In finance, we know that risk is defined as uncertainty since we are unable to predict the future more accurately. Based on the assumption that prices follow a lognormal distribution and returns follow a normal distribution, we could define risk as standard deviation or variance of the returns of a security. We call this our conventional definition of volatility (uncertainty). Since a normal distribution is symmetric, it will treat a positive deviation from a mean in the same manner as it would a negative deviation. This is against our conventional wisdom since we treat them differently. To overcome this, Sortino (1983) suggests a lower partial standard deviation. Up to now, we assume that the volatility of a time series is a constant. Obviously this is not true. Another observation is volatility clustering, which means that high volatility is usually followed by a high-volatility period, and this is true for low volatility that is usually followed...