FINC3017 · Investments And Portfolio Management
Modern Portfolio Theory & the Efficient Frontier
Modern Portfolio Theory & the Efficient Frontier (Week 4) generalises the two-asset case to many assets using matrix algebra. You trace the efficient frontier (the upper branch of the opportunity set), find the global minimum-variance portfolio, and add a risk-free asset to get the capital allocation line, whose slope is the Sharpe ratio. The line tangent to the frontier touches the tangency (maximum-Sharpe) portfolio, which everyone holds in some combination with the risk-free asset — the two-fund theorem — and an investor's optimal risky weight w* depends on the tangency Sharpe and their risk aversion.
What this chapter covers
- 01Markowitz mean-variance: E[R_p] = w'μ, σ_p² = w'Σw
- 02The efficient frontier (upper branch) and the global minimum-variance (GMV) portfolio
- 03GMV weights w_GMV = Σ⁻¹¹/(¹'Σ⁻¹¹) and σ²_GMV = 1/C
- 04The capital allocation line (CAL): E[R_p] = rf + w(E[R_A] − rf), with slope = Sharpe
- 05The tangency (max-Sharpe) portfolio and the two-fund theorem
- 06The capital market line (CML) from rf through the tangency portfolio
- 07Optimal risky weight w* = (E[R_T] − rf)/(A·σ_T²)
- 08Markowitz (full Σ) vs the single-index model
CAL slope and the optimal risky weight
- 2 marks(a) The CAL slope equals the tangency Sharpe ratio: (E[R_T] − rf)/σ_T = (12% − 4%)/20% = 8/20 = 0.40.
- 3 marks(b) Optimal risky weight: w* = (E[R_T] − rf)/(A·σ_T²) = 0.08/(3 × 0.20²) = 0.08/(3 × 0.04) = 0.08/0.12 = 0.6667.
- 1 mark(c) Resulting risk: σ_p = w*·σ_T = 0.6667 × 20% = 13.33%.
- 1 mark(c cont.) Resulting mean: E[R_p] = rf + w*(E[R_T] − rf) = 4% + 0.6667 × 8% = 4% + 5.33% = 9.33%.
Key terms
- Global minimum-variance (GMV) portfolio
- The single portfolio with the lowest possible variance among all combinations of the risky assets, with weights w_GMV = Σ⁻¹¹/(¹'Σ⁻¹¹). It sits at the leftmost tip of the frontier; everything above it on the upper branch is efficient.
- Capital allocation line (CAL)
- The straight line of risk-return combinations from the risk-free asset through a chosen risky portfolio. Its slope is that portfolio's Sharpe ratio, so the steepest CAL touches the tangency portfolio and represents the best risk-reward trade available.
- Tangency portfolio
- The risky portfolio with the maximum Sharpe ratio — where the CAL is tangent to the efficient frontier. With a risk-free asset, every investor holds the tangency portfolio plus borrowing or lending, regardless of their risk aversion (the two-fund theorem).
- Two-fund theorem
- With a risk-free asset and homogeneous expectations, all investors split their money between just two funds — the risk-free asset and the single tangency portfolio — differing only in the mix. Risk aversion sets the proportions, not the choice of risky portfolio.
- Optimal risky weight (w*)
- The fraction of wealth a utility-maximising investor places in the risky tangency portfolio, w* = (E[R_T] − rf)/(A·σ_T²). It rises with the excess return and falls with risk aversion A and with tangency variance.
Modern Portfolio Theory & the Efficient Frontier FAQ
What is the difference between the CAL and the CML?
The capital allocation line connects the risk-free asset to ANY chosen risky portfolio, while the capital market line is the specific CAL drawn through the tangency (market) portfolio — the one with the highest Sharpe ratio. Under CAPM equilibrium the tangency portfolio is the market portfolio, so the CML is the best possible CAL everyone can access.
Why does everyone hold the same tangency portfolio?
The two-fund theorem says that once a risk-free asset exists, the highest-Sharpe risky portfolio dominates all others, so every investor wants that same mix of risky assets. They differ only in how much they borrow or lend at the risk-free rate, which is set by their risk aversion, not by a different choice of risky portfolio.
How does risk aversion change my optimal allocation?
A higher coefficient of risk aversion A lowers the optimal risky weight w* = (E[R_T] − rf)/(A·σ_T²), so a more cautious investor lends more to the risk-free asset and holds less of the tangency portfolio. The tangency portfolio itself does not change — only your position along the capital allocation line does.
Exam move
Be able to move from a tangency portfolio plus risk-free rate to the CAL slope and the optimal weight w* in seconds, and know that the GMV is the leftmost frontier point. Understand the two-fund theorem conceptually — that everyone holds the same risky mix and only the borrowing/lending split varies — because the exam tests this idea as both a calculation and a concept.
