Limitation of using black's shortcut to portfolio optimization in excel
Limitation of using Black's Shortcut to
Portfolio Optimization in Excel
Given return matrix of securities in the universe
Step 1: setup data
Let = , be the Variance/Covariance matrix and = , 1 be the Return column vector,
The weight of securities in the portfolio can be obtained by =∑
The efficient frontier can be obtained by forming linear combination of two portfolios and
Portfolio standard deviation is = ∑ +∑ ∑ , ,
Portfolio return is =∑
Sharpe ratio is ℎ =
Step 2: plot the efficient frontier
To plot the efficient frontier, one lets weight runs within a range, let's say -1 (short-sell 100%) to 1
(100%). Respective and are calculated appropriately.
The efficient frontier is a scattered plot with as x-axis and as y-axis
The optimal portfolio tangent with Capital Market Line is the one with highest Sharpe ratio. That is
One issue arises here. The position that produces Sharpe ratio equal 1.30872 may be close, but not
exactly the optimal one.
The range has a spread (in this case 0.05). If we makes the spread smaller, we may get to a Sharpe
ratio even higher than 1.30872
Indeed, at = 0.64, Sharpe ratio hits 1.30873
The smaller the spread we make, the closer we get to the optimal portfolio.
The solution to the issue is to use a solver that would maximize Sharpe ratio, with weights as
This way, we utilize Excel's computation power to solve the tangency portfolio problem for us.