Annualized Returns

perfana.monte_carlo.returns.annualized_returns_m(data, weights, freq, geometric=True, rebalance=True)[source]

Calculates the annualized returns from the Monte Carlo simulation

The formula for annualized geometric returns is formulated by raising the compound return to the number of periods in a year, and taking the root to the number of total observations. For the rebalance, geometric returns, the annualized returns is derived by:

\[\begin{split}&y = M / s \\ &\frac{1}{N}\sum^N_i \left[\prod_j^T \left(1 + \sum^A_k (r_{ijk} \cdot w_k \right) \right]^{\frac{1}{y}} - 1\end{split}\]

where s is the number of observations in a year, and M is the total number of observations, N is the number of trials in the simulation, T is the number of trials in the simulation and A is the number of assets in the simulation.

For simple returns (geometric=FALSE), the formula for the rebalanced case is:

\[\frac{s}{NM} \left[\sum^N_i \sum^T_j \sum^A_k (r_{ijk} \cdot w_k) \right]\]
Parameters

  • data (ndarray) – Monte carlo simulation data. This must be 3 dimensional with the axis representing time, trial and asset respectively.

  • weights (Union[Iterable[Union[int, float]], ndarray, Series]) – Weights of the portfolio. This must be 1 dimensional and must match the dimension of the data’s last axis.

  • freq (Union[str, int]) – Frequency of the data. Can either be a string (‘week’, ‘month’, ‘quarter’, ‘semi-annual’, ‘year’) or an integer specifying the number of units per year. Week: 52, Month: 12, Quarter: 4, Semi-annual: 6, Year: 1.

  • geometric (bool) – If True, calculates the geometric mean, otherwise, calculates the arithmetic mean.

  • rebalance (bool) – If True, portfolio is assumed to be rebalanced at every step.

Returns

Annualized returns of the portfolio

Return type

float

Examples

>>> from perfana.datasets import load_cube
>>> from perfana.monte_carlo import annualized_returns_m
>>> cube = load_cube()[..., :7]
>>> weights = [0.25, 0.18, 0.13, 0.11, 0.24, 0.05, 0.04]
>>> annualized_returns_m(cube, weights, 'month')
0.02111728739277985