The efficient frontier and the global minimum-variance portfolio

What is the investment opportunity set?

Every combination of available assets is a point on the risk–return plane. The investment opportunity set is the collection of all portfolios you can actually build.

With two assets the possibilities form a curve. If the assets are perfectly correlated the curve is a straight line between them. With any correlation below +1 the portfolio’s risk is less than the weighted average of the two risks, so the curve bulges to the left. With hundreds of assets, every point on and to the right of an outer curve can be reached by some mix, and the whole region is the opportunity set.

What are the minimum-variance frontier and the global minimum-variance portfolio?

Risk-averse investors want the least risk for any given return. For each target expected return, one portfolio has the lowest variance. Joining all of them draws the minimum-variance frontier , the left edge of the opportunity set. Its left-most point is the global minimum-variance portfolio (GMV): no portfolio of the available risky assets has lower risk.

Investment opportunity setevery portfolio you can build from the available risky assets

• Interior points Use it not the lowest risk for their return; dominated
• Minimum-variance frontier Memorize lowest variance for each expected return (the left edge)
  • Lower branch below the GMV; dominated, inefficient
  • Global minimum-variance portfolio left-most point; lowest risk of any risky portfolio
  • Upper branch = Markowitz efficient frontier from the GMV upward; more return for the same risk is impossible

The GMV is usually a mix, not the single least risky asset. Low correlations let a blend of assets be less risky than any one of them: with equity and bond funds of 20% and 10% volatility and a correlation of 0.2, the least risky mix holds some equities.

Which portfolios are efficient?

The frontier has two branches. The upper branch, from the GMV upward, is the Markowitz efficient frontier . Each portfolio on it offers the highest return available for its risk. The lower branch is not efficient: for each portfolio on it, the portfolio directly above has the same risk and a higher return. We say the lower one is a dominated portfolio .

The efficient frontier is concave. Moving right along it, each extra unit of risk buys a smaller increase in return.

Click or tap anywhere on the chart to test a portfolio: the readout says whether it is efficient or names the portfolio that dominates it. The solid line is the efficient frontier, the dashed line the dominated branch.

Exam tip “Minimum variance for its return” is not the same as “efficient”. Efficient also needs a position at or above the GMV.

Go deeper Finding the GMV weights for two assets

For two assets the weight in asset 1 that minimizes variance follows from setting the slope of the variance to zero. Equity and bond funds with σ of 20% and 10% and a correlation of 0.2 have a covariance of 0.004. The weight in equities is (0.01 − 0.004) ÷ (0.04 + 0.01 − 0.008) = 14.29%, and the GMV’s σ is 9.56%, below the bond fund’s own 10%. The formula is not required by the learning outcomes; it is useful as intuition for how a mix beats its parts.

$$w_1 = \frac{\sigma_2^{2} - \mathrm{Cov}_{1,2}}{\sigma_1^{2} + \sigma_2^{2} - 2\,\mathrm{Cov}_{1,2}}$$

On your calculator

TI BA II Plus HP 12C

1. (0.1x²−0.004)÷(0.2x²+0.1x²−(2×0.004))= 0.1429

x² square

1. 0.1ENTER×0.004−0.2ENTER×0.1ENTER×+0.008−÷ 0.1429

ENTER enter (on the HP, puts the number on the stack)

Rearrange itTwo-asset minimum-variance portfolio: solve for the weight or the risk

The question asks for

the weight in asset 1the risk of the minimum-variance portfolio

Two risky assets have standard deviations of 20.00% and 10.00% and a correlation of 0.2. The weight in the first asset that gives the minimum-variance portfolio is closest to:

Standard deviation, asset 1%Standard deviation, asset 2%Correlation, ρ

$$w_1=\frac{{\sigma }_2^2-{\mathrm{Cov}}_{1,2}}{{\sigma }_1^2+{\sigma }_2^2-2{\mathrm{Cov}}_{1,2}}$$

