Quick Recap: What VaR Measures
In the first article of this series, we defined Value at Risk (VaR) as a threshold: a currency figure that a portfolio is not expected to lose more than, over a given time horizon, at a chosen confidence level. We also established that VaR is a modelling choice — the same portfolio can produce different VaR numbers depending on which of three standard methods is used to calculate it.
This article covers the first and most widely taught of those three methods: the Parametric method, also called the Variance-Covariance method.
What Is the Parametric (Variance-Covariance) Method?
The Parametric method calculates VaR using a formula built from two statistical inputs: the volatility (standard deviation) of an asset’s returns, and, for a portfolio, the covariances between every pair of assets held.
Instead of looking at thousands of historical scenarios one by one (as the Historical Simulation method does, covered in the next article), the Parametric method assumes that returns follow a known, well-behaved statistical pattern — specifically, a normal distribution — and uses that assumption to calculate VaR directly through a formula, in seconds, without simulating anything.
The Core Assumption: The Normal Distribution
The normal distribution is the familiar symmetric “bell curve” — most outcomes cluster near the average, with progressively fewer, more extreme outcomes as you move further from it in either direction.
The Parametric method assumes daily portfolio returns behave this way. This assumption is powerful because it lets us describe an asset’s entire range of possible outcomes using just two numbers: its mean return and its standard deviation. Everything else — including the VaR threshold — can be derived mathematically from those two inputs alone.
This is both the method’s greatest strength (speed, simplicity) and its best-known weakness, which we return to in the disadvantages section below.
https://en.wikipedia.org/wiki/Normal_distribution
The Building Blocks: Standard Deviation and Z-Scores
Standard deviation measures how much daily returns typically swing around their average — in other words, volatility. A higher standard deviation means bigger day-to-day price swings and, all else equal, a higher VaR.
A Z-score tells you how many standard deviations away from the mean a given confidence level sits on a normal distribution. Risk managers use a fixed set of Z-scores for the confidence levels used across the industry:
| Confidence Level | Z-Score (One-Tailed) |
| 90% | 1.2816 |
| 95% | 1.6449 |
| 97.5% | 1.9600 |
| 99% | 2.3263 |
| 99.9% | 3.0902 |
These Z-scores come from the standard normal distribution and are the same for every asset and every portfolio — only the standard deviation changes from one calculation to the next.
The Parametric VaR Formula — Single Asset
For a single position, the formula is:
| VaR = V × Z × σ × √t Where: V = Position Value (currency) Z = Z-score for the chosen confidence level σ (sigma) = Standard deviation of daily returns t = Time horizon in days (t = 1 for a 1-day VaR) |
This formula converts a statistical concept (volatility) directly into a currency figure (VaR) in one step — no simulation, no iteration.
Worked Example: Single-Asset VaR
This example follows exactly the same numbers used in the accompanying Excel workbook (Sheet ‘1. Single-Asset VaR’), so you can cross-check every step.
Step 1 — Gather the data. We start with 31 days of daily closing prices for a single position (illustrative sample data), which produces 30 daily returns.
Step 2 — Calculate the statistics.
| Statistic | Value |
| Number of return observations | 30 |
| Mean daily return | -0.2410% |
| Daily volatility (standard deviation) | 1.4399% |
| Worked Example — Single Asset Position Value (V) = ₹50,00,000 Daily Volatility (σ) = 1.4399% Z-score at 99% confidence = 2.3263 1-Day VaR (99%) = ₹50,00,000 × 2.3263 × 1.4399% = ₹1,67,484 Reading this: under normal market conditions, there is a 99% chance this position will not lose more than ₹1,67,484 in one trading day. 10-Day VaR (99%) = ₹1,67,484 × √10 = ₹5,29,630 At the 95% confidence level, the same position gives a 1-Day VaR of ₹1,18,420 — noticeably lower, because a 95% threshold is less strict than 99% (Z = 1.6449 vs 2.3263). | |
Notice something worth flagging: the Parametric formula above does not use the mean return at all — only the standard deviation. This is a common simplification for short horizons like 1 day, where the mean daily return is tiny compared to the volatility and has almost no effect on the result. Over longer horizons, some practitioners use a mean-adjusted version of the formula; we keep the simplified version here since it is the industry-standard convention for short-horizon VaR.
