Portfolio management: Markowitz, CAPM and APT
Diversification cuts risk unless returns move together. Markowitz finds efficient portfolios, CAPM prices only beta risk, and APT uses several factors.
Worked example 1: a two-asset portfolio
Invest 60% in X (expected return 12%, σ 10%) and 40% in Y (expected return 18%, σ 20%). The correlation is 0.2. Find the portfolio's return and risk. What if the correlation were +1 or −1?
- E(Rp) = 0.6 × 12 + 0.4 × 18 = 7.2 + 7.2 = 14.4%.
- σp² = (0.6 × 10)² + (0.4 × 20)² + 2 × 0.6 × 0.4 × 0.2 × 10 × 20 = 36 + 64 + 19.2 = 119.2.
- σp = √119.2 = 10.92%.
- ρ = +1: σp = 0.6 × 10 + 0.4 × 20 = 14% (no diversification benefit). ρ = −1: σp = |6 − 8| = 2%.
Return 14.4%, risk 10.92%, below the 14% weighted average of the two risks. The lower the correlation, the greater the risk reduction.
CML vs SML
Both lines start at the risk-free rate. The CAPM equation is the SML.
Worked example 2: CAPM and mispricing
Rf = 6%, Rm = 14% and a share's beta is 1.5. Analysts expect it to return 20%. Is it fairly priced?
- Required return = 6 + 1.5 × (14 − 6) = 6 + 12 = 18%.
- Expected 20% > required 18%, so alpha = +2%.
- It plots above the SML, so it is undervalued: buy. Buying pushes its price up until its expected return falls to 18%.
Required return 18%; the share is undervalued by 2 percentage points.
Worked example 3: beta, two ways
A share's σ is 30%, the market's σ is 15% and their correlation is 0.6. Find the share's beta.
- Method 1: β = ρ × σi ÷ σm = 0.6 × 30 ÷ 15 = 1.2.
- Method 2: Cov = ρ × σi × σm = 0.6 × 0.30 × 0.15 = 0.027; Var(m) = 0.15² = 0.0225; β = 0.027 ÷ 0.0225 = 1.2.
β = 1.2: the share tends to move 1.2% for each 1% move in the market (aggressive). A beta below 1 is defensive.
Worked example 4: Sharpe, Treynor and Jensen
A fund earned 16% with σ = 20% and β = 1.2. Rf = 6% and the market returned 12%. Find the three performance measures.
- Sharpe = (Rp − Rf) ÷ σp = (16 − 6) ÷ 20 = 0.50.
- Treynor = (Rp − Rf) ÷ βp = 10 ÷ 1.2 = 8.33%.
- Jensen's alpha = Rp − [Rf + βp(Rm − Rf)] = 16 − (6 + 1.2 × 6) = 16 − 13.2 = 2.8%.
Sharpe 0.50, Treynor 8.33%, alpha +2.8%: the manager beat the CAPM benchmark.
Models and authors
| Model or measure | Author and year | Key point |
|---|---|---|
| Portfolio selection | Markowitz, 1952 | Mean-variance analysis; efficient frontier; covariances matter |
| Single-index model | Sharpe, 1963 | Links each security to one market index; far fewer inputs |
| CAPM | Sharpe 1964, Lintner 1965, Mossin 1966 | Only systematic risk (beta) is priced |
| Treynor ratio | Treynor, 1965 | Excess return per unit of beta |
| Sharpe ratio | Sharpe, 1966 | Excess return per unit of σ |
| Jensen's alpha | Jensen, 1968 | Actual return minus the CAPM-required return |
| Efficient market hypothesis | Fama, 1970 | Weak, semi-strong and strong forms |
| APT | Ross, 1976 | Multi-factor, no-arbitrage; factors such as inflation, industrial production and interest-rate spreads |
CAPM assumptions and the portfolio process
- Investors are rational, risk-averse and plan over a single period.
- They can borrow and lend any amount at the risk-free rate.
- Expectations are the same for all; there are no taxes or transaction costs; assets are divisible and markets perfect.
- Portfolio beta is the weighted average of the betas; portfolio σ is a weighted average only when ρ = +1.
- Process: set objectives and constraints, analyse securities, build the portfolio, revise it and evaluate performance.
- Active management tries to beat the market; passive management (index funds) tries to match it.
NET asks CAPM, portfolio-beta and two-asset numericals, the CML vs SML difference, Sharpe vs Treynor vs Jensen, and model-author matches (Markowitz, Sharpe, Ross).
Typical MCQ: “β = 0.8, Rf = 5% and Rm = 15%. What is the required return?” 5 + 0.8 × (15 − 5) = 13%. Trap: 0.8 × 15 = 12% forgets the risk-free rate.
Typical MCQ: “Which measure uses total risk: Sharpe or Treynor?” Sharpe (it divides by σ). Treynor divides by beta, so it suits well-diversified portfolios.
CML: sigma, efficient portfolios only. SML: beta, everything. Sharpe divides by Sigma, Treynor by beTa, and Jensen gives the Jump above CAPM (alpha).