Using Student’s t Returns to Model Fat-Tailed Portfolio Growth

# Is there an easily implementable alternative to lognormal growth (something with fatter tails)?


# Is there an easily implementable alternative to lognormal growth (something with fatter tails)?












I have a toy model in Excel for the growth of a investment portfolio. I assume iid lognormal annual growth factors:

`=EXP(mu+sigma*NORM.S.INV(RAND()))`

where mu and sigma are calculated to get a given mean and standard deviation right.

I would really like to play around with something that has fatter tails, but with the same mean and standard deviation.

Do you have a suggestion that is easily implementable (in one cell)? Can you also point out how to compute its parameters given mean and standard deviation? And, finally, if there is an additional parameter, could you provide a 'reasonable' value for it (annual returns, bunch of asset classes in one portfolio)? That would be great.

## Answer by Marco Breitig (score 2, accepted)

https://quant.stackexchange.com/a/11302

If you mean by fat tails just fatter tails than the gaussian distribtuion, i.e. a distribution with finite variance, for instance the Student's t-distribution has fatter tails than the normal distribution. If you mean distributions with infinite variance, you have to have a look at Lévy distribution. In a first attempt you could just substitute the standard normal distribution with Student's t-distribution. Your formular would look like this:

```
=EXP(mu+sigma*T.INV(RAND(),DoF))
```

For degrees of freedom ($\operatorname{DoF}$) you have to specify an integer, preferably $\operatorname{DoF}>2$ so that the variance is finite. For $\operatorname{DoF}\to\infty$ Student's t-distribution converges against the standard normal distribution. The smaller the $\operatorname{DoF}$, the fatter the tails (to the extreme that for $\operatorname{DoF}\leq 2$ the variance does not exist and for $\operatorname{DoF}\leq 1$ the expectation does not exist).

Some words of caution: You can scale Student's t-distribution like you do with the standard normal distribution, but the rescaled parameter $\sigma$ is not the standard deviation. Have a look at the scaling behaviour of Student's t-distribution. To get the right variance, you have to scale with $$ \sigma := \sqrt{\operatorname{Var}\left[X\right]\frac{\operatorname{DoF}-2}{\operatorname{DoF}}} \textrm{,} $$ where $\operatorname{Var}\left[X\right]$ denotes the variance of your rescaled Student's t-distribution $X:=\mu+\sigma T$ with $T\sim t\left(\operatorname{DoF}\right)$. The expecated value $\mu$ is the same. Reasonable values for $\operatorname{DoF}$ are $3,\ldots,30$, the smaller the fatter are the tails.

The $\operatorname{exp}$ function in your model transforms instantaneous returns to normal growth factors.

A second word of caution: Normally you would simulate a realisation of a stochastic process which has incemrents distributed like $\mathcal{N}\left(\mu,\sigma^{2}\right)$. You can add up this log-returns to get to the overall return. With $\operatorname{exp}$ you get back to the value process of your portfolio. This works, because the sum of gaussian random variables stays gaussian. In fact, because of the central limit theorem the sum of random variables with finite variance which are not too dependent will converge in distribution to the gaussian distribution. So if you'd replicate the described strategy with Student's t-distributed random variables instead of gaussian random variables and $\operatorname{DoF}>2$ (finite variance), the resulting sum will converge in distribution to a gaussian one and you'd lose the fat tail property of Student's t-distribution.