Historical simulation

The parametric method works out Value at Risk from a formula. It needs the returns to follow a normal, bell-shaped distribution. Real market returns often do not.

Historical simulation sidesteps the assumption. It asks a different question: what would today's portfolio have done in the past?

So the manager takes the portfolio as it stands, holds the weights fixed, and runs it through a long stretch of history — say five years of daily returns. That produces one return for every day in the window.

Then the returns are simply sorted, and the loss at the chosen percentile is read off. For a 5% VaR, the worst 5% of those days are what matter.

No bell curve is needed anywhere. The distribution is whatever history actually was.

The method (section 17.4.2.2). The portfolio's return is simulated assuming the same composition existed for a long period, say 5 years. The daily return — or weekly, or monthly — is calculated for each period. Those returns are arranged in descending or ascending order, and the 1%, 5%, 10% or any other value for which VaR is needed is calculated using the percentile approach. For a 5% VaR, the lowest 5 percentile values are the ones taken into consideration.

The result it tends to give. The workbook observes that VaR estimated using historical simulation is generally lower than the parametric method's estimate, because the true price change does not follow a normal distribution — which was a primary condition of the parametric method.

What it shares with the parametric method. In both cases, the portfolio weights are kept constant throughout the period of estimation and calculation. That is an assumption, and a strong one: the portfolio being tested against five years of history is today's portfolio, not the one actually held then.

Where it sits among the three methods. Section 17.4.2 gives parametric first, historical simulation second, and Monte Carlo simulation third. Monte Carlo also sorts its outcomes and reads a percentile, which the workbook says is similar to the historical simulation method — the difference is that Monte Carlo's returns are randomly generated rather than observed.

The limitation the exam asks for. Chapter 17's sample question 3 asks which of three statements is a limitation of historical simulation, and the answer is I only — the past may not repeat itself. The other two options, that the distribution is assumed to be normal and that mean-variance estimates can be biased, belong to the parametric method, not to this one.

Illustrative figures, following the workbook's method. A manager holds a Rs 100 crore portfolio and wants a one-day 95% VaR by historical simulation.

She takes 5 years of daily returns, which at about 250 trading days a year gives 1,250 observations, and applies each day's returns to today's weights. She then sorts all 1,250 simulated returns from worst to best.

The 5% percentile falls at the 62nd worst day (5% of 1,250 = 62.5). Suppose the return on that day is −1.74%.

95% one-day VaR = 1.74% of Rs 100 crore = Rs 1,74,00,000.

For a 99% VaR she reads the 12th worst day instead (1% of 1,250 = 12.5). If that day is −3.05%, the 99% VaR is Rs 3,05,00,000.

Compare this with the parametric route on the same portfolio. Using the workbook's own illustrative inputs — a 15% expected annual return and 20% annual standard deviation, giving a daily return of 0.06% and daily volatility of 1.26% — the parametric 95% VaR is 0.06% − (1.645 x 1.26%) ≈ 2.01%, or Rs 2,01,00,000.

The historical figure of Rs 1,74,00,000 is the lower of the two, exactly as the workbook says it generally will be.

And the weakness is visible in the same arithmetic. Her 1,250 days contain no event worse than −3.05%. If the next month delivers a −7% day, the model never suggested one was possible — because it never happened in the window she chose.

Chapter 17 (Risk), section 17.4.2.2 (Historical Simulation Method), is the second of the three VaR methods and is examined mostly against the other two.

The workbook's own sample question 3 asks for the limitation of the method — the past may not repeat itself — and deliberately offers the normal-distribution assumption as a distractor, because that belongs to the parametric method. Expect also a question on the percentile approach, on the five-year window, and on why the historical estimate usually comes out lower than the parametric one.

• Historical simulation assumes nothing about the distribution. The normality assumption is the parametric method's. Being offered normality as a limitation of historical simulation is the workbook's own trap, and the answer is no.
• Its real limitation is that the past may not repeat itself. A loss larger than anything in the window is invisible to the model.
• It usually gives a lower VaR than the parametric method, precisely because real returns are not normal.
• Weights are held constant. Today's portfolio is run through old market data; the portfolio actually held five years ago is irrelevant to the calculation.
• The percentile is taken from the loss tail. For a 5% VaR you want the worst 5% of observations, not the middle or the best.
• Do not confuse it with stress testing. Stress testing asks what one extreme scenario would do; historical simulation uses the whole observed distribution to read a percentile.