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?

  • $\begingroup$ As far as I remember, there exists no 'nice' closed form density or even characteristic function for the student distribution. There exists no 'simple' way to extend the 1day-case to an N-day case under the t-distribution, to my knowledge. $\endgroup$ – Kermittfrog Apr 15 at 9:42
  • $\begingroup$ and how about a more 'well behaved' gaussian distribution? or in general, without specifying the distribution line shape? $\endgroup$ – Luigi Apr 15 at 10:06
  • 2
    $\begingroup$ @Luigi You can simply add normally distributed log-returns using the convolution stability of that distribution, i.e. that sum is still normal. The sum of Student $t$ distribution is not Student $t$ though, see here. $\endgroup$ – Kevin Apr 15 at 10:12
  • $\begingroup$ @Kevin, so I should calculate the log-returns for 1 day and then add them n times to get the compounded returns for n days? $\endgroup$ – Luigi Apr 15 at 10:24

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.



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