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Chapter 12: Portfolios without Short Sales
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Classical problem: Unconstrained optimization
is the Sharpe ratio
Unconstrained: Implementation in Excel
Constrained maximization: No short sales
Constant c is important!
Different c’s will give different porttolios
xi >= 0, I = 1,…,N
Constrained: Implementation in Excel
Start with arbitrary solution
Use Solver, adding two constraints:
Non-negativity
Portfolio sums to 1
Solution with constraints
Changing c changes the portfolio
Short sale constraints negatively impact returns
Excel/Solver can accommodate more complicated constraints
And the answer is …
Variance-covariance matrix Means
0.100.03-0.080.058%
0.030.200.020.039%
-0.080.020.300.2010%
0.050.030.200.9011%
c3.0%<-- This is the constant
Optimal portfolio without short sale restrictions (Chapter 9, Proposition 1)
0.6219<-- {=MMULT(MINVERSE(B3:E6),G3:G6-C8)/SUM(MMULT(MINVERSE(B3:E6),G3:G6-C8))}
Total 1<-- =SUM(B11:B14)
Portfolio mean 8.62%<-- {=MMULT(TRANSPOSE(B11:B14),G3:G6)}
Portfolio sigma 19.39%<-- {=SQRT(MMULT(TRANSPOSE(B11:B14),MMULT(B3:E6,B11:B14)))}
(mean-constant)/sigma
28.99%<-- =(B17-C8)/B18
PORTFOLIO OPTIMIZATION ALLOWING SHORT SALES
Follows Proposition 1, Chapter 9
Variance-covariance matrix Means
0.100.03-0.080.058%
0.030.200.020.039%
-0.080.020.300.2010%
0.050.030.200.9011%
c5.0%<-- This is the constant
Here we start with an arbitrary feasible portfolio and use Solver
Total 1<-- =SUM(B11:B14)
Portfolio mean 9.02%<-- {=MMULT(TRANSPOSE(B11:B14),G3:G6)}
Portfolio sigma 23.81%<-- {=SQRT(MMULT(TRANSPOSE(B11:B14),MMULT(B3:E6,B11:B14)))}
(mean-constant)/sigma
16.89%<-- =(B17-C8)/B18
PORTFOLIO OPTIMIZATION
WITH MORE COMPLICATED CONSTRAINTS
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