Factor Models 101

Problem

The number of unique terms in a covariance matrix is given by:

$$\begin{align} \underbrace{N}_{\text{variance terms}}+\underbrace{\frac{N^2-N}{2}}_{\text{covariance terms}} = \frac{N^2+N}{2} \end{align}$$

It isn’t feasible to calculate these reliably. Therefore a portfolio strategy based on a naive approximation of the covariance matrix won’t be reliable.

Factor Models to the Rescue

Single Factor Model (Intuition)

Consider the time-series regression for stock $i$.

$$\begin{align} r_{i,t} = a_i + \underbrace{b_i}_{\text{factor loading}}\underbrace{(r_m,t -r_{f,t})}_{\text{factor}} + u_{i,t} \end{align}$$

Note that variance of return for the stock can be described by:

$$\begin{align} Var(r_{i,t}) &= Var(a_i + b_i(r_{m,t} -r_{f,t}) + u_{i,t}) \\ &= b_i^2 Var(r_{m,t} -r_{f,t}) + Var(u_{i,t}) \end{align}$$

Furthermore the covariance for stocks $i$ and $j$ is given by:

$$\begin{align} Cov(r_{i,t}, r_{j,t}) &= Cov\big[(a_i + b_i(r_{m,t} -r_{f,t}) + u_{i,t}), (a_j + b_j(r_{m,t} -r_{f,t}) + u_{j,t})\big] \\ &= b_ib_jVar(r_{m,t} - r_{f,t}) + Cov(u_{j,t},u_{i.t}) \\ &= b_ib_jVar(r_{m,t} - r_{f,t}) \end{align}$$

The last line follows only if the error terms are truly independent between stocks.

Assuming this is true, the covariance between two stocks is known.

How many terms do we need to approximate? Just $b_i$ and $Var(u_i)$ for all $i$ and the estimate for $(r_{m,t} - r_f)$.

$$\begin{align} \underbrace{N}_{b_i} + \underbrace{N}_{Var(u_i)} + \underbrace{1}_{(r_{m,t} - r_f)} = 2N +1 \end{align}$$

The goal is to construct a factor model that yields $Cov(u_{j,t},u_{i.t}) = 0$.

Approximation

Time series regressions assume constant loadings. We don’t like this assumption so we instead emphasize a cross sectional approach.

For the cross-sectional regression, let there be $N$ stocks and $K$ factors of interest. Gather $N$ returns on the day. Additionally, collect $K$ characteristics for each stock. These characteristics should be chosen in a way that they are good approximations for the loading on the factor of interest. These $K$ characteristics from $N$ stocks form an $N\times K$ matrix.

We now run a regression of the following form, solving for $f_t$.

$$\begin{align} \underbrace{\begin{bmatrix}r_1 \\ r_2 \\ \vdots \\ r_n \end{bmatrix}}_{\text{Day $t$ Returns}} &= \underbrace{\begin{bmatrix} b_{1,1} & b_{1,2} & \dots & b_{1,k} \\ b_{2,1} & b_{2,2} & \dots & b_{2,k} \\ \vdots & \vdots & \ddots & \vdots \\ b_{n,1} & b_{n,2} & \dots & b_{n,k} \end{bmatrix}}_{\text{Proxied loadings}} \underbrace{\begin{bmatrix} f_1 \\ f_2 \\ \vdots \\ f_k \end{bmatrix}}_{\text{Factor Sensitivities}} + \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{bmatrix} \\ \pmb r_t &= B_t \pmb f_t + \pmb u_t \end{align}$$

Interpreting $f_t$

Normalize the columns of $B_t$ (excluding the dummy variable columns) to be mean $0$ and unit variance. Furthermore, append a column of 1s. Define this new matrix:

$$\begin{align} \tilde B_t = \begin{bmatrix} \pmb{1} & \text{Normed}(B_t) \end{bmatrix} \end{align}$$

Let the normalized $B_t$ with an appended column of $\pmb1$s equal $\tilde B_t$.

If we run the regression:

$$\begin{align} \pmb r_t &= \tilde B_t \pmb f_t + \pmb u_t \end{align}$$

We can do some analysis to find that $f_k$ describes the returns to the portfolio where the weight on stock $i$ in the portfolio is given by $\tilde b_{i,k}/N$. Succinctly:

$$\begin{align} f_k &= \sum_{i=1}^N r_{i} \tilde b_{i,k} / N \end{align}$$

Technically the result is actually:

$$\begin{align} f_k &= \sum_{i=1}^N r_{i} \ddot b_{i,k} / N \end{align}$$

Where $\ddot b_k$ is the unique variation in $\tilde b$.

The coefficient corresponding to the vector of ones, $f_1$, has $b_i=1\ \forall i \in N$. Plugging this into the formula for $f_k$ we see the constant represents the return of the equal weighted portfolio. Thus, $f_1$ is referred to as the country factor. Factors that aren’t composed of dummy variables are called style factors.

$$\begin{align} \pmb r_t = B_t \pmb f_t + \pmb u_t \end{align}$$

The return to asset $r_i$ is given by $\pmb b_i' \pmb f_t + u_i$.

