Comparing Variance and Volatility Penalties in Portfolio Optimization
Summary
The document compares the classical mean–variance portfolio objective with an alternative that penalizes return standard deviation directly. The proposed objective may be easier to interpret because volatility is expressed in the same units as returns. The author notes that it remains convex, allowing numerical optimization to retain the global-optimum guarantee when solved correctly, although it loses the quadratic form and the associated analytical solutions available in some standard settings.
It also discusses expressing the classical objective as maximizing expected return subject to a risk constraint. Because variance and standard deviation are monotonically related for nonnegative risk levels, a volatility limit can be framed in intuitive return units while corresponding to a suitable risk-aversion parameter. The document poses questions about practical use and numerical properties but does not resolve them or present empirical tests, solver comparisons, or evidence that either formulation performs better in practice.
Key ideas
- The standard Markowitz objective rewards expected return and penalizes portfolio return variance.
- A proposed alternative penalizes standard deviation, making the risk penalty easier to interpret in return units.
- The alternative remains convex but is not quadratic and generally requires numerical optimization.
- A volatility constraint can express a risk target directly and can correspond to a suitable risk-aversion setting.
- The document offers no empirical comparison or resolution of numerical-performance concerns.
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# The Viability/Usefulness of Mean Standard Deviation Optimization?
# The Viability/Usefulness of Mean Standard Deviation Optimization?
The classical Markowitz objective is: $$ f(w) = w^T \mu - \frac{\lambda}{2} w ^T \Sigma w = \mathbb{E}[r^Tw] - \frac{\lambda}{2} \sigma^2(r^Tw) $$ where $\mu$ is the vector of mean returns. This is a quadratic function, and we can directly/analytically solve for the solution in the unconstrained case, as well as in cases with various reasonable constraints (e.g. the full investment constraint $\sum_{i} w_i = 1)$. It rewards increased expected returns, and at the same rate, depending on $\lambda$, penalizes variance of returns.
I am interested in another objective, the same as above, but instead of penalizing the variance penalizes the standard-deviation of returns. I am interested in this objective mostly because the standard-deviation of returns is much easier to interpret than the variance of returns - particularly because it is in the same units. As such, we would consider: $$ g(w) = w^T \mu - \lambda \sqrt{w ^T \Sigma w} = \mathbb{E}[r^Tw] - \lambda \sigma(r^Tw) $$ Like the Markowitz objective, this function is convex, although not quadratic, so all of the theoretical goodness that we get with convex problems is retained. However, we don't get analytical solutions, but if say a numerical solver converges to a solution we are guaranteed to have a global optimum.
Is there a reason this objective isn't used more frequently? Does it have poor numerical properties?
One reason that I initially thought of is that the problem above with $f$ can be re-formulated as:
- Maximize the mean return
- Given an equality constraint on the variance, which is in turn an equality constraint on volatility (take square-roots). This constrained problem is equivalent to the unconstrained problem with a correctly chosen value of $\lambda$. The constraint can be set in terms of volatility rather than variance, which is intuitive as it is in terms of returns. Thus, if we wish to set a volatility constraint, rather than use a mysterious risk aversion, we could get explicit and interpretable solutions.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.