portfolio VaR = Z × σp × V
Portfolio VaR is at most the sum of individual VaRs when ρ ≤ 1 and positions are long.
Historical simulation cut-off
VaR = loss at the (1 − confidence level) percentile of ranked outcomes
With 100 observations at 95%, the 5th worst outcome is a common reading. Follow the convention in the question.
Expected shortfall
ES = average of losses beyond the VaR cut-off
ES is always at least as large as VaR at the same confidence level.

How to solve Value at Risk (VaR) Methods questions

Use this order for any VaR question. It keeps the method, the units and the interpretation straight.

  1. 1Read the three inputs: portfolio value, holding period and confidence level. Note which method the question wants.
  2. 2For variance-covariance, find the standard deviation for the holding period. Convert daily σ using √t if needed. For a portfolio, compute σp using weights, σ and correlation.
  3. 3Pick Z for the confidence level from the question. If none is given, use 1.645 for 95% and 2.33 for 99%.
  4. 4Compute VaR = Z × σ × V. Keep money and percentage units separate.
  5. 5For historical simulation, apply each past return to the current value, sort the outcomes from worst to best, and read the loss at the required percentile.
  6. 6For Monte Carlo, state that scenarios are generated from an assumed distribution and then ranked the same way. Numerical data is usually given as simulated outcomes.
  7. 7Write the interpretation in words: with X% confidence, the loss over t days will not exceed ₹VaR.
  8. 8Add one line on limitations if the question asks for comment, such as normality, fat tails or no information beyond VaR.

Quickest way: Fast parametric VaR in three lines

When to use it: Use when σ and Z are given or standard, and the question is a direct single-asset or two-asset VaR.

  1. Get the σ for the full holding period: σ(1 day) × √t.
  2. Multiply: Z × σ × V. This is the VaR in rupees.
  3. Write the sentence: at the stated confidence, loss over the period will not exceed this amount.
  4. For two assets, compute σp first and only then multiply. Never add σs directly.

Common mistakes in Value at Risk (VaR) Methods

  • Scaling the daily σ by t instead of √t.

    Students treat risk like return, which grows in proportion to time.

    Fix: Variance grows with time, so σ grows with √t. For 10 days multiply by √10 ≈ 3.162.

  • Adding individual VaRs to get portfolio VaR.

    It feels natural to total the risks of the two assets.

    Fix: Use σp with the correlation. Simple addition is correct only when ρ = +1.

  • Using the wrong Z, such as 1.96 for a one-tailed 95% VaR.

    1.96 is memorised from two-tailed confidence intervals.

    Fix: VaR looks at one tail only. Use 1.645 for 95% and 2.33 for 99% unless the question states otherwise.

  • Calling VaR the maximum possible loss.

    The word "maximum" in the definition is read without the confidence level.

    Fix: Say the loss will not exceed VaR with the stated confidence. Losses beyond VaR can and do occur.

  • Reading the wrong percentile in historical simulation.

    Students rank best to worst, or confuse 95% with the 95th worst.

    Fix: Rank losses worst first. At 95% confidence, you look at the worst 5% of outcomes. With 100 observations that is around the 5th worst.

  • Mixing percentage VaR and rupee VaR.

    σ is in percent but V is in rupees, and the conversion is skipped.

    Fix: Convert σ to decimal (2% = 0.02) before multiplying by V.

Worked examples

Example 1

A treasury manager holds a portfolio worth ₹10,00,00,000. The daily standard deviation of returns is 1.5%. Assuming normal returns, calculate the 10-day VaR at 99% confidence (Z = 2.33). Take √10 = 3.162.

Show the solution
  1. Daily σ = 1.5% = 0.015.
  2. 10-day σ = 0.015 × 3.162 = 0.04743, or 4.743%.
  3. VaR = Z × σ × V = 2.33 × 0.04743 × ₹10,00,00,000.
  4. 2.33 × 0.04743 = 0.110512.
  5. 0.110512 × ₹10,00,00,000 = ₹1,10,51,200 (approx.).
  6. Interpretation: with 99% confidence, the loss over 10 days should not exceed about ₹1.11 crore.

Answer: 10-day 99% VaR ≈ ₹1,10,51,200.

Example 2

A firm holds ₹60,00,000 in Stock A and ₹40,00,000 in Stock B. Daily σ of A is 2% and of B is 3%. Correlation between them is 0.5. Calculate the 1-day 95% VaR of the portfolio (Z = 1.645) and compare it with the sum of the individual VaRs.

Show the solution
  1. Total value V = ₹1,00,00,000. Weights: wA = 0.6, wB = 0.4.
  2. Variance = (0.6)²(0.02)² + (0.4)²(0.03)² + 2 × 0.6 × 0.4 × 0.5 × 0.02 × 0.03.
  3. First term: 0.36 × 0.0004 = 0.000144.
  4. Second term: 0.16 × 0.0009 = 0.000144.
  5. Third term: 2 × 0.6 × 0.4 × 0.5 = 0.24; 0.24 × 0.0006 = 0.000144.
  6. Variance = 0.000144 + 0.000144 + 0.000144 = 0.000432.
  7. σp = √0.000432 ≈ 0.020785, or 2.0785%.
  8. Portfolio VaR = 1.645 × 0.020785 × ₹1,00,00,000 = ₹3,41,913 (approx.).
  9. VaR of A alone = 1.645 × 0.02 × ₹60,00,000 = ₹1,97,400.
  10. VaR of B alone = 1.645 × 0.03 × ₹40,00,000 = ₹1,97,400.
  11. Sum of individual VaRs = ₹3,94,800.
  12. Diversification benefit = ₹3,94,800 − ₹3,41,913 ≈ ₹52,887.

Answer: Portfolio 1-day 95% VaR ≈ ₹3,41,913, lower than the undiversified sum of ₹3,94,800 because correlation is below 1.

Exam tips

  • Write the interpretation sentence every time. Many marks are given for stating VaR with confidence level and period.
  • Show σp working in full for two-asset questions. Part marks are awarded for the variance step even if the final figure is off.
  • If the question gives Z or √t values, use them exactly as given, even if your calculator differs slightly.
  • For theory questions, give a short comparison of the three methods covering assumption, strength and weakness. Add expected shortfall as the answer to VaR's tail blindness.
  • In case-scenario MCQs, check whether the question asks for rupee VaR or percentage VaR and whether the period is 1 day or longer.

Practice questions from 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 VaR and expected shortfall?

VaR is the loss cut-off at a confidence level. Expected shortfall is the average loss in the cases that go beyond that cut-off. So expected shortfall tells you how bad the tail is, while VaR only tells you where the tail starts.

Which VaR method is best?

None is best in every case. Variance-covariance is fast but assumes normal returns. Historical simulation uses actual data but assumes the past repeats. Monte Carlo handles complex and non-linear positions but needs a model and more computing.

Why do we use the square root of time?

If daily returns are independent, variance adds up across days. So variance over t days is t times daily variance, and σ is σ(1 day) × √t. This rule holds only under those assumptions.

Why is VaR criticised?

It ignores losses beyond the cut-off and can understate risk when returns have fat tails. It also depends on the data window and model chosen. Expected shortfall and stress testing are used alongside it.