Forecasting $f_t$

1. Collect $T$ approximations for the vector $\pmb f_t$.
2. Compute the factor covariance matrix using an exponential weighted average. The formula looks like this:

$$\begin{align} \hat \Sigma_{f,T} = \text{EMA}\left[(\pmb f_t - \pmb {\bar f_t})(\pmb f_t - \pmb {\bar f_t})^\prime\right] \end{align}$$

3. Recall that that each index of the cross-sectional residual vector $u_t$ corresponds to a particular asset. Collect these residuals for a particular asset and approximate its variance using an exponentially weighted average. Gather the approximations to form the diagonal matrix $\hat \Upsilon_T$.
4. Finally:

$$\begin{align} \hat V_{T+1} = B_{T+1} \hat \Sigma_{f,T} B_{T+1}^\prime + \hat \Upsilon_T \end{align}$$

In case the last equation doesn’t look familiar:

$$\begin{align} Var(\pmb r_t) &= Var(B_t \pmb f_t + \pmb u_t) \\ &= B_t Var(\pmb f_t)B_t^\prime + Var(\pmb u_t) \\ &= B_{T+1} \hat \Sigma_{f,T} B_{T+1}^\prime + \hat \Upsilon_T \end{align}$$

Industry Factors

Let there be $J$ industry factors and $S$ style factors. Then the linear regression problem is:

$$\begin{align} r_i = \underbrace{f_1}_{\text{Country Factor}} + \sum_j \underbrace{b_{j,i}}_{\text{Industry Dummy\ \ }} \underbrace{f_j}_{\text{Return to Industry Factor}} + \sum_s \underbrace{b_{s,i}}_{\text{Style\ \ }}\underbrace{f_s}_{\text{Exposure}} + u_i \end{align}$$

The dummy variable trap leads to an unidentified solution to the least squares problem. Therefore, the problem is usually formed as constrained so that the market cap weighted industry factor returns sum to $0$:

$$\begin{align} \sum_j m_j f_j =0 \end{align}$$

Where the market cap weighting is:

$$\begin{align} m_j = \cfrac{ \sum_{i \in j} mcap_i}{\text{Total}(mcap)} \end{align}$$

Obviously, constrained optimization influences our interpretation of $f_t$.

Problem Setup

Our goal is to solve a constrained weighted least squares problem where we weight the variances of returns appropriately.

Usually our weight matrix is $\Omega = \sigma^2 \text{diag}(1/mcap_i)$.

Formally, the problem being solved is:

$$\begin{align} \argmax_{\pmb f_t} \ (\pmb r_t - \tilde B_t \pmb f_t)^\prime \Omega^{-1}(\pmb r_t - \tilde B_t \pmb f_t)\ \ s.t.\ \ C\pmb f_t = \pmb d \end{align}$$

Where $\pmb f_t \in \mathbb{R}^{K \times 1}$, $\pmb r_t \in \mathbb R^{N \times 1}$, $B_t \in \mathbb R^{N \times K}$, $C \in \mathbb R^{P \times K}$, $\pmb d \in \mathbb R^{P \times 1}$, and $\Omega \in \mathbb R^{N \times N}$. Note that $P=1+J+S$ where $J$ is the number of industries and $S$ is the number of style factors.

Final point to emphasize is that the column of $C$ describing the constraint must correspond to the columns of $B_t$. We’ve described the problem as imposing the constraint $m_j = \cfrac{ \sum_{i \in j} mcap_i}{\text{Total}(mcap)}$.

If this were the desired goal, given a $\tilde B$ matrix that has this form:

$$\begin{align} \tilde B = \begin{bmatrix} \pmb{1} & \pmb{1}(\text{ind}_1) & \dots & \pmb{1}(\text{ind}_J) & \pmb s_1 & \dots & \pmb s_S \end{bmatrix} \end{align}$$

Which would imply a $C$ of this form:

$$\begin{align} C = \begin{bmatrix} 0 & m_1 & \dots & m_J & 0 & \dots & 0 \end{bmatrix} \end{align}$$

Solution

After solving the associated Lagrangian for $\pmb f_t$ we see

$$\begin{align} \pmb f_t = W_t \pmb r_t \end{align}$$

Where the linear estimator $W_t$ is:

$$\begin{align} W_t = (I - Q)(\tilde B_t^\prime \Omega^{-1}\tilde B_t)^{-1} \tilde B_t^\prime \Omega^{-1} \end{align}$$

and

$$\begin{align} Q = (\tilde B_t^\prime \Omega^{-1} \tilde B_t)^{-1} C'(C(\tilde B_t' \Omega^{-1} \tilde B_t)^{-1} C')^{-1} C \end{align}$$

Perhaps a slightly easier way to read these formula is to define $A = \tilde B_t^\prime \Omega^{-1} \tilde B_t$.

Now:

$$\begin{align} W &= (I-Q)A^{-1}\tilde B_t' \Omega^{-1} \\ \\ Q &= A^{-1} C' (CA^{-1}C')^{-1}C \end{align}$$

Interpreting $W_t$

Hopefully giving the linear estimator that maps $\pmb r_t \mapsto \pmb f_t$ the name $W_t$ and that it has $N$ columns is suggestive that $W_t$ has something to do with the weights of a portfolio. In fact, each row of $W_t$ gives the weights of assets in the portfolio which generates the return to the corresponding row in $\pmb f_t$. Recall that $W_t \in \mathbb R^{K \times N}$:

$$\begin{align} \underbrace{f_k}_{\text{Portfolio Return}} = \underbrace{\pmb w_k^\prime}_{\text{Portfolio Weights}} \quad\underbrace{\pmb r_t}_{\text{Asset Returns}} \end{align}$$