Calculating ex ante returns & probability of a negative return over some horizon

Question

One way to go on about this is to parametrically calculate the returns, i.e. hold the exposure constant and backtest against the factor changes over that horizon. This is not forward looking per se but it's using assumption based model to arrive at that distribution (so no longer assuming a normal distribution). The other way is to simulate returns through a Black Scholes Model.

Would the following statement be correct: in an ex ante framework there is a 50% chance of returns being either negative or position, assuming normal distribution and a mean of zero. Whereas in an ex-post framework where we have a realised mean return of 3% per year and run approximately 3% of risk then there is a probability of 16% of being negative over a one year period. Does this make any sense at all to you? How would you back out that 16%?

Answer 1

For a normal random variable $\xi$ with mean 0, then \begin{align*} P(\xi < 0) = P(\xi > 0) = 50\,\%. \end{align*}

For a normal random variable $\eta$ with mean (i.e., realized mean return) $\mu=3\,\%$ and risk (i.e., standard deviation) $\sigma = 3\,\%$, then \begin{align*} P(\eta < 0) &= P\left(\frac{\eta - \mu}{\sigma} < \frac{ - \mu}{\sigma} \right)\\ &= P\left(\frac{\eta - \mu}{\sigma} < -1 \right)\\ &\approx 15.87\,\%. \end{align*}