Why Portfolio Variance Is Convex in Weights
Summary
The document resolves a confusion between variance as a function of a probability parameter and portfolio variance as a function of asset weights. A Bernoulli variable’s variance can be concave in its probability parameter, but portfolio variance, expressed as weights multiplied through the covariance matrix, is convex in the weights when the covariance matrix is positive semidefinite. These are different functions of different inputs.
In mean-variance optimization, expected return is linear and the variance penalty is a convex quadratic form, making the return-minus-risk objective concave for maximization. Equivalently, one can negate the objective and minimize a convex function. The response also notes that coherent risk measures such as expected shortfall are convex, and that some constrained problems may need only quasiconvexity. The explanation is conceptual; it does not discuss numerical solvers or constraints in depth.
Key ideas
- Variance can have different curvature depending on the variable it is defined over.
- Portfolio variance is a convex quadratic form in asset weights when the covariance matrix is positive semidefinite.
- A linear return term minus a convex variance penalty gives a concave objective for maximization.
- Negating a concave maximization objective yields an equivalent convex minimization problem.
- Coherent risk measures such as expected shortfall are convex.
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# Why is portfolio optimization a convex problem if variance is concave?
# Why is portfolio optimization a convex problem if variance is concave?
Variance is concave, so portfolio risk must be too.
The mean-variance model employs quadratic programming to optimize (minimize) portfolio risk. My understanding is that quadratic programming requires a convex objective function. So if the objective function in portfolio optimization is concave, how can it be solved with a routine that requires a convex one?
Below are my guesses:
- Portfolio risk is in fact convex somehow due to it being weighted variance
- It complies to the optimizer because it is not variance, but a positive-definite covariance matrix, that satisfies the convexity requirement
## Answer by kurtosis (score 1, accepted)
https://quant.stackexchange.com/a/57404
First, a correction is in order: the math question you cite is the variance for a Bernoulli random variable as a function of the parameter $p$. That is, indeed, concave in $p$. However, the variance of a portfolio, $w^T\Sigma w$, is not concave in $w$. So your initial presumption of concavity is not correct.
For a Bernoulli random variable, the uncertainty of outcomes is most uncertain for outcomes that are equally likely. That is very different from a portfolio where weights of $1/N$ diversify our exposure to multiple sources of risk and thus tend to reduce the total variance.
For a mean-variance portfolio optimization, we have the following problem: $$ \begin{align} \max_w &~w^T R - \frac{lambda}{2} w^T\Sigma w \\ \text{s.t.} &~||w||_1 = 1. \end{align} $$
Here, the objective function is a linear function minus a quadratic form; that is concave.
If we instead use a coherent measure of risk, the objective function just becomes $w^T R - \frac{\lambda}{2} \text{Risk}(w)$. Note that coherent risk measures (like CVaR/ES/TCE/ETL) are convex as discussed in Föllmer and Schied (2008).
Both of these objectives are concave. However, as Arshdeep's existing answer notes, a concave function can be made convex by multiplying by -1. Finally I should note that we do not even need convexity but often merely quasi-convexity (which might be the case for constrained optimizations).
## Answer by Arshdeep (score 3)
https://quant.stackexchange.com/a/57387
I'm in no way a portfolio theory expert, but the negative of a convex function is concave and vice versa. You can look at minimizing a concave function as maximizing a convex function and vice versa.
Also, the optimization problem is over the weights, and not over densities (which variance is concave in as your link shows). Portfolio variance is convex in the weights.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.