Parametric VaR
Also written Parametric method · Variance-covariance VaR · Parametric Value at Risk
Value at Risk estimated from just two parameters — expected return and standard deviation — assuming returns are normally distributed.
In plain language
Value at Risk puts one number on a bad day: "at 95% confidence, the most this portfolio should lose in one day is ₹X."
There are three ways to estimate it. The parametric method is the simplest. It assumes returns follow a normal distribution, so the whole distribution is described by two parameters: the mean (expected return) and the standard deviation (risk). Knowing those, the loss at any confidence level can be read off the normal curve with a z-score.
The other two methods are historical simulation (replay past returns) and Monte Carlo simulation (generate thousands of random scenarios).
How it works
Step 1 — convert to daily figures. The workbook uses 250 (or 252) trading days, not 365.
- Daily return = annual return ÷ 250
- Daily standard deviation = annual standard deviation ÷ √250
Step 2 — choose the one-tailed z-score. VaR cares only about losses, so all of (1 − confidence) sits in the left tail:
| Confidence | z-score |
|---|---|
| 90% | 1.282 |
| 95% | 1.645 |
| 99% | 2.33 |
(For a two-sided 95% range, the z-value would be 1.96 — not the right one for VaR.)
Step 3 — scale to n days: return × n, standard deviation × √n.
The workbook's table — annual return 15%, annual SD 20%, portfolio 100 million: daily return 0.06%, daily SD 1.26%, giving VaR of 2.876%, 2.013% and 1.555% at 99%, 95% and 90%.
The formula
VaR (currency) = [ E(P) × n − z × σ(P) × √n ] × Portfolio value
E(P) = expected daily return σ(P) = daily standard deviation
z = 1.282 (90%), 1.645 (95%), 2.33 (99%)
n = number of days (a week = 5)
A worked example
Illustrative figures, using the workbook's method. A PMS strategy manages ₹50 crore, with expected annual return 12% and annual standard deviation 18%.
Daily return = 12% ÷ 250 = 0.048%
Daily SD = 18% ÷ √250 = 18 ÷ 15.811 = 1.1384%
| Confidence | Calculation | VaR % | VaR on ₹50 crore |
|---|---|---|---|
| 90% | 0.048 − 1.282 × 1.1384 | −1.41% | ≈ ₹70.6 lakh |
| 95% | 0.048 − 1.645 × 1.1384 | −1.82% | ≈ ₹91.2 lakh |
| 99% | 0.048 − 2.33 × 1.1384 | −2.60% | ≈ ₹1.30 crore |
Read the 95% line as the workbook does: on 95 days out of 100, the one-day loss should not exceed about ₹91 lakh; on 5 days out of 100, it will.
Ten-day 95% VaR: return 0.048 × 10 = 0.48%; SD 1.1384 × √10 = 3.600%.
VaR = 0.48 − 1.645 × 3.600 = −5.44% → about ₹2.72 crore
Why NISM asks about it
Chapter 17 (Risk), section 17.4.2.1, explains the parametric method with the three-confidence-level table, and the Chapter 17 caselet asks for a 10-day 95% VaR. Sample questions test the interpretation of a VaR statement, the one-tailed z-values, VaR's advantages and limitations, and how historical simulation and Monte Carlo differ from the parametric approach.
Common exam traps
- One tail, not two. 95% VaR uses 1.645, not 1.96.
- Returns scale with n; standard deviation scales with √n. Multiplying the SD by 10 for a 10-day VaR is the classic error.
- 250 or 252 trading days, not 365.
- The caselet's answer depends on rounding. The workbook rounds the daily SD to 1.26%, giving −5.83% for the 10-day 95% VaR. Unrounded (20 ÷ √250 = 1.2649%) the same calculation gives about −5.86%. Choose the option closest to −5.83%.
- VaR is not the maximum possible loss. 5% of the time the loss is larger. The workbook lists underestimating extreme losses and gains as a limitation, and notes VaR does not capture liquidity — hence liquidity-adjusted VaR.
- The normality assumption is the weak point. The workbook says real asset returns are far from normally distributed, and that VaR by historical simulation is generally found to be lower than parametric VaR for that reason.
Check yourself
1.Which VaR method randomly generates portfolio returns nearly 10,000 times and can accommodate any distribution pattern?
- a)Parametric method
- b)Historical simulation method
- c)Monte Carlo simulation
- d)Bid-ask spread method
Show the answer
Answer: (c) Monte Carlo simulation
Monte Carlo simulation randomly generates values with given parameters, repeats nearly 10,000 times, sorts the outcomes and reads the percentile. It works best for complex portfolios and can accommodate any distribution.
Parametric VaR assumes normal returns. Historical simulation uses actual past returns, not random ones. The bid-ask spread measures liquidity, not VaR.
Where this is taught
Free preparation for NISM Series XXI-BRelated terms
- Credit riskThe risk that a borrower fails to meet its obligations on a debt instrument — the risk credit rating agencies exist to grade, and the one that triggers a segregated portfolio in a mutual fund.
- Standard deviationA measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
- Systematic riskThe part of an investment's risk that comes from economy-wide forces moving every asset at once — it cannot be diversified away, and it is the only risk the market pays you to carry.
- Loss Given DefaultIn the XXI-B workbook, the expected money loss from a borrower's default, calculated as probability of default × exposure at default. Textbooks often use LGD differently.