Bootstrap: future scenarios from your own history

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The bootstrap is a method for generating many possible paths from a single observed history: you draw past returns at random and with replacement and put them together into new paths. This preserves the true distribution of returns, fat tails included, without having to assume a normal distribution. The block bootstrap draws entire stretches and thereby also preserves relationships over time.

The idea: drawing with replacement

The name comes from the expression about pulling yourself out of a swamp by your own bootstraps: the method derives statements about uncertainty from the data itself. Efron (1979) introduced it as a general method for determining the distribution of an estimator without deriving it mathematically.

For investors, the bootstrap matters above all in Monte Carlo simulation. Instead of rolling the dice for monthly returns from an assumed normal distribution, you take a portfolio's real monthly returns and draw them again in random order. Every simulated month is then a month that actually happened, including a crash month such as October 2008.

The simple bootstrap step by step

Suppose a portfolio had six monthly returns: 3%, −2%, 1%, 4%, −5% and 2%. Together they come to 2.74% over six months, an average of 0.50% a month. For a new path, you draw a month number between 1 and 6 six times. Each number may occur more than once.

With 2,000 or 10,000 such paths over 30 years, you get a distribution of possible final wealth. Its percentiles show what could come out in an unfavorable and in a favorable case. In the process, the bootstrap preserves the shape of the distribution: the skewness and fat tails of the real returns are retained, unlike with a normal distribution.

Block bootstrap: relationships over time

The simple bootstrap treats each month as independent. Real returns are not quite independent: calm and turbulent phases alternate (volatility clustering), and crashes often stretch over several months. Cont (2001) counts this clustering among the stylized facts of financial returns. A bootstrap that shuffles months individually tears such phases apart: long downturns such as 2000 to 2002 occur less often in the simulation than in reality.

The block bootstrap following Kunsch (1989) therefore draws entire blocks of consecutive observations. Within a block, the sequence is preserved; only the blocks are reshuffled. For the method to work in theory, the block length has to grow with the amount of data, but more slowly than the data.

This wrapping around past the end is called the circular block bootstrap. Without it, months at the beginning and end of the history would occur less often than months in the middle, because fewer blocks contain them.

Advanced: the stationary bootstrap

Fixed blocks have a blemish: the generated path is “broken” at the block boundaries, and its statistical properties depend on where within a block you are. Politis and Romano (1994) therefore proposed the stationary bootstrap: the block length is itself random, following a geometric distribution. After each month, a new block is started at a random position with probability p; otherwise, the path continues with the next month.

With p = 1/12, the blocks are twelve months long on average; with a probability of 12.4%, a block is longer than 24 months. The path generated in this way is stationary, and the results are less sensitive to the exact choice of block length than with fixed blocks.

Several assets at once

If you simulate several assets or portfolios, you draw the same months for all of them. This preserves the correlations between them, including in crisis months when almost everything falls at the same time. If you drew separate months for each asset, the relationships would disappear, and the simulation would overstate diversification.

Bootstrapping metrics

The bootstrap is also suitable for estimating the uncertainty of a metric: you draw many samples from the history, calculate for each one, for example, the Sharpe ratio, and read off a range from their dispersion. Efron and Tibshirani (1993) describe the methods for such confidence intervals in detail.

Reading the results correctly

A bootstrap simulation does not deliver a forecast but a range. Percentiles are customary: the 10% percentile of final wealth is the value that nine out of ten paths exceed; the median is the middle path. For withdrawal plans, the success rate also counts, i.e. the share of paths in which the money lasts until the end.

Two subtleties are often overlooked. First, the order of the returns makes no difference to final wealth as long as no money is added or withdrawn: a product stays the same however you arrange its factors. The dispersion of the paths then arises only because some months are drawn several times and others not at all. Second, the order becomes decisive as soon as money flows in or out: a crash shortly after withdrawals begin hits much harder than the same crash at the end. It is precisely this sequence-of-returns risk that the simulation makes visible.

As for the number of paths: more paths make the percentiles more precise but change nothing about the assumptions. With 2,000 paths, the random error (standard error) of a success rate of around 90% is below one percentage point. That is enough to compare plans. The greater uncertainty lies in the history you choose.

Bootstrap or normal distribution?

No method generates the future. The bootstrap repeats the past in a new order: a month worse than the worst in the history never occurs, and a history from an exceptionally good phase leads to optimistic results. Parametric methods, by contrast, allow deliberately more cautious assumptions. It often makes sense to run both side by side.

Limitations and common mistakes

• Short history: five years of monthly data contain only 60 different months. The simulation can only vary what these months contain.
• Past return as the future: the bootstrap adopts the average return of the history. If the history was a boom (such as equities from 2010 to 2021), the future is painted too rosy.
• Block length: blocks that are too short lose relationships; blocks that are too long repeat entire years almost unchanged and narrow the range.
• Structural breaks: interest rate levels, inflation and valuations change. A history of falling interest rates says little about bonds when rates are rising.

How Portfolio Lab calculates

The Monte Carlo simulation offers four methods: “Historical (bootstrap)” draws the portfolio's monthly returns individually with replacement (the default), and “Historical (block bootstrap, 12 months)” draws blocks of twelve consecutive months with a random starting month and wraps around from the end of the history to the start. In addition, there are the normal distribution and a Student-t distribution with five degrees of freedom, scaled to the same volatility. There is no stationary bootstrap with a random block length.

Financial goals and ALM use the block bootstrap by default and draw the same months or blocks for all portfolios so that the relationship is preserved. At least twelve months of history are required. The random number generator starts from a fixed seed, so the same input always produces the same result. Details in the Monte Carlo methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.