Practical Choices for Building a Multi-Factor APT Model
Summary
The document lays out practical questions involved in estimating a multi-factor Arbitrage Pricing Theory model for Swiss equities. It considers whether factors should be unexpected innovations or can instead be observed macroeconomic series such as changes in industrial production, energy prices, or consumer confidence. It also asks whether a decade of monthly data is adequate and whether estimating stock-level sensitivities differs materially from estimating portfolio sensitivities.
For implementation, the author proposes constructing portfolios that mimic individual factors by minimizing portfolio variance subject to beta exposure constraints and a fully invested constraint. A small matrix example shows how the desired factor can be targeted while other factor exposures are set to zero. The document is a set of research design questions, not a resolved method: it supplies no estimates, empirical comparison, or guidance on sample length, and does not establish that the proposed optimization is sufficient. Its value is in identifying choices researchers must examine when translating a factor model into investable portfolios.
Key ideas
- APT factors are theoretically expressed as unexpected components, raising questions about using observed macroeconomic changes instead.
- The author asks whether monthly data spanning a decade provides enough observations for estimation.
- The proposed analysis estimates factor sensitivities for individual Swiss-listed stocks.
- A variance-minimizing portfolio with beta constraints is suggested for mimicking each factor.
- The document presents these as open design questions and provides no empirical resolution.
Tags
Full text
# Multi-factor APT model in practice: non-zero mean factors, observations needed and portfolios
# Multi-factor APT model in practice: non-zero mean factors, observations needed and portfolios
I'm going to build a multi-factor APT model for the Swiss market starting from the work made by Chen, Roll and Ross (to which I will add and test some additional factors). I have some doubts though:
- The APT prescribes that each factor should be expressed as its unexpected component:$$\tilde{F_t}=F_{t}-E[F_t \mid t-1]$$ What if it is not the case? Are results highly affected? I was thinking to put for example consumer confidence, changes in oil or electricity prices, etc. for which it is difficult to find forecasts or for which it is not that cleaver to assume that individuals use simple forecasting techniques like the naïve or drift method. Chen, Roll and Ross use for example changes in monthly industrial production, which is not expressed as its unexpected component.
- Are 10 years of monthly observations enough to make a good estimate or should I use more/less observations?
- My idea was to apply my model on every stock that is actually registered on the SIX Swiss Exchange. Is it possible? I've read that generally this models are applied to portfolios of stocks. Does this make a huge difference?
- Once the estimates are done my idea was to build $K$ factors portfolios (with $K$ being the number of factors). The procedure I had in mind was to solve a quadratic programming problem of the type: $$\min_w w'\Sigma w$$ $$s.t. \ B w \leq b$$ where: $w$ is a vector of weights, $\Sigma$ is the variance-covariance matrix, $B$ is a matrix of $K+1$ rows (where the $K+1$th row is filled with ones) and $N$ columns (with $N$ being the number of stocks analyzed), $b$ is a vector of constraints and $\beta_{k, j}$ is the sensitivity of stock $j$ to factor $k$. As an example: to find weights of the portfolio which mimics the 2nd factor (with $K=3$ and $N=4$): $$B =\begin{bmatrix} \beta_{1,1} & \beta_{1,2} & \beta_{1,3} & \beta_{1,4}\\ \beta_{2,1} & \beta_{2,2} &\beta_{2,3} & \beta_{2,4}\\ \beta_{3,1} & \beta_{3,2}& \beta_{3,3} & \beta_{3,4}\\ 1 & 1 & 1 & 1 \end{bmatrix} \qquad b=\begin{bmatrix} 0\\ 1\\ 0\\ 1\\ \end{bmatrix}$$ Is this the right procedure?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.