To calculate the probability of losing money in six months, we need to consider the distribution of logarithmic returns for the stock. Given that it follows a normal distribution with an annualized mean of 12% and an annualized standard deviation of 50%, we can use this information to determine the probability.
Step 1: Convert the annualized mean and standard deviation to the corresponding mean and standard deviation for six months:
Since we want to calculate the probability of losing money in six months, we need to adjust the mean and standard deviation accordingly. The mean for six months is half of the annual value, and the standard deviation is adjusted by taking the square root of half of the annual value.
Mean for six months = (12% annual mean) * (6/12)
Standard deviation for six months = √((0.5 / 12) * 50% annual standard deviation)
Mean for six months = 0.06 (6%)
Standard deviation for six months = √((0.5 / 12) * 0.5) ≈ 0.125
Step 2: Determine the probability of losing money using the normal distribution:
To find the probability of losing money in six months, we need to calculate the area under the normal distribution curve to the left of zero (negative returns).
Using a standard normal distribution table or a statistical software, we can find the probability by calculating the cumulative distribution function (CDF) at the mean return value of zero and the adjusted standard deviation of 0.125.
Let's denote this probability as P(X < 0), where X represents the logarithmic returns variable.
Therefore, the probability of losing money can be calculated as P(X < 0).
Step 3: Calculate the probability of losing money:
Using a standard normal distribution table or statistical software, the probability of losing money in six months can be determined based on the calculated mean and standard deviation.
To support the calculation, we can use a statistical software or refer to a standard normal distribution table.
Please note, however, that the specific numerical value of the probability cannot be determined without the exact mean and standard deviation values.