Modeling Positive-Return Fees and Net P&L Distributions

# Distribution when positive values are rescaled?


# Distribution when positive values are rescaled?












Suppose I have a series of gross P&L values, which are normally distributed with mean $\mu$, variance $\sigma^2$.

For positive P&L values, there is a $x\%$ commission. For example, $x=5\%$.

So the net pnl, $p_i$ is:

- $p_i$ if $p_i$<0

- $p_i * (1-x)$ if $p_i$>0

Given different $\mu%$ and $\sigma$, how can I calculate whether the net P&L over $n$ trades will be positive?

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/79736

As the payoff function is continuous, we can find the new density function by the change of variable for probability densities:

$$ \begin{align} y&=r(x)\equiv\begin{cases} x & x\leq 0 \\ (1-\alpha)x & x> 0 \end{cases} \\ \end{align} $$

Thus

$$ \begin{align} f_Y(y)&=f_X(r^{-1}(y))\left|\frac{d}{dy}(r^{-1}(y))\right|\\ &=\begin{cases} f(y) & y\leq 0 \\ \frac{f\left(\frac{y}{1-\alpha}\right)}{1-\alpha} & y> 0 \end{cases} \end{align} $$

At closer expection, this is the weighted sum of the densities of two truncated normal variates. The MGF exists in closed form, thus we can easily produce mean, variance etc. As the MGF of a sum of independent RVs is the product of their individual MGFs, we could calculate the moments of the cumulative PnL over a number of individual PnLs using approximate methods.

It feels as if the derivation of the characteristic function would be too cumbersome, though...

So to answer your question: Either

- simulate for varying parameters, or

- calculate the first moments and use an Edgeworth expansion or similar.

## Answer by MrLCh (score 0)

https://quant.stackexchange.com/a/79388

I believe there is no "simple" answer to your question. To calculate the expected value of net return $n$ you could do the following:

$$\begin{align}\mathbb{E}(n) &= \int_{-\infty}^{\infty} t * f_n(t) dt \\ &= \int_{-\infty}^{0} t * f_p(t) dt + \int_{0}^{\infty} t * (1-x) * f_p(t) dt \\ &= \int_{-\infty}^{0} t * f_p(t) dt + \int_{0}^{\infty} (t- tx) * f_p(t) dt \\ &= \int_{-\infty}^{0} t * f_p(t) dt + \int_{0}^{\infty} t * f_p(t) dt - x \int_{0}^{\infty} t * f_p(t) dt \\ &= \mathbb{E}(p) - x \int_{0}^{\infty} t * f_p(t) dt \end{align}$$

where $f_n$ denotes the density of the $n$ and $f_p$ denotes the density of $p$ (so the density of the normal distribution with mean $\mu$ and variance $\sigma^2$).