5 Factor models

The purpose of factor models is to impose a structure on financial variables and their covariance matrix by explaining them through a small number of common factors. This can help to overcome estimation error by reducing the number of parameters, i. e., the dimensionality of the estimation problem, making portfolio optimization more robust against noise in the data. Factor models also provide a decomposition of financial risk to systematic and security specific components [CM13].

Let \(Z_t\) be an \(N\)-dimensional random vector representing the analyzed financial variable at time \(t\). We can write \(Z_t\) as a linear function of components as

\[Z_t = \beta F_t + \theta_t,\]

where \(F_t\) is a \(K\)-dimensional random vector representing the common factors at time \(t\), \(\beta\) is the \(N\times K\) matrix of factor exposures (or loadings, betas), and \(\theta_t = \alpha + \varepsilon_t\) is the specific component such that \(\mathbb{E}(\theta_t) = \alpha\) and \(\varepsilon_t\) is white noise 1 with covariance \(\Sigma_\theta\). If \(Z_t\) represents security returns, then \(\beta\mathbb{E}(F_t)\) is the expected explained return and \(\alpha\) is the expected unexplained return.

The factors \(F_t\) should account for all common movement between pairs of variables. Thus in the model the covariance of \(Z_t\) is determined only by the covariance of \(F_t\). This requires that \(\mathrm{Cov}(\varepsilon_{t_1}, F_{t_2}) = \mathbf{0}\) for all \(t_1, t_2\) and that \(\mathrm{Cov}(\varepsilon_t) = \Sigma_\varepsilon\) is diagonal. We call this an exact factor model. If \(\Sigma_\varepsilon\) is not diagonal then we have an approximate factor model.

The above model is also static because \(F_t\) has only a contemporaneous effect on \(Z_t\). 2 Another common assumption is that the sequence of random variables \(F_t\) are weakly stationary, so that \(\mathbb{E}(F_t) = \mu_F\) and \(\mathrm{Cov}(F_t) = \Sigma_F\). This allows us to estimate the model from (historical) scenarios.

With these assumptions, we can write the expected value of \(Z_t\) as

\[\mu_Z = \alpha + \beta\mu_F.\]

The covariance matrix will be given by

\[\Sigma_Z = \beta\Sigma_F\beta^\mathsf{T} + \Sigma_\theta.\]

This way the covariance matrix \(\Sigma_Z\) has only \(NK + N + K(K+1)/2\) parameters, which is linear in \(N\), instead of having \(N(N+1)/2\) covariances, which is quadratic in \(N\).

Note that because the common factors and the specific components are uncorrelated, we can separate the risks associated with each. For a portfolio \(\mathbf{x}\) the portfolio level factor exposures are given by \(\mathbf{b} = \beta^\mathsf{T}\mathrm{x}\) and we can write the total portfolio variance as \(\mathbf{b}^\mathsf{T}\Sigma_F\mathbf{b} + \mathbf{x}^\mathsf{T}\Sigma_\theta\mathbf{x}\).

While the factor structure may introduce a bias on the covariance estimator, it will also lower its variance owing to the reduced number of parameters to estimate.

5.1 Explicit factor models

In these models the factors are predefined based on financial or economic theory, independently from the data. They are observed from e. g. macroeconomic quantities (inflation, economic growth, etc.) or from security characteristics (industry, dividend yield, market capitalization, etc.) [Con95]. The goal of these models is typically to explain security returns.

In the case of macroeconomic factor models the historical factor time-series \(\mathbf{F}\) can be observed. Then we can apply time-series regression to get estimators for \(\beta\), \(\alpha\), and \(\varepsilon\). However, enough stationary data might not be available for an accurate regression.

The simplest macroeconomic factor models use a single factor as common risk component. In Sharpe’s single factor model this common factor is the market risk. In practice, we can approximate the market risk factor by the returns of a stock index. We can either form an equally-weighted or a market capitalization-weighted portfolio of the index constituents. The covariance matrix with single factor model structure has only \(2N+1\) parameters to estimate.

