The Global Minimum Variance Problem

Mathematically, choose portfolio weights to solve the following constrained optimization problem:

\[ \underset{w_1,w_2,\dots,w_N}{\text{min}} \text{var}[r_{p}] \]

subject to constraints: \(\sum_i w_i=1\)

Optimization in Python

The function cvxopt.solvers.qp solves problems of the general form: \[\begin{align} \underset{w}{\text{min }}& \frac{1}{2} w' Q w + p'w \\ \text{subject to } & Gw \le h \\ & Aw = b \\ \end{align}\]

• We can write portfolio variance as \(w' V w\) for weights \(w\) and covariance matrix \(V\).

GMV in Python

def gmv(means, cov):
    n = len(means)
    Q = matrix(cov, tc="d")
    p = matrix(np.zeros(n), (n, 1), tc="d")
    # Constraint: short-sales allowed
    G = matrix(np.zeros((n,n)), tc="d")
    h = matrix(np.zeros(n), (n, 1), tc="d")
    # Constraint: fully-invested portfolio
    A = matrix(np.ones(n), (1, n), tc="d")
    b = matrix([1], (1, 1), tc="d")
    sol = Solver(Q, p, G, h, A, b)
    wgts = np.array(sol["x"]).flatten() if sol["status"] == "optimal" else np.array(n * [np.nan])
    return wgts

Short-sales constraints: GMV

def gmv_ssc(means, cov):
    n = len(mns)
    Q = matrix(cov, tc="d")
    p = matrix(np.zeros(n), (n, 1), tc="d")
    # Constraint: short-sales not allowed
    G = matrix(-np.identity(n), tc="d")
    h = matrix(np.zeros(n), (n, 1), tc="d")
    # Constraint: fully-invested portfolio
    A = matrix(np.ones(n), (1, n), tc="d")
    b = matrix([1], (1, 1), tc="d")
    sol = Solver(Q, p, G, h, A, b)
    wgts = np.array(sol["x"]).flatten() if sol["status"] == "optimal" else np.array(n * [np.nan])
    return wgts

Upper and Lower position limits: GMV

def constrained_gmv(means, cov, min_wgt, max_wgt):
    n = len(means)
    Q = matrix(cov, tc="d")
    p = matrix(np.zeros(n), (n, 1), tc="d")
    # Constraint: minimum and maximum position limits
    G = matrix(np.vstack((-np.identity(n), np.identity(n))), tc="d")
    h = matrix(np.append(min_wgt*np.ones(n), max_wgt*np.ones(n)), (2*n, 1), tc="d")        
    # Constraint: fully-invested portfolio
    A = matrix(np.ones(n), (1, n), tc="d")
    b = matrix([1], (1, 1), tc="d")
    sol = Solver(Q, p, G, h, A, b)
    wgts = np.array(sol["x"]).flatten() if sol["status"] == "optimal" else np.array(n * [np.nan])
    return wgts  

Portfolio optimization inputs

• Set of expected returns for assets

• Set of std deviations (variances) for assets

• Set of correlations (covariances) across assets

Strategy 1: Est-All

• use historical data to estimate expected returns, standard deviations, and correlations.
  
• optimal risky portfolio is the tangency portfolio
• scale tangency up or down depending on risk aversion or target expected return

Strategy 2: Est-SD-Corr

• use historical data to estimate standard deviations and correlations
• assume expected returns are the same across all assets.
  
• optimal risky portfolio is the global minimum variance portfolio.
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

Strategy 3: Est-SD

• use historical data to estimate standard deviations only
• assume correlations across assets are zero
• assume expected returns are the same across all assets
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

Strategy 4: Est-None

• do not use historical data to estimate expected returns, standard deviations, or correlations.
• the optimal portfolio is an equal-weighted portfolio of the assets (\(1/N\) portfolio).
• for the purposes of determining optimal capital allocation, use the cross-sectional average of the historical time-series average return as the expected return input.

Alternatives (for a later date)

• Better method to forecast volatility?
• Use a factor model to estimate cross-asset correlations?
• Estimate within- and across-industry correlations?
• Model industry correlations using a factor model?

Today’s notebooks contain backtests comparing estimation choices and position limit choices for portfolios of

• stocks, bonds, and gold
• Fama-French 48 industry portfolios.