|The Theory \ Details \ appendices \ appendix C|
Fix some minimum acceptance excess rate . In the following context the latter is also is also called Target Excess Rate.
Definition. Information Ratio , corresponding to , is equal to the difference between expected excess growth rate and , divided by volatility:
Note. Widely recognized Sharpe Ratio is a particular case of information ratio corresponding to .
Definition. Sortino Ratio , corresponding to , is equal to the difference between expected excess growth rate and , divided by downside volatility with MAR equal to :
Normalized Sortino Ratio
Definition. Normalized Sortino Ratio, corresponding to , is similar to Sortino ratio, but with normalized downside volatility in denominator:
Under the assumptions of the analytical model, the normalized Sortino ratio coincides with information ratio.
Definition. STARR Ratio is equal to the difference between expected excess growth rate and , divided by the Conditional Value-at-Risk, transformed to logarithmic return:
Normalized STARR Ratio
Definition. Normalized STARR Ratio (NSTARR) is the STARR ratio, corrected in such a way that in case of normally distributed logarithmic returns it coincides with information ratio.
where denotes Normalized CVaR.
Definition. Normalized CVaR is a measure, based on CVaR, corrected in such a way that under the assumptions of the analytical model it coincides with volatility measure.
where and are standard normal density and distribution function respectively.
Under the assumptions of the analytical model, all above performance measures are equivalent when used to sort the list of available portfolios according to their investment attractiveness. Otherwise, because of the
properties of downside volatility and CVaR respectively, the Sortino Ratio and the STARR ratio might become more relevant measures of performance.
Standard performance measures vs. their instantaneous counterparts
Below we focus on the Sharpe ratio, but the same logic holds true for all other performance measures, presented above.
Definition. Sharpe Ratio is equal to the expected excess growth rate divided by volatility:
Definition. Instantaneous Sharpe Ratio is equal to excess Mu divided by volatility:
Both the Sharpe ratio and the instantaneous Sharpe ratio sort assets according to their relative performance in the past. However, there is an essential distinction in the information the corresponding rankings reflect.
as separate alternative investments, thus depriving the investor of the continuous rebalancing advantages. For further details see [Nielsen, Vassalou; 2004].
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