For fundamental factor models there are two methods:

  • BARRA method: We determine the estimate \(\EBeta\) of the matrix \(\beta\) from the observed security specific attributes (assumed time invariant). Then we do cross-sectional regression for each \(t\) to obtain the factor time-series \(\mathbf{F}\) and its estimated covariance matrix \(\ECov_F\): \(\mathbf{F}_t = (\EBeta^\mathsf{T}\ECov_\theta^{-1}\EBeta)^{-1}\EBeta^\mathsf{T}\ECov_\theta^{-1}\mathbf{Z}_t\), where \(\ECov_\theta\) is the diagonal matrix of estimated specific variances accounting for the cross-sectional heteroskedasticity in the model.

  • Fama–French method: For each time \(t\) we order the securities into quintiles by a given security attribute. Then we long the the top quintile and short the bottom quintile. The factor realization at time \(t\) corresponding to the attribute will be the return of this hedged portfolio. After obtaining the factor time-series corresponding to each attribute, we can estimate \(\beta\) using time-series regression.

A drawback of explicit factor models is the misspecification risk. The derivation of the covariance matrix \(\Sigma_Z\) strongly relies on the orthogonality between the factors \(F_t\) and the residuals \(\theta_t\), which might not be satisfied if relevant factors are missing from the model.

5.2 Implicit factor models

Implicit factor models derive both the factors and the factor exposures from the data, it does not need any external input. However, the statistical estimation procedures involved are prone to discovering spurious correlations, and they can only work if we assume the factor exposures time invariant over the estimation period. Mainly two methods are used to derive the factors.

5.2.1 Factor analysis

This method assumes a more specific structure, a normal factor model with the additional requirements \(\mathbb{E}(F_t) = \mathbf{0}\) and \(\mathrm{Cov}(F_t) = \mathbf{I}\).

  • For the first condition, we redefine \(\alpha' = \alpha + \beta\mathbb{E}(F_t)\).

  • For the second condition, note that the variables \(\beta\) and \(\mathbf{F}_t\) can only be determined up to an invertible matrix \(\mathbf{H}\), because \(\beta\mathbf{F}_t = \beta\mathbf{H}^{-1}\mathbf{H}\mathbf{F}_t\). Thus by decomposing the factor covariance as \(\Sigma_F = \mathbf{Q}\mathbf{Q}^\mathsf{T}\) and choosing \(\mathbf{H} = \mathbf{Q}^{-1}\), we arrive at the desired structure.

The covariance matrix will then become \(\Sigma_Z = \beta\beta^\mathsf{T} + \Sigma_\theta\). The matrix \(\mathbf{Q}\) is still determined only up to a rotation. This freedom can help in finding interpretation for the factors.

Assuming that the variables \(Z_t\) are independent and identically normally distributed, we can estimate \(\alpha\), \(\beta\), and \(\Sigma_\theta\) using maximum likelihood method, yielding the estimations \(\EAlpha\), \(\EBeta\), and \(\ECov_\theta\). Then we use cross-sectional regression (accounting for the cross-sectional heteroskedasticity in \(\varepsilon_t\)) to estimate the factor time-series: \(\mathbf{F}_t = (\EBeta^\mathsf{T}\ECov_\theta^{-1}\EBeta)^{-1}\EBeta^\mathsf{T}\ECov_\theta^{-1}(\mathbf{z}_t - \EAlpha)\).

The number of factors can be determined e. g. using likelihood ratio test. The drawback of factor analysis is that it is not efficient for large problems.

5.2.2 Principal Component Analysis

The simplest way to estimate an implicit factor model is by principal component analysis (PCA), a dimension reduction method having the form of an affine transformation.

We are looking for the best approximation \(\tilde{Z}_t = \alpha + \beta F_t\) assuming \(F_t =\mathbf{d} + \mathbf{A} Z_t\), \(\mathbb{E}(F_t) = \mathbf{0}\), and \(\mathrm{dim}(F_t) = K < \mathrm{dim}(Z_t) = N\).

We can obtain this by eigendecomposition of \(\Sigma = \mathrm{Cov}(Z_t)\):

\[\Sigma = \mathbf{V}\Lambda\mathbf{V}^\mathsf{T},\]

where the matrix \(\Lambda\) is diagonal with the eigenvalues \(\lambda_1,\dots,\lambda_N\) in it ordered from largest to smallest, and the matrix \(\mathbf{V}\) contains the corresponding eigenvectors \(\mathbf{v}_1,\dots,\mathbf{v}_N\) as columns.

