Portfolio Management and Asset Allocation

Modern Portfolio Theory (MPT)

Harry Markowitz (Nobel 1990) showed that combining assets can reduce risk without proportionally reducing return.

Diversification: Combining assets with low or negative correlation reduces portfolio volatility.

$$\begin{aligned} \text{Portfolio Variance} &= w_{1}^2\sigma_{1}^2 + w_{2}^2\sigma_{2}^2 + 2w_{1}w_{2}\sigma_{1}\sigma_{2}\rho_{12} \\ \rho &= -1: \text{Maximum diversification benefit} \\ \rho &= +1: \text{No diversification benefit} \\ \rho &= 0: \text{Partial diversification} \end{aligned}$$

---

Efficient Frontier

The set of portfolios that offer the maximum expected return for a given level of risk (or minimum risk for a given return).

Minimum Variance Portfolio: Lowest possible volatility point on the efficient frontier.

Optimal Portfolio: Tangency point where the Capital Market Line (CML) touches the efficient frontier.

---

CAPM (Capital Asset Pricing Model)

$$\begin{aligned} E(R_{i}) &= \text{Rf} + \beta_{i} \times [E(\text{Rm}) - \text{Rf}] \\ E(R_{i}) &= \text{Expected return of asset} i \\ \text{Rf} &= \text{Risk}-\text{free rate} \\ \beta_{i} &= \text{Beta} (\text{sensitivity to market movements}) \\ E(\text{Rm}) - \text{Rf} &= \text{Equity risk premium} (ERP) \end{aligned}$$

Beta interpretation:

• β = 1.0: Moves with the market
• β > 1.0: More volatile than market (e.g., growth stocks)
• β < 1.0: Less volatile (e.g., utilities)
• β = 0: Uncorrelated to market

---

Performance Measurement

Sharpe Ratio = (Portfolio return - Risk-free rate) / Portfolio standard deviation

→ Return per unit of total risk. Higher = better.

Treynor Ratio = (Portfolio return - Rf) / Beta

→ Return per unit of systematic risk.

Jensen’s Alpha (α) = Actual return − CAPM expected return

→ Positive alpha = outperformed risk-adjusted benchmark.

Information Ratio = Alpha / Tracking error

→ Consistency of active management.

---

Asset Allocation

Strategic Asset Allocation (SAA): Long-term target mix based on investor’s goals, risk tolerance, and time horizon.

Tactical Asset Allocation (TAA): Short-term deviations from SAA to exploit market opportunities.

Asset classes: Equities, fixed income, alternatives (real estate, commodities, private equity), cash.

---

Key Concept Cards

Efficient Frontier ★★★★★ : The set of optimal portfolios. No feasible portfolio can offer higher return for the same risk. Adding assets expands the frontier.

CAPM ★★★★★ : E(R) = Rf + β × ERP. Beta measures systematic risk. The market only pays for non-diversifiable risk.

Sharpe Ratio ★★★★★ : Excess return per unit of total risk. The standard risk-adjusted performance benchmark.

---

Practice Quiz

Q1. Portfolio A has Sharpe ratio 0.8, Portfolio B has Sharpe ratio 0.6. Which portfolio is better risk-adjusted?

Portfolio A. Higher Sharpe ratio means more return per unit of total risk. If both have the same risk tolerance, Portfolio A delivers better risk-adjusted performance.

Q2. Why can’t diversification eliminate all portfolio risk?

Systematic risk — market-wide factors (recessions, interest rate changes, inflation) — affects all assets simultaneously. These cannot be diversified away because all assets respond (to varying degrees) to the same market factors. Only unsystematic (company-specific) risk is eliminated through diversification.