Performance Attribution : Annualizing alpha & factor return contributions

Question

Let's say I have a factor model which I am using for Performance Attribution. I'd like to separate returns from alpha vs. returns from exposure to various risk factors.

For each date, the factor model decomposes the returns of a security into its daily alpha (i.e. idiosyncratic component not explained by the factor model), as well as the daily return contribution from its component factors. The table below is a mock example with 1-factor and five non-overlapping periods. For simplicity let's assume the 5 periods span one year.

Notice that the 'arithmetic annual return' is 10.34% (11.03/10 - 1). Or equivalently via compounding, it is (1+security return period 1) * (1+security return period 2) ... (1+ security return period 5) - 1 = 10.34%

I'd like to generate an annualized view of the performance attribution. In other words, I'd like to de-compose the 10.34% as the sum of : annualized Alpha + annualized Returns from Factor Exposures + annualized Interaction Term contribution. (The interaction effect measures the combined impact of manager selection and allocation decisions. Algebraically, it is a consequence of binomial expansion: $$\textrm{Price(t+1)} = \textrm{Price}(t)*[ 1 + \alpha ]*[ 1 + \textrm{net factor contribution}]$$

How do I map the matrix of daily alphas, daily factor returns, and daily security returns into a sum of annualized alpha component + annualized factor return component + annualized interaction effect? I imagine there is a clever matrix algebra solution to this problem.

2-period solution

Notation:

Security Return in period 1 = $R_1$; Alpha in period 1 = $\alpha_1$; Net Factor Contribution in period 1 = $NFC_1$

We know by definition: $R_1$ = $\alpha_1$ + $NFC_1$ and $R_2$ = $\alpha_2$ + $NFC_2$

and Annual arithmetic return is = (1 + $R_1$) * (1 + $R_2$) - 1

Expanding terms we can find we can express annual arithmetric return as the sum of component annualized alpha, annualized net factor returns, and annualized interaction terms:

Annual arithmetic return (as a sum of the three components) =

( $\alpha_1$ + $\alpha_1$$\alpha_2$ + $\alpha_2$ ) + ( $NFC_2$ + $NFC_1$ + $NFC_1$*$NFC_2$ ) + ( $NFC_1$$\alpha_2$ + $\alpha_1$$NFC_2$ )

The same solution in matrix algebra terms:

vector1 = (1 + $R_1$) = (1 + $\alpha_1$ + $NFC_1$)

vector2 = (1 + $R_2$) = (1 + $\alpha_2$ + $NFC_2$)

The annual arithmetic return is then the sum of the elements of the 3x3 matrix resulting from the outer cross-product of vector1 and vector2 - 1. The upper left portion of the matrix (see below) consists of the alpha contributions, and the top-right,bottom-left, and bottom-right contain the factor return contributions which can now be grouped with another matrix operation:

$$ \begin{matrix} 1 & a{2} & NFC{2}\\ a{1} & a{1}*a{2} & a{1}*NFC{2}\\ NFC{1} & NFC{1}*a{2} & NFC{1}*NFC{2}\\ \end{matrix} $$

So -- what is a computationally clever way to scale this to 251 periods (assuming 251 trading days = 1 year) while grouping terms into the sum of 3 annualized components?

Answer 1

A simple top-down shortcut calculation :

1. Set annualized alpha = compounded alpha = 1 + a1 + a2 + a1*a2 + ... = $\Pi$ (1 + $\alpha_t$)
2. Set annualized return from factors = compounded factor return = $\Pi$ (1 + $factorReturn_t$)
3. Interaction Term contribution is then = Compounded Security Return - Compounded alpha - Compounded factor return

Therfore the annualized arithmetic return is de-composed into these three components.

Update: There is the text "Market Risk Management for Hedgefunds" (Duc & Schorderet) that addresses the issue of annualizing various risk measures in chapter 8 "The Annualization Problem".

Update 2 (3/12/2012): Also, the following papers cover the annualization of performance attribution statistics: Bonafede, Forest, and Matheos (2002) and Carino (2002). Since the interaction term is significant, and various annualization methods satisfy different critieria (i.e. commutativity, metric consistency, etc.) the problem is not as straightforwards as it might seem.