**Question 928** mean and median returns, mode return, return distribution, arithmetic and geometric averages, continuously compounding rate, no explanation

The arithmetic average continuously compounded or log gross discrete return (AALGDR) on the ASX200 accumulation index over the 24 years from 31 Dec 1992 to 31 Dec 2016 is **9.49**% pa.

The arithmetic standard deviation (SDLGDR) is **16.92** percentage points pa.

Assume that the log gross discrete returns are normally distributed and that the above estimates are true population statistics, not sample statistics, so there is no standard error in the sample mean or standard deviation estimates. Also assume that the standardised normal Z-statistic corresponding to a one-tail probability of **2.5**% is exactly **-1.96**.

If you had a $1 million fund that replicated the ASX200 accumulation index, in how many years would the **mode** dollar value of your fund first be expected to lie outside the **95**% confidence interval forecast?

Note that the mode of a log-normally distributed future price is: ##P_{T \text{ mode}} = P_0.e^{(\text{AALGDR} - \text{SDLGDR}^2 ).T} ##