PYTHON:

By definition, the probability of obtaining one return of this period that is less than the value at x% of its underlying distribution is x%. For instance, if the underlying distribution of returns follows a standard normal distribution, the probability of obtaining a return that is less than the value at 2.5% is 2.5%, and this value at 2.5% is -1.96; the probability of obtaining a return that is less than the value at 5% is 5%, and this value at 5% is -1.645. When x% is relatively small, we can call these returns "bad events".

Now, what about the chance of observing consecutive bad events?

For simplicity, what is the probability of observing a period where returns during this period, the last period and the period before the last period are all less than the value at x% of its underlying distribution?

Please follow the suggested steps below.

Simulate one path (list) of 100,000 periods of returns from a normal distribution with mean = 0.05 and variance = 0.01. (For simplicity, assume returns are i.i.d.., that is you don't need to worry about autocorrelation.) What is the value at 10%? Please print out your answer in one complete sentence.

What is the probability (in percents) of obtaining a period where the returns during this period, the last period and the period before the last period are all less than the value at 10% (from 2a)? Please print out your answer in one complete sentence.