Extending to a Portfolio: Why a Single Asset Formula Isn’t Enough
Real portfolios hold more than one position. If you simply added up each asset’s individual VaR, you would overstate the portfolio’s true risk — because assets rarely move in perfect lockstep. When one position loses money, another may hold steady or even gain, partially offsetting the loss. This offsetting effect is diversification, and a correct portfolio VaR calculation must account for it.
This is where the “Variance-Covariance” half of the method’s name comes in. To calculate portfolio VaR properly, we need not just each asset’s individual volatility, but also how each pair of assets moves relative to one another — their covariance.
The Variance-Covariance Matrix Explained
A variance-covariance matrix is a grid that shows, for every pair of assets in a portfolio, how their returns move together. The diagonal of the matrix holds each asset’s own variance (standard deviation squared); the off-diagonal cells hold the covariance between each pair.
A related, more intuitive measure is correlation, which rescales covariance to always fall between -1 and +1. A correlation near +1 means two assets move almost identically; near -1 means they move in opposite directions; near 0 means their movements are unrelated. Correlation doesn’t feed directly into the VaR formula, but it’s the easiest way to sanity-check whether your covariance numbers make sense.
The Portfolio VaR Formula
For a 3-asset portfolio with weights wA, wB, wC (each asset’s share of total portfolio value), the portfolio variance is calculated by expanding every pairwise weight-covariance term:
| Portfolio Variance = wA² × CovAA + wB² × CovBB + wC² × CovCC + 2×wA×wB × CovAB + 2×wA×wC × CovAC + 2×wB×wC × CovBC Portfolio Std. Dev. = √(Portfolio Variance) Portfolio VaR = Portfolio Value × Z × Portfolio Std. Dev. × √t |
Note that CovAA is simply Asset A’s own variance (σA²), and CovAB = CovBA (the matrix is symmetric). The logic scales to any number of assets — a 10-asset portfolio just has more cross-terms to sum.
Worked Example: 3-Asset Portfolio VaR
This example matches the accompanying Excel workbook exactly (Sheet ‘2. Portfolio VaR’). The portfolio holds three illustrative positions: Equity A, Equity B, and a Bond Fund C.
Step 1 — Individual asset statistics (from 30 days of illustrative daily returns):
| Statistic | Equity A | Equity B | Bond Fund C |
| Mean Daily Return | -0.1585% | 0.1328% | -0.0155% |
| Standard Deviation | 0.8701% | 1.9208% | 0.4472% |
Step 2 — Position values and weights:
| Asset | Position Value (₹) | Weight |
| Equity A | 40,00,000 | 40% |
| Equity B | 35,00,000 | 35% |
| Bond Fund C | 25,00,000 | 25% |
| Total Portfolio Value | 1,00,00,000 | 100% |
Step 3 — Variance-covariance matrix (in decimal return-squared units, matching the Excel workbook):
| Equity A | Equity B | Bond Fund C | |
| Equity A | 0.0000757 | 0.0000430 | -0.0000131 |
| Equity B | 0.0000430 | 0.0003689 | -0.0000094 |
| Bond Fund C | -0.0000131 | -0.0000094 | 0.0000200 |
Step 3b — Correlation matrix (for reference — easier to interpret than raw covariance):
| Equity A | Equity B | Bond Fund C | |
| Equity A | 1.00 | 0.26 | -0.34 |
| Equity B | 0.26 | 1.00 | -0.11 |
| Bond Fund C | -0.34 | -0.11 | 1.00 |
Equity A and Equity B are moderately positively correlated (0.26) — they tend to move somewhat together. Both equities are negatively correlated with the Bond Fund (-0.34 and -0.11) — the bond tends to hold up or move opposite when the equities fall. This negative correlation is exactly what creates a diversification benefit, as we’ll see in Step 6.
