# Risk Exposure

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

It can be concluded that:

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.Thus, the balanced state of hAMM system equals to "

**zero naked position**":$N=X+Y=0$

The key to achieve the balance of hAMM system is to

**restrict naked position**.