Quick Recap: One Method Down, Two To Go
The previous article in this series covered the Parametric (Variance-Covariance) method — fast, formula-driven, but built on an assumption that returns follow a normal distribution. We flagged that this assumption is also its biggest weakness: real markets have “fat tails,” and a method that assumes a bell curve can understate the true chance of a severe loss.
This article covers the second method: Historical Simulation. It takes the opposite philosophy — instead of assuming a distribution, it lets actual history speak for itself.
What Is Historical Simulation VaR?
Historical Simulation calculates VaR by taking a portfolio’s actual historical returns and applying them directly to today’s position values, generating a full set of hypothetical “what if that day happened again” outcomes. VaR is then read directly off this set — no formula, no distributional assumption, just a straightforward count through history.
Where the Parametric method reduces an asset’s behaviour to two numbers (mean and standard deviation), Historical Simulation keeps every observed data point intact — including whatever unusual, extreme, or skewed patterns actually occurred.
The Core Idea: No Distribution Assumption
This is the method’s defining feature. It does not assume returns are normally distributed, symmetric, or well-behaved in any particular way. If the historical data contains a crash, a spike, or a fat tail, that event is captured exactly as it happened and will show up in the VaR calculation. This makes Historical Simulation especially useful for portfolios where the normal-distribution assumption is shaky — including those with options or other non-linear instruments.
The trade-off, as we’ll see in the disadvantages section, is that the method is entirely dependent on the specific historical window used — it can only “see” the kinds of events that happened to occur in your data.
https://en.wikipedia.org/wiki/Normal_distribution
The Historical Simulation Process — Three Steps
- Step 1 — Build hypothetical P&L. For each historical day, apply that day’s actual return to today’s current position value, producing a hypothetical rupee gain or loss: “if that day’s market move happened again tomorrow, what would I gain or lose?”
- Step 2 — Rank the outcomes. Sort every hypothetical P&L from worst (most negative) to best (most positive).
- Step 3 — Read off the threshold. VaR at a given confidence level is simply the loss sitting at the corresponding rank. With 100 historical observations, the 99% VaR is the 1st worst day; the 95% VaR is the 5th worst day.
The Historical Simulation VaR Formula
Unlike the Parametric method, there is no algebraic formula to solve — the “formula” is really a procedure. In Excel, it is implemented using the SMALL() function, which extracts the Nth smallest value from a range:
| Hypothetical Daily P&L = Historical Daily Return × Current Position Value VaR (Confidence Level) = -SMALL(P&L Range, Rank) Where Rank = N × (1 – Confidence Level) N = number of historical observations e.g. with N=100 at 99% confidence: Rank = 100 × 0.01 = 1st worst day e.g. with N=100 at 95% confidence: Rank = 100 × 0.05 = 5th worst day |
The negative sign converts a loss (a negative P&L) into a positive VaR figure, consistent with the convention used throughout this series.
Worked Example: Single-Asset Historical VaR
This example follows exactly the same numbers used in the accompanying Excel workbook (Sheet ‘1. Single-Asset Historical VaR’), using 100 days of illustrative daily prices for a single position.
Step 1 — Gather the data and build hypothetical P&L. We start with 101 days of daily closing prices (illustrative sample data), producing 100 daily returns. Each day’s return is multiplied by the position value to get that day’s hypothetical P&L.
| Statistic | Value |
| Position Value | ₹50,00,000 |
| Number of historical observations | 100 |
| Mean daily return (reference only) | 0.3062% |
| Daily volatility (reference only) | 1.6625% |
Note that the mean and standard deviation are shown here for reference and comparison only — Historical Simulation does not use them in the VaR calculation itself. This is a key structural difference from the Parametric method.
