Equivalent Mean-Variance Portfolio Optimization Formulations
Summary
The document compares three standard mean-variance portfolio formulations: maximizing expected return subject to a risk limit, minimizing variance subject to a return target, and minimizing a risk-adjusted objective with a tradeoff parameter. It assumes fully invested portfolios with nonnegative weights, then asks whether adding return or risk constraints to the tradeoff formulation is meaningful.
The answer treats the parameterized objective as a complete formulation: changing the parameter traces the efficient frontier, while the constraints in the other two formulations play a similar role in selecting a point on that frontier. The document states that each frontier portfolio has minimum variance for its expected return and describes solving across parameter values, target returns, or risk levels as equivalent approaches. It provides no derivation or discussion of practical complications such as estimation error, transaction costs, or additional portfolio constraints.
Key ideas
- Mean-variance optimization can maximize return under a risk cap or minimize variance for a return target.
- A risk-return tradeoff parameter selects preferences within a combined objective.
- Sweeping that parameter traces the efficient frontier.
- The constrained and parameterized formulations produce the same frontier under the stated setup.
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Full text
# Different ways of portfolio optimization
# Different ways of portfolio optimization
There are different ways to optimize portfolios:
$$ \max R^Tw\tag{1}$$
or
$$ \min w^T \Sigma w\tag{2}$$
and finally using a risk tolerance $\lambda$:
$$ \min{(w^T\Sigma w-\lambda R^T w)}\tag{3}$$
Suppose we have the constraint $\sum w_i = 1$, $w_i\ge 0$ for all the optimization problems.
Additionally, we can define further constraints for problems $(1)$ and $(2)$:
For $(1)$: $w^T\Sigma w\le \sigma$, i.e. the risk should not exceed a certain level $\sigma$.
The same is possible for $(2)$ with return, adding the constraint: $R^T w\ge r$, for a minimal target return $r$.
My question is, in the optimization problem $(3)$, does it make sense to add a constraint like $w^T \Sigma w \le \sigma$ or $R^Tw \ge r$?
Am I right to say that adding such a constraint we would discard the solution (a efficient frontier portfolio) which does not satisfy this constraint?
## Answer by James (score 2, accepted)
https://quant.stackexchange.com/a/14820
It doesn't make sense because (3) is a complete set up: it defines the (minus utility) function that you have to minimize wrt $w$. The parameter $\lambda$ allows one to assess the tradeoff between risk and return explicitly. On the other hand, in (1) and (2) such parameter is absent but the constraints on risk in (1) and on return in (2) perform a function similar to that of $\lambda$.
There is only one efficient frontier. Each point on the frontier is a portolio that has minimal variance for a given expected return.
There are 3 equivalent ways to obtain the frontier, and Wikipedia mentions two of them explicitly: I) Solve (3) for all positive values of $\lambda$ II) Solve (2) for all possible values of expected return. One can also solve (1) for all possible values of portfolio variance. In any case, the solution is the very same efficient frontier.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.