Total, Factor, and Alpha Returns
The portfolio's daily return is taken from the weighted returns of each security in the portfolio. The weight of each security is the dollar position size divided by the total gross market value of the portfolio — or reference equity, NAV, or AUM, if specified — displayed as “% Equity” in the application. The daily security returns are taken directly from the risk model and include corporate action effects, such as splits and/or dividends.
The daily portfolio return can be decomposed into a daily factor and alpha contributors. Factor contributors are a function of the portfolio weights, the exposures of the assets in the portfolio, and the behavior of the factor (the 'factor return'). Alpha contributors are a function of the portfolio weights and the portion of an asset's return that cannot be explained by factors.
The daily portfolio factor return is calculated from the dot product of portfolio factor exposures and risk model factor returns.
The daily portfolio alpha return (or idiosyncratic return) is taken as the difference between the daily total portfolio return and the daily portfolio factor return.
Factor exposures
The portfolio factor exposures are calculated by multiplying the portfolio weights
from the weighted factor exposures for each security in the portfolio. The factor exposures for each security are taken directly from the model and are displayed under Analyze/Exposure in the application.
Factor returns
The daily factor returns are taken directly from the risk model.
Date Conventions
Portfolio position sizes on a particular date refer to the end of day position sizes. The return calculation assumes that the closing portfolio positions are held until the end of the next day. For dates with cash in/out flows, a start of day reference equity may be provided, as described in this cashflows support article.
To calculate the return on the next date, the position weights and security factor exposures are taken from the same date as the holdings date when calculating the portfolio factor exposures. The daily factor returns are taken from the next day.
Trading
For more precise PnL beyond what is possible for a pure holdings based attribution, users can always import their own PnLs based on realized transactions beyond market close. In this case total returns are the client specified total return, and factor and alpha contributions are subtracted from the client specified total return. Any difference between the client specified total return and the risk model total return will be attributed to 'Trading'. Click for more information: how to implement your own PnLs.
Cumulative Returns
The cumulative percent return over a period is the compound sum of the daily portfolio returns.
For daily portfolio returns of r1, r2, ... rT,
the compound sum over T days is (1+r1)*(1+r2)*...*(1+rT) - 1
The cumulative dollar return over a period is the algebraic sum over the dollar portfolio returns. The dollar portfolio return is calculated by taking the dollar position size as the weight without dividing by equity.
Cumulative active percent return over a period is the compound sum of the daily portfolio returns less the compound sum of the daily index returns.
Linking
The geometric linking of decomposed cumulative returns has issues: although in a single period all contributions add up in a single period, the decomposed returns geometrically compounded won't sum to the total portfolio's cumulative return across time periods. By using a Linking algorithm, we can guarantee that the sum of decomposed cumulative returns across time periods equals the portfolio's actual geometric return. Linking is performed to decompose both Total and Active cumulative active return. Omega Point uses the 'Frongello' approach to Linking.
Sample excel with return linking implemented for a return time-series is attached to the bottom of this article.
Annualized Returns
The geometric return, also known as the geometric mean return, is typically used to calculate the average rate of return of an investment over multiple periods, accounting for compounding. Since Omega Point uses the Frongello linking method to break down the cumulative return of a stock into two factors (A and B), and then annualize the returns for A and B, the sum of the annualized returns of A and B will not necessarily equal the annualized return of the stock.
Here’s why:
Geometric Nature of Returns: Geometric returns take into account the compounding effect over time. Therefore, the relationship between individual component returns and the total return is not additive but multiplicative.
Annualization Process: Annualizing a return involves converting the return over a period into an equivalent annual return. This process is done using the geometric mean, which does not simply sum up the returns but rather adjusts for compounding.
To illustrate this, let's say you have a stock with a cumulative return over a period, broken down into two factors A and B using the Frongello method. When you annualize these returns:
Annualized Return for A = (1 + Return A)^(1/n) - 1
Annualized Return for B = (1 + Return B)^(1/n) - 1
The combined effect of these two annualized returns is not simply the sum of the annualized Return A and Return B. Instead, the total annualized return would be calculated considering the compounding effect over the entire period, which might be different from the simple sum of the annualized returns of the components.
Therefore, you should not expect the sum of the annualized returns of A and B to be equivalent to the annualized return of the stock.
Additional references for non-additive annualized returns:
Sample excel with return linking implemented for a return time-series is attached below.