If we partition the eigenvector matrix as \(\mathbf{V}=[\mathbf{V}_{K}, \mathbf{V}_{N-K}]\), where \(\mathbf{V}_{K}\) consists of the first \(K\) eigenvectors, we can construct the solution: \(\beta = \mathbf{V}_{K}\), \(\alpha = \mathbb{E}(Z_t)\), and the principal components \(\mathbf{F}_t = \mathbf{V}_{K}^\mathsf{T}(Z_t - \mathbb{E}(Z_t))\). For the residuals \(\varepsilon_t = Z_t- \tilde{Z}_t = \mathbf{V}_{N-K}\mathbf{V}_{N-K}^\mathsf{T}(Z_t - \mathbb{E}(Z_t))\) we have \(\mathbb{E}(\varepsilon_t) = \mathbf{0}\) and \(\mathrm{Cov}(F_t, \varepsilon_t) = \mathbf{0}\), as required in the factor model definition. Also the factors will be uncorrelated: \(\Sigma_F = \frac{1}{T}F_tF_t^\mathsf{T} = \Lambda_K\), containing the first \(K\) eignevalues in the diagonal.

The intuition behind PCA is that it does an orthogonal projection of \(Z_t\) onto the hyperplane spanned by the \(K\) directions corresponding to the \(K\) largest eigenvalues. The \(K\)-dimensional projection \(\tilde{Z}_t\) contains the most randomness of the original variable \(Z_t\), that an affine transformation can preserve. Eigenvalue \(\lambda_i\) measures the randomness of principal component \(F_{t,i}\), the randomness of \(Z_t\) along the direction \(\mathbf{v}_i\). The ratio \(\sum_{i=1}^K \lambda_i / \sum_{i=1}^N \lambda_i\) is the fraction of randomness explained by the factor model approximation.

We can also do PCA on the correlation matrix. We get the correlation matrix \(\Omega\) from the covariance matrix \(\Sigma\) by \(\Omega = \mathbf{D}^{-1/2}\Sigma\mathbf{D}^{-1/2}\), where \(\mathbf{D} = \mathrm{Diag}(\Sigma)\). Then we keep only the \(K\) largest eigenvalues: \(\tilde{\Omega} = \mathbf{V}_K\Lambda_K\mathbf{V}_K^\mathsf{T}\). To guarantee that this approximation will also be a correlation matrix, we add a diagonal correction term \(\mathrm{Diag}\left(\mathbf{I} - \tilde{\Omega}\right)\). Finally, we transform it back into a covariance matrix. By this method we can get better performance when the scale of data variables is very different.

Advantages of PCA based factor models is that they are computationally simple, they need no external data, and that they avoid misspecification risk because the residuals and the factors are orthogonal by construction. Also they yield factors ranked according to their fraction of explained variance, which can further help determining the number of factors \(K\) to use. The drawback of implicit factors is that they do not have financial interpretation, and the factor exposures can be unstable. Factor models can be generalized in many ways; see in [BS11].

5.2.2.1 Number of factors

In practice, the number of factors \(K\) is unknown and has to be estimated. Too few factors reduce the explanatory power of the model, too many could lead to retaining insignificant factors. In [BN01] the authors propose an information criteria to estimate \(K\):

\[K^* = \mathrm{argmin}_K \mathrm{log}\left(\varepsilon_K^T\varepsilon_K\right) + K\left(\frac{N+T}{NT}\right)\mathrm{log}\left(\frac{NT}{N+T}\right),\]

where \(\varepsilon_K\) is the vector of residuals in the case of \(K\) factors.

If we wish to keep the number of factors time invariant, then a heuristic method can help: \(K = \lfloor\mathrm{log}(N)\rfloor\).

5.2.2.2 Practical considerations

In practice, PCA relies on the assumption that the number of observations \(T\) is large relative to the number of securities \(N\). If this does not hold, i. e., \(N > T\), then we can do PCA in a different way. Suppose that \(\mathbf{Z}\) is the \(N\times T\) data matrix, and \(\EMean\) is the mean of the data.

