We have eight years of monthly price data for nine passively-managed funds we're considering for our retirement portfolio. How might we use the historical data to determine the optimal investment allocation in each fund?

This is a problem with a non-linear objective and one equality constraint. We can solve analytically using method of Lagrange multipliers.

Define:
\(\sigma_{x,P}^2=\) portfolio variance
\(\mathbf{x}=\) vector of portfolio weights
\(\Sigma=\) covariance matrix
\(\lambda=\) Lagrange multipler

Objective:

\[\min_{\mathbf{x}}\sigma_{x,p}^2=\mathbf{x^T}\Sigma\mathbf{x~~\textrm{s.t.}~~\mathbf{x^T1}=1}\]

Set up the Lagrangian function as follows:

\[L(\mathbf{x},\lambda) = \mathbf{x^T}\Sigma\mathbf{x} + \lambda(\mathbf{x^T1}-1)\]

Take the partial derivatives with respect to each asset, \(x_i\) and set equal to zero:
\[0=\frac{\partial L}{\partial x_i}=2x_i\sigma_i^2 + 2\sum_{i \neq j}\sigma_{ij} + \lambda\] Also take the partial derivative with respect to \(\lambda\) and set equal to zero:

\[0 = \frac{\partial L}{\partial \lambda}=\mathbf{x^T1}-1\]
Solve 10 equations with 10 unknowns:

 VLCAX  VVIAX   VBR VGSLX   EFV    SCZ  VEMAX     SHV    BWZ lambda
 0.90% -0.65% 0.14% 0.02% 0.11% -0.11% -0.22% 100.11% -0.30% -0.00%

Solution implies shorting several assets–not realistic for retirement account.

• Simulation using sampling with replacement.
  • Can't simulate each asset's returns independently because of strong pairwise correlations.
  • Solution: randomly sample from historical sets of monthly returns for all assets.
• Obective: maximize return given upper bound on standard deviaton.
  
• Nonlinear objective and nonlinear constraints.
• Good brute force technique that requires minimal theory.
  
• Enumerate over universe of possible weights. Keep universe small using simplying constraints.
  • Assume no asset gets more than 40% allocation.
    
  • assume discrete asset allocations in 10% increments(i.e. 0%, 10%, etc.).

\(\mathbf{\mu=}\) mean vector
\(\sigma_{p,x}=\) portfolio standard deviation, scalar
\(x_i=\) portfolio weight for asset \(i\)
\(S=\) constant, max allowed standard deviation

Objective:
\[\max_{\mathbf{x}}\mathbf{x^T\mu}\]
subject to
\[\sum_{i=1}^Nx_i=1\] \[0\le x_i\le0.4~\textrm{for all }i\] \[100x_i\bmod10=0~\textrm{for all }i\] \[\sigma_{p,x} \le S \]

Maximize return assuming s.d. \(\le\) 1.5%

$max_ret
[1] 0.003847719

$sd
[1] 0.01473267

$weights
VLCAX VVIAX   VBR VGSLX   EFV   SCZ VEMAX   SHV   BWZ 
  0.2   0.0   0.1   0.0   0.0   0.0   0.0   0.4   0.3 

New terms:
\(R_i=\) return for asset \(i\)
\(\mathbf{R}=\) vector of returns, all assets
\(\mu_{p,x}=\) mean portfolio return

Assume:
\[R_i \sim iid~ N(\mu_i,\sigma_i^2)\] \[cov(R_i,R_j) = \sigma_{ij}\]
Then the return of the portfolio is

\[\mu_{p,x}=E[\mathbf{x'R}]=\mathbf{x^T}E[\mathbf{R}]=\mathbf{x^T \mu}\]
The variance of the portfolio is:

\[\sigma^2_{p,x}=\mathbf{x^T}\Sigma\mathbf{x}\]

• Assume individual assets are normally distributed random variables. Use historical historical means, variances as and covariances as parameter inputs.
• No need for simulation, just calculated portfolio mean and variance given a set of weights.
  
• Objective: maximize return given upper bound on standard deviation.
  
• Same constraints used in MC problem.
  
• Enumerate over universe of possible weights, using simplifying constraints.

Maximize return assuming s.d. \(\le\) 1.5%

$max_ret
[1] 0.003339794

$sd
[1] 0.01497017

$weights
VLCAX VVIAX   VBR VGSLX   EFV   SCZ VEMAX   SHV   BWZ 
  0.2   0.0   0.0   0.1   0.0   0.0   0.0   0.4   0.3 

• Cannot use variance or standard deviation as constraint because non-linear.
  

• Compromise: start with mean absolute deviation as constraint:
  \[\frac{1}{T}\sum_{t=1}^T\Big|\sum_ix_i\big(R_i(t)-\mu_i\big)\Big| \le M\textrm{ where M constant}\]

• Break out constraint above to satisfy linearity requirement:
  \[\sum_ix_i\big(R_i(t)- \mathbf{\mu_i}\big) \le y_t \textrm{ for all }t\]
  \[\sum_ix_i\big(R_i(t)- \mathbf{\mu_i}\big) \ge -y_t \textrm{ for all }t\]

\[\frac{1}{T}\sum_{t=1}^{T}y_t \le M\textrm{ }\]

\[\max \mu_{p,x}= \sum_i x_i \mu_i\]
subject to
\[-y_t \le \sum_ix_i\big(R_i(t)- \mathbf{\mu_i}\big) \le y_t \textrm{ for all }t\]
\[\frac{1}{T}\sum_{t=1}^{T}y_t \le M\textrm{ }\]
\[\sum_{i}x_i = 1\]
\[0 \le x_i \le 0.4,~\textrm{for all }i,~~y_t \ge0,~\textrm{for all }t\]

• Maximize return assuming mean absolute deviation \(\le\) 1.5%.
  
• Use lp() from lpSolve package.

$max_ret
[1] 0.006768058

$mad
[1] 0.015

$weights
VLCAX VVIAX   VBR VGSLX   EFV   SCZ VEMAX   SHV   BWZ 
 0.40  0.11  0.00  0.06  0.00  0.00  0.00  0.40  0.03 

Define:
\(q\ge 0\), constant

Objective:

\[\min~ \mathbf{x^T}\Sigma \mathbf{x} - q \mathbf{\mu^Tx}\]
subject to
\[\sum_{i}x_i = 1\]
\[0 \le x_i \le 0.4,~\textrm{for all }i\]

Optimize with \(q = 0.5\).

$max_ret
[1] 0.01198807

$sd
[1] 0.03607641

$weights
VLCAX VVIAX   VBR VGSLX   EFV   SCZ VEMAX   SHV   BWZ 
  0.4   0.2   0.2   0.2   0.0   0.0   0.0   0.0   0.0