- Global Frameworks: CFA Level 1 (Portfolio Management), US CMA Part 2.
1. Historical vs. Expected Risk and Return
Investment returns are quantified as the percentage change in value over a specific period. Expected return is the probability-weighted average of potential future returns:
E(R) = Σ [ Pᵢ × Rᵢ ] for i = 1 to m
E(R) = Σ [ Pᵢ × Rᵢ ] for i = 1 to m
Risk is defined as the volatility or uncertainty of these returns, measured statistically via variance (σ²) and standard deviation (σ):
σ = √ { Σ [ Pᵢ × (Rᵢ − E(R))² ] } for i = 1 to m
σ = √ { Σ [ Pᵢ × (Rᵢ − E(R))² ] } for i = 1 to m
2. Portfolio Risk and Diversification Dynamics
When individual assets are combined into a portfolio, the portfolio’s expected return is a simple weighted average:
E(Rₚ) = Σ [ wⱼ × E(Rⱼ) ] for j = 1 to n
E(Rₚ) = Σ [ wⱼ × E(Rⱼ) ] for j = 1 to n
However, portfolio risk (standard deviation) is not a weighted average of individual risks. It depends heavily on the covariance (σ₁₂) and correlation (ρ₁₂) between asset returns:
σₚ = √ [ w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂ ]
σₚ = √ [ w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂ ]
- Diversification: If correlation (ρ) is less than +1.0, the portfolio standard deviation drops below the weighted average of individual risks. Combining assets with low or negative correlations dampens overall portfolio volatility without sacrificing expected returns.
3. Systematic Risk vs. Unsystematic Risk
Total risk can be divided into two distinct categories:
- Unsystematic Risk (Idiosyncratic/Specific Risk): Risks unique to a single company or industry (e.g., management changes, strikes, product recalls). This risk can be completely diversified away by holding a sufficiently large portfolio (typically 20–30 randomly selected stocks).
- Systematic Risk (Market/Macro Risk): Market-wide risks that affect all businesses (e.g., GDP shifts, inflation, changes in central bank interest rates). This risk cannot be diversified away.
4. The Capital Asset Pricing Model (CAPM) and Beta
Because rational investors diversify away unsystematic risk, the market only compensates them for bearing systematic risk. Systematic risk is measured by Beta (β), which scales an individual stock’s volatility relative to the broader market portfolio:
βᵢ = Covariance(Rᵢ, Rₘ) / Variance(Rₘ)
βᵢ = Covariance(Rᵢ, Rₘ) / Variance(Rₘ)
The Capital Asset Pricing Model (CAPM) establishes the required rate of return for an asset based on its systematic risk profile:
E(Rᵢ) = R_f + βᵢ × [E(R_m) − R_f]
E(Rᵢ) = R_f + βᵢ × [E(R_m) − R_f]
Where:
R_f= Risk-free rate (e.g., US Treasury bills or European Bunds)
E(R_m) − R_f = Market Risk Premium (MRP)