Build in Correlation VS Auto Correlation
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Aleksander Hansen
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Not sure if I'm reading your question correctly, but I'll take a shot:
Correlation:
1. Bootstrapping, which is just random draws, with replacement, from say historical returns data, will produce returns data with correlations that may be representative for the returns data over the horizon. That is, the historical returns implicitly define the correlations, and when drawing from this sample of historical returns you obtain some correlation.
2. Contrast this with a Monte Carlo simulation, where you have to specify the correlation structure explicitly using such as the Cholesky decomposition for a simple Geometric Brownian motion.
[Strictly speaking, if you simulate with MC using copulas, this would not be the case, though you would still have to go trough the trouble of generating the copulas (marginal distribution + copula = joint distribution) for each returns series; think of the copula as what glues them together in one sense]
Autocorrelation:
3. Since bootstrapping involves random draws from the, e.g., historical returns series, it will fail to capture autocorrelation (if it is iid there is no autocorrelation though).
Why? Because the draws are random rather than consecutive. Autocorrelation means that today's return depends on yesterday's return, however if you draw the returns at random you will not be able to capture this as the series you draw might look like: [return2, return5, return30, return 7...]. Clearly it would no longer make any sense to look for autocorrelation (how would you interpret that return7 depends on return30? Obviously that would be absurd)
4. Similarly, for Monte Carlo, you would have to estimate and then specify any autocorrelation structure for your simulation - it's not automatically built in. [in the case of copulas you would solve this be frequent recalibration]
Correlation:
1. Bootstrapping, which is just random draws, with replacement, from say historical returns data, will produce returns data with correlations that may be representative for the returns data over the horizon. That is, the historical returns implicitly define the correlations, and when drawing from this sample of historical returns you obtain some correlation.
2. Contrast this with a Monte Carlo simulation, where you have to specify the correlation structure explicitly using such as the Cholesky decomposition for a simple Geometric Brownian motion.
[Strictly speaking, if you simulate with MC using copulas, this would not be the case, though you would still have to go trough the trouble of generating the copulas (marginal distribution + copula = joint distribution) for each returns series; think of the copula as what glues them together in one sense]
Autocorrelation:
3. Since bootstrapping involves random draws from the, e.g., historical returns series, it will fail to capture autocorrelation (if it is iid there is no autocorrelation though).
Why? Because the draws are random rather than consecutive. Autocorrelation means that today's return depends on yesterday's return, however if you draw the returns at random you will not be able to capture this as the series you draw might look like: [return2, return5, return30, return 7...]. Clearly it would no longer make any sense to look for autocorrelation (how would you interpret that return7 depends on return30? Obviously that would be absurd)
4. Similarly, for Monte Carlo, you would have to estimate and then specify any autocorrelation structure for your simulation - it's not automatically built in. [in the case of copulas you would solve this be frequent recalibration]
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