Calculating the Gain Needed to Recover Trading Losses

# How to calculate necessary gain to compensate a loss in a financial transaction?


# How to calculate necessary gain to compensate a loss in a financial transaction?












(Feel free to suggest the correct Stackexchange community - or otherwise - if this is not the correct one)

When trading financial markets, a gain of `x`%, won't recover a loss of `x`% (same applies to any gain-loss scenario, in any market, I assume)

e.g. I start with 100, I lose 2% (98% or original capital), I gain 2%. I'm still in loss (99.96% of original capital)

Is there a formula to calculate the necessary % gain of a single transaction over `k` loss of x% gain after y losses of x%? (if the formula is generalized to `z` gains of x% and `k` losses of x% is preferable - x% is fixed)

## Answer by Magic is in the chain (score 5, accepted)

https://quant.stackexchange.com/a/46105

Let x represent the percent change-e.g. 2%, let k represent the number of decreases, and z the number of increases. Something like this? We want to find z such that:

$\left(1-x\right)^k\left(1+x\right)^z=1$

Rearrange,

$\left(1+x\right)^z=\frac{1}{\left(1-x\right)^k}$

And take log:

$z \ln \left(1+x\right)=-k \ln \left(1-x\right)$

and solve for z:

$z =-k \frac{\ln \left(1-x\right)}{\ln \left(1+x\right)}$

Or you want z to represent the percent increase such that:

$\left(1-x\right)^k\left(1+z\right)^k=1$

$z=\frac{1}{1-x}-1=\frac{x}{1-x}$

## Answer by David Addison (score 3)

https://quant.stackexchange.com/a/46118

This is a common question type on the GMAT.

Suppose you lose x% of your initial funds, $V_0$, so you now have $V_0(1-x/100)$. What y% return on the new funds do you need to return to your initial funds?

Stated this way, we express our problem as:

$V_0(1-x/100)(1+y/100)=V_0 $, or $(1-x/100)(1+y/100)=1$

Solving for y, we have:

$y= \frac{100}{1-x/100}-100$, or

$y=\frac{x/100}{1-x/100}$