Автор: Kell

# Moving block bootstrap method of investing

**PLUS AND MINUS IN BETTING SPREAD**

There's no single solution, but a critical piece of the analysis for estimating return and risk, including the potential for drawdowns and fat tails , is generating synthetic performance histories with a process called bootstrapping. The idea is based on simulating returns by drawing on actual results to see thousands of alternative histories to consider how the future may unfold.

The dirty little secret in this corner of Monte Carlo analysis is that there's more than one way to execute bootstrapping tests. To cut to the chase, block bootstrapping is a superior methodology for asset pricing because it factors in the reality that market returns exhibit autocorrelation.

The bias for momentum - positive and negative - in the short run, in other words, can't be ignored, as it is in standard bootstrapping. There's a tendency for gains and losses to persist - bear and bull markets are the obvious examples, although shorter, less extreme runs of persistence also mark the historical record as well.

Conventional bootstrapping ignores this fact by effectively assuming that returns are independently distributed. They're not, which is old news. The empirical literature demonstrates rather convincingly a strong bias for autocorrelation in asset returns. Designing a robust bootstrapping test on historical performance demands that we integrate autocorrelation into the number crunching to minimize the potential for generating misleading results.

The key point is recognizing that sampling historical returns for analysis should focus on multiple periods. Let's assume that we're looking at monthly performance data. A standard bootstrap would reshuffle the sequence of actual results and generate alternative return histories - randomly, based on monthly returns in isolation from one another. That would be fine if asset returns weren't highly correlated in the short run.

But as we know, positive and negative returns tend to persist for a stretch, sometimes in the extreme. The solution is sampling actual histories in blocks of time in this case several months to preserve the autocorrelation bias. The question is how to choose the length for the blocks, along with some other parameters. Much depends on the historical record, the frequency of the data, and the mandate for the analysis. There's a fair amount of nuance here. Fortunately, R offers several practical solutions, including the meboot package "Maximum Entropy Bootstrap for Time Series".

To make this illustration clear in the charts, we'll ignore the basic rules of bootstrapping and focus on a ridiculously short period: the 12 months through March If this was an actual test, I'd crunch the numbers as far back as history allows, which runs across decades. I'm also generating only ten synthetic return histories; in practice, it's prudent to create thousands of data sets.

Figure 3. Example of circular block bootstrap. The solution proposed by Politis and Romano visually consists in wrapping the observed sample of data around in a circle 7. Stationary Block Bootstrap The stationary block bootstrap method, also introduced by Politis and Romano 15 , uses as resampling scheme a uniform sampling with replacement of circular overlapping blocks of random length.

This bootstrap method is illustrated in Figure 4. Figure 4. Example of stationary block bootstrap. How to choose the block length? It has been shown 2 both theoretically and empirically that the performances of block bootstrap methods depend significantly on the chosen block length In order to choose an appropriate block length, it is possible to refer to: Theoretical results on the asymptotic optimal block length that have been established for example by Hall et al.

In Portfolio Optimizer, the default block length resp. When the bootstrap method fails Politis et al. Below are two examples of situations in which bootstrap methods can fail. First, bootstrap methods are fundamentally dependent on the quality of the observed sample of data. If this sample is not a reasonable representation of the underlying population too small sample, biased sample, sample with an incorrect dependence structure… , the sampling distribution of any statistic of interest computed on the bootstrap samples will not be accurate.

Second, bootstrap methods are known to be inconsistent in some specific cases. For example 21 , in the case of the i. The sampling distribution of the mean of heavy tailed random variables converges to a random c.

As a side note, Portfolio Optimizer allows to generate bootstrap samples of any number of observations. Example of usage - Financial planning Since the seminal paper Determining Withdrawal Rates Using Historical Data of Bengen 23 , financial planning using computer-based simulations, also known as Monte Carlo simulations, has become standard practice, c.

Although the future is impossible to predict, such simulations help to set realistic expectations about the potential performances of financial portfolios over long horizons and under real-life constraints 25 e. As an illustration of how Portfolio Optimizer could be integrated into a financial planning software, I will use the methodology of Anarkulova et al.

The results of step 2, that is, 10, projected cumulative wealth paths over months, are illustrated in Figure 5. Figure 5. Example of Monte Carlo simulation of asset returns bootstrap with Portfolio Optimizer. For the sake of comparison, the results of an arbitrary Monte Carlo simulation performed by the Moneytree financial planning solution, taken from a presentation of Moneytree, are illustrated in Figure 6.

Figure 6.

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