FNCE30001 Investments
Portfolio Theory
A portfolio's expected return is a plain weighted average of its assets — but its risk is not, because returns co-move imperfectly and the covariance term drags portfolio variance below the weighted average. The whole of Markowitz theory is the geometry of harvesting that gap: the bullet-shaped feasible set of risky portfolios, its leftmost tip the global minimum-variance portfolio (GMV), and its efficient upper arm. Adding a risk-free asset turns every mix with a risky portfolio into a straight line — the Capital Allocation Line (CAL) — whose slope is the Sharpe ratio, reward per unit of total risk. The investor's job splits cleanly in two by the separation theorem: first pick the steepest-CAL (maximum-Sharpe) risky portfolio, which is the same for everyone; then dial risk by mixing that portfolio with the risk-free asset according to personal risk aversion. In equilibrium that tangency portfolio is the market portfolio, and the CAL through it is the Capital Market Line (CML). The AU framing proxies the market with the S&P/ASX 200.
What this chapter covers
- 015.1 Two-asset building blocks and the role of correlation
- 025.2 Why any ρ < 1 bends the frontier left
- 035.3 The global minimum-variance portfolio (GMV)
- 045.4 N assets and the covariance input explosion
- 055.5 The risk-free asset, the CAL and the Sharpe ratio
- 065.6 Optimal allocation y* and mean-variance utility
- 075.7–5.8 The separation theorem, the market portfolio and the CML
Worked example: the CAL, the Sharpe ratio and the optimal allocation
- +1Identify. Risk-free plus one risky portfolio — every mix lies on the CAL; the slope is the Sharpe ratio and y* comes from mean-variance utility.
- +1(a) Sharpe ratio (the CAL slope): S = (E[r_p] − r_f)/σ_p = (12 − 4)/20 = 0.40.
- +1(b) Optimal weight: y* = (E[r_p] − r_f)/(Aσ_p²) = (0.12 − 0.04)/(3 × 0.20²) = 0.08/0.12 = 0.667.
- +1(c) Complete portfolio: E[r_C] = 0.04 + 0.667(0.08) = 9.33%; σ_C = 0.667(0.20) = 13.33%. Cross-check on the CAL: 0.04 + 0.40(0.1333) = 9.33%, consistent.
- +1Interpret: the cautious investor holds ~67% fund / 33% cash. A less risk-averse investor (A = 1.5) would get y* = 1.333 > 1 — borrowing at r_f to lever up — but both ride the same CAL.
Key terms
- Global minimum-variance portfolio (GMV)
- The leftmost tip of the feasible set — the combination with the lowest possible standard deviation, whatever the return. For two assets, w_D^min = (σ_E² − cov)/(σ_D² + σ_E² − 2cov). Because ρ < 1, the GMV's σ can sit below both individual assets' SDs.
- Capital Allocation Line (CAL)
- The straight line of (σ, E[r]) combinations from the risk-free rate r_f through a risky portfolio P: E[r_C] = r_f + [(E[r_p] − r_f)/σ_p]σ_C. Points left of P mix in lending; points right borrow at r_f to lever up.
- Sharpe ratio
- The slope of the CAL: S = (E[r_p] − r_f)/σ_p — reward per unit of total risk. The best risky portfolio is the one whose CAL is steepest, i.e. the maximum Sharpe ratio. Sharpe uses total σ, not beta, because P is the investor's whole risky holding.
- Optimal risky weight y*
- With mean-variance utility U = E[r] − ½Aσ², the investor puts y* = (E[r_p] − r_f)/(Aσ_p²) into the risky portfolio and the rest in cash. Higher risk aversion A lowers y* (more cash). A sets how much, never which risky portfolio.
- Separation theorem and the CML
- Portfolio choice splits into two independent steps: find the maximum-Sharpe (tangency) portfolio — the same for everyone given common beliefs — then mix it with r_f by personal A. In equilibrium the tangency portfolio is the market portfolio M, and the CAL through M is the Capital Market Line: E[r_C] = r_f + [(E[r_M] − r_f)/σ_M]σ_C.
Portfolio Theory FAQ
Does negative correlation guarantee zero risk?
No — that is a common slogan trap. Only ρ = −1, and only with the specific hedging weights, produces a genuinely zero-variance portfolio. For any −1 < ρ < 1, the portfolio SD is below the weighted average and can sit below the less-risky asset, but never reaches zero. Quote the variance formula, not the slogan.
Why is the GMV's risk below both individual assets?
Because the correlation is below 1, the two assets partly offset, so mixing them produces a combination whose standard deviation is lower than either asset alone. The GMV sits to the left of both single-asset points on the frontier — the signature payoff of imperfect correlation. The closed-form GMV weight tilts toward the lower-σ asset but holds some of the riskier one because the diversification benefit outweighs its extra variance.
What is the separation theorem, and why does it matter?
Choice splits into two steps that don't interact. Step one: find the optimal risky portfolio — the point where a ray from r_f is tangent to the efficient frontier, the maximum-Sharpe portfolio. Given common beliefs this is identical for every investor. Step two: mix it with the risk-free asset by personal risk aversion A. So the claim 'the optimal risky weights depend on the investor's risk aversion' is false: A changes the split between the tangency portfolio and cash, not the risky mix itself.
When do I use total risk σ versus beta?
In the CAL/CML/Sharpe world you measure reward against total standard deviation σ, because the portfolio P is the investor's whole risky holding. Beta and the security market line come later, for an asset added to an already-diversified portfolio, where only systematic risk survives. Plotting a single stock against σ on a CML, or expecting an inefficient stock to lie on the CML, is the classic mix-up — the efficient frontier (curved) is the best you can do with risky assets only, while the CML (straight, tangent at M) dominates it once a risk-free asset exists.
Exam move
Carry the four objects in order: the frontier, the GMV, the CAL/Sharpe, and the CML. For the GMV, use the closed-form two-asset weight and check the answer tilts toward the lower-σ asset and beats both SDs. For allocation, compute the Sharpe ratio as the CAL slope, then y* = (E[r_p] − r_f)/(Aσ_p²), and read y > 1 as borrowing to lever. Keep separation crisp: the tangency portfolio is the same for everyone; only the cash split depends on A. And never confuse the CAL/CML (total risk σ) with the SML to come (systematic β) — that single distinction settles a large share of the conceptual marks.