Step 2 — Rank all 100 hypothetical P&L outcomes from worst to best. The ten worst days looked like this:
| Rank | Hypothetical P&L (₹) |
| Worst #1 | -1,69,309 |
| Worst #2 | -1,59,452 |
| Worst #3 | -1,55,834 |
| Worst #4 | -1,34,497 |
| Worst #5 | -1,15,379 |
| Worst #6 | -1,08,651 |
| Worst #7 | -95,867 |
| Worst #8 | -90,758 |
| Worst #9 | -88,892 |
| Worst #10 | -88,738 |
| Worked Example — Single Asset 99% Confidence: Rank = 100 × (1 – 0.99) = 1st worst day 1-Day VaR (99%) = -SMALL(P&L range, 1) = ₹1,69,309 95% Confidence: Rank = 100 × (1 – 0.95) = 5th worst day 1-Day VaR (95%) = -SMALL(P&L range, 5) = ₹1,15,379 Reading this: under the conditions actually observed in our 100-day history, there is a 99% chance this position would not have lost more than ₹1,69,309 in a single day. | |
Cross-Checking with PERCENTILE
The workbook also calculates VaR using Excel’s PERCENTILE() function as a cross-check, and the two methods give slightly different answers:
| Method | 99% VaR | 95% VaR |
| -SMALL() — discrete order statistic | ₹1,69,309 | ₹1,15,379 |
| -PERCENTILE() — interpolated | ₹1,59,550 | ₹1,08,987 |
The difference comes down to convention. SMALL() picks an actual observed data point — the literal Nth worst day that happened. PERCENTILE() interpolates between two neighbouring data points to estimate a smoother threshold, which is why its result sits between adjacent ranked observations rather than landing exactly on one of them. Both are used in practice; SMALL() is more common because it ties VaR to an actual historical event that can be pointed to and explained (“this is what happened on day X”), which many risk committees find more intuitive and defensible than an interpolated number.
Extending to a Portfolio: Applying Historical Returns to Current Weights
For a multi-asset portfolio, the principle stays identical, but the P&L calculation must combine every asset’s historical return, applied to that asset’s current position value, on each historical day:
| Portfolio Hypothetical P&L (Day t) = (Return_A,t × Position Value_A) + (Return_B,t × Position Value_B) + (Return_C,t × Position Value_C) Portfolio VaR (Confidence Level) = -SMALL(Portfolio P&L Range, Rank) |
This is a crucial methodological point: each day’s portfolio P&L uses that day’s actual historical returns for every asset simultaneously — preserving whatever correlation or co-movement genuinely existed between the assets on that specific day. Unlike the Parametric method, there is no separate covariance matrix to build; the historical data already embeds the relationships between assets.
Worked Example: 3-Asset Portfolio Historical VaR
This example matches the accompanying Excel workbook exactly (Sheet ‘2. Portfolio Historical VaR’), using the same illustrative 3-asset portfolio structure as the Parametric VaR article, now recalculated using 100 days of historical data.
| 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 1 — Build the portfolio’s daily hypothetical P&L by combining all three assets’ returns on each historical day against current position values, then rank the resulting 100 portfolio outcomes.
| Rank | Portfolio Hypothetical P&L (₹) |
| Worst #1 | -2,72,784 |
| Worst #2 | -2,58,989 |
| Worst #3 | -2,47,311 |
| Worst #4 | -2,32,540 |
| Worst #5 | -2,20,624 |
| Worked Example — 3-Asset Portfolio 99% Confidence: 1-Day Portfolio VaR = -SMALL(Portfolio P&L range, 1) = ₹2,72,784 95% Confidence: 1-Day Portfolio VaR = -SMALL(Portfolio P&L range, 5) = ₹2,20,624 10-Day Portfolio VaR (99%), √10 scaling = ₹2,72,784 × √10 = ₹8,62,619 | |
Diversification Benefit Revisited
As with the Parametric method, we can check how much diversification is helping by comparing the sum of each asset’s own worst day against the portfolio’s actual worst day:
| Asset | Individual 1-Day VaR (99%) |
| Equity A | ₹1,73,199 |
| Equity B | ₹1,99,143 |
| Bond Fund C | ₹27,329 |
| Sum (Undiversified VaR) | ₹3,99,670 |
The properly calculated portfolio VaR is ₹2,72,784 — ₹1,26,886 (31.8%) lower than the undiversified sum of ₹3,99,670. There’s a subtlety worth calling out here that doesn’t arise in the Parametric method: each asset’s *individual* VaR above uses that asset’s own single worst day, which may not be the same calendar day as the portfolio’s worst combined day. This timing mismatch is itself part of the diversification story — an asset’s worst day often isn’t as damaging to the whole portfolio if the other assets weren’t having a bad day at the same time.
