# Risk Exposure

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When $X+Y\neq0$, the system has **risk exposure** because of the constant price fluctuation.

At any given moment $t$, we have the state of a derivative $S_t$, the sum of total long positions $X_t$, the sum of total short positions $Y_t$ and the current price index $I_t$. Assuming that we have a new price $I_{t+1}$ at the moment $t+1$, then the risk exposure $V{exposure}$ can be calculated as follows:

$V_{exposure}=\sum\Big((X_t+Y_t)*(I_{t+1}-I_t)\Big)$

$X_t$

*is always positive and*$Y_t$*is always negative.*

Consider that $I_{t+1}-I_t$ from external markets is unpredictable and cannot be controlled, the only way to reduce the risk exposure is to restrict the value of $X_t+Y_t$.

Obviously, if $X_t+Y_t=0$, the risk exposure will not increase at moment $t$.

Thus, the hAMM system reaches a balanced state (i.e. $X+Y=0$), which is so called "**risk exposure closed**".

Naked Position

At any moment, the value of $X+Y$ means the "unhedged" or **"naked" position** $N$ of a certain derivative at that moment:

$N=X+Y$

When $N=0$, the sum of total long positions equal to the sum of total short positions, and there are no naked positions.

When $N>0$, the sum of total long positions are greater than the sum of total short positions, and the naked positions are long positions.

It can be concluded that:

Thus, the balanced state of hAMM system equals to "**zero naked position**":

The key to achieve the balance of hAMM system is to **restrict naked position**.

When $N<0$, the sum of total short positions are greater than the sum of total long positions, and the naked positions are short positions.

$V_{exposure}$ can be simplified as follows:

$V_{exposure}=\sum\Big(N_t*(I_{t+1}-I_t)\Big)$

When $N=0$, no matter how the index was fluctuated, the net profit/loss of all positions equals zero, and the net loss of the system will not increase.

When $N\neq0$, if the sign of $I_{t+1}-I_t$ and $N$ are the same (i.e., naked positions and the index fluctuation has the same direction), the net loss of the system may increase.

$N=X+Y=0$