Then we do eigendecomposition of the \(T\times T\) matrix \(\frac{1}{N}(\mathrm{Z} - \EMean\mathbf{1}^\mathsf{T})^\mathsf{T}(\mathrm{Z} - \EMean\mathbf{1}^\mathsf{T}) = \mathbf{W}\Lambda\mathbf{W}^\mathsf{T}\). Then we get the principal component matrix as \(\mathbf{F} = \Lambda\mathbf{W}^\mathsf{T}\).

The most efficient way to compute PCA is to use singular value decomposition (SVD) directly on the data matrix \(\mathbf{Z}\). Then we get \(\mathbf{Z} = \mathbf{V}\Lambda\mathbf{W}^\mathsf{T}\). Now it is easy to see the connection between the above two PCA approaches:

\[\mathbf{F} = \mathbf{V}^\mathsf{T}\mathbf{Z} = \mathbf{V}^\mathsf{T}\mathbf{V}\Lambda\mathbf{W}^\mathsf{T} = \Lambda\mathbf{W}^\mathsf{T}.\]

5.3 Modeling considerations

We have arrived at the decomposition of the covariance matrix \(\ECov_Z=\EBeta\ECov_F\EBeta^\mathsf{T}+\ECov_\theta\), where \(\ECov_F\) is the \(K\times K\) factor sample covariance matrix and \(K\) is the number of factors. Typically we have only a few factors, so \(K \ll N\) and thus \(\ECov_F\) is much lower dimensional than the \(N\times N\) matrix \(\ECov_Z\). This makes it much cheaper to find the decomposition \(\ECov_F=\mathbf{F}\mathbf{F}^\mathsf{T}\), and consequently the factorization \(\ECov_Z=\mathbf{G}\mathbf{G}^\mathsf{T}\), where

(5.1)\[\mathbf{G} = \begin{bmatrix} \EBeta\mathbf{F} & \ECov_\theta^{1/2} \end{bmatrix}.\]

This form of \(\mathbf{G}\) is typically very sparse. In practice sparsity is more important than the number of variables and constraints in determining the computational efficiency of the optimization problem. Thus it can be beneficial to focus on the number of nonzeros in the matrix \(\mathbf{G}\) and try to reduce it. Indeed, the \(N \times (N+K)\) dimensional \(\mathbf{G}\) is larger than the \(N\times N\) dimensional Cholesky factor of \(\ECov_Z\) would be, but in \(\mathbf{G}\) there are only \(N(K + 1)\) nonzeros in contrast to the \(N(N + 1)/2\) nonzeros in the Cholesky factor. Because in practice \(K\) tends to be a small number independent of \(N\), we can conclude that the usage of a factor model reduced the number of nonzeros, and thus the storage requirement by a factor of \(N\). This will in most cases also lead to a significant reduction in the solution time.

In the risk minimization setting (2.1) of the Markowitz problem, according to Sec. 8.1.1.6 (Quadratic form) we can model the factor risk term \(\mathbf{x}^\mathsf{T}\EBeta\ECov_F\EBeta^\mathsf{T}\mathbf{x} \leq t_1\) as \((\frac{1}{2}, t_1, \mathbf{F}^\mathsf{T}\EBeta^\mathsf{T}\mathbf{x}) \in \mathcal{Q}^{K+2}\), and the specific risk term \(\mathbf{x}^\mathsf{T}\ECov_\theta\mathbf{x} \leq t_2\) as \((\frac{1}{2}, t_2, \mathrm{Diag}(\ECov_\theta^{1/2})^\mathsf{T}\mathbf{x}) \in \mathcal{Q}^{N+2}\). The resulting risk minimization problem will then look like

(5.2)\[\begin{split}\begin{array}{lrcl} \mbox{minimize} & t_1 + t_2 & &\\ \mbox{subject to} & (\frac{1}{2}, t_1, \mathbf{F}^\mathsf{T}\EBeta^\mathsf{T}\mathbf{x}) & \in & \mathcal{Q}^{K+2},\\ & (\frac{1}{2}, t_2, \mathrm{Diag}(\ECov_\theta^{1/2})^\mathsf{T}\mathbf{x}) & \in & \mathcal{Q}^{N+2},\\ & \EMean^\mathsf{T}\mathbf{x} & \geq & r_\mathrm{min},\\ & \mathbf{x} & \in & \mathcal{F}. \end{array}\end{split}\]

