6.4Lesson 6.4 · Module 6 — Portfolio Theory

The Efficient Frontier

Many portfolios are feasible, but only the upper minimum-variance boundary is efficient. Watch one canvas build the frontier from scratch.

Dominance: northwest is better
Feasible set from weight vectors
Minimum-variance boundary
Global minimum-variance portfolio
Only upper branch is efficient

Start

Learning objectives

By the end of this lesson, you should be able to:

1Define portfolio dominance and explain why northwest is better.
2Connect a single weight vector to a single point on the risk-return graph.
3Describe the feasible set as the collection of all achievable portfolios.
4Construct the minimum-variance boundary by finding the lowest-σ portfolio for each target return.
5Identify the global minimum-variance portfolio as the leftmost point.
6Explain why only the upper branch of the boundary is efficient.
7Recognize that the efficient frontier narrows choice but does not select one portfolio.

6.4.1Section 1 · Dominance

Northwest is better

Before building the frontier, we need a rule for comparing two portfolios: a portfolio that offers more return at the same risk, or less risk at the same return, dominates.

Portfolio dominance

A dominates B if E [R_{A}] \geq E [R_{B}] and σ_{A} \leq σ_{B}

With at least one strict inequality. On the graph, the dominating portfolio sits to the upper-left (northwest).

$E [R_{A}], E [R_{B}]$: Expected returns of portfolios A and B.
$σ_{A}, σ_{B}$: Standard deviations of A and B.

Interaction · Dominance check

For each pair, decide whether one dominates the other — or neither. Remember: northwest is better.

A	E[R] = 8%	σ = 12%
B	E[R] = 8%	σ = 9%

Which is correct?

C	E[R] = 10%	σ = 11%
D	E[R] = 7%	σ = 11%

Which is correct?

E	E[R] = 6%	σ = 5%
F	E[R] = 10%	σ = 12%

Which is correct?

G	E[R] = 9%	σ = 8%
H	E[R] = 8%	σ = 10%

Which is correct?

6.4.2Section 2 · The construction

Building the frontier on one canvas

The same coordinate system develops through eight stages. Step through to watch the feasible set, the boundary, the GMV, and finally the efficient frontier emerge — rather than meeting several disconnected charts.

Stage 1

Two points and a rule

Plot two portfolios. The one up and to the left — more return at no more risk, or less risk at no less return — dominates. Northwest is better.

Historical monthly instructional estimates · MIT 15.401 · 1946–2001 — not current estimates.

6.4.3Interaction · Find the leftmost point

Minimum-variance boundary builder

Each round presents three candidates at the same target return. Select the lowest σ. After four rounds the points connect into the boundary.

Round 1 of 4 · Target E[R] = 1.20%

Three candidates all hit the target return. Pick the one with the lowest σ — the leftmost point at this return level.

α	σ = 5.55%
β	σ = 5.26%
γ	σ = 5.72%

6.4.4Section 4 · The optimization

What 'leftmost' means mathematically

For each target return level, the boundary point solves a constrained minimization. You do not solve it by hand — the key idea is 'leftmost feasible point for each return.'

Minimum-variance optimization

w min w^{T} Σ w s.t. w^{T} μ = μ^{*}, 1^{T} w = 1

For a target return μ*, find the weight vector minimizing portfolio variance subject to achieving that return and weights summing to 1.

$w$: Portfolio weight vector.
$Σ$: Covariance matrix.
$μ$: Vector of expected returns.
$μ^{*}$: Target expected return.

6.4.5Section 5 · Global minimum-variance

The leftmost point of all

The global minimum-variance portfolio (GMV) has the lowest volatility of any feasible portfolio — no allocation achieves less risk.

Asset	GMV weight
GM	48.58%
IBM	45.34%
Motorola	6.08%
E[R] / σ	1.23% / 5.25%

The GMV is a single point, not a curve. The boundary wraps around the left edge of the feasible set; the GMV is the leftmost point on that curve.

6.4.6Section 6 · Why only the upper branch is efficient

Eliminate the dominated branch

The boundary has two branches meeting at the GMV. Every point on the lower branch is dominated by an upper-branch twin at the same σ. Classify each pair below.

P1	σ = 5.80	E[R] = 1.45
P2	σ = 5.80	E[R] = 0.95

P3	σ = 5.50	E[R] = 1.35
P4	σ = 5.50	E[R] = 0.85

P5	σ = 6.00	E[R] = 1.50
P6	σ = 8.00	E[R] = 1.80

6.4.7Section 7 · The efficient frontier

The efficient frontier

The efficient frontier is the upper portion of the minimum-variance boundary, from the GMV upward. No portfolio on it can be improved upon.

Definition · Efficient frontier

The set of portfolios offering the highest expected return for each level of risk. No other feasible portfolio offers more return at the same or lower volatility.

The frontier narrows the choice set, but it does not pick a single best portfolio. Multiple points are efficient — different risk-return trade-offs.

Efficient point	E[R] (%)	σ (%)
GMV (lowest risk)	1.23	5.25
Moderate	1.40	5.76
Higher risk / return	1.50	6.46

Different investors may choose different efficient points depending on risk tolerance. The frontier shows what is available — the choice depends on preferences.

Assumptions and limitations

Which assets are in the universe — add or remove one and the frontier changes.
Estimated expected returns, volatilities, and correlations — all uncertain.
Portfolio constraints — long-only, short-selling allowed, or other restrictions.
The time horizon — monthly estimates may not match annual goals.
Standard deviation as the risk measure — it treats upside and downside symmetrically.

Efficiency is relative to a model. Change the inputs and the frontier shifts.

6.4.8Final check · Classify five portfolios

Which are efficient?

Five portfolios: one in the interior, one on the lower branch, one is the GMV, two are on the upper frontier. Classify each.

Port.

σ %

7.00

E[R] %

1.10

Port.

σ %

5.50

E[R] %

0.90

Port.

σ %

5.25

E[R] %

1.23

Port.

Z₁

σ %

5.76

E[R] %

1.40

Port.

Z₂

σ %

6.46

E[R] %

1.50

6.4.9Transition · A point on the axis

What if you can also hold something risk-free?

We now have a curve of efficient choices. But what changes if an investor can also hold an asset whose return is known in advance?

A risk-free asset sits at $(0, r_{f})$ — zero volatility, known return. The next lesson explores what happens when this point enters the picture.

6.4.10Optional · Common questions

Questions you may still have

Try itMastery check

Pass with 4 of 6 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
Portfolio B has the same expected return as A but lower σ. What is the relationship?
02
The minimum-variance boundary contains:
03
Why is the lower branch of the boundary NOT efficient?
04
Using the MIT dataset, the GMV portfolio has approximately:
05
A portfolio is efficient when:
06
Does the efficient frontier select one universally best portfolio?

Lesson summary

1Portfolio dominance: northwest is better.
2The feasible set contains every achievable risk-return combination.
3The minimum-variance boundary holds the lowest-σ portfolio for each target return.
4The global minimum-variance portfolio is the leftmost point.
5Only the upper branch of the boundary is efficient.
6The efficient frontier narrows choice but does not select one portfolio.
7Efficiency depends on model inputs, which are estimated and uncertain.

Replay this lesson Continue to Risk-Free Asset, Tangency, and Sharpe