Performance and Risk Calculations
Within PerformaGlobal, we have developed a comprehensive approach to include a number of industry recognised calculations and risk measures which will help to both monitor and control the performance returns generated. It is currently stated as a recommendation within the GIPS standards that relevant risk measures, such as tracking error, beta, information ratio, sharpe ratio, treynor ratio, etc, be presented along with total returns for both benchmarks and composites. The formulae for both these and associated measures used in performance calculations are as follows:
Performance Calculations
The system calculates asset-weighted composite returns based on underlying portfolio market values and returns. Asset weighted returns are calculated using the formula:
Where
Wi is the Weight of Portfolio
i at the start of the month
Ri is the Return of Portfolio
i for the month
The system can also calculate equal-weighted composite returns based on underlying portfolio returns. Although equal weighted composite returns are generally reported as supplemental information, the equal weighted return is calculated as above by setting each of the portfolio weights to one, thus giving:
Where
Ri is the Return of Portfolio
i for the month
n is the number of portfolios in the composite
for the month
Both portfolio and benchmark data may be loaded into the system for any time periods up to annual (e.g. real estate) and down to daily market values, inflows, outflows etc. Annual composite returns are then calculated based on whatever underlying data is available, regardless of its frequency (daily, monthly, quarterly). As per GIPS, the system will not perform any annualisation for periods of less than a year.
Both gross and net returns are calculated by the system as standard. If gross and net returns are available for the underlying portfolio data, the composite calculations are performed based on the relevant gross and net portfolio returns. If only one return series is available, a fee scale can be attached to the composite (which must define the highest fee schedule of any member portfolio) and the relevant return series is calculated using that. The formula used to calculate gross for net (or vice versa) is:
Where
F is the annual fee rate (calculated from
the banding structure)
n is the number of days in the period
Dispersion Calculations
The system calculates a variety of dispersion measures allowing the user to report on their preferred measures.
Asset weighted dispersion is the most widely accepted measure of dispersion.The
formula for its calculation is as follows:
Where
Wiis the weight of the ith portfolio
at the beginning of the period
Ri is the annual return for
portfolio I
Equal weighted dispersion is calculated as above by setting each of the portfolio weights to one, thus giving:
Where
Ri is Annual Return for Portfolio I
n is the number of portfolios in the composite
Risk Calculations
Standard Deviation
The system calculates the standard deviation of monthly composite returns (a system parameter determines whether to use the simple returns or the logged return series). The annualized standard deviation is calculated by multiplying this monthly figure by the square root of 12. It is understood that statistically, 36 periods should be used for the calculation of risk, but the system will calculate the figure for smaller periods and it is up to the user to determine whether or not to report on this data. The formula for the calculation of standard deviation is as follows:
Where
s is the Monthly Standard Deviation
Ri is the Return for month i
n is the number of months
Alpha
Where
Rp is the return of the portfolio (or composite) for month i
RB is the return of the benchmark for month i
n is the number of months
Beta
Where
Rp is the return of the portfolio (or composite) for month i
RB is the return of the benchmark for month i
n is the number of months
Correlation Coefficient
Covariance
Where
Rp is the return of the portfolio (or composite) for month i
RB is the return of the benchmark for month i
n is the number of months
Tracking Error
Where
Ri is the out-performance n is the number of time periods
Information Ratio
Sharpe Ratio
Treynor Ratio
Additional calculations can be found in the PerformaGlobal Help System. This is available to all Performa clients.