Scaling to Multiple Days: Two Approaches
The previous article used the square-root-of-time rule to scale a 1-day VaR to a 10-day VaR. That shortcut is a Parametric-method convention — it assumes returns are independent day to day, which is a normal-distribution-style assumption. Historical Simulation offers a genuinely more rigorous alternative: build actual 10-day cumulative returns directly from history, using overlapping windows, and apply the same ranking procedure to those.
| 10-Day Cumulative Return (Window starting Day t) = [(1+Return_t) × (1+Return_t+1) × … × (1+Return_t+9)] – 1 10-Day Hypothetical P&L = 10-Day Cumulative Return × Position Value True 10-Day VaR (Confidence Level) = -SMALL(10-Day P&L Range, Rank) |
| Worked Example — True 10-Day VaR vs √10 Approximation From 100 daily observations, there are 91 overlapping 10-day windows (Day 0→10, Day 1→11, … Day 90→100). True 10-Day VaR (99%), from overlapping windows = ₹2,50,852 √10-Scaled 10-Day VaR (99%), from the 1-day figure = ₹1,69,309 × √10 = ₹5,35,402 These two numbers differ substantially — the √10 approximation here is more than double the historically observed 10-day loss. |
This gap is a genuinely important, and often overlooked, lesson. The square-root-of-time rule assumes daily losses compound independently in the worst-possible sequence, which tends to overstate risk when a real 10-day worst-case window actually occurred with some daily moves offsetting others. The reverse can also happen — during a sustained crisis, real losses can compound in the same direction for many days running, making the √10 shortcut *understate* risk instead. Because the true relationship depends entirely on the data, this is exactly the kind of question Historical Simulation is well suited to answer directly, rather than assuming an answer through a formula. One caveat: overlapping windows reuse the same daily observations multiple times, so the 91 window-level outcomes are not independent of each other — treat this as a genuinely more data-driven estimate, not a statistically “clean” one.
Advantages of Historical Simulation
- No distributional assumption — captures fat tails, skewness, and any unusual pattern actually present in the data
- Simple to explain: VaR corresponds to an actual historical event, not an abstract statistical construct
- Naturally captures whatever correlation existed between assets on each historical day, without needing a separate covariance matrix
- Handles non-linear instruments (like options) more accurately than the Parametric method, since actual historical payoffs can be used directly
- Easy to audit — a risk manager can point to the specific historical day driving the VaR figure
Disadvantages and Limitations
1. Entirely dependent on the historical window chosen
If the chosen history doesn’t contain a severe event, the method won’t predict one. A calm 100-day window will produce a reassuringly low VaR that may understate the portfolio’s true risk the moment market conditions change.
2. “This time is different” risk
Markets evolve — new instruments, new regulations, new macroeconomic regimes. Historical Simulation implicitly assumes the future will resemble the past; structural shifts (a new product, a new competitor, a regime change) won’t be reflected in historical returns at all.
3. Requires a substantial amount of clean historical data
Unlike the Parametric method, which can be estimated from a relatively small sample, meaningful tail percentiles need a long history — a 99% VaR from only 100 observations rests on just a single data point (the worst day). Institutions typically use 250 to 500+ days precisely to make that tail estimate more robust.
4. Every historical observation is weighted equally
A market move from 300 days ago counts exactly as much as yesterday’s move, even though more recent conditions may be far more relevant to today’s risk. (A variant called Weighted Historical Simulation addresses this by giving more recent observations greater weight — a refinement beyond the scope of this article.)
5. Ghosting effect
A single extreme historical event can dominate the VaR figure for the entire length of the data window used, then abruptly disappear from the calculation the day it rolls out of the window — causing VaR to jump for no reason connected to current market conditions.
Historical Simulation vs Parametric: When to Use Which
| Consideration | Favours Parametric | Favours Historical Simulation |
| Speed of calculation | ✓ | |
| Small data sample available | ✓ | |
| Portfolio has options/non-linear instruments | ✓ | |
| Concerned about fat tails / extreme events | ✓ | |
| Need an explainable, auditable VaR figure | ✓ | |
| Returns genuinely close to normally distributed | ✓ |
In practice, many institutions calculate VaR using more than one method and compare the results — a material divergence between methods is itself a useful risk signal, prompting a closer look at what’s driving the difference.
https://decode-finance.com/category/market-risk-credit-risk-and-operational-risks