Different ways of portfolio optimization

Question

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 1

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.