Simulation Fundamentals

The starting point is a matrix of historical time series $D \in R^{T \times N}$ and associated probability vector $p \in R^{T}$ . The time series realizations can include any variable of interest, e.g., asset prices, macro variables, and risk factors.

To simulate new synthetic paths for these variables, we likely have to perform some data transformation to get better statistical properties, i.e., the variables that we project must be stationary in some way, so we can learn from the data. We call these transformed stationary time series $S T$ , and denote the transformation $f : D \mapsto S T \in R^{\tilde{T} \times \tilde{N}}$ .

Once we have a stationary projection $\tilde{S T} \in R^{S \times \tilde{N} \times H}$ of $S$ samples with horizon $H$ , we can compute a simulation for the variables of interest $g : \tilde{S T} \mapsto R \in R^{S \times N \times H}$ .

Computing good stationary data is by no means trivial, and one likely has to have some knowledge about the nature of the time series to know which transformations are helpful for achieving stationarity and preserving signal.

Another important point about the transformation is that we should be able to accurately invert it, i.e., we must be able to accurately compute both $f$ and $g$ . For most common investment time series, we have some suggestions for transformations one can make, but there is full flexibility in relation to these as long as they achieve the desired outcomes (stationarity and signal preservation). We note that when using our suggested transformations, the transformed data $S T$ will have the same number of columns and one less row than the historical time series $D$ , i.e. $\tilde{T} = T - 1$ and $\tilde{N} = N$ .

Note that there is a third dimension $H$ to the synthetic paths. Hence, the simulations are not restricted to having the same horizon as the original time series, but this would be equivalent to setting $H = 1$ . If one wishes to work with an intermediate horizon $1 \leq h \leq H$ , one can select the desired horizon and work with $R_{h} \in R^{S \times N}$ .

For further details, see the Portfolio Construction and Risk Management book.