Estimating Compounded Returns with Simulation and Characteristic Functions
Summary
The document considers how to obtain a distribution for returns accumulated across multiple days when daily returns are modeled with a heavy-tailed Student’s t distribution. It presents two general methods for sums of independent continuous random variables, which can support analysis of log returns or other additive quantities related to compounding.
The first method is Monte Carlo simulation: repeatedly draw from the daily distribution, combine observations over the desired horizon, and estimate quantities such as quantiles. The answer cautions that estimates need enough simulations, especially for distributions with poorly behaved or undefined moments. The second method multiplies the daily characteristic functions to represent a sum, then numerically inverts the result to recover a density or distribution. Numerical inversion can be demanding when the distribution is not smooth and may require integration over a wide frequency range. The response does not show a worked calculation or resolve the distinction between summing daily returns and compounding simple returns, so the appropriate aggregation model must be chosen separately.
Key ideas
- Simulation can estimate the distribution of a sum by repeatedly drawing daily observations and aggregating them.
- Quantile and moment estimates from simulation require enough draws, especially for heavy-tailed distributions.
- The characteristic function of a sum of independent variables is the product of their individual characteristic functions.
- Numerical inversion can recover a density or distribution, but may be cumbersome for irregular distributions.
- The aggregation method must reflect whether the inputs are additive returns or simple returns that compound multiplicatively.
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Full text
# Compounded returns over $n$ days
# Compounded returns over $n$ days
I have calculated the daily returns of a certain stock. They can be analytically described by a 3 parameter Student's $t$ distribution whose density function is
$$f(x)=\frac{\left(\frac{\nu}{\nu+\left(\frac{x-\mu}{\sigma}\right)^2}\right)^{\frac{\nu+1}{2}}}{\sqrt{\nu}\sigma B\left(\frac\nu2,\frac12\right)},$$ where $\mu=0.11$, $\sigma=0.54$ and $\nu=1.52$.
With this starting point: how can I calculate the distribution function of (compounded) returns over $n$ days?
## Answer by Kermittfrog (score 5)
https://quant.stackexchange.com/a/63401
In addition to @Kevin's very helpful comment + link, if you need to calculate the distribution / density of the sum of (any) independent continuous random variables, e.g. sum of Student t distributed vars, or some other mixture, you can go two ways:
### Option A: Simulation
Simply simulate the target distribution $N$ number of times. You can then estimate moments, quantiles etc. from that.
N.B: Make sure that $N$ is sufficiently large, e.g. through resampling and testing the tightness of your estimators. For distributions whose moments are not defined, (e.g. with $d.o.f.\leq 4$ for a Student t distribution, if you require first four moments) you might need lots of simulations to get sufficient results.
### Option B: Numerical integration of characteristic functions
We know that the characterstic function of a random variable,
$$ \varphi_X(t)\equiv \int f(x)e^{itx}\mathrm{d}x $$
exists always, and that the characteristic function of the sum of independent random variables is the product of their individual characteristic functions. Finally, thru Gil-Pelaez or the inversion relationship,
$$ f(x)=\frac{1}{2\pi}\int e^{-itx}\phi_X(t)\mathrm{d}t, $$
or more exactly through a numerical approximation of this inversion, we can always try to recover the distribution or density of our convoluted variable.
N.B.: The numerical inversion can become quite cumbersome if your distribution is not sufficiently smooth, e.g. if it is has a strong 'bend' somewhere, implying that you need a very large region of integration to accumulate all material frequency contributions.
HTH?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.