Ch3. Risk and Return — Portfolio Theory and the Capital Asset Pricing Model

The Risk-Return Relationship

Risk: The possibility of unexpected variability in returns.
Risk-return trade-off: The higher the risk, the higher the return investors require.

Expected Return = Risk-Free Rate + Risk Premium

Measuring Returns

Expected Return

E(R) = Σ P_i × R_i

P_i: Probability of each scenario
R_i: Return in each scenario

Example:
Boom   (30%): +20% return
Normal (50%):  +8% return
Recession (20%): −5% return

E(R) = 0.3×20% + 0.5×8% + 0.2×(−5%)
     = 6% + 4% + (−1%) = 9%

Variance and Standard Deviation

Variance = Σ P_i × [R_i − E(R)]^2
Standard Deviation = √Variance

Using the example above:
Var = 0.3×(20−9)^2 + 0.5×(8−9)^2 + 0.2×(−5−9)^2
    = 0.3×121 + 0.5×1 + 0.2×196
    = 36.3 + 0.5 + 39.2 = 76
Std Dev = √76 ≈ 8.72%

Portfolio Theory

Portfolio Expected Return

E(R_p) = Σ w_i × E(R_i)

w_i: Weight of asset i in the portfolio

Portfolio Variance

Two-asset portfolio:

Var(R_p) = w_A^2 × σ_A^2 + w_B^2 × σ_B^2 + 2 × w_A × w_B × σ_AB

σ_AB = Cov(A, B) = ρ_AB × σ_A × σ_B

ρ_AB: Correlation coefficient (−1 ≤ ρ ≤ +1)

Diversification Effect:

ρ = +1: No diversification benefit (risk unchanged)
ρ = −1: Perfect diversification possible (risk fully eliminated)
0 < ρ < +1: Partial diversification (the realistic case)

Systematic vs. Unsystematic Risk

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk (Market Risk):
- Cannot be eliminated through diversification
- Driven by economy-wide factors: business cycles,
  interest rates, inflation
- Measured by Beta (β)

Unsystematic Risk (Idiosyncratic Risk):
- Can be eliminated through diversification
- Affects only a specific company or industry
- Disappears in a well-diversified portfolio

CAPM (Capital Asset Pricing Model)

CAPM: A model that determines the required return on an asset based on its systematic risk (beta).

E(R_i) = R_f + β_i × [E(R_m) − R_f]

R_f:        Risk-free rate (e.g., 10-year US Treasury yield)
β_i:        Beta of asset i (measure of systematic risk)
E(R_m):     Expected return of the market portfolio (e.g., S&P 500)
[E(R_m) − R_f]: Market risk premium

Interpreting Beta (β):

β = 1: Same risk as the market (moves with the market)
β > 1: Higher risk than the market (aggressive stock)
β < 1: Lower risk than the market (defensive stock)
β = 0: Risk-free asset
β < 0: Moves opposite to the market (hedge asset)

CAPM Application:

Example:
R_f = 3%, Market risk premium = 6%, β = 1.5
Required return = 3% + 1.5 × 6% = 3% + 9% = 12%

The Security Market Line (SML)

SML: A straight line that shows the relationship between beta and expected return in the CAPM framework.

SML:
y-axis = Expected return
x-axis = Beta

Slope    = Market risk premium = E(R_m) − R_f
y-intercept = R_f

Asset above SML: Undervalued (actual return > required return)
Asset below SML: Overvalued (actual return < required return)

Key Concept Cards

CAPM Formula ★★★★★ : E(R) = R_f + β × (R_m − R_f). Risk-free rate + Beta × market excess return. Memory tip: CAPM = Risk-free + β × (Market − Risk-free)

Beta Interpretation ★★★★★ : β = 1 same as market, β > 1 aggressive, β < 1 defensive. Measure of systematic risk. Memory tip: β > 1 aggressive, β < 1 defensive

Diversification and Correlation ★★★★☆ : Diversification benefit is greatest when correlation is closest to −1. No benefit when correlation = +1. Memory tip: Lower correlation → greater diversification benefit

Practice Quiz

Q. R_f = 2%, market return = 10%, β = 0.8. What is the CAPM required return?

E(R) = 2% + 0.8 × (10% − 2%) = 2% + 6.4% = 8.4%.

Q. Two assets have a correlation of −0.5. What is the benefit of combining them in a portfolio?

Negative correlation → large diversification benefit. The portfolio’s standard deviation will be less than the weighted average of the individual assets’ standard deviations.

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