Cov = 0.2 × 0.2 × 0.1 = 0.004; w₁ = (0.1² − 0.004) ÷ (0.2² + 0.1² − 2 × 0.004) = 0.142857, so 14.29%

Weight in asset 114.29%

On your calculator

TI BA II PlusHP 12C

1. 0.2×0.2×0.1=STO1the covariance
2. (0.1x²−RCL1)÷(0.2x²+0.1x²−(2×RCL1))=STO2the weight as a decimal

STO store · x² square · RCL recall a stored number

1. 0.2ENTER0.2×0.1×STO1the covariance
2. 0.1ENTER×RCL1−0.2ENTER×0.1ENTER×+RCL1ENTER+−÷STO2the weight as a decimal

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Go deeper Why the frontier is only as good as its inputs, and what short selling does

The frontier is built from expected returns, standard deviations and correlations, all of which are estimates. Small errors in the inputs can move the frontier a lot, so a frontier drawn from last decade’s returns is a picture of the past more than a guide to the future. Rules on short selling and limits on holdings also change its shape: forbidding short sales removes the portfolios that need them.

How does adding assets improve the frontier?

If a new asset class is not perfectly correlated with the existing holdings, the opportunity set expands to the north-west: more return for each level of risk, or less risk for each return. Adding international assets to a domestic-only set is the usual example. The new frontier lies above and to the left of the old one. A high expected return on the new asset is not required; low correlation does the work.

Keep adding asset classes until a new one no longer improves the trade-off. A practical test compares Sharpe ratios. A new asset helps the existing portfolio when

$$\frac{E(R_{new}) - R_f}{\sigma_{new}} > \frac{E(R_p) - R_f}{\sigma_p} \times \rho_{new,p}$$

Each side is a Sharpe ratio, the extra return per unit of risk. The lower the new asset’s correlation with the portfolio, the lower the bar it has to clear.

What are the avenues for diversification?

Correlations among the major asset classes are generally not high, so there are many ways to diversify. Within one country, correlations among asset classes have been fairly stable. Between countries they have been rising with globalization. The practical routes are these.

Ways to cut risk without giving up returneach works by adding assets that do not move together

• Asset classes Memorize stocks, bonds, cash, real estate, commodities
• Index funds and ETFs Memorize cheap diversification for small portfolios
• Countries and currencies Memorize different economies and currency returns
• Not owning your employer’s stock Memorize your wages already depend on it
• Test each addition Use it trading and tracking costs; the Sharpe ratio rule
• Insurance and negatively correlated assets Use it put options, gold; a small or negative return can pay for lower risk

• Across asset classes. Domestic and foreign stocks and bonds, large and small companies, growth and value, real estate, commodities and cash. Industries and sectors can also be used.
• Through index funds or exchange-traded funds (ETFs). Spreading a small portfolio across many asset classes one security at a time means buying dozens of holdings in each class, which is expensive to trade and to keep track of. A fund that tracks an index does the job for less.
• Across countries and currencies. Economies differ in industry mix, policy and politics, and currency returns are largely uncorrelated with stock returns, which can lower the risk of a foreign holding even when the local market is volatile.
• By not concentrating in your employer. Your pay, promotions and job already depend on your employer. Holding its shares as well concentrates your wealth in one source.
• By testing each addition. Every new holding has trading and tracking costs, and a fund with a new name may duplicate exposure you already have. Use the Sharpe ratio test above.
• By buying insurance. Insurance, put options and sometimes gold have a negative correlation with a portfolio. Their expected return is often small or negative, but they cut the chance of a severe loss, which can be worth the price. This is the exception to the rule that a risk premium is positive.

Go deeper How these ideas link forward

The case for diversifying with real assets and other alternative investments rests on the same low correlations. The point about not over-weighting your employer’s shares returns as a behavioral bias in the behavioral finance lesson. And the efficient frontier is only half the story: the next lesson adds a risk-free asset and shows which single risky portfolio every investor should hold.