We can also have prespecified limits \(\gamma_1\) for the factor risk and \(\gamma_2\) for the specific risk. Then we can use the return maximization setting:

(5.3)\[\begin{split}\begin{array}{lrcl} \mbox{maximize} & \EMean^\mathsf{T}\mathbf{x} & &\\ \mbox{subject to} & (\frac{1}{2}, \gamma_1, \mathbf{F}^\mathsf{T}\EBeta^\mathsf{T}\mathbf{x}) & \in & \mathcal{Q}^{K+2},\\ & (\frac{1}{2}, \gamma_2, \mathrm{Diag}(\ECov_\theta^{1/2})^\mathsf{T}\mathbf{x}) & \in & \mathcal{Q}^{N+2},\\ & \mathbf{x} & \in & \mathcal{F}. \end{array}\end{split}\]

5.4 Example

In this chapter there are two examples. The first shows the application of macroeconomic factor models on the case study developed in the preceding chapters. The second example demonstrates the performance gain that factor models can yield on a large scale portfolio optimization problem.

5.4.1 Single factor model

In the example in Sec. 4.5 (Example) we have seen that the effective dimension \(\tilde{N}\) is low, indicating that there are only one or two independent sources of risk for all securities. This suggests that a single factor model might give a good approximation of the covariance matrix. In this example we will use the returns of the SPY ETF as market factor. The SPY tracks the S&P 500 index and thus is a good proxy for the U.S. stock market. Then our model will be:

\[R_t = \alpha + \beta R_{\mathrm{M}, t} + \varepsilon_t,\]

where \(R_{\mathrm{M}}\) is the return of the market factor. To estimate this model using time-series regression, we need to obtain scenarios of linear returns on the investment time horizon (\(h=1\) year in this example), both for the individual securities and for SPY.

To get this, we add SPY as the ninth security, and proceed exactly the same way as in Sec. 3.3 (Example) up to the point where we have the expected yearly logarithmic return vector \(\EMean_h^\mathrm{log}\) and covariance matrix \(\ECov_h^\mathrm{log}\). Then we generate Monte Carlo scenarios using these parameters.

scenarios_log = np.random.default_rng().multivariate_normal(m_log, S_log, 100000)

Next, we convert the scenarios to linear returns:

scenarios_lin = np.exp(scenarios_log) - 1

Then we do the linear regression using the OLS class of the statsmodels package. The independent variable X will contain the scenarios for the market returns and a constant term. The dependent variable y will be the scenarios for one of the security returns.

params = []
resid = []
X = np.zeros((scenarios_lin.shape[0], 2))
X[:, 0] = scenarios_lin[:, -1]
X[:, 1] = 1
for k in range(N):
    y = scenarios_lin[:, k]
    model = sm.OLS(y, X, hasconst=True).fit()
    resid.append(model.resid)
    params.append(model.params)
resid = np.array(resid)
params = np.array(params)

After that we derive the estimates \(\EAlpha\), \(\EBeta\), \(s_\mathrm{M}\), and \(\ECov_\theta\) of the parameters \(\alpha\), \(\beta\), \(\sigma_\mathrm{M}\), and \(\Sigma_\theta\) respectively. We also drop the off-diagonal elements of \(\ECov_\theta\).

a = params[:, 1]
B = params[:, 0]
s_M = np.var(X[:, 0])
S_theta = np.cov(resid)
S_theta = np.diag(np.diag(S_theta))

Finally, we compute the decomposition (5.1) of the covariance matrix:

G = np.block([[B[:, np.newaxis] * np.sqrt(s_M), np.sqrt(S_theta)]])

We can see that in this \(8 \times 9\) matrix there are only \(16\) nonzero elements:

G = np.array([
    [0.174, 0.253, 0,      0,      0,      0,      0,      0,      0     ],
    [0.189, 0,     0.205,  0,      0,      0,      0,      0,      0     ],
    [0.171, 0,     0,      0.181,  0,      0,      0,      0,      0     ],
    [0.180, 0,     0,      0,      0.190,  0,      0,      0,      0     ],
    [0.274, 0,     0,      0,      0,      0.314,  0,      0,      0     ],
    [0.225, 0,     0,      0,      0,      0,      0.201,  0,      0     ],
    [0.221, 0,     0,      0,      0,      0,      0,      0.218,  0     ],
    [0.191, 0,     0,      0,      0,      0,      0,      0,      0.213 ]
])

This sparsity can considerably speed up the optimization. See the next example in Sec. 5.4.2 (Large scale factor model) for a large scale problem where this speedup is apparent.

