Historical simulation

What Is Historical Simulation?

Historical simulation is a non-parametric approach used in financial risk management to estimate potential future losses of a portfolio based on actual past market movements. Unlike parametric methods that assume specific statistical distributions for asset returns, historical simulation directly uses a defined period of historical data to construct a distribution of potential future outcomes. This method is widely employed to calculate Value at Risk (VaR) and Expected Shortfall (ES), providing insights into the magnitude of losses that could be experienced under adverse market conditions. Its reliance on observed data makes it a straightforward and intuitive risk management technique, particularly for complex portfolios where underlying asset return distributions may not be easily modeled.

History and Origin

The concept of using historical data to forecast potential future outcomes has been implicitly present in financial analysis for a long time. However, the formalization and widespread adoption of historical simulation as a quantitative risk measure gained significant traction in the 1990s, particularly with the rise of Value at Risk (VaR). VaR models were developed as a way to aggregate risk positions across various product groups and asset classes, providing a single number to senior management indicating the potential loss a firm could face during a "bad trading day."¹⁶

The Basel Committee on Banking Supervision, an international body that sets standards for bank regulation, played a pivotal role in the proliferation of VaR and, consequently, historical simulation. In 1996, the Basel Committee extended its original "Basel I" framework to allow banks to use their own internal models, including VaR, for calculating capital requirements for market risk.¹⁵ This regulatory endorsement spurred many financial institutions to develop their own VaR engines, with a considerable number opting for systems based on historical simulation due to its simplicity and direct use of empirical data.¹⁴ The method became a "gold standard" for a time, used to guide decisions by financial firms and regulators monitoring their soundness.¹³

Key Takeaways

• Historical simulation is a non-parametric method for estimating financial risk, directly using past market data.
• It is commonly used to calculate Value at Risk (VaR) and Expected Shortfall (ES).
• The method does not assume specific statistical distributions for asset returns, relying instead on empirical observations.
• Its simplicity and intuitive nature have made it popular in financial institutions, especially for complex portfolios.
• While transparent, historical simulation can be slow to react to sudden changes in market volatility and may underestimate tail risk.

Formula and Calculation

Historical simulation calculates Value at Risk (VaR) by observing the actual past changes in portfolio value over a specified historical period. There isn't a complex mathematical formula in the traditional sense, but rather a methodology to re-evaluate a current portfolio's value under historical scenarios.

To calculate VaR using historical simulation for a portfolio, the following steps are typically followed:

1. Define a historical look-back period (e.g., 250 days, 500 days).
2. Collect daily percentage changes for all assets in the portfolio over this period.
3. Apply each day's historical percentage changes to the current portfolio's market values. This creates a hypothetical portfolio value for each day in the look-back period, effectively simulating what the current portfolio's value would have been if that historical day's market movements were to recur today.
4. Calculate the hypothetical profit or loss (P/L) for each of these simulated days.
5. Rank these hypothetical P/L values from worst (largest loss) to best (largest gain).
6. Identify the VaR at a given confidence level. For example, a 99% VaR would correspond to the P/L at the 1st percentile of the ranked distribution (i.e., the 2.5th worst loss in a 250-day sample).

If we denote the portfolio value at time (t) as (V_t) and the historical daily returns of the assets in the portfolio as (\Delta p_i), where (i = 1, \dots, N) for (N) historical observations, the steps can be summarized as:

1. Calculate hypothetical portfolio values:
   [V_{t, i}^{\text{hypothetical}} = V_t \times (1 + \text{daily return based on historical day } i)]
   where the daily return for historical day (i) is applied to the current portfolio's composition.

2. Calculate hypothetical P/L:
   [\text{P/L}i = V{t, i}^{\text{hypothetical}} - V_t]

3. Sort the (\text{P/L}_i) values in ascending order.

4. The VaR at a confidence level of (1-\alpha) (e.g., 99% confidence means (\alpha = 0.01)) is the ((\alpha \times N))-th value in the sorted list. For example, for 250 observations and 99% VaR, it would be the (0.01 \times 250 = 2.5)-th observation. In practice, this often means taking the 3rd worst observation (rounding up).

The direct use of historical data in this financial modeling approach avoids assumptions about specific return distributions, capturing empirical fat tails and skewness often observed in financial markets.

Interpreting the Historical Simulation

Interpreting the results of a historical simulation, particularly a VaR figure, involves understanding what the number represents and its inherent limitations. A Value at Risk (VaR) number derived from historical simulation, such as "$10 million VaR at 99% confidence over a 1-day horizon," indicates that, based on the historical data used, there is a 1% chance the portfolio could lose $10 million or more over the next trading day. Conversely, there is a 99% chance that the loss will not exceed $10 million.

This interpretation emphasizes the potential maximum loss under "normal" market conditions, as defined by the historical period observed. It's a percentile-based measure, meaning it identifies a specific point in the distribution of historical returns. However, it does not provide information about losses beyond that threshold, known as "tail risk." For instance, if the actual loss exceeds the 99% VaR, historical simulation VaR doesn't tell you how much greater that loss might be. For insights into losses beyond the VaR threshold, Expected Shortfall is often used in conjunction with or as an alternative to VaR, as it represents the expected loss given that the VaR threshold has been breached.

Regulators and risk managers often look at VaR figures generated by historical simulation to assess a firm's exposure and determine adequate capital requirements. However, it is crucial to remember that the accuracy of the historical simulation relies heavily on the chosen historical period's relevance to future market conditions.

Hypothetical Example

Consider a simplified portfolio consisting of two assets: Stock A and Stock B.

Current Portfolio Value:

• Stock A: $500,000
• Stock B: $500,000
• Total Portfolio Value: $1,000,000

Objective: Calculate the 1-day, 99% VaR using historical simulation with the last 10 days of market data.

Historical Daily Returns (last 10 days):

Step-by-Step Calculation:

1. Calculate hypothetical daily P/L for the current portfolio for each historical day:

   • For each historical day, apply the corresponding asset returns to the current holdings of Stock A and Stock B.
   • Day 1: ( (500,000 \times 0.0050) + (500,000 \times 0.0020) = 2,500 + 1,000 = +3,500 )
   • Day 2: ( (500,000 \times -0.0120) + (500,000 \times -0.0080) = -6,000 - 4,000 = -10,000 )
   • Day 3: ( (500,000 \times 0.0080) + (500,000 \times 0.0110) = 4,000 + 5,500 = +9,500 )
   • Day 4: ( (500,000 \times -0.0030) + (500,000 \times 0.0060) = -1,500 + 3,000 = +1,500 )
   • Day 5: ( (500,000 \times 0.0150) + (500,000 \times 0.0070) = 7,500 + 3,500 = +11,000 )
   • Day 6: ( (500,000 \times -0.0200) + (500,000 \times -0.0150) = -10,000 - 7,500 = -17,500 )
   • Day 7: ( (500,000 \times 0.0010) + (500,000 \times -0.0020) = 500 - 1,000 = -500 )
   • Day 8: ( (500,000 \times -0.0070) + (500,000 \times -0.0040) = -3,500 - 2,000 = -5,500 )
   • Day 9: ( (500,000 \times 0.0100) + (500,000 \times 0.0090) = 5,000 + 4,500 = +9,500 )
   • Day 10: ( (500,000 \times -0.0090) + (500,000 \times -0.0100) = -4,500 - 5,000 = -9,500 )

2. Rank the hypothetical daily P/L values from worst to best:

   • -$17,500 (Day 6)
   • -$10,000 (Day 2)
   • -$9,500 (Day 10)
   • -$5,500 (Day 8)
   • -$500 (Day 7)
   • +$1,500 (Day 4)
   • +$3,500 (Day 1)
   • +$9,500 (Day 3)
   • +$9,500 (Day 9)
   • +$11,000 (Day 5)

3. Determine the 99% VaR. With 10 observations, a 99% confidence level means we are looking for the loss that is exceeded 1% of the time. For 10 observations, 1% of 10 is 0.1. Rounding up to the nearest whole number (or choosing the next worst observation if the percentile falls between two values), the 1st worst loss is -$17,500.

Therefore, the 1-day, 99% VaR for this portfolio, using this historical simulation, is $17,500. This means that based on the last 10 days of market performance, the portfolio is not expected to lose more than $17,500 on any given day 99% of the time. This hypothetical example highlights how historical data is directly used to estimate potential downside exposure, a key aspect of risk analysis.

Practical Applications

Historical simulation is a foundational technique with various practical applications in finance, primarily within the realm of financial risk management.

1. Market Risk Measurement: One of its most common uses is in calculating Value at Risk (VaR) and Expected Shortfall (ES) for trading portfolios. Banks and investment firms use these measures to quantify potential losses from adverse movements in financial markets over specific time horizons and confidence levels.¹²
2. Regulatory Compliance: Regulatory bodies, such as the Basel Committee on Banking Supervision and the Federal Reserve¹¹, have incorporated VaR into capital adequacy frameworks for financial institutions. While the frameworks have evolved to include more sophisticated methods and stressed scenarios, historical simulation remains a component or a benchmark for validating other models.
3. Stress Testing: Beyond standard VaR, historical simulation can be adapted for stress testing, where specific past periods of extreme market events (e.g., the 2008 financial crisis, the dot-com bust) are used as scenarios to assess a portfolio's resilience. This helps institutions understand their vulnerabilities under severe, but historically observed, conditions. The Federal Reserve also uses historical simulation models for certain supervisory purposes, such as projecting operational risk losses.¹⁰
4. Portfolio Management: Portfolio managers use historical simulation to understand the potential downside risk of their investment strategies. By evaluating the historical performance of similar portfolios, they can make informed decisions about asset allocation and hedging strategies to manage their risk exposure.
5. Risk Budgeting: Firms can allocate "risk budgets" to different trading desks or business units based on their historical loss profiles. Historical simulation provides the empirical data needed to set and monitor these limits, promoting disciplined risk-taking across an organization.
6. Model Validation: Even when more complex parametric models are used, historical simulation often serves as a benchmark for backtesting and validating their performance. By comparing the results of the complex model against the straightforward historical simulation, analysts can identify discrepancies and potential model deficiencies.

Limitations and Criticisms

While straightforward and intuitive, historical simulation has several notable limitations and has faced significant criticism, particularly in the wake of major financial crises.

One primary criticism is its inherent backward-looking nature. Historical simulation assumes that past market behavior is representative of future market behavior.⁹ This assumption can be problematic, especially during periods of rapid market change or unprecedented events. If the historical period used does not include extreme events, the model may significantly underestimate potential losses, failing to capture "tail risk" or unforeseen future scenarios.⁸ For example, many traditional Value at Risk models, including those based on historical simulation, were criticized for underestimating risk leading up to and during the 2007-2009 Global Financial Crisis because they did not adequately account for extreme, correlated movements that had not been observed in recent history.⁷,⁶,⁵

Another drawback is the "ghosting" effect. Older, but potentially relevant, extreme market events eventually drop out of the defined historical window, causing the risk measure to suddenly decrease, even if current market conditions remain fragile. Conversely, a single extreme event within the window can disproportionately inflate risk estimates for an extended period, leading to spurious volatility in the risk numbers.⁴

Furthermore, the method's reliance on a finite sample size can lead to large standard errors, especially when calculating risk at very high confidence levels (e.g., 99.9%). With limited historical data, the estimates of extreme quantiles (like the 99.9th percentile loss) can be unreliable.³ Research also suggests that historical simulation may systematically underestimate Expected Shortfall because observed sample datasets are finite and cannot capture the virtually infinite possible values of continuous probability distributions.²

Finally, historical simulation can be slow to adapt to changes in market volatility and correlations. It responds only when new data points enter the historical window, meaning it may not quickly reflect shifts in market dynamics or new derivatives that alter risk profiles. This "under-responsiveness" can lead to risk estimates that are too low when risk is actually increasing and too high when markets stabilize.¹

Historical Simulation vs. Monte Carlo Simulation

Historical simulation and Monte Carlo simulation are both simulation-based approaches used in quantitative finance to model potential future outcomes, particularly for risk measurement. However, they differ fundamentally in how they generate future scenarios.

The core distinction lies in how the "future" is envisioned. Historical simulation looks backward, effectively replaying past scenarios, while Monte Carlo simulation builds forward from assumed statistical properties, generating entirely new, plausible scenarios. While historical simulation offers the advantage of simplicity and the absence of explicit distributional assumptions, Monte Carlo simulation provides greater flexibility to explore a wider range of potential outcomes, especially those in the extreme tails of the distribution.

FAQs

What is the main advantage of historical simulation?

The main advantage of historical simulation is its simplicity and transparency. It does not require making assumptions about the underlying statistical distribution of asset returns, which can be complex and prone to error. Instead, it directly uses actual past market data, making its results intuitive and easy to understand for financial professionals and regulators. It inherently captures non-normal phenomena like "fat tails" and skewness if they were present in the historical period.

How far back should historical data be used for historical simulation?

The length of the historical look-back period is a critical decision. A shorter period (e.g., 250 days or one year of trading days) may be more responsive to recent market conditions but can suffer from high sampling error and might not include rare, extreme events. A longer period (e.g., 5 years) captures more data points, potentially including more extreme events, but may incorporate "stale" data that is no longer relevant to current market dynamics. There is a trade-off between capturing enough data for statistical validity and ensuring the data is still relevant, particularly regarding market volatility.

Can historical simulation predict black swan events?

No, historical simulation cannot predict "black swan events" (unforeseen, rare, high-impact events). By its nature, historical simulation relies solely on past observations. If an event has never occurred in the historical dataset, the model will not account for it. This is a significant limitation, as true black swan events are by definition outside historical experience. For this reason, historical simulation is often complemented by stress testing and other qualitative risk assessment methods to address scenarios not captured by historical data.

Is historical simulation still used by financial institutions today?

Yes, historical simulation is still used by financial institutions, though often in conjunction with, or as a component of, more sophisticated financial models. Regulators like the Basel Committee have moved towards more advanced risk measures like Expected Shortfall (ES) and stressed VaR (sVaR) that incorporate longer liquidity horizons and stress scenarios. However, historical simulation remains valuable for its transparency, ease of implementation, and as a benchmark for validating other models through backtesting. Its simplicity makes it a useful tool for understanding baseline risk exposures.