Portfolio Optimization and the Efficient Frontier
How should you mix stocks, bonds and gold? Plot every combination of weights, and the top-left edge is the efficient frontier: the highest return for each level of risk.
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Text version · Quant Basics, lesson 4: “Portfolio Optimization and the Efficient Frontier”
How should you mix stocks, bonds and gold? Plot every combination of weights, and the top-left edge is the efficient frontier: the highest return for each level of risk.
Every allocation on one chart
Work out the return and volatility of every mix of stocks, bonds and gold and plot them as a cloud of points. The top-left edge is the efficient frontier: the highest return for each level of risk.
Two special points
The minimum variance portfolio is the leftmost point in the whole cloud. The maximum-Sharpe ratio portfolio is where a line from the risk-free rate just touches the frontier: it earns the most excess return for each unit of risk.
Small input changes, big answer changes
Optimal weights are extremely sensitive to estimates of expected returns and correlations, and those numbers are hard to estimate in the first place. In practice, weights are often capped, or people switch to risk parity, which only looks at risk.
Key takeaways
- Portfolios on the efficient frontier have the highest return for a given risk, or the lowest risk for a given return.
- The minimum variance portfolio can be less volatile than any single asset in it, as long as correlations aren't high.
- Sharpe ratio = (return − risk-free rate) ÷ volatility: how much excess return you get for each unit of risk.
- Optimal weights are very sensitive to the inputs, so in practice people often add weight caps or use more robust methods.
Try it yourself
You can do these exercises in this lesson's interactive simulation:
- Pick “Minimum variance”: what's the portfolio volatility? Is it lower than the 6% of 100% bonds? What's the allocation?
- Pick “Max Sharpe”: how much goes to stocks, bonds and gold? What's the Sharpe ratio?
- Raise the stock × bond correlation from −0.2 to 0.8: what does the max-Sharpe allocation become?
- Set all three correlations to −0.8: why does it say they “can't all be true at once”?
Quiz and explanations
- 1. What does it mean if a portfolio sits below the efficient frontier?
- Answer: Another portfolio offers a higher return for the same risk.It's "dominated" by a portfolio on the frontier, so it isn't an optimal choice.
- 2. Return is 8%, volatility 16% and the risk-free rate 2%. What's the Sharpe ratio?
- Answer: 0.375.(8% − 2%) ÷ 16% = 0.375.
- 3. 50% stocks and 50% bonds, with volatilities of 18% and 6% and a correlation of 0. Roughly what's the portfolio volatility?
- Answer: About 9.5%.√(0.5² × 18%² + 0.5² × 6%²) = √0.009 ≈ 9.5%, below the 12% average of the two.
Terms in this lesson
- efficient frontier (Efficient Frontier): The curve joining the portfolios with the highest return for each level of risk.
- minimum variance portfolio (Minimum Variance Portfolio): The portfolio with the lowest volatility of all.
- risk parity (Risk Parity): Allocating by risk instead of money so each asset class contributes equally to portfolio swings; low-volatility assets like bonds get more money.
- Sharpe ratio (Sharpe Ratio): (Return − risk-free rate) ÷ volatility: how much excess return you earn per unit of risk.