| Worked Example — Portfolio Variance and VaR Portfolio Variance = (0.40²×0.0000757) + (0.35²×0.0003689) + (0.25²×0.0000200) + (2×0.40×0.35×0.0000430) + (2×0.40×0.25×-0.0000131) + (2×0.35×0.25×-0.0000094) Portfolio Variance = 0.0000663 Portfolio Standard Deviation = √0.0000663 = 0.8144% Total Portfolio Value = ₹1,00,00,000 Z-score at 99% confidence = 2.3263 1-Day Portfolio VaR (99%) = ₹1,00,00,000 × 2.3263 × 0.8144% = ₹1,89,466 10-Day Portfolio VaR (99%) = ₹1,89,466 × √10 = ₹5,99,144 |
Understanding the Diversification Benefit
Here’s where the portfolio approach proves its value. If we ignored covariance entirely and simply added up each asset’s own VaR:
| Asset | Individual 1-Day VaR (99%) |
| Equity A | ₹80,965 |
| Equity B | ₹1,56,392 |
| Bond Fund C | ₹26,008 |
| Sum (Undiversified VaR) | ₹2,63,365 |
The properly calculated portfolio VaR is ₹1,89,466 — about ₹73,899 (28.1%) lower than the undiversified sum of ₹2,63,365. This gap is the diversification benefit, and it exists specifically because the Bond Fund’s negative correlation with the two equities offsets some of their combined risk. This is precisely the property we described in the foundations article: VaR properly aggregated across a portfolio is not simply additive — it depends entirely on how the components move together.
Scaling VaR Over Time: The Square-Root-of-Time Rule
Both worked examples above scaled the 1-day VaR to a 10-day VaR by multiplying by the square root of 10 (≈3.1623). This shortcut, the square-root-of-time rule, assumes daily returns are independent and identically distributed from one day to the next. It’s a reasonable approximation for short horizons under normal conditions, but it tends to understate risk during periods of market stress, when volatility itself changes and daily returns become more correlated with each other — precisely the situations where an accurate VaR matters most.
Advantages of the Parametric Method
- Fast to calculate — a closed-form formula, not a simulation
- Requires relatively little historical data compared to simulation-based methods
- Easy to update — recalculating with new volatility and correlation estimates is straightforward
- Widely understood and taught, making results easy to communicate and audit
- Scales cleanly to very large portfolios once the covariance matrix is built
Disadvantages and Limitations
1. It assumes returns are normally distributed
Real financial returns typically have “fat tails” — extreme moves happen more often than a normal distribution predicts. This means the Parametric method can understate the true probability of large losses, particularly during market stress.
2. It struggles with non-linear instruments
Options and other instruments with non-linear payoffs don’t respond to market moves in a simple, proportional way. The Parametric method’s reliance on standard deviation and covariance — both linear measures — makes it a poor fit for portfolios with significant options exposure.
3. The covariance matrix can become unstable
Correlations between assets are not fixed — they shift over time and often spike toward 1 during a crisis (assets that normally move independently start falling together). A covariance matrix built from calm historical data can understate risk precisely when diversification is needed most.
4. Sensitive to the historical window chosen
A longer data window smooths out noise but may miss a genuine recent shift in volatility; a shorter window reacts quickly to new conditions but is noisier. There is no universally correct choice, and different windows can produce meaningfully different VaR figures for the same portfolio.
When Should You Use Parametric VaR?
The Parametric method is best suited to portfolios of straightforward, linear instruments — equities, bonds, currencies — where speed and simplicity matter and a normal-distribution approximation is unlikely to be badly wrong. It is less suitable for portfolios with significant options or other derivatives exposure, or for institutions that need to capture fat-tail risk precisely — cases better served by the Historical Simulation or Monte Carlo methods, which we cover in the next two articles of this series.
https://decode-finance.com/category/market-risk-credit-risk-and-operational-risks