We compute the efficient frontier in the following points:

deltas = np.logspace(start=-1, stop=1.5, num=20)[::-1]

If we plot the efficient frontier on Fig. 5.1, and the portfolio composition on Fig. 5.2 we can compare the results obtained with and without using a factor model.

_images/eff_frontier_factor.png

Fig. 5.1 The efficient frontier.

_images/portfolio_composition_factor.png

Fig. 5.2 Portfolio composition \(\mathbf{x}\) with varying level if risk-aversion \(\delta\).

5.4.2 Large scale factor model

In this example we will generate Monte Carlo based return data and compare optimization runtime between the Cholesky decomposition based and factor model based representation of the covariance matrix.

The following function generates the data, and the factor model:

def random_factor_model(N, K, T):
    # Generate K uncorrelated, zero mean factors, with weighted variance
    S_F = np.diag(range(1, K + 1))
    Z_F = np.random.default_rng(seed=1).multivariate_normal(np.zeros(K), S_F, T).T

    # Generate random factor model parameters
    B = np.random.default_rng(seed=2).normal(size=(N, K))
    a = np.random.default_rng(seed=3).normal(loc=1, size=(N, 1))
    e = np.random.default_rng(seed=4).multivariate_normal(np.zeros(N), np.eye(N), T).T

    # Generate N time-series from the factors
    Z = a + B @ Z_F + e

    # Residual covariance
    S_theta = np.cov(e)
    S_theta = np.diag(np.diag(S_theta))

    # Optimization parameters
    m = np.mean(Z, axis=1)
    S = np.cov(Z)

    return m, S, B, S_F, S_theta

The following code computes the comparison by increasing the number of securities \(N\). The factor model in this example will have \(K=10\) common factors, which is kept constant, because it typically does not depend on \(N\).

# Risk limit
gamma2 = 0.1

# Number of factors
K = 10

# Generate runtime data
list_runtimes_orig = []
list_runtimes_factor = []
for n in range(5, 15):
    N = 2**n
    T = N + 2**(n-1)
    m, S, B, S_F, S_theta = random_factor_model(N, K, T)

    F = np.linalg.cholesky(S_F)
    G_factor = np.block([[B @ F, np.sqrt(S_theta)]])
    G_orig = np.linalg.cholesky(S)

    optimum_orig, runtime_orig = Markowitz(N, m, G_orig, gamma2)
    optimum_factor, runtime_factor = Markowitz(N, m, G_factor, gamma2)
    list_runtimes_orig.append((N, runtime_orig))
    list_runtimes_factor.append((N, runtime_factor))

tup_N_orig, tup_time_orig = list(zip(*list_runtimes_orig))
tup_N_factor, tup_time_factor = list(zip(*list_runtimes_factor))

The function call Markowitz(N, m, G, gamma2) runs the Fusion model, the same way as in Sec. 2.4 (Example). To get the runtime of the optimization, we add the following line to the model:

time = M.getSolverDoubleInfo("optimizerTime")

The results can be seen on Fig. 5.3, showing that the factor model based covariance matrix modeling can result in orders of magnitude faster solution times for large scale problems.

_images/runtime_comparison.png

Fig. 5.3 Runtimes with and without using a factor model.

Footnotes

1

For a white noise process \(\varepsilon_t\) we have \(\mathbb{E}(\varepsilon_t) = \mathbf{0}\) and \(\mathrm{Cov}(\varepsilon_{t_1}, \varepsilon_{t_2}) = \Omega\) if \(t_1 = t_2\), where \(\Omega\) does not depend on \(t\), otherwise \(\mathbf{0}\).

2

Additional lags of \(F_t\) in the \(Z_t\) equations can be easily allowed, then we obtain a dynamic factor model.