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Markowitz frontier lab

Three risky assets. Change their returns, risks and correlations and watch the frontier, the minimum-variance portfolio G, the tangency portfolio P and the capital allocation line respond.

051015202530358101315Standard deviation σ (%)Expected return E[R] (%)ABCGRfP+0.70%
Possible portfoliosEfficient frontierInefficient branchCAL from the risk-free rateYour portfolio
Start from
Asset A
Asset B
Asset C
Risk-free asset
Rules and display
G

Global minimum variance

Standard deviation σ
10.05%
Expected return E[R]
8.86%
Weights
A70%
B21%
C9%
The leftmost point of the frontier. Everything on the dashed branch below it is dominated.
P

Tangency portfolio

Standard deviation σ
11.14%
Expected return E[R]
9.98%
Sharpe ratio
0.537
Weights
A45%
B25%
C30%
The highest Sharpe ratio. Every investor holds this P, mixed with the risk-free asset.
You

Your portfolio

Standard deviation σ
12.54%
Expected return E[R]
9.90%
Sharpe ratio
0.470
Weights
A20%
B70%
C10%
Inefficient. A frontier portfolio with the same σ earns 10.60%, which is 0.70 points more.
The maths behind your point
E[R] = 0.20 × 8.0% + 0.70 × 10.0% + 0.10 × 13.0% = 9.90%
Expected return is the weighted average of the assets' expected returns.
Own terms = 0.202 × 11.02 + 0.702 × 15.02 + 0.102 × 21.02 = 119.50 %2
Each asset's own risk: w2σ2, added up.
Pair terms = 2 × 0.20 × 0.70 × 0.35 × 11.0 × 15.0 + 2 × 0.20 × 0.10 × 0.20 × 11.0 × 21.0 + 2 × 0.70 × 0.10 × 0.45 × 15.0 × 21.0 = 37.86 %2
Each pair counts twice: 2 wiwjρijσiσj. Negative correlation makes this part shrink, or go below zero.
σ2 = 119.50 + 37.86 = 157.36 %2, so σ = √157.36 = 12.54%
Variance is the two parts together; the standard deviation is its square root.
Weights as decimals, in the order A, B, C; returns and SDs in percent.
Things to try
  • Switch C off. The dots collapse onto one curve: with two assets every mix lies on it. Switch C back on and the frontier moves left.
  • Drag ρAC toward −0.5 and watch G move left: lower correlation, less risk.
  • Turn short sales off. The frontier now stops at the highest-return asset instead of running on forever.
  • Raise Rf. The CAL pivots and P slides up the frontier toward riskier mixes.
  • Set your weights so your dot sits below G. The arrow shows how much return you leave on the table.

Exam coming up and the efficient frontier still feels slippery?

Bring your problem sets. We work through them together until every type feels routine.

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