# mean-variance optimization === max sharpe ratio portfolio?

Noobie here. I just wanna ask a simple question:

in the context of portfolio optimization, is Mean-Variance optimization the same as the max sharpe ratio portfolio?

• While they typically yield the same portfolio, they have different objective functions. In some literature you will see an analysis of the sensitivity of the optimal objective with respect to estimation error in the input. This is a different analysis for MVO than for max-Sharpe. Jan 5, 2022 at 17:44

Basically the answer is yes, although we can also give a slightly more complicated answer:

In Mean Variance Optimization we traditionally consider two problems:

First the slightly simpler problem when there are N risky assets. In this case the solution is a curve, the famous "efficient frontier".

Then, in the next chapter of the textbook, we consider that there are N risky assets and one risk-free asset, so a total of N+1 assets. In this case we can go a little further and the solution concept involves a single point on the frontier, the famous "tangency portfolio" which is also the point that achieves the "maximum sharpe ratio". And mixes of risk free and tangency portfolio also have this Sharpe ratio and are valid solutions.

(So in this version of the problem the answer to your question is a definite yes. But you will also find people who will say that Mean Variance Optimization is equivalent to finding the efficient frontier; that is another way to look at it, when you don't assume a risk-free asset).

• (+1) I'd call finding the maximum Sharpe ratio portfolio a special case of mean-variance portfolio optimization. Jan 5, 2022 at 19:37

To be precise, no! Mean-variance optimization and the maximum Sharpe Ratio portfolio are related but different concepts.

• When someone says "mean-variance optimization" I think of someone formulating a portfolio choice problem where the investor chooses a portfolio return $$R_p$$ from a feasible set $$\mathcal{S}$$, and the investor only has preferences over the mean and variance of the portfolio return. As an expected utility maximization problem, it would take the form:

$$$$\begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over R_p)} & \operatorname{E}[u(\mu, \sigma^2)] \\ \mbox{subject to} & \operatorname{E}[R_p] = \mu \\ & \operatorname{Var}(R_p) = \sigma^2 \\ & R_p \in \mathcal{S} \end{array}$$$$

• If investor preferences are such that they like higher returns and dislike higher variance, then a mean-variance investor will choose a portfolio such that its return is somewhere on the efficient side of the mean-variance frontier. (I define these terms below.)

• The maximum Sharpe Ratio portfolio (aka tangency portfolio) is a particular portfolio on the efficient side of the mean-variance frontier.

• The maximum Sharpe Ratio portfolio comes up a lot, but that an investor only cares about mean and variance does not on its own imply that he/she will buy the maximum Sharpe Ratio portfolio. You need additional assumptions.

• An investor that only cares about mean and variance (likes a higher mean return and dislikes higher variance) will choose a portfolio along the mean-variance frontier: the frontier of minimum volatility for any given expected return. When someone says "mean-variance optimization," they may be referring to optimization problem of finding the mean-variance frontier.

• Note that this kind of mean-variance investor doesn't care about hedging risks from their job, weather, etc...; doesn't care about skewness, maximum loss, etc.... It's a strong assumption.

## The mean-variance frontier (minimum variance for any given mean)

Let $$R$$ be a random vector denoting the return of $$n$$ different assets. $$\operatorname{E}[R]$$ is a vector of expected returns. Let $$\Sigma = \operatorname{Var}(R)$$ be the covariance matrix for the assets. Let $$\mathbf{x}$$ be a vector of portfolio weights. Let's say that buying and shorting is allowed so that feasible set $$\mathcal{S} = \left\{ \mathbf{x} \cdot R \; |\; \mathbf{x} \in \mathbb{R}^n \text{ and } \sum x_i = 1 \right\}$$. Note this will achieve portfolio return $$\mathbf{x}' R$$. The expected return of the portfolio is $$\mathbf{x}' \operatorname{E}[R]$$ (i.e. $$\sum_i x_i \operatorname{E}[R_i]$$). The variance for the portfolio return is $$\mathbf{x}'\Sigma \mathbf{x}$$.

The point $$\left( \mu , \mathbf{x}'\Sigma \mathbf{x} \right)$$ lies on the mean-variance frontier if it's a solution to:

$$$$\begin{array}{*2{>{\displaystyle}r}} \mbox{minimize (over \mathbf{x})} & \mathbf{x}'\Sigma \mathbf{x} \\ \mbox{subject to} & \sum x_i = 1 \\ & \mathbf{x}'\operatorname{E}[R] = \mu \end{array}$$$$

That is, portfolio return $$\mathbf{x}'R$$ achieves expected return $$\mu$$ at minimum possible variance. It turns out the mean-variance frontier has two sides:

• The so called efficient side where a portfolio has maximum expected return for a given variance.

• The inefficient side where a portfolio has minimum expected return for a given variance.

• Practical comment: Stepping away from theory and back to practical reality, the problem here is of garbage in and garbage out: you don't know $$\operatorname{E}[R]$$ or $$\Sigma$$ with any precision, and garbage inputs lead to garbage portfolio choice outputs. A quant approach to portfolio choice will involve trying to generate somewhat less ridiculously noisy estimates of expected returns and covariance and formulating a more humble portfolio choice problem cognizant of how difficult it is to estimate those objects.

## Mean variance efficient portfolios

A highly related (but different) problem is to achieve maximum expected return for a given variance $$\sigma^2$$. A portfolio is said to be mean-variance efficient if it gives the maximum expected return achievable for a given level of variance:

$$$$\begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over \mathbf{x})} & \mathbf{x}'\operatorname{E}[R] \\ \mbox{subject to} & \sum x_i = 1 \\ & \mathbf{x}'\Sigma\mathbf{x} = \sigma^2 \end{array}$$$$

Mean-variance efficient portfolios give returns that lie along higher expected return (i.e. efficient) side of the mean-variance frontier.

## Mean-variance frontier constructed from risky assets

If $$R$$ includes all risky assets and $$\Sigma$$ is full rank (i.e. random vector $$R$$ does NOT include the risk-free rate or redundant assets that allow construction of a risk free rate), then the corresponding mean-variance frontier is called the mean-variance frontier of risky assets.

The distinction here is that one is excluding the risk free rate from the allowed investments.

## Maximum Sharpe ratio portfolio (i.e. tangency portfolio)

A particular portfolio on the efficient side of the mean-variance frontier constructed using risky assets is the tangency portfolio. This portfolio lies on the mean-variance frontier of risky assets and achieves the maximum possible Sharpe ratio.

$$$$\begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over \mathbf{x})} & \frac{\mathbf{x}'\left( \operatorname{E}[R] - r_f \right)}{\sqrt{\mathbf{x}'\Sigma\mathbf{x}}} \\ \mbox{subject to} & \sum x_i = 1 \end{array}$$$$

### Mean-variance efficient frontier of all assets

If there exists a risk free rate, it can be shown that the efficient side of the mean-variance frontier over all assets (as opposed to just risky assets) is produced by investing in varying combinations of the tangency portfolio and the risk free rate.

Max Sharpe portfolio is a special case in Mean Variance optimization. Special in the way that by setting the right risk aversion parameter, you will get the same result from both optimization methods. However it is not right to say they are the same. Mean Variance optimization is much more flexible as the user can tailor the optimization set-up by setting the suitable risk aversion parameter.