The minimum-variance portfolio and the efficient frontier

Which portfolios can you build?

The mixes of the available assets that you could hold are the feasible portfolios . With two assets the mixes lie on one curve. With many assets they fill a region in the risk and return plane. Points above and to the left of that region cannot be built from these assets.

The Markowitz efficient frontier is the region’s upper-left edge: for each level of risk, the highest expected return (equivalently, the lowest risk for each return). The minimum-variance frontier is the whole left edge, including its lower part. That lower part is inefficient, because a portfolio on the upper part has the same risk and a higher return. Such a dominated mix is an inefficient portfolio .

Feasible portfoliosevery mix you can build

• Minimum-variance frontier Memorize lowest risk for each return
  • Efficient frontier Memorize upper part, from the minimum-variance portfolio up
  • Inefficient part Memorize below the minimum-variance portfolio; dominated
• Minimum-variance portfolio Memorize leftmost point; return plays no part
• Optimal risky (tangency) portfolio Memorize highest Sharpe ratio; needs a risk-free asset

The optimization behind the frontier says: minimize variance for a target return k, invest everything, and allow no negative weights.

$$\min \sum_{i=1}^{N}\sum_{j=1}^{N} w_i w_j \sigma_{i,j} \quad\text{subject to}\quad \sum_i w_i = 1,\ w_i \ge 0,\ E[r_P] = k$$

The condition $w_i \ge 0$ is the no-short-selling constraint . Short selling means borrowing a security and selling it, to buy it back later, which gives a negative weight. With no short selling, k can only lie between the lowest and the highest expected return of the assets. With assets returning 6% and 13%, a target of 10% is feasible, but 5% and 14% are not.

With more than two assets this problem needs numerical methods, such as a spreadsheet solver or an optimizer, and the exam does not ask you to solve it by hand. The matrix version (minimize $w^{\top}Vw$) is background only.

How do you find the minimum-variance mix of two assets?

Set the weights so the portfolio’s variance is as low as it can go. Its expected return plays no part. The minimum-variance portfolio , also called the global minimum-variance portfolio , is the leftmost point of the frontier.

$$w_A = \dfrac{\sigma_B^2 - \sigma_{A,B}}{\sigma_A^2 + \sigma_B^2 - 2\sigma_{A,B}}, \quad w_B = 1 - w_A$$

When the correlation is given instead of the covariance, replace $\sigma_{A,B}$ with $\rho_{A,B}\sigma_A\sigma_B$.

$$w_A = \dfrac{\sigma_B^2 - \rho_{A,B}\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho_{A,B}\sigma_A\sigma_B}$$

Why is holding only the lower-risk fund a poor choice? 100% in Fund A gives 5% at 10% risk. A 75/25 mix of A and B gives 0.75 × 5% + 0.25 × 12% = 6.75% at exactly the same 10% risk. Holding only Fund A leaves 1.75 points of return on the table: it sits on the lower, inefficient part of the frontier.

At a correlation of −1 the minimum-variance mix has zero risk, with $w_A = \sigma_B \div (\sigma_A + \sigma_B)$ = 20 ÷ 30 = 66.7% (the lesson on portfolio expected return and variance shows why).

How do you hit a target return?

With two assets the target fixes the weights. Then compute the risk.

$$w_A = \dfrac{r_p - r_B}{r_A - r_B}, \quad w_B = \dfrac{r_A - r_p}{r_A - r_B}$$

These target-return weights come from the portfolio return being a weighted average, solved for the weight.

Doing this for every target return traces the two-asset minimum-variance frontier. Its part above the minimum-variance portfolio (here, weights in A from 87.5% down to 0%) is the efficient frontier.

Go deeper The frontier in one formula

Substituting the target-return weights into the variance gives the frontier in terms of the target return alone.

$$\sigma_p^2 = \dfrac{(r_p - r_B)^2\sigma_A^2 + (r_A - r_p)^2\sigma_B^2 + 2(r_p - r_B)(r_A - r_p)\sigma_{A,B}}{(r_A - r_B)^2}$$

At 9% it returns 0.017347 again. The formula solves the problem without the no-short-selling constraint and then keeps only the weights between 0 and 1.

Minimum variance or best risk-adjusted?

Once a risk-free asset exists, the best risky portfolio is not the minimum-variance one. It is the frontier portfolio with the highest Sharpe ratio , the optimal risky portfolio . It is also called the tangency portfolio , because the line from the risk-free rate touches the frontier there (the last lesson of this module draws that line).

$$\text{Sharpe ratio} = \dfrac{E[r_P] - r_f}{\sigma_P}$$

A cautious investor sits near the lower left of the efficient frontier and a risk-tolerant one near the upper right. Nobody should pick a point below the minimum-variance portfolio.

How do optimizers choose?

There are three kinds of optimization.

Kind	Objective	Constraint
Return-constrained optimization	minimize risk	a target expected return
Risk-constrained optimization	maximize expected return	a target level of risk
Risk-adjusted optimization	maximize the Sharpe ratio	none beyond the budget (and no short sales)

With Funds A and B and a 15% risk budget, the best return is about 9.88% (about 30% in Fund A), found by searching the efficient part of the frontier. Return-constrained runs also trace the lower, inefficient boundary of the feasible region.

Optimization

• Return-constrained Know the idea minimize risk for a target return
• Risk-constrained Know the idea maximize return for a target risk
• Risk-adjusted Know the idea maximize the Sharpe ratio

The optimization inputs are expected returns, volatilities, and correlations or covariances, plus the investor’s risk tolerance, capacity and willingness to bear losses, time horizon and cash needs. Small changes in expected returns can swing the weights a lot, because the inputs are estimates (the last lesson of this module returns to this).