# 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?

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.

• Thanks for you answer. I would agree if the following is true: What is then the relationship of solution of the problems? For example are the efficient frontiers the same? Does this imply, if I have a maximal risk or minimal target return I should you use $(1)$ or $(2)$. Otherwise $(3)$? What is then the advantage of $(3)$?
– math
Sep 20 '14 at 18:53
• For example it is stated on en.wikipedia.org/wiki/… that every solutino of $(3)$ leads to a efficient frontier portfolio. why is this true?
– math
Sep 20 '14 at 19:10
• @user8 I think James is pretty clear. For these problems, it is pretty easy to analytically show that they all produce the same efficient frontier. So for a $\lambda$ you could find a target return or target volatility problem that will produce the same optimal portfolio. Whether you want to use one or the other depends on what you're trying to do and what optimization software you have. Nevertheless, contra James I still could imagine a situation where I would optimize utility and still constrain variance or tracking error.
– John
Nov 19 '14 at 22:41