BAFI6010 · Advanced Investment Management
Modern Portfolio Theory & CAPM
This is the theoretical engine room of BAFI 6010. Modern Portfolio Theory (Markowitz 1952) turns expected returns and a covariance matrix into an efficient frontier of best risk-return trade-offs, and a risk-free asset unlocks the Capital Market Line through the market portfolio. The CAPM (Sharpe 1964) then prices a single asset off its systematic risk (beta) alone, because diversification removes firm-specific risk for free. Master the two diagrams — the frontier and the Security Market Line — plus the four performance ratios and the Black-Litterman input fix, and most of the Topic 2 exam falls out.
What this chapter covers
- 011. Diversification — how the correlation ρ between assets sets the size of the risk reduction
- 022. MPT (Markowitz 1952) — E(rp) = Σ wi E(ri) and σp² = Σ wi²σi² + ΣΣ wiwj Covij
- 033. Efficient frontier & GMV — the upper arm dominates; below the minimum-variance tip is inferior
- 044. Capital Market Line — rf tangent to the frontier at the market portfolio; slope = Sharpe ratio
- 055. Systematic vs unsystematic risk — only undiversifiable market risk is rewarded
- 066. CAPM & the SML — E(ri) = rf + βi(E(rm) − rf), βi = Cov(i,m)/Var(m)
- 077. Alpha & the sign map — above the SML = underpriced = buy; below = overpriced = avoid
- 088. Ratios & Black-Litterman — Sharpe/Treynor/information/alpha; equilibrium returns blended with views
Two-asset portfolio risk and return
- +2Expected return is the weighted average: E(rp) = 0.60(14%) + 0.40(9%) = 8.4% + 3.6% = 12.0%.
- +1Write the full two-asset variance including the cross-term: σp² = wG²σG² + wV²σV² + 2 wG wV ρ σG σV.
- +1Own-variance terms: 0.60²(0.22)² = 0.017424 and 0.40²(0.12)² = 0.002304.
- +1Covariance term: 2(0.60)(0.40)(0.30)(0.22)(0.12) = 0.0038016. Sum σp² = 0.0235296, so σp = √0.0235296 = 15.3%.
Key terms
- Efficient frontier
- The set of portfolios giving the highest expected return for each level of risk (equivalently the lowest risk for each return). Portfolios below it are dominated and never chosen.
- Global minimum-variance portfolio (GMV)
- The leftmost tip of the frontier — the lowest-risk mix of the risky assets. The efficient arm is everything above it.
- Capital Market Line (CML)
- The straight line from the risk-free rate rf tangent to the frontier at the market portfolio. It plots reward against TOTAL risk σ; its slope is the market Sharpe ratio.
- Systematic vs unsystematic risk
- Systematic (market-wide) risk cannot be diversified away and is measured by beta; unsystematic (firm-specific) risk diversifies away as you add assets. Only systematic risk is rewarded.
- Beta (β)
- An asset's sensitivity to the market, β = Cov(i, market)/Var(market), estimated by regressing the asset's returns on the market's. It is the CAPM's single risk measure for one asset.
- Security Market Line (SML)
- The CAPM plotted as required return vs beta: E(ri) = rf + β(E(rm) − rf). In equilibrium every asset sits on it; it plots against SYSTEMATIC risk, not total σ.
- Alpha (α)
- The vertical gap between an asset's forecast return and its SML-required return. Positive alpha (above the line) = underpriced = buy; negative alpha (below) = overpriced = avoid.
- Black-Litterman model
- A method that blends market-equilibrium (cap-weight-implied) returns with the manager's private views to produce stable return inputs for the SAME mean-variance optimiser — not a new optimiser.
Modern Portfolio Theory & CAPM FAQ
What is the difference between the CML and the SML?
The CML plots reward against TOTAL risk (σ) and only describes efficient portfolios — blends of the risk-free asset and the market portfolio. The SML plots required return against SYSTEMATIC risk (β) and holds for EVERY asset, efficient or not. Swapping the x-axis is the single most common Topic 2 mistake.
Why are investors only rewarded for systematic risk?
Because unsystematic (firm-specific) risk can be removed for free simply by diversifying, the market will not pay a premium to carry it. Only systematic (market-wide) risk survives diversification, so only beta earns a risk premium — which is exactly what the CAPM prices.
How do I tell if a stock is a buy under the CAPM?
Compute the required return from the SML, rf + β(E(rm) − rf), and compare it with the forecast return. If the forecast is higher, alpha is positive, the stock plots above the SML and is underpriced (a buy); if lower, it is overpriced (avoid). Remember: above the line = underpriced — do not read it backwards.
What is the difference between the Sharpe and Treynor ratios?
Both are reward-to-risk measures but differ in the denominator: Sharpe uses total risk σ, Treynor uses systematic risk β. For a well-diversified portfolio they rank similarly; a large gap flags poor diversification. Sharpe is meant to compare portfolios as a relative measure — not to score a single security.
Is Black-Litterman a replacement for mean-variance optimisation?
No. It is a better way to GENERATE the expected-return inputs — anchoring on market-equilibrium returns and gently tilting toward the manager's views — which are then fed into the same MV optimiser. It fixes MPT's extreme sensitivity to return inputs and its tendency to produce concentrated, corner-solution portfolios.
Why does portfolio volatility fall below the weighted-average of the component volatilities?
Whenever the correlation ρ between assets is below 1, the covariance cross-term in the variance formula is smaller than it would be under perfect correlation, so some risk cancels. That gap is the diversification benefit; if ρ = 1 there is no benefit and σp equals the weighted-average volatility.
Exam move
Topic 2 is tested in both assessments, so split your prep. For the closed-book mid-semester test (no formula sheet) drill the calculations until they are automatic: two-asset variance WITH the covariance cross-term, CAPM/SML required return and alpha, and the four ratios (Sharpe, Treynor, information, Jensen's alpha) — write each from memory and check you can take the square root last. For the final, where roughly 80% of marks are applied discussion, practise the 'criticise MPT' and 'criticise CAPM' answers: state each assumption as something to challenge (frictionless efficient markets, unlimited borrow-lend at rf, σ as the only risk measure, an unobservable market portfolio, betas measured with error) and name the fixes (Black-Litterman, APT, Fama-French). Keep the two diagrams and the CML-vs-SML distinction crystal clear, memorise the alpha sign map (above the line = underpriced = buy), and you will bank both the calculation and the discussion marks.
