We don’t need a numerical solver

• We can find the tangency portfolio without using any type of numerical solver (if short sales are allowed).
• We don’t need cvxopt.
• We do need to solve some equations, but that is fairly easy.
• Can do similar for GMV portfolio and frontier.

Equations to solve

• Solve \[\begin{pmatrix} \text{var}_1 & \cdots & \text{cov}_{1n} \\ \vdots & \vdots & \vdots \\ \text{cov}_{1n} & \cdots & \text{var}_n \end{pmatrix} \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} \mu_1 - r_f \\ \vdots \\ \mu_n - r_f \end{pmatrix}\]
• The solution minimizes (1/2) * variance minus risk premium. It is on the capital allocation line.
• Divide by sum of weights to get tangency portfolio.

Solve in Excel

• The equations are solved in Excel with the MINVERSE and MMULT functions.
• MMULT(RPREM, MINVERSE(COV))
• RPREM should be in a row
• Result will be in a row

Global minimum variance portfolio

• Solve \[\begin{pmatrix} \text{var}_1 & \cdots & \text{cov}_{1n} \\ \vdots & \vdots & \vdots \\ \text{cov}_{1n} & \cdots & \text{var}_n \end{pmatrix} \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} 1 \\ \vdots \\ 1 \end{pmatrix}\]
• Divide by sum of weights

Frontier (hyperbola)

• Solve

\[\begin{pmatrix} \text{var}_1 & \cdots & \text{cov}_{1n} \\ \vdots & \vdots & \vdots \\ \text{cov}_{1n} & \cdots & \text{var}_n \end{pmatrix} \begin{pmatrix} w_1 \\ \vdots \\ w_n \end{pmatrix} = \begin{pmatrix} \mu_1 \\ \vdots \\ \mu_n \end{pmatrix}\]

• Divide by sum of weights
• Put some weight x on the GMV portfolio and 1-x on this portfolio. Vary x and trace out frontier.

Excel Example

• portfolios.xlsx
• U.S., developed, and emerging from Applied Finance

Julius Example

• Ask Julius to use yfinance to get Yahoo adjusted closing prices for
  • SPY = S&P 500
  • VBR = Vanguard small-cap value
  • IEF = Treasury bonds
  • UUP = U.S. dollar bullish
• Ask Julius to downsample prices to end-of-month and compute monthly returns as percent changes in the downsampled prices.

• Ask Julius to compute means and covariance matrix as numpy arrays.
• Tell Julius the risk-free rate is 0.05/12 and ask Julius to compute the risk premia.
• Ask Julius to multiply the risk premia by the inverse of the covariance matrix.
• Ask Julius to divide the result by the sum of its elements.