Question

what is lower partial standard deviation of return and how to calculate? Are lower partial standard deviation of return and downside deviation same thing?

How to annualized weekly lower partial standard deviation of return?

Answer 1

Lower Partial Standard Deviation of Return

For the measurement of risk, the use of standard deviation creates problems like asymmetry of distribution and an alternative to risk portfolio for deviation of returns provides negative outcomes. To overcome these problems, Sortino (1983) suggest the Lower Partial Standard Deviation (LPSD).

LPSD is defined as the average of squared deviations from the risk free rate conditional on negative excess returns. It is calculated by squaring the negative deviations to obtain an analogue to variance and takes a square root to obtain a "left tail standard deviation".

The formula of LPSD is,

where, Ri < Rf

Where, LPSD = Lower Partial Standard Deviation

R_f = Risk- free rate of return for period f

R_i = Risk- free return for period i

n = number of quantity (or) time

Downside Deviation of Return

Downside deviation is the measurement of downside risk and price volatility. It aims on returns that falls below a minimum acceptable return. This deviation is used in the calculation of Sortino ratio. It solves the issues faced on downside risk. It is also used to compare multiple potential investments.

The formula for downside deviation is,

Downside Deviation = (R_b - MAR) ² / N

Where, B = Total quantity of periodic returns below the MAR

R _b = Return for period b

MAR = Minimum Acceptable Return

N = Total quantity of all periods

Lower Partial Standard Deviation and Downside Deviation

LPSD is used to measure expected risk and volatility of the trader or firm. But, Downside deviation only aims on the volatility of negative returns. Hence, it is clear that lower partial standard deviation and downside deviation are different from each other.

Annualizing Weekly Lower Partial Standard Deviation of Return

Generally, a stock trader will access to daily, weekly, monthly or quarterly data of stock price or stock portfolio price. He will use this data to calculate returns and standard deviations. The annual standard deviation is used to calculate the volatility of the firm or trader. The annualized lower partial standard deviation of return will be calculated for weekly duration is,

Annualized Standard Deviation = Standard Deviation of Weekly Returns * Square Root (52)

For monthy calculation, the square root is 12; for daily calculation, the square root is 250; for quarterly calculation, the square root is 4. And for LPSD annualization, LPSD of weekly returns will be taken for calculation.