Investments, tenth edition

  Lower Partial Standard Deviation and the Sortino Ratio 

 The use of standard deviation as a measure of risk when the return distribution is non-

normal presents two problems: (1) the asymmetry of the distribution suggests we should 

look at negative outcomes separately; and (2) because an alternative to a risky portfolio is 

a risk-free investment, we should look at deviations of returns from the risk-free rate rather 

than from the sample average, that is, at negative excess returns. 

 A risk measure that addresses these issues is the    lower  partial  standard  deviation 

(LPSD)    of excess returns, which is computed like the usual standard deviation, but using 

only “bad” returns. Specifically, it uses only negative deviations from the risk-free rate 

(rather than negative deviations from the sample average), squares those deviations to obtain 

an analog to variance, and then takes the square root to obtain a “left-tail standard devia-

tion.” The LPSD is therefore the square root of the average squared deviation,  conditional   on 

a negative excess return. Notice that this measure ignores the frequency of negative excess 

returns, that is, portfolios with the same average squared negative excess returns will yield 

the same LPSD regardless of the relative frequency of negative excess returns. 

 Practitioners who replace standard deviation with this LPSD typically also replace the 

Sharpe ratio (the ratio of average excess return to standard deviation) with the ratio of aver-

age excess returns to LPSD. This variant on the Sharpe ratio is called the    Sortino  ratio.