Index Investment Return Distribution

This page performs a Monte Carlo simulation of index investment returns and displays their probability distribution. The x-axis represents the return, while the y-axis shows the corresponding probability.

Average Annual Return (0.01 ~ 50)

Risk (0.01 ~ 50)

Leverage (0.01 ~ 5)

Number of Years (1 ~ 50)

years

Number of Simulations (10 ~ 5000)

runs

Bin Width (0.1 ~ 10)

Please enter the average annual return and risk of the index you are interested in. Press the calculate button to see their probability distribution. The simulation will be performed for the specified number of years. You can also simulate returns with leverage.

Monte Carlo Simulation Formula

Return $S_t$ from time $t-1$ to $t$ is simulated using the following formula.

$$ S_t = S_{t-1} \exp { \left( \left( L \mu - 0.5 L^2 \mu ^2 \right) \delta t + \epsilon \sqrt{ \delta t } L \sigma \right) } $$

where $\mu$ is the average annual return, $\sigma$ is the risk, and $L$ is the leverage. $\epsilon$ is a normally distributed random variable with mean 0 and standard deviation 1.