Historical simulation and historical value at risk

What is historical simulation?

Historical simulation takes the market moves that actually happened on each past day and applies them to the positions you hold today. With a year of daily data you get about 250 scenarios: “what today’s portfolio would have made or lost on each of those days”.

The implicit assumption of historical simulation is that past price changes are a reasonable guide to future ones. The method treats the past data as the whole population of possible outcomes.

No distribution is assumed, so whatever the data contain is kept: fat tails (excess kurtosis), skew and volatility clustering (calm and turbulent spells). Each past date is used once, as it happened. There is no random draw and no replacement.

Every result is tied to a date. “The portfolio would lose EUR 610,000 if markets moved as they did on day 202” is a sentence a board understands. That explainability is the method’s best-known strength.

The inputs are the daily log returns for shares, $\tilde r_t = \ln(P_t \div P_{t-1})$, and, for bonds, changes in benchmark yields (see below).

How do you turn a past day into today’s profit or loss?

Use it

Multiply each position by the percentage change that day’s log return implies, and add the positions. The result is that day’s scenario profit and loss , written P/L.

$$\text{P/L} = D\left(e^{\tilde r} - 1\right)$$

Here $D$ is today’s position value and $\tilde r$ the day’s log return. The exponential turns a log return back into a percentage change. Multiplying the position by the log return itself is a small but common slip.

How do you read VaR from the ranked results?

Use it

Sort the scenario results from the worst loss upward. Historical value at risk (VaR) for a confidence level $c$ and $n$ scenarios sits at rank $(1 - c) \times n$ from the worst. For 95% that is rank $0.05 \times n$.

$$k = (1 - c)\,n$$

If the rank is not a whole number, either take the nearer neighbor or use rank and linear interpolation : take a weighted mix of the two neighboring losses, with weights that follow the fractional part of the rank.

$$\text{VaR} = L_{\lfloor k \rfloor} + (k - \lfloor k \rfloor)\left(L_{\lceil k \rceil} - L_{\lfloor k \rfloor}\right)$$

Here $L_j$ is the $j$th largest loss. In plain words: at rank 12.6, take 40% of the 12th loss and 60% of the 13th.

Taking the 12th or the 13th loss without interpolating (EUR 226,201 or EUR 224,432) is also accepted practice. The options in a question will show which rule is meant.

How does it compare with parametric VaR?

Parametric value at risk assumes normal returns and uses only the standard deviations and the correlation. Historical VaR uses the actual tail. When returns have fat tails, the two agree near the middle and part company far out in the tail.

For the same portfolio, the daily standard deviations are 1.34% for fund A and 1.79% for fund B, with a correlation of 0.67 (rounded from the data). In euros, the portfolio’s standard deviation combines the two positions as in Learning Module 8.

$$\sigma_p = \sqrt{D_1^2\sigma_1^2 + D_2^2\sigma_2^2 + 2D_1D_2\rho\,\sigma_1\sigma_2}$$

• Position risks: $D_A\sigma_A$ = 6,000,000 × 0.0134 = 80,400 and $D_B\sigma_B$ = 4,000,000 × 0.0179 = 71,600.
• Variance: 80,400² + 71,600² + 2 × 0.67 × 80,400 × 71,600 = 6,464,160,000 + 5,126,560,000 + 7,713,897,600 = 19,304,617,600, so $\sigma_p$ = EUR 138,941.
• Parametric 95% VaR = 1.645 × 138,941 = EUR 228,558. Parametric 99% VaR = 2.33 × 138,941 = EUR 323,733. These are one-tailed z-values, as in Learning Module 7.

	95%	99%
Historical	225,140	418,261
Parametric (normal)	228,558	323,733

At 95% the two are within 2% of each other. At 99% the history shows a loss about 29% larger than the normal curve allows. The worst day, EUR 609,595, is 4.4 standard deviations of the portfolio’s daily result. A normal curve gives such a day about once in 175,000 days, yet it happened once in a single year of data.

Parametric VaR is normally computed with a mean daily return of zero, the zero-mean convention . A zero mean usually cannot be statistically rejected for daily data, and the daily standard deviation is far larger than the daily mean, so best practice leaves drift out of the parametric calculation. Historical simulation, by contrast, automatically contains whatever drift the data had (here the average daily P/L was −EUR 484, close to zero).

Exam tip A positive average return in the history does not tell you which VaR is larger. VaR lives in the loss tail, and a few large gains do not move it.

Not just VaR: replaying past forecast errors

Know the idea

The same idea works for any quantity with a track record. A private equity team has 40 past deals and, for each, the gap between forecast and actual revenue. Sorted, the five worst gaps are −31.2%, −24.5%, −20.1%, −18.4% and −15.9%. The 10th percentile sits at rank 0.10 × 40 = 4: −18.4%.

If the new deal’s base-case net present value (NPV) is USD 2,000,000 and each 1% revenue shortfall cuts it by USD 60,000, the 10th-percentile case is 2,000,000 − 18.4 × 60,000 = USD 896,000. History tempers over-optimistic forecasts: the downside comes from what actually happened, not from the model’s hopes.

What can go wrong with the data?

Know the idea

History is rarely complete. Asynchronous markets trade on different days and close on different holidays (or for unplanned reasons). Some securities did not exist for the whole window. Thinly traded securities have sparse or stale prices. The usual fix is a proxy : a similar instrument, or a weighted mix of several, stands in for the missing history.

Data gapswhat can go wrong with the history

• Asynchronous markets different holidays and trading days, unplanned closures
• Securities that did not exist listed or issued after the window starts
• Thinly traded securities sparse or stale prices
• Fix: a proxy Use it a similar instrument, or a weighted mix of several, stands in

Bonds need special care. Do not replay a bond’s own past price changes. A bond maturing next week would inherit last year’s price swings, when today it has almost no price risk left. Instead take the benchmark yield changes (for example the 1-month, 6-month, 1-year and 2-year yields) or changes in zero-coupon prices, apply them to today’s yield curve and reprice the bond’s remaining cash flows (CF): present value = $\sum CF_t \div (1 + y_t)^t$.

A two-year history of a bond that now matures in three weeks would overstate its risk. A bond that has only existed for a month gives too few scenarios, so a proxy is needed.

Strengths and weaknesses

The strengths and weaknesses of historical simulation follow from one fact: it replays the data and nothing else.

Historical simulationreplay the past on today’s positions

• Strengths Memorize
  • Simple replays history; still used for complex securities
  • No distribution assumed fat tails and skew kept
  • Real-world data includes past extremes
  • No correlation model co-movements kept as they happened
  • Low computing cost no new scenarios generated
• Weaknesses Memorize
  • Backward looking blind to shocks history never showed
  • Limited scenario variety only the window you chose
  • Data limitations missing, short or stale histories; new instruments

Because every day counts once, a single extreme day in the window moves the result a lot: historical simulation is sensitive to outliers and to the window you pick. It handles non-linear positions such as options without difficulty, since each scenario simply reprices them.