Change in the Probability Distribution of Index-Investing Returns

Displays how the probability distribution of returns for index investing changes. With the starting value set to 1, you can see how returns vary by investment horizon.

Average Annual Return (0.01 ~ 50)

Risk (0.01 ~ 50)

Leverage (0.01 ~ 5)

Number of Years (1 ~ 50)

years

Enter the index's annual average return and risk. Then press Calculate to see how the return distribution changes. The probability distribution up to the period set in Elapsed Years will be computed. You can also compute returns with leverage applied.

Formulas

Mode

The mode $mode$ is computed using the formula below.

$$ mode = \exp \left( ( \mu L - 1.5 \sigma ^2 L^2 ) t \right) $$

Here, $\mu$ is the annual average return, $\sigma$ is the risk, and $L$ is the leverage. $t$ is the elapsed years.

Median

The median $median$ is computed using the formula below.

$$ median = \exp \left( ( \mu L - 0.5 \sigma ^2 L^2 ) t \right) $$

95% Confidence Interval

The 95% confidence interval bounds are from lower $lower$ to upper $upper$.

$$ lower = \exp \left( ( \mu L - 0.5 \sigma ^2 L^2 ) t - 1.96 \sigma L \sqrt{t} ) \right) $$

$$ upper = \exp \left( ( \mu L - 0.5 \sigma ^2 L^2 ) t + 1.96 \sigma L \sqrt{t} ) \right) $$