Tangency Portfolios with Long–Short Weight Bounds
Summary
The document asks how to find a tangency portfolio when weights must lie between negative one and one while summing to one. It frames the objective as maximizing the portfolio’s expected return divided by its volatility, and asks whether the added bounds allow a closed-form solution. The question also reveals a mistaken candidate formula for the unconstrained portfolio.
The answer gives the standard unconstrained maximum-Sharpe-ratio weights using the inverse covariance matrix and expected returns in excess of the risk-free rate, normalized to sum to one. It explains that allowing short sales is part of the unconstrained formulation, so the proposed lower and upper bounds are not needed to express that case. By contrast, forbidding short sales requires nonnegative weights and generally calls for numerical optimization rather than the same closed form. The discussion does not solve the bounded optimization problem explicitly or examine its numerical methods.
Key ideas
- The unconstrained tangency portfolio uses inverse covariance and excess expected returns, normalized to sum to one.
- The proposed covariance-times-mean formula is not the standard maximum-Sharpe solution.
- Allowing short positions distinguishes the unconstrained portfolio from a long-only portfolio.
- A long-only maximum-Sharpe portfolio generally requires optimization rather than the unconstrained closed form.
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Full text
# Tangency portfolio with two additional constraints so that portfolio weights are unconstrained
# Tangency portfolio with two additional constraints so that portfolio weights are unconstrained
I know that the formula for determining the weights of the Tangency portfolio is given as $w_{tan}$ = $\frac{\Sigma \mu}{\iota^{\prime}\Sigma\mu }$, but I was wondering how to derive the weights in case we add the constraints that the weights should be larger than or equal to -1, and smaller than or equal to 1.
I was wondering whether there is a closed form solution available, and/or what the derivation looks like?
I guess the optimisation problem would look something like this: $$ \frac{w^{\prime}\mu}{\sqrt{w^{\prime}\Sigma w}} $$ s.t. $$ w^{\prime}\iota = 1 $$ $$ w_{i} \geq -1, \forall i = 1, \dots, N $$ $$ w_{i} \leq 1, \forall i = 1, \dots, N $$
Please correct me if this is the wrong representation of the problem.
## Answer by develarist (score 4)
https://quant.stackexchange.com/a/58913
Yes there are two ways to solve the tangency portfolio:
- closed-form analytical solution
- optimization problem (maximization of the Sharpe ratio)
The closed-form analytical solution you incorrectly wrote is actually
$$w_{\text{tan}} = \frac{\Sigma^{-1} \left(\mu - r_f \cdot \iota\right)}{\iota^{\prime}\Sigma^{-1}\left(\mu - r_f \cdot \iota\right)}$$
This is already the unconstrained maximum Sharpe ratio portfolio, where "unconstrained" means short-selling is allowed so that weights can be larger than or equal to $-1$ (in other words, less than $0$), and smaller than or equal to $1$. For the equivalent optimization problem we therefore don't need to impose additional constraints (besides the sum-to-1 rule) in order to "unconstrain" the weights since the unconstrained portfolio is an unconstrained problem.
On the other hand, the constrained max Sharpe portfolio, where short-sales are forbidden, does not have a simple closed-form analytical solution like the unconstrained portfolio, and does require an additional constraint of non-negativity, $w_i \geq 0 \enspace \forall i=